Finite Group Databasebeta

← #29702300

Subgroups — #29702300

Order 15616.

1034 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839980C211yesmaximal, characteristic
2C2×C4×#11758201C211yesmaximal, characteristic, complemented
3C2×#16839844C211yesmaximal, characteristic
4#21448886C211yesmaximal, characteristic, complemented
5#21448728C211yesmaximal, characteristic, complemented
6#21446627C211yesmaximal, characteristic, complemented
7#21446584C211yesmaximal, characteristic, complemented
8#21446501C211yesmaximal, characteristic, complemented
9#21446497C211yesmaximal, characteristic, complemented
10#21446166C211yesmaximal, characteristic
11#21446150C211yesmaximal, characteristic, complemented
12#21446081C211yesmaximal, characteristic, complemented
13#21445680C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#1863C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×D488C2211yescharacteristic
18C2×#14519766C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16839980C418yescomplemented
24C4×#11758201—24no—
25#16839844—24no—
26C2×#14519708C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839669C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839653—21no—
41#16839652—21no—
42#16839636—21no—
43#16839633—21no—
44#16839633—21no—
45#16839627C2211yescharacteristic
46#16839619—21no—
47C4⋊C4×Dic61C2211yescharacteristic
48C4×#11758196C2211yescharacteristic
49C4×#11758198—21no—
50#95×D61—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#107×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×D244C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201C2×C414yescomplemented
71C2×D488—22no—
72C2×D488—22no—
73C2×D488C2×C414yes—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519766—22no—
77#14519766C2×C414yescomplemented
78C4×D244—22no—
79C4×D244—41no—
80#14519739—22no—
81#14519739—22no—
82#14519739—22no—
83#14519739—41no—
84#14519739—41no—
85#14519739C2×C414yescomplemented
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×#11758196D411yescharacteristic
90C2×#11758196D411yescharacteristic
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102#14519708—22no—
103#14519706C2×C414yescomplemented
104#14519701—22no—
105#14519699—22no—
106#14519699—41no—
107#14519697—22no—
108#14519697C2×C414yescomplemented
109#14519697D414yes—
110C4×#92418—22no—
111C8×Dic61—22no—
112C8×Dic61—22no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684D411yescharacteristic
120#14519684D411yescharacteristic
121C22×Q8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4×Q8×C61—22no—
125C4×Q8×C61—41no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#107×C61—22no—
134#103×C61—41no—
135#103×C61C2×C414yescomplemented
136#95×C61—21no—
137C23×D61—21no—
138C23×D61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C22×C412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244C22×C412yes—
167C2×D244C22×C412yes—
168C2×D244C2×D412yes—
169C2×D244#4112yes—
170C2×C4×D61—24no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—81no—
178C2×C4×D61—81no—
179C2×C4×D61—81no—
180C2×C4×D61—81no—
181C2×C4×D61—81no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61C22×C412yes—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—42no—
193C4⋊C4×C61—81no—
194C4⋊C4×C61C22×C412yes—
195C4⋊C4×C61C2×D412yes—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—42no—
199#11758201—28no—
200#11758198—24no—
201#11758198—24no—
202#11758198—24no—
203#11758198—42no—
204#11758198—42no—
205#11758198—42no—
206#11758198—42no—
207#11758198—81no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—81no—
213#11758196—22no—
214#11758196—22no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—81no—
218C4×Dic61C22×C412yes—
219C4×Dic61#4112yes—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418C22×C412yes—
225C2×#92418C2×D412yes—
226D488—28no—
227C23×C61C22×D411yescharacteristic
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—41no—
242Q8×C61—22no—
243Q8×C61—22no—
244Q8×C61—41no—
245Q8×C61—41no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61#13512yes—
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×C4×C612−1+411yescharacteristic
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61C4×D412yes—
286C2×Dic61C4×D412yes—
287C2×Dic61#13512yes—
288D244—22no—
289D244—22no—
290D244—22no—
291D244—22no—
292D244—22no—
293D244—41no—
294D244—41no—
295D244C4×D414yes—
296C4×D61—84no—
297C4×D61—84no—
298C4×D61—84no—
299C4×D61—84no—
300C8×C61—22no—
301C8×C61—41no—
302#92418—22no—
303#92418—22no—
304#22999—611nomaximal, Sylow
305C22×C61C2×C4×D411yescharacteristic
306C22×C61D4211yescharacteristic
307C22×C61#54811yescharacteristic
308C22×C61C2×#13511yescharacteristic
309C22×C61C2×#13511yescharacteristic
310C22×C61#57611yescharacteristic
311C22×C61#57711yescharacteristic
312C2×D61—24no—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—42no—
322C2×D61—81no—
323C2×D61—81no—
324C2×D61—81no—
325C2×D61—81no—
326C4×C61—22no—
327C4×C61—42no—
328C4×C61—42no—
329C4×C61—42no—
330C4×C61—41no—
331C4×C61—81no—
332C4×C61C2×C4×D411yescharacteristic
333C4×C61#52011yescharacteristic
334C4×C61#54711yescharacteristic
335C4×C61#55311yescharacteristic
336Dic61—22no—
337Dic61—22no—
338Dic61—22no—
339Dic61—41no—
340Dic61—81no—
341Dic61—81no—
342C2×#463—611no—
343C2×C4×SD16—611no—
344C2×C4×D8—611no—
345#2367—611no—
346#2349—611no—
347#1958—611no—
348#1949—611no—
349#1934—611no—
350#1933—611no—
351#1879—611no—
352#1877—611no—
353#1864—611no—
354#1863D6111yescharacteristic, complemented
355C8×C4⋊C4—611no—
356C4×#102—611no—
357C2×C61#236111yescharacteristic
358C2×C61C4×#13511yescharacteristic
359C2×C61#294611yescharacteristic
360C2×C61#328011yescharacteristic
361C2×C61#328211yescharacteristic
362C2×C61#336011yescharacteristic
363C2×C61#336111yescharacteristic
364D61—44no—
365D61—82no—
366C22×SD16—611no—
367C22×D8—611no—
368C2×#124—611no—
369C2×C4×Q8C2×D6111yescharacteristic
370C2×C4×D4—611no—
371C2×C4×D4—611no—
372C2×C4×D4—611no—
373C2×C4×D4—611no—
374#463—618no—
375C4×SD16—1224no—
376C4×D8—1224no—
377C2×#107C2×D6111yescharacteristic
378C2×#106—611no—
379C2×C4⋊C8—611no—
380C2×#103—611no—
381C2×#103C2×D6111yescharacteristic
382C2×#103C2×D6111yescharacteristic
383C2×#102—611no—
384C2×#102—611no—
385C2×#102—611no—
386C2×#102—611no—
387C2×#102—611no—
388C2×C4×C8—611no—
389C2×C4×C8—611no—
390C2×C4×C8—611no—
391#391—611no—
392#390—611no—
393#389—611no—
394#389C2×D6111yescharacteristic
395#388—1221no—
396#388—1221no—
397#385—1221no—
398#385—1221no—
399#385—1221no—
400#382—611no—
401#380—1221no—
402C4×C4⋊C4—611no—
403C4×C4⋊C4—611no—
404C4×C22⋊C4—1221no—
405C2×#95—1221no—
406#340—611no—
407#340—611no—
408#340—611no—
409#340C2×D6111yescharacteristic
410#336—611no—
411#336C2×D6111yescharacteristic
412C61#2299911yescharacteristic, complemented, Sylow
413C22×Q8C22×D6111yescharacteristic
414C22×D4—611no—
415C22×D4—611no—
416C23×C4—1221no—
417C23×C4—1221no—
418C23×C4—1221no—
419C23×C4—1221no—
420C2×SD16—614no—
421C2×SD16—1222no—
422C2×SD16—1222no—
423C2×D8—614no—
424C2×D8—1222no—
425C2×D8—1222no—
426C22×C8—611no—
427C22×C8—611no—
428C22×C8C22×D6111yescharacteristic
429C22×C8C22×D6111yescharacteristic
430#124—614no—
431#124—1222no—
432C4×Q8—22no—
433C4×Q8—41no—
434C4×D4—614no—
435C4×D4—1222no—
436C4×D4—1222no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4—2441no—
440C4×D4—2441no—
441C4×D4—2441no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C42—21no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—1221no—
464C2×C42C22×D6111yescharacteristic
465#107—22no—
466#106—1222no—
467C4⋊C8—614no—
468#103—41no—
469#103—1222no—
470#103C4×D6114yescomplemented
471#102—614no—
472#102—614no—
473#102—1222no—
474#102—1222no—
475#102—2441no—
476C4×C8—1222no—
477C4×C8—1222no—
478C4×C8—1222no—
479#95—21no—
480#95—611no—
481#95—611no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488C24—1221no—
489C24—1221no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8C2×C4×D6112yes—
496C2×D4—612no—
497C2×D4—612no—
498C2×D4—612no—
499C2×D4—612no—
500C2×D4—1221no—
501C2×D4—1221no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—1221no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—1224no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—4881no—
520C22×C4—4881no—
521C22×C4—4881no—
522C22×C4—4881no—
523C22×C4—4881no—
524C22×C4—2442no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4—2442no—
529C22×C4—2442no—
530C22×C4D4×D6111yescharacteristic
531C22×C4D4×D6111yescharacteristic
532C22×C4C23×D6111yescharacteristic
533SD16—1228no—
534D8—1228no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—612no—
541C2×C8—612no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8C2×C4×D6112yes—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—42no—
552C4⋊C4—81no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—4881no—
560C4⋊C4C2×C4×D6112yes—
561C4⋊C4D4×D6112yes—
562C22⋊C4—1224no—
563C22⋊C4—1224no—
564C22⋊C4—1224no—
565C22⋊C4—4881no—
566C22⋊C4—2442no—
567C22⋊C4—2442no—
568C22⋊C4—2442no—
569C22⋊C4—2442no—
570C42—21no—
571C42—21no—
572C42—42no—
573C42—612no—
574C42—612no—
575C42—1221no—
576C42—1221no—
577C42—4881no—
578C23—1222no—
579C23—1222no—
580C23—1222no—
581C23—1222no—
582C23—2441no—
583C23—2441no—
584C23—2441no—
585C23—2441no—
586C23—2441no—
587C23—2441no—
588C23—2441no—
589C23—2441no—
590C23—2441no—
591C23—2441no—
592C23C2×D4×D6111yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—41no—
596Q8—41no—
597D4—614no—
598D4—1222no—
599D4—1222no—
600D4—1222no—
601D4—1222no—
602D4—1222no—
603D4—2441no—
604D4—2441no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—612no—
623C2×C4—612no—
624C2×C4—612no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—4884no—
635C2×C4—4884no—
636C2×C4—4884no—
637C2×C4—4884no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4#1451980012yes—
645C2×C4C22×C4×D6111yescharacteristic
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4#41×D6111yescharacteristic
649C2×C4#1451982211yescharacteristic
650C2×C4#1451982511yescharacteristic
651C8—22no—
652C8—41no—
653C8—1222no—
654C8—1222no—
655C22—1224no—
656C22—4881no—
657C22—4881no—
658C22—4881no—
659C22—4881no—
660C22—2442no—
661C22—2442no—
662C22—2442no—
663C22—2442no—
664C22—2442no—
665C22—2442no—
666C22—2442no—
667C22—2442no—
668C22—2442no—
669C22C4×D4×D6111yescharacteristic
670C22#1684039411yescharacteristic
671C22#1684042211yescharacteristic
672C22#135×D6111yescharacteristic
673C22C2×#1451980011yescharacteristic
674C22#1684056611yescharacteristic
675C22#1684057111yescharacteristic
676C4—22no—
677C4—42no—
678C4—41no—
679C4—42no—
680C4—42no—
681C4—81no—
682C4—1222no—
683C4—1222no—
684C4—1222no—
685C4—4881no—
686C4—4881no—
687C4—2441no—
688C4C4×D4×D6111yescharacteristic
689C4#1684024311yescharacteristic
690C4#1684041411yescharacteristic
691C4#1684045711yescharacteristic
692C2—4882no—
693C2—2444no—
694C2#2144883811yescharacteristic
695C2C4×#1451980011yescharacteristic
696C2#2145465011yescharacteristic
697C2#2145750211yescharacteristic
698C2#2145751911yescharacteristic
699C2#2145847111yescharacteristic
700C2#2145847811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1863C2
socleC23×C61C22×D4
prime radicalC22×D488C22
cosocleC22×C4×C61C24
hypercenter#1863D61
hypocenterC61#22999
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×#11758201C2semidirect
4. #21448886C2semidirect
5. #21448728C2semidirect
6. #21446627C2semidirect
7. #21446584C2semidirect
8. #21446501C2semidirect
9. #21446497C2semidirect
11. #21446150C2semidirect
12. #21446081C2semidirect
13. #21445680C2semidirect
14. #21445633C2semidirect
15. C61×#1863C2semidirect
23. #16839980C4semidirect
70. C2×#11758201C2×C4semidirect
77. #14519766C2×C4semidirect
85. #14519739C2×C4semidirect
103. #14519706C2×C4semidirect
108. #14519697C2×C4semidirect
135. #103×C61C2×C4semidirect
354. #1863D61semidirect
412. C61#22999semidirect
470. #103C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399802
C2×C4×#117582012
C2×#168398442
#214488862
#214487282
#214466272
#214465842
#214465012
#214464972
#214461662
#214461502
#214460812
#214456802
#214456332
C61×#18632
#2299961