Finite Group Databasebeta

← #16822809

Subgroups — #16822809

Order 3712.

1056 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#90805C211yesmaximal, characteristic, complemented
2C2×#11745244C211yesmaximal, characteristic, complemented
3C2×#11745299C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D29C211yesmaximal, characteristic, complemented
5#14271076C211yesmaximal, characteristic, complemented
6#14271075C211yesmaximal, characteristic, complemented
7#14271072C211yesmaximal, characteristic, complemented
8#14271071C211yesmaximal, characteristic, complemented
9#14271042C211yesmaximal, characteristic, complemented
10#14271041C211yesmaximal, characteristic, complemented
11#14271041C211yesmaximal, characteristic, complemented
12#14271039C211yesmaximal, characteristic, complemented
13#14271039C211yesmaximal, characteristic, complemented
14#14271025C211yesmaximal, characteristic, complemented
15C29×#386C211yesmaximal, characteristic, complemented
16C24×D29C2211yescharacteristic
17C22×D4×C29C2211yescharacteristic
18C23×C4×C29C2211yescharacteristic, complemented
19C22×C22⋊D29C2211yescharacteristic
20C22×C22⋊D29C2211yescharacteristic
21C22×C22⋊D29C2211yescharacteristic
22C22×C22⋊D29C2211yescharacteristic, complemented
23C2×#90805—24no—
24C2×#90805—24no—
25C2×#90805—24no—
26C22×D116C2211yescharacteristic
27C22×D116C2211yescharacteristic
28C22×C4×D29C2211yescharacteristic
29C22×C4×D29C2211yescharacteristic
30C22×C4×D29C2211yescharacteristic, complemented
31C22×Dic58C2211yescharacteristic, complemented
32C2×C22⋊C4×C29C2211yescharacteristic
33C2×C22⋊C4×C29C2211yescharacteristic
34C2×C22⋊C4×C29C2211yescharacteristic, complemented
35C2×C22⋊Dic29C2211yescharacteristic
36C2×C22⋊Dic29C2211yescharacteristic
37C2×C22⋊Dic29C2211yescharacteristic, complemented
38#11745299—24no—
39C2×#90781C2211yescharacteristic
40C2×#90781C2211yescharacteristic
41C2×#90781C2211yescharacteristic
42C2×#90781C2211yescharacteristic
43C2×#90781C2211yescharacteristic
44C2×#90781C2211yescharacteristic
45C2×#90781C2211yescharacteristic, complemented
46C2×#90781C2211yescharacteristic, complemented
47#11745244—24no—
48C22⋊C4×D29—24no—
49C29×#95C2211yescharacteristic, complemented
50C29×#95C2211yescharacteristic, complemented
51#11745179C2211yescharacteristic, complemented
52#11745179C2211yescharacteristic, complemented
53#11745179C2211yescharacteristic, complemented
54#11745179C2211yescharacteristic, complemented
55#11745179C2211yescharacteristic, complemented
56#11745179C2211yescharacteristic, complemented
57C24×C29C2311yescharacteristic
58C24×C29C2311yescharacteristic
59C23×D29—22no—
60C23×D29—22no—
61C23×D29—22no—
62C23×D29—22no—
63C23×D29—22no—
64C23×D29—41no—
65C23×D29—41no—
66C23×D29C2311yescharacteristic
67C23×D29C2311yescharacteristic
68C23×D29C2311yescharacteristic
69C2×D4×C29—22no—
70C2×D4×C29—22no—
71C2×D4×C29—41no—
72C22×C4×C29—22no—
73C22×C4×C29—22no—
74C22×C4×C29—22no—
75C22×C4×C29C2311yescharacteristic
76C22×C4×C29C2311yescharacteristic
77C22×C4×C29C2311yescharacteristic
78C22×C4×C29C2311yescharacteristic
79C22×C4×C29C2311yescharacteristic
80C2×C22⋊D29—22no—
81C2×C22⋊D29—22no—
82C2×C22⋊D29—22no—
83C2×C22⋊D29—22no—
84C2×C22⋊D29—22no—
85C2×C22⋊D29—22no—
86C2×C22⋊D29—22no—
87C2×C22⋊D29—22no—
88C2×C22⋊D29—22no—
89C2×C22⋊D29—41no—
90C2×C22⋊D29—41no—
91C2×C22⋊D29D414yescomplemented
92C22×Dic29C2311yescharacteristic
93C22×Dic29C2311yescharacteristic
94C22×Dic29C2311yescharacteristic
95C22×Dic29C2311yescharacteristic
96C22×Dic29C2311yescharacteristic
97#90805—232no—
98C2×D116—22no—
99C2×D116—22no—
100C2×D116—22no—
101C2×D116—22no—
102C2×D116—41no—
103C2×D116D414yescomplemented
104C2×C4×D29—22no—
105C2×C4×D29—22no—
106C2×C4×D29—22no—
107C2×C4×D29—22no—
108C2×C4×D29—22no—
109C2×C4×D29—22no—
110C2×C4×D29—41no—
111C2×C4×D29—41no—
112C2×C4×D29D414yescomplemented
113C2×Dic58—22no—
114C2×Dic58—22no—
115C2×Dic58—22no—
116C22⋊C4×C29—22no—
117C22⋊C4×C29—22no—
118C22⋊C4×C29—41no—
119C22⋊Dic29—22no—
120C22⋊Dic29—22no—
121C22⋊Dic29—41no—
122#90781—22no—
123#90781—22no—
124#90781—22no—
125#90781—22no—
126#90781—22no—
127#90781—22no—
128#90781—41no—
129#90781—41no—
130C23×C29—21no—
131C23×C29—21no—
132C23×C29—21no—
133C23×C29—21no—
134C23×C29—21no—
135C23×C29—21no—
136C23×C29—21no—
137C23×C29—21no—
138C23×C29—21no—
139C23×C29—21no—
140C23×C29—21no—
141C23×C29—21no—
142C23×C29C2×D412yes—
143C23×C29#4112yescomplemented
144C23×C29C2411yescharacteristic
145C22×D29—21no—
146C22×D29—21no—
147C22×D29—21no—
148C22×D29—21no—
149C22×D29—28no—
150C22×D29—21no—
151C22×D29—21no—
152C22×D29—21no—
153C22×D29—21no—
154C22×D29—21no—
155C22×D29—21no—
156C22×D29—21no—
157C22×D29—21no—
158C22×D29—21no—
159C22×D29—21no—
160C22×D29—21no—
161C22×D29—21no—
162C22×D29—44no—
163C22×D29—44no—
164C22×D29—44no—
165C22×D29—44no—
166C22×D29—44no—
167C22×D29—44no—
168C22×D29C2×D412yes—
169C22×D29C2×D412yes—
170C22×D29C2×D412yes—
171C22×D29C2×D412yes—
172C22×D29#4112yescomplemented
173D4×C29—44no—
174C2×C4×C29—28no—
175C2×C4×C29—21no—
176C2×C4×C29—21no—
177C2×C4×C29—21no—
178C2×C4×C29—21no—
179C2×C4×C29—21no—
180C2×C4×C29—21no—
181C2×C4×C29—21no—
182C2×C4×C29—21no—
183C2×C4×C29—21no—
184C2×C4×C29—21no—
185C2×C4×C29—21no—
186C2×C4×C29—21no—
187C2×C4×C29—21no—
188C2×C4×C29C2×D412yes—
189C2×C4×C29C2×D412yes—
190C22⋊D29—28no—
191C22⋊D29—28no—
192C22⋊D29—44no—
193C22⋊D29—44no—
194C2×Dic29—21no—
195C2×Dic29—21no—
196C2×Dic29—21no—
197C2×Dic29—21no—
198C2×Dic29—21no—
199C2×Dic29—21no—
200C2×Dic29—21no—
201C2×Dic29—21no—
202C2×Dic29—21no—
203C2×Dic29—21no—
204C2×Dic29—21no—
205C2×Dic29—21no—
206C2×Dic29—21no—
207C2×Dic29C2×D412yes—
208C2×Dic29C2×D412yes—
209D116—28no—
210D116—44no—
211C4×D29—28no—
212C4×D29—28no—
213C4×D29—44no—
214Dic58—28no—
215#2404—291nomaximal, Sylow
216C22×C29—22no—
217C22×C29—22no—
218C22×C29—22no—
219C22×C29—22no—
220C22×C29—22no—
221C22×C29—22no—
222C22×C29—41no—
223C22×C29—41no—
224C22×C29—41no—
225C22×C29—41no—
226C22×C29—41no—
227C22×C29—41no—
228C22×C29—41no—
229C22×C29—41no—
230C22×C29C22×D411yescharacteristic
231C22×C29C22×D411yescharacteristic
232C22×C29C22×D411yescharacteristic
233C22×C29C2×#4111yescharacteristic
234C22×C29C2×#4111yescharacteristic
235C22×C29H(5,𝔽2)11yescharacteristic
236C22×C29H(5,𝔽2)11yescharacteristic
237C2×D29—22no—
238C2×D29—22no—
239C2×D29—22no—
240C2×D29—22no—
241C2×D29—22no—
242C2×D29—22no—
243C2×D29—22no—
244C2×D29—22no—
245C2×D29—22no—
246C2×D29—22no—
247C2×D29—22no—
248C2×D29—22no—
249C2×D29—416no—
250C2×D29—41no—
251C2×D29—41no—
252C2×D29—41no—
253C2×D29—41no—
254C2×D29—41no—
255C2×D29—41no—
256C2×D29—41no—
257C2×D29—41no—
258C2×D29#12014yes—
259C4×C29—22no—
260C4×C29—22no—
261C4×C29—41no—
262C4×C29—41no—
263C4×C29—41no—
264Dic29—22no—
265Dic29—22no—
266Dic29—41no—
267Dic29—41no—
268Dic29—41no—
269C22×#41—291no—
270C2×#120—291no—
271C2×#120—291no—
272C22×C22⋊C4—291no—
273#393—291no—
274#393—291no—
275#392—291no—
276#392—291no—
277#386—291no—
278#386—291no—
279#386—291no—
280#386—291no—
281#386—291no—
282#386D2911yescharacteristic, complemented
283#381—291no—
284C2×C29—42no—
285C2×C29—42no—
286C2×C29C2×#12011yescharacteristic
287C2×C29#52511yescharacteristic
288C2×C29#53511yescharacteristic
289C2×C29#54611yescharacteristic
290C2×C29#54611yescharacteristic
291C2×C29#54611yescharacteristic
292C2×C29#54611yescharacteristic
293D29—24no—
294D29—42no—
295D29—42no—
296C25—291no—
297C2×#41—584no—
298C2×#41—584no—
299C2×#41—584no—
300C22×Q8—291no—
301C22×D4—291no—
302C22×D4—291no—
303C22×D4—291no—
304C22×D4—291no—
305C22×D4—291no—
306C22×D4—291no—
307C22×D4C2×D2911yescharacteristic
308C23×C4—291no—
309C23×C4—291no—
310C23×C4—291no—
311C23×C4C2×D2911yescharacteristic, complemented
312#120—584no—
313#120—584no—
314C2×C22⋊C4—291no—
315C2×C22⋊C4—291no—
316C2×C22⋊C4—291no—
317C2×C22⋊C4—291no—
318C2×C22⋊C4—291no—
319C2×C22⋊C4—291no—
320C2×C22⋊C4—291no—
321C2×C22⋊C4—291no—
322C2×C22⋊C4—291no—
323C2×C22⋊C4—291no—
324C2×C22⋊C4—291no—
325C2×C22⋊C4—584no—
326C2×C22⋊C4C2×D2911yescharacteristic
327C2×C22⋊C4C2×D2911yescharacteristic
328C2×C22⋊C4C2×D2911yescharacteristic, complemented
329#95—291no—
330#95—291no—
331#95—291no—
332#95—291no—
333#95—291no—
334#95—291no—
335#95C2×D2911yescharacteristic, complemented
336#95C2×D2911yescharacteristic, complemented
337C29#240411yescharacteristic, complemented, Sylow
338C24—291no—
339C24—291no—
340C24—291no—
341C24—582no—
342C24—582no—
343C24—582no—
344C24—582no—
345C24—582no—
346C24—1161no—
347C24—1161no—
348C24C22×D2911yescharacteristic
349C24C22×D2911yescharacteristic
350#41—5832no—
351C2×Q8—582no—
352C2×Q8—582no—
353C2×Q8—582no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—294no—
358C2×D4—294no—
359C2×D4—582no—
360C2×D4—582no—
361C2×D4—582no—
362C2×D4—582no—
363C2×D4—582no—
364C2×D4—582no—
365C2×D4—582no—
366C2×D4—582no—
367C2×D4—582no—
368C2×D4—582no—
369C2×D4—582no—
370C2×D4—582no—
371C2×D4—582no—
372C2×D4—1161no—
373C2×D4—1161no—
374C2×D4—1161no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—291no—
379C22×C4—291no—
380C22×C4—294no—
381C22×C4—291no—
382C22×C4—291no—
383C22×C4—291no—
384C22×C4—582no—
385C22×C4—582no—
386C22×C4—582no—
387C22×C4—582no—
388C22×C4—582no—
389C22×C4—582no—
390C22×C4—1161no—
391C22×C4—1161no—
392C22×C4C22×D2911yescharacteristic
393C22×C4C22×D2911yescharacteristic
394C22×C4C22×D2911yescharacteristic
395C22×C4C22×D2911yescharacteristic
396C22×C4C22×D2911yescharacteristic
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—582no—
401C22⋊C4—582no—
402C22⋊C4—582no—
403C22⋊C4—582no—
404C22⋊C4—582no—
405C22⋊C4—582no—
406C22⋊C4—582no—
407C22⋊C4—582no—
408C22⋊C4—1161no—
409C22⋊C4—1161no—
410C22⋊C4—1161no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—292no—
424C23—292no—
425C23—292no—
426C23—292no—
427C23—292no—
428C23—588no—
429C23—581no—
430C23—581no—
431C23—581no—
432C23—581no—
433C23—581no—
434C23—581no—
435C23—581no—
436C23—581no—
437C23—581no—
438C23—581no—
439C23—581no—
440C23—581no—
441C23—581no—
442C23—581no—
443C23—581no—
444C23—581no—
445C23—1164no—
446C23—1164no—
447C23—1164no—
448C23—1164no—
449C23—1164no—
450C23—1164no—
451C23#9080512yescomplemented
452C23D4×D2912yes—
453C23C23×D2911yescharacteristic
454Q8—588no—
455D4—44no—
456D4—588no—
457D4—588no—
458D4—588no—
459D4—1164no—
460D4—1164no—
461D4—1164no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—292no—
477C2×C4—292no—
478C2×C4—588no—
479C2×C4—588no—
480C2×C4—581no—
481C2×C4—581no—
482C2×C4—581no—
483C2×C4—581no—
484C2×C4—581no—
485C2×C4—581no—
486C2×C4—581no—
487C2×C4—581no—
488C2×C4—581no—
489C2×C4—581no—
490C2×C4—581no—
491C2×C4—581no—
492C2×C4—581no—
493C2×C4—1164no—
494C2×C4D4×D2912yes—
495C2×C4D4×D2912yes—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—294no—
511C22—582no—
512C22—582no—
513C22—582no—
514C22—582no—
515C22—582no—
516C22—582no—
517C22—582no—
518C22—582no—
519C22—582no—
520C22—582no—
521C22—582no—
522C22—582no—
523C22—11616no—
524C22—1161no—
525C22—1161no—
526C22—1161no—
527C22—1161no—
528C22—1161no—
529C22—1161no—
530C22—1161no—
531C22—1161no—
532C22C2×#9080511yescharacteristic
533C22C2×D4×D2911yescharacteristic
534C22C2×D4×D2911yescharacteristic
535C22C2×D4×D2911yescharacteristic
536C22#1174535911yescharacteristic
537C22#41×D2911yescharacteristic
538C22#1174536411yescharacteristic
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—582no—
545C4—582no—
546C4—1161no—
547C4—1161no—
548C4—1161no—
549C2—42no—
550C2—42no—
551C2—584no—
552C2—1162no—
553C2—1162no—
554C2#120×D2911yescharacteristic
555C2#1427167211yescharacteristic
556C2#1427172911yescharacteristic
557C2#1427180311yescharacteristic
558C2#1427180511yescharacteristic
559C2#1427180911yescharacteristic
560C2#1427181011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#386C2
hypercenter#386D29
hypocenterC29#2404
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#90805C2semidirect
2. C2×#11745244C2semidirect
3. C2×#11745299C2semidirect
4. C2×C22⋊C4×D29C2semidirect
5. #14271076C2semidirect
6. #14271075C2semidirect
7. #14271072C2semidirect
8. #14271071C2semidirect
9. #14271042C2semidirect
10. #14271041C2semidirect
11. #14271041C2semidirect
12. #14271039C2semidirect
13. #14271039C2semidirect
14. #14271025C2semidirect
15. C29×#386C2semidirect
18. C23×C4×C29C22semidirect
22. C22×C22⋊D29C22semidirect
30. C22×C4×D29C22semidirect
31. C22×Dic58C22semidirect
34. C2×C22⋊C4×C29C22semidirect
37. C2×C22⋊Dic29C22semidirect
45. C2×#90781C22semidirect
46. C2×#90781C22semidirect
49. C29×#95C22semidirect
50. C29×#95C22semidirect
51. #11745179C22semidirect
52. #11745179C22semidirect
53. #11745179C22semidirect
54. #11745179C22semidirect
55. #11745179C22semidirect
56. #11745179C22semidirect
91. C2×C22⋊D29D4semidirect
103. C2×D116D4semidirect
112. C2×C4×D29D4semidirect
143. C23×C29#41semidirect
172. C22×D29#41semidirect
282. #386D29semidirect
311. C23×C4C2×D29semidirect
328. C2×C22⋊C4C2×D29semidirect
335. #95C2×D29semidirect
336. #95C2×D29semidirect
337. C29#2404semidirect
451. C23#90805semidirect

Maximal subgroups

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

subgroupindex
C22×#908052
C2×#117452442
C2×#117452992
C2×C22⋊C4×D292
#142710762
#142710752
#142710722
#142710712
#142710422
#142710412
#142710392
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C29×#3862
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