Finite Group Databasebeta

← #29702315

Subgroups — #29702315

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839987C211yesmaximal, characteristic
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3C2×#16839849C211yesmaximal, characteristic
4#21448886C211yesmaximal, characteristic
5#21448738C211yesmaximal, characteristic, complemented
6#21446627C211yesmaximal, characteristic
7#21446588C211yesmaximal, characteristic, complemented
8#21446501C211yesmaximal, characteristic
9#21446497C211yesmaximal, characteristic, complemented
10#21446173C211yesmaximal, characteristic
11#21446157C211yesmaximal, characteristic, complemented
12#21446067C211yesmaximal, characteristic
13#21445680C211yesmaximal, characteristic, complemented
14#21445638C211yesmaximal, characteristic, complemented
15C61×#1865C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519766C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16839987C418yescomplemented
24C4×#11758202—24no—
25#16839849—24no—
26C2×#14519710C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839669C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839659—21no—
41#16839657—21no—
42#16839653—21no—
43#16839652—21no—
44#16839633—21no—
45#16839627C2211yescharacteristic
46#16839626—21no—
47C4⋊C4×Dic61—21no—
48C4⋊C4×Dic61C2211yescharacteristic
49C4×#11758196C2211yescharacteristic
50C4×#11758198—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202C2×C414yescomplemented
68C2×#11758190—22no—
69C2×#11758190—22no—
70C2×#11758190C2×C414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519766—22no—
74#14519766C2×C414yes—
75C4×Dic122—22no—
76C4×Dic122—41no—
77#14519742—22no—
78#14519742—22no—
79#14519742—41no—
80#14519742C2×C414yescomplemented
81#14519739—22no—
82#14519739—41no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196D411yescharacteristic
90C2×#11758196D411yescharacteristic
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102#14519710—22no—
103#14519706C2×C414yescomplemented
104#14519702—22no—
105#14519701—22no—
106#14519700D414yes—
107#14519699—41no—
108#14519697—22no—
109#14519697C2×C414yes—
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#106×C61—22no—
134#103×C61—41no—
135#103×C61C2×C414yescomplemented
136#95×C61—21no—
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
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×D244C22×C412yes—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—81no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C22×C412yes—
171C2×Dic122C2×D412yes—
172C2×Dic122#4112yes—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C22×C412yes—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61C22×C412yes—
186C4⋊C4×C61C2×D412yes—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—42no—
190#11758202—28no—
191#11758198—24no—
192#11758198—24no—
193#11758198—42no—
194#11758198—42no—
195#11758198—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—42no—
201#11758196—22no—
202#11758196—24no—
203#11758196—22no—
204#11758196—42no—
205#11758196—42no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—24no—
209C4×Dic61—42no—
210C4×Dic61—42no—
211C4×Dic61—81no—
212C4×Dic61C22×C412yes—
213C4×Dic61#4112yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C22×C412yes—
219C2×#92418C2×D412yes—
220#11758190—28no—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—41no—
232Q8×C61—41no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61#13512yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61C4×D412yes—
276C2×Dic61C4×D412yes—
277C2×Dic61#13512yes—
278D244—22no—
279D244—22no—
280D244—41no—
281D244—41no—
282C4×D61—84no—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122C4×D414yes—
287C8×C61—22no—
288C8×C61—41no—
289#92418—22no—
290#92418—22no—
291#23000—611nomaximal, Sylow
292C22×C61C2×C4×D411yescharacteristic
293C22×C61D4211yescharacteristic
294C22×C61#54811yescharacteristic
295C22×C61C2×#13511yescharacteristic
296C22×C61C2×#13511yescharacteristic
297C22×C61#57711yescharacteristic
298C22×C61#57811yescharacteristic
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—41no—
311C4×C61—81no—
312C4×C61C2×C4×D411yescharacteristic
313C4×C61#52011yescharacteristic
314C4×C61#54711yescharacteristic
315C4×C61#55311yescharacteristic
316Dic61—24no—
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—41no—
321Dic61—81no—
322Dic61—81no—
323C2×#464—611no—
324C2×C4×Q16—611no—
325C2×C4×SD16—611no—
326#2367—611no—
327#2351—611no—
328#1958—611no—
329#1949—611no—
330#1934—611no—
331#1933—611no—
332#1880—611no—
333#1878—611no—
334#1865D6111yescharacteristic, complemented
335#1862—611no—
336C8×C4⋊C4—611no—
337C4×#103—611no—
338C2×C61#236111yescharacteristic
339C2×C61C4×#13511yescharacteristic
340C2×C61#294711yescharacteristic
341C2×C61#328211yescharacteristic
342C2×C61#328511yescharacteristic
343C2×C61#336411yescharacteristic
344C2×C61#336511yescharacteristic
345D61—82no—
346C22×Q16—611no—
347C22×SD16—611no—
348C2×#124—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8C2×D6111yescharacteristic
353C2×C4×D4—611no—
354#464—618no—
355C4×Q16—1224no—
356C4×SD16—1224no—
357C2×#107—611no—
358C2×#106C2×D6111yescharacteristic
359C2×C4⋊C8—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103C2×D6111yescharacteristic
364C2×#103C2×D6111yescharacteristic
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×C4×C8—611no—
369C2×C4×C8—611no—
370C2×C4×C8—611no—
371#391—611no—
372#390—611no—
373#389—611no—
374#389—1221no—
375#389—1221no—
376#389C2×D6111yescharacteristic
377#388—1221no—
378#388—1221no—
379#385—1221no—
380#382—611no—
381#382—1221no—
382C4×C4⋊C4—611no—
383C4×C4⋊C4—611no—
384C4×C4⋊C4—1221no—
385C4×C22⋊C4—1221no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340C2×D6111yescharacteristic
390#336—611no—
391#336C2×D6111yescharacteristic
392C61#2300011yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×Q8C22×D6111yescharacteristic
395C22×D4—611no—
396C23×C4—1221no—
397C2×Q16—614no—
398C2×Q16—1222no—
399C2×Q16—1222no—
400C2×SD16—614no—
401C2×SD16—1222no—
402C2×SD16—1222no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407#124—614no—
408#124—1222no—
409C4×Q8—22no—
410C4×Q8—41no—
411C4×Q8—614no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—2441no—
416C4×Q8—2441no—
417C4×D4—1222no—
418C4×D4—2441no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—21no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#106—22no—
447C4⋊C8—614no—
448#103—41no—
449#103—614no—
450#103—1222no—
451#103—1222no—
452#103C4×D6114yescomplemented
453#102—614no—
454#102—1222no—
455#102—2441no—
456C4×C8—1222no—
457C4×C8—1222no—
458C4×C8—1222no—
459#95—21no—
460#95—611no—
461#95—611no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468C24—1221no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—612no—
475C2×Q8—612no—
476C2×Q8—612no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8C2×C4×D6112yes—
481C2×D4—612no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504Q16—1228no—
505SD16—1228no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—612no—
512C2×C8—612no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8C2×C4×D6112yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—1222no—
525C4⋊C4—1224no—
526C4⋊C4—1222no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C4⋊C4C2×C4×D6112yes—
535C4⋊C4D4×D6112yes—
536C22⋊C4—1224no—
537C22⋊C4—1224no—
538C22⋊C4—4881no—
539C22⋊C4—2442no—
540C22⋊C4—2442no—
541C42—21no—
542C42—21no—
543C42—42no—
544C42—612no—
545C42—612no—
546C42—1224no—
547C42—1221no—
548C42—1221no—
549C42—4881no—
550C42—2442no—
551C42—2442no—
552C23—1222no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23C2×D4×D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—614no—
565Q8—1222no—
566Q8—1222no—
567Q8—1222no—
568D4—1222no—
569D4—1222no—
570D4—2441no—
571D4—2441no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—4884no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4#1451979812yes—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4#1451982211yescharacteristic
617C2×C4#1451982511yescharacteristic
618C8—22no—
619C8—41no—
620C8—1222no—
621C8—1222no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—2442no—
627C22—2442no—
628C22—2442no—
629C22C4×D4×D6111yescharacteristic
630C22#1684039411yescharacteristic
631C22#1684042211yescharacteristic
632C22#135×D6111yescharacteristic
633C22C2×#1451979811yescharacteristic
634C22#1684057111yescharacteristic
635C22#1684057811yescharacteristic
636C4—22no—
637C4—41no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—1222no—
643C4—1222no—
644C4—1224no—
645C4—1222no—
646C4—4881no—
647C4—4881no—
648C4—2441no—
649C4C4×D4×D6111yescharacteristic
650C4#1684024311yescharacteristic
651C4#1684041411yescharacteristic
652C4#1684045711yescharacteristic
653C2—4882no—
654C2#2144883811yescharacteristic
655C2C4×#1451979811yescharacteristic
656C2#2145466111yescharacteristic
657C2#2145752411yescharacteristic
658C2#2145755311yescharacteristic
659C2#2145850711yescharacteristic
660C2#2145851411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1865C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1865D61
hypocenterC61#23000
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758202C2semidirect
5. #21448738C2semidirect
7. #21446588C2semidirect
9. #21446497C2semidirect
11. #21446157C2semidirect
13. #21445680C2semidirect
14. #21445638C2semidirect
15. C61×#1865C2semidirect
23. #16839987C4semidirect
67. C2×#11758202C2×C4semidirect
80. #14519742C2×C4semidirect
103. #14519706C2×C4semidirect
135. #103×C61C2×C4semidirect
334. #1865D61semidirect
392. C61#23000semidirect
452. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399872
C2×C4×#117582022
C2×#168398492
#214488862
#214487382
#214466272
#214465882
#214465012
#214464972
#214461732
#214461572
#214460672
#214456802
#214456382
C61×#18652
#2300061