Finite Group Databasebeta

← #29702265

Subgroups — #29702265

Order 15616.

1030 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839966C211yesmaximal, characteristic
2C2×C4×#11758201C211yesmaximal, characteristic
3C2×#16839849C211yesmaximal, characteristic
4#21448873C211yesmaximal, characteristic, complemented
5#21448720C211yesmaximal, characteristic
6#21446619C211yesmaximal, characteristic, complemented
7#21446579C211yesmaximal, characteristic, complemented
8#21446497C211yesmaximal, characteristic, complemented
9#21446494C211yesmaximal, characteristic, complemented
10#21446152C211yesmaximal, characteristic
11#21446150C211yesmaximal, characteristic
12#21446088C211yesmaximal, characteristic, complemented
13#21445680C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic
15C61×#1865C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×C4⋊C4×D61—21no—
24#16839966C418yescomplemented
25C4×#11758201—24no—
26#16839849—24no—
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×C4×#92418C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38#16839669C2211yescharacteristic
39#16839668C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839657—21no—
42#16839646—21no—
43#16839643—21no—
44#16839636—21no—
45#16839633—21no—
46#16839631C2211yescharacteristic
47#16839621—21no—
48C4⋊C4×Dic61C2211yescharacteristic
49C4×#11758196—21no—
50C4×#11758196C2211yescharacteristic
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—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758190—22no—
68C2×#11758190—22no—
69C2×#11758190C2×C414yes—
70C2×#11758201—22no—
71C2×#11758201—22no—
72C2×#11758201C2×C414yes—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519775—22no—
76#14519775C2×C414yescomplemented
77#14519742—22no—
78#14519742—41no—
79C4×D244—22no—
80C4×D244—41no—
81#14519739—22no—
82#14519739—22no—
83#14519739—41no—
84#14519739C2×C414yes—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196D411yescharacteristic
94C2×#11758196D411yescharacteristic
95C4⋊C4×D61—44no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103#14519708—22no—
104#14519706C2×C414yescomplemented
105#14519702—41no—
106#14519701—22no—
107#14519700—22no—
108#14519700C2×C414yescomplemented
109#14519699—22no—
110#14519697D414yes—
111C4×#92418—22no—
112C8×Dic61—22no—
113C8×Dic61—22no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684D411yescharacteristic
121#14519684D411yescharacteristic
122C22×Q8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C4×Q8×C61—22no—
126C4×Q8×C61—41no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#106×C61—22no—
135#103×C61—41no—
136#103×C61C2×C414yescomplemented
137#95×C61—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×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×D244C2×D412yes—
164C2×D244#4112yes—
165C2×C4×D61—24no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—81no—
173C2×C4×D61—81no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C22×C412yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—42no—
191C4⋊C4×C61—42no—
192C4⋊C4×C61C22×C412yes—
193C4⋊C4×C61C2×D412yes—
194C42×C61—21no—
195C42×C61—21no—
196C42×C61—42no—
197#11758201—28no—
198#11758198—24no—
199#11758198—42no—
200#11758198—42no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206#11758196—24no—
207#11758196—24no—
208#11758196—22no—
209#11758196—42no—
210#11758196—42no—
211#11758196—81no—
212#11758196SD1614yescomplemented
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C4×Dic61C22×C412yes—
218C4×Dic61#4112yes—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418C22×C412yes—
224C2×#92418C2×D412yes—
225#11758190—28no—
226C23×C61C22×D411yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61C2×SD1612yes—
256C2×C4×C61C23×C411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61C4×D412yes—
279C2×Dic61C4×D412yes—
280C2×Dic61C2×SD1612yes—
281C2×Dic61C2×SD1612yes—
282C2×Dic61#13512yes—
283D244—22no—
284D244—22no—
285D244—22no—
286D244C4×D414yes—
287C4×D61—48no—
288C4×D61—84no—
289C4×D61—84no—
290Dic122—22no—
291Dic122—22no—
292Dic122—41no—
293Dic122—41no—
294C8×C61—22no—
295C8×C61—41no—
296#92418—22no—
297#92418—22no—
298#22996—611nomaximal, Sylow
299C22×C61C2×C4×D411yescharacteristic
300C22×C61D4211yescharacteristic
301C22×C61#54811yescharacteristic
302C22×C61C22×SD1611yescharacteristic
303C22×C61C2×#13511yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57811yescharacteristic
306C2×D61—24no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C4×C61—22no—
314C4×C61—42no—
315C4×C61—42no—
316C4×C61—41no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61C2×C4×D411yescharacteristic
320C4×C61#52011yescharacteristic
321C4×C61#54711yescharacteristic
322C4×C61#55311yescharacteristic
323Dic61—22no—
324Dic61—22no—
325Dic61—42no—
326Dic61—41no—
327Dic61—81no—
328Dic61—81no—
329Dic61C4×SD1614yescomplemented
330C2×#461—611no—
331C2×C4×SD16—611no—
332C2×C4×SD16—611no—
333#2365—611no—
334D4×C4⋊C4—611no—
335#1956—611no—
336#1948—611no—
337#1933—611no—
338#1933—611no—
339#1877—611no—
340#1877—611no—
341#1865—611no—
342#1865D6111yescharacteristic, complemented
343C8×C4⋊C4—611no—
344C4×#102—611no—
345C2×C61#236111yescharacteristic
346C2×C61C2×C4×SD1611yescharacteristic
347C2×C61#294711yescharacteristic
348C2×C61D4×SD1611yescharacteristic
349C2×C61#328311yescharacteristic
350C2×C61#335911yescharacteristic
351C2×C61#336011yescharacteristic
352D61—44no—
353C22×SD16—611no—
354C22×SD16—611no—
355C2×C4⋊Q8—611no—
356C2×C4×Q8—611no—
357C2×C4×Q8C2×D6111yescharacteristic
358C2×C4×D4—611no—
359C2×C4×D4—611no—
360C2×C4×D4—611no—
361C22×C4⋊C4—1221no—
362#461—618no—
363C4×SD16—1224no—
364C4×SD16—1224no—
365C2×#106—611no—
366C2×#106C2×D6111yescharacteristic
367C2×C4⋊C8—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103C2×D6111yescharacteristic
373C2×#103C2×D6111yescharacteristic
374C2×#102—611no—
375C2×#102—611no—
376C2×C4×C8—611no—
377C2×C4×C8—611no—
378C2×C4×C8—611no—
379#391—611no—
380#391—611no—
381#389—611no—
382#389—1221no—
383#389C2×D6111yescharacteristic
384#387—1221no—
385#387—1221no—
386#385—1221no—
387#385—1221no—
388#384—611no—
389#380—1221no—
390C4×C4⋊C4—611no—
391C4×C4⋊C4—611no—
392C4×C4⋊C4—1221no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340C2×D6111yescharacteristic
397#336—611no—
398#336C2×D6111yescharacteristic
399C61#2299611yescharacteristic, complemented, Sylow
400C22×Q8—611no—
401C22×Q8C22×D6111yescharacteristic
402C22×D4—611no—
403C23×C4—1221no—
404C23×C4—1221no—
405C23×C4—1221no—
406C2×SD16—614no—
407C2×SD16—614no—
408C2×SD16—1222no—
409C2×SD16—1222no—
410C2×SD16—1222no—
411C2×SD16—1222no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—614no—
417C4⋊Q8—1222no—
418C4×Q8—22no—
419C4×Q8—41no—
420C4×Q8—1222no—
421C4×Q8—2441no—
422C4×D4—614no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—2441no—
427C4×D4—2441no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—2444no—
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—21no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453#106—22no—
454#106—1222no—
455C4⋊C8—614no—
456#103—41no—
457#103—614no—
458#103—1222no—
459#103—1222no—
460#103—2441no—
461#103C4×D6114yescomplemented
462#102—614no—
463#102—1222no—
464C4×C8—1222no—
465C4×C8—1222no—
466C4×C8—1222no—
467#95—21no—
468#95—611no—
469#95—611no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—1221no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8C2×C4×D6112yes—
489C2×D4—612no—
490C2×D4—612no—
491C2×D4—612no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—1224no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—4881no—
508C22×C4—4881no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4D4×D6111yescharacteristic
516C22×C4D4×D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518SD16—1228no—
519SD16—1228no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—612no—
526C2×C8—612no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8C2×C4×D6112yes—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—614no—
539C4⋊C4—1222no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—4881no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C4⋊C4C2×C4×D6112yes—
551C4⋊C4D4×D6112yes—
552C22⋊C4—1224no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—612no—
559C42—612no—
560C42—1221no—
561C42—1221no—
562C42—2442no—
563C42—2442no—
564C23—1222no—
565C23—1222no—
566C23—1222no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23C2×D4×D6111yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—41no—
575Q8—41no—
576Q8—1222no—
577Q8—1222no—
578Q8—2441no—
579Q8—2441no—
580D4—614no—
581D4—1222no—
582D4—1222no—
583D4—1222no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—4884no—
615C2×C4—4884no—
616C2×C4—2448no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4SD16×D6112yes—
626C2×C4C22×C4×D6111yescharacteristic
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4#41×D6111yescharacteristic
630C2×C4#1451982211yescharacteristic
631C2×C4#1451982511yescharacteristic
632C8—22no—
633C8—41no—
634C8—1222no—
635C8—1222no—
636C22—1224no—
637C22—2442no—
638C22—2442no—
639C22—2442no—
640C22—2442no—
641C22—2442no—
642C22—2442no—
643C22C4×D4×D6111yescharacteristic
644C22#1684039411yescharacteristic
645C22#1684042211yescharacteristic
646C22C2×SD16×D6111yescharacteristic
647C22#135×D6111yescharacteristic
648C22#1684057111yescharacteristic
649C22#1684057811yescharacteristic
650C4—22no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—81no—
656C4—614no—
657C4—1222no—
658C4—1222no—
659C4—4881no—
660C4—4881no—
661C4—2441no—
662C4—2442no—
663C4C4×D4×D6111yescharacteristic
664C4#1684024311yescharacteristic
665C4#1684041411yescharacteristic
666C4#1684045711yescharacteristic
667C2—2444no—
668C2#2144883811yescharacteristic
669C2C4×SD16×D6111yescharacteristic
670C2#2145466111yescharacteristic
671C2#2145750811yescharacteristic
672C2#2145753411yescharacteristic
673C2#2145846011yescharacteristic
674C2#2145847111yescharacteristic
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#22996
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
4. #21448873C2semidirect
6. #21446619C2semidirect
7. #21446579C2semidirect
8. #21446497C2semidirect
9. #21446494C2semidirect
12. #21446088C2semidirect
13. #21445680C2semidirect
15. C61×#1865C2semidirect
24. #16839966C4semidirect
76. #14519775C2×C4semidirect
104. #14519706C2×C4semidirect
108. #14519700C2×C4semidirect
136. #103×C61C2×C4semidirect
212. #11758196SD16semidirect
329. Dic61C4×SD16semidirect
342. #1865D61semidirect
399. C61#22996semidirect
461. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399662
C2×C4×#117582012
C2×#168398492
#214488732
#214487202
#214466192
#214465792
#214464972
#214464942
#214461522
#214461502
#214460882
#214456802
#214456332
C61×#18652
#2299661