Finite Group Databasebeta

← #32158569

Subgroups — #32158569

Order 18176.

2390 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986481C211yesmaximal, characteristic
2#22591311C213yesmaximal, complemented
3#22591291C213yesmaximal, complemented
4#22591261C211yesmaximal, characteristic, complemented
5#22591216C213yesmaximal, complemented
6#22591208C213yesmaximal, complemented
7#22591178C211yesmaximal, characteristic
8#22591178C211yesmaximal, characteristic
9#22590979C213yesmaximal, complemented
10#22590948C213yesmaximal, complemented
11#22590833C213yesmaximal, complemented
12#22590798C213yesmaximal, complemented
13#22590705C211yesmaximal, characteristic, complemented
14C4×Q8×Dic71C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C71C211yesmaximal, characteristic, complemented
16C22×#11765866C2211yescharacteristic
17#17986481C4116yescomplemented
18C2×Q8×Dic71C2211yescharacteristic
19C2×Q8×Dic71C2211yescharacteristic
20C2×Q8×Dic71C2211yescharacteristic
21C2×C4×D284C2213yes—
22C2×#14556090C2213yes—
23C2×#14556086C2216yes—
24C2×#14556085C2213yes—
25C2×C4⋊C4×D71C2213yes—
26#17985901C2213yes—
27#17985901C2213yes—
28#17985900C2213yes—
29#17985900C2213yes—
30#17985898C2213yes—
31#17985898C2213yes—
32#17985891C2213yes—
33#17985891C2213yes—
34#17985889C2216yes—
35#17985889C2213yes—
36#17985889C2216yes—
37#17985889C2213yes—
38#17985886C2213yes—
39#17985886C2213yes—
40#17985880C2211yescharacteristic
41#17985880C2213yes—
42#17985880C2216yes—
43#17985865C2213yes—
44#17985861C2216yes—
45#17985856C2216yes—
46#17985856C2213yes—
47#17985856C2213yes—
48C4⋊C4×Dic71C2213yes—
49C4⋊C4×Dic71C2213yes—
50C4⋊C4×Dic71C2213yes—
51C4⋊C4×Dic71C2213yes—
52C4×C4⋊Dic71C2213yes—
53C4×C4⋊Dic71C2213yes—
54C4×C4⋊Dic71C2213yes—
55C4×(C2×D71)⋊C4C2213yes—
56C4×(C2×D71)⋊C4C2213yes—
57#17985846C2216yes—
58#17985845C2213yes—
59C42×Dic71C2213yes—
60C2×C4×Q8×C71C2211yescharacteristic
61C2×C4×Q8×C71C2211yescharacteristic
62C2×C4×Q8×C71C2211yescharacteristic
63#389×C71C2213yes—
64#389×C71C2213yes—
65#384×C71C2213yes—
66C4×C4⋊C4×C71C2213yes—
67C2×#11765866—24no—
68C2×#11765866—24no—
69C2×#11765866C2×C418yes—
70C22×D284C2313yes—
71C22×C4×D71C2313yes—
72C22×C4×D71C2316yes—
73Q8×Dic71—24no—
74Q8×Dic71—24no—
75Q8×Dic71C2×C418yes—
76C4×D284C2×C4124yes—
77#14556090C2×C4124yes—
78#14556086—412no—
79#14556085C2×C4124yes—
80C2×C4⋊Dic71C2313yes—
81C2×C4⋊Dic71C2313yes—
82C2×C4⋊Dic71C2313yes—
83C2×C4⋊Dic71C2313yes—
84C2×C4⋊Dic71C2313yes—
85C2×Dic71⋊C4C2311yescharacteristic
86C2×Dic71⋊C4C2313yes—
87C4⋊C4×D71C2×C4124yes—
88C2×(C2×D71)⋊C4C2313yes—
89C2×(C2×D71)⋊C4C2313yes—
90C2×(C2×D71)⋊C4C2316yes—
91C2×(C2×D71)⋊C4C2313yes—
92C2×(C2×D71)⋊C4C2313yes—
93C2×(C2×D71)⋊C4C2316yes—
94C2×(C2×D71)⋊C4C2313yes—
95C2×(C2×D71)⋊C4C2313yes—
96C2×C4×Dic71C2311yescharacteristic
97C2×C4×Dic71C2313yes—
98C2×C4×Dic71C2313yes—
99C2×C4×Dic71C2313yes—
100C2×C4×Dic71C2313yes—
101C2×C4×Dic71C2313yes—
102C2×C4×Dic71C2316yes—
103C2×C4×Dic71C2313yes—
104C71⋊#95C2311yescharacteristic
105C71⋊#95C2313yes—
106C71⋊#95C2313yes—
107C71⋊#95C2313yes—
108C71⋊#95C2316yes—
109C71⋊#95C2311yescharacteristic
110C71⋊#95C2311yescharacteristic
111C71⋊#95C2313yes—
112C71⋊#95C2313yes—
113C71⋊#95C2311yescharacteristic
114C71⋊#95C2316yes—
115C71⋊#95C2313yes—
116C22×Q8×C71C2311yescharacteristic
117C4×Q8×C71—24no—
118C4×Q8×C71—24no—
119C4×Q8×C71C2×C418yes—
120C2×C4⋊C4×C71C2313yes—
121C2×C4⋊C4×C71C2313yes—
122C2×C4⋊C4×C71C2311yescharacteristic
123C2×C4⋊C4×C71C2316yes—
124C2×C4⋊C4×C71C2313yes—
125C2×C4⋊C4×C71C2313yes—
126C2×C42×C71C2313yes—
127C2×C42×C71C2313yes—
128C2×C42×C71C2313yes—
129#95×C71C2313yes—
130#95×C71C2313yes—
131C23×D71C2413yes—
132#11765866—232no—
133C2×D284—26no—
134C2×D284—26no—
135C2×D284C22×C4112yes—
136C22×Dic71C2411yescharacteristic
137C22×Dic71C2413yes—
138C22×Dic71C2413yes—
139C22×Dic71C2411yescharacteristic
140C22×Dic71C2413yes—
141C22×Dic71C2411yescharacteristic
142C22×Dic71C2411yescharacteristic
143C2×C4×D71—26no—
144C2×C4×D71—26no—
145C2×C4×D71—46no—
146C2×C4×D71—46no—
147C2×C4×D71C22×C4112yes—
148C2×C4×D71C22×C4124yes—
149C4⋊Dic71—26no—
150C4⋊Dic71—26no—
151C4⋊Dic71—26no—
152C4⋊Dic71—43no—
153C4⋊Dic71C22×C4112yes—
154Dic71⋊C4C22×C4112yes—
155Dic71⋊C4C22×C414yes—
156(C2×D71)⋊C4—26no—
157(C2×D71)⋊C4—26no—
158(C2×D71)⋊C4—46no—
159(C2×D71)⋊C4—43no—
160(C2×D71)⋊C4—43no—
161(C2×D71)⋊C4—46no—
162(C2×D71)⋊C4C22×C4112yes—
163(C2×D71)⋊C4C22×C4112yes—
164C4×Dic71—26no—
165C4×Dic71—26no—
166C4×Dic71—22no—
167C4×Dic71—26no—
168C4×Dic71—43no—
169C4×Dic71—43no—
170C4×Dic71—46no—
171C4×Dic71C22×C4112yes—
172C2×Q8×C71C22×C414yes—
173C2×Q8×C71#4114yes—
174C2×Q8×C71#4114yes—
175C22×C4×C71C2411yescharacteristic
176C22×C4×C71C2413yes—
177C22×C4×C71C2413yes—
178C22×C4×C71C2413yes—
179C22×C4×C71C2411yescharacteristic
180C22×C4×C71C2411yescharacteristic
181C22×C4×C71C2413yes—
182C4⋊C4×C71—26no—
183C4⋊C4×C71—26no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—43no—
186C4⋊C4×C71—46no—
187C4⋊C4×C71C22×C4112yes—
188C42×C71—26no—
189C42×C71—26no—
190C42×C71C22×C4112yes—
191C23×C71C2511yescharacteristic
192C22×D71—23no—
193C22×D71—23no—
194C22×D71—23no—
195C22×D71—23no—
196C22×D71—23no—
197C22×D71—23no—
198C22×D71C23×C416yes—
199Q8×C71#117116yescomplemented
200C2×C4×C71—21no—
201C2×C4×C71—23no—
202C2×C4×C71—21no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—23no—
206C2×C4×C71—21no—
207C2×C4×C71—23no—
208C2×C4×C71—23no—
209C2×C4×C71—23no—
210C2×C4×C71—23no—
211C2×C4×C71—21no—
212C2×C4×C71—23no—
213C2×C4×C71—23no—
214C2×C4×C71—21no—
215C2×C4×C71C23×C412yes—
216C2×C4×C71C23×C416yes—
217C2×C4×C71C23×C416yes—
218C2×C4×C71C2×#4116yes—
219C2×C4×C71C2×#4116yes—
220C2×C4×C712−1+412yes—
221C2×Dic71—21no—
222C2×Dic71—21no—
223C2×Dic71—21no—
224C2×Dic71—23no—
225C2×Dic71—21no—
226C2×Dic71—21no—
227C2×Dic71—21no—
228C2×Dic71—23no—
229C2×Dic71—21no—
230C2×Dic71—23no—
231C2×Dic71—23no—
232C2×Dic71—23no—
233C2×Dic71—23no—
234C2×Dic71—23no—
235C2×Dic71—23no—
236C2×Dic71—21no—
237C2×Dic71—21no—
238C2×Dic71C23×C412yes—
239C2×Dic71C23×C416yes—
240C2×Dic71C23×C412yes—
241C2×Dic71H(5,𝔽2)12yes—
242D284—224no—
243C4×D71—224no—
244C4×D71—424no—
245C22×C71C24×C411yescharacteristic
246C22×C71C22×#4111yescharacteristic
247C22×C71C22×#4111yescharacteristic
248C22×C71C2×H(5,𝔽2)11yescharacteristic
249C22×C71C2×2−1+411yescharacteristic
250C22×C71#58511yescharacteristic
251C22×C71#58511yescharacteristic
252C2×D71—26no—
253C2×D71—26no—
254C2×D71—26no—
255C2×D71—43no—
256C2×D71—43no—
257C2×D71—43no—
258C2×D71—43no—
259C4×C71—26no—
260C4×C71—22no—
261C4×C71—22no—
262C4×C71—41no—
263C4×C71—43no—
264C4×C71—43no—
265C4×C71C2×#117112yes—
266Dic71—22no—
267Dic71—26no—
268Dic71—22no—
269Dic71—22no—
270Dic71—43no—
271Dic71—41no—
272Dic71—43no—
273#34263—711nomaximal, Sylow
274C2×C71C22×#11711yescharacteristic
275C2×C71C4×H(5,𝔽2)11yescharacteristic
276C2×C71C4×2−1+411yescharacteristic
277C2×C71#351611yescharacteristic
278C2×C71#351711yescharacteristic
279C2×C71#352211yescharacteristic
280C2×C71#352411yescharacteristic
281D71—46no—
282C2×#520—711no—
283#2357—713no—
284#2354—713no—
285#2351—711no—
286Q8×C4⋊C4D7111yescharacteristic, complemented
287#2345—713no—
288#2344—713no—
289#2341—711no—
290#2341—711no—
291#2319—713no—
292#2314—713no—
293#2298—713no—
294#2293—713no—
295#2276—711no—
296C42×Q8—711no—
297C71#3426311yescharacteristic, complemented, Sylow
298C22×#41—711no—
299#520—7116no—
300C2×C4×Q8—711no—
301C2×C4×Q8—711no—
302C2×C4×Q8—711no—
303C2×C4×Q8C2×D7111yescharacteristic
304C2×C4×Q8C2×D7111yescharacteristic
305C2×C4×Q8C2×D7111yescharacteristic
306C2×C4×D4—713no—
307C2×C4×D4—713no—
308C2×#117—716no—
309C2×#117—713no—
310C22×C4⋊C4—713no—
311#390—713no—
312#390—713no—
313#390—713no—
314#390—713no—
315#389—713no—
316#389—713no—
317#389C2×D7113yes—
318#389C2×D7113yes—
319#388—716no—
320#388—716no—
321#388—713no—
322#388—713no—
323#388—713no—
324#388—713no—
325#387—713no—
326#387—716no—
327#387—711no—
328#387—713no—
329#387—713no—
330#384C2×D7113yes—
331#382—713no—
332#381—716no—
333#380—713no—
334#380—713no—
335#380—716no—
336C4×C4⋊C4—713no—
337C4×C4⋊C4—713no—
338C4×C4⋊C4—713no—
339C4×C4⋊C4—713no—
340C4×C4⋊C4—713no—
341C4×C4⋊C4—713no—
342C4×C4⋊C4—713no—
343C4×C4⋊C4C2×D7113yes—
344C4×C22⋊C4—713no—
345C4×C22⋊C4—713no—
346C4×C22⋊C4—716no—
347#376—713no—
348C43—713no—
349C2×#41—718no—
350C2×#41—1424no—
351C2×#41—1424no—
352C22×Q8C22×D7111yescharacteristic
353C22×D4—713no—
354C23×C4—716no—
355C23×C4—713no—
356C4×Q8—24no—
357C4×Q8—24no—
358C4×Q8—718no—
359C4×Q8—1424no—
360C4×Q8—1424no—
361C4×Q8C4×D7118yes—
362C4×D4—7124no—
363C4×D4—7124no—
364#117—28412no—
365#117—7124no—
366C2×C4⋊C4—713no—
367C2×C4⋊C4—713no—
368C2×C4⋊C4—713no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—7124no—
371C2×C4⋊C4—713no—
372C2×C4⋊C4—713no—
373C2×C4⋊C4—713no—
374C2×C4⋊C4C22×D7113yes—
375C2×C4⋊C4C22×D7113yes—
376C2×C4⋊C4C22×D7111yescharacteristic
377C2×C4⋊C4C22×D7116yes—
378C2×C4⋊C4C22×D7113yes—
379C2×C4⋊C4C22×D7113yes—
380C2×C22⋊C4—716no—
381C2×C22⋊C4—713no—
382C2×C22⋊C4—716no—
383C2×C22⋊C4—713no—
384C2×C22⋊C4—713no—
385C2×C22⋊C4—713no—
386C2×C22⋊C4—713no—
387C2×C22⋊C4—713no—
388C2×C42—716no—
389C2×C42—713no—
390C2×C42—713no—
391C2×C42—713no—
392C2×C42—713no—
393C2×C42—713no—
394C2×C42—713no—
395C2×C42—711no—
396C2×C42C22×D7113yes—
397C2×C42C22×D7113yes—
398C2×C42C22×D7113yes—
399#95—711no—
400#95—713no—
401#95—713no—
402#95—713no—
403#95—711no—
404#95—711no—
405#95—713no—
406#95—713no—
407#95—716no—
408#95—716no—
409#95—711no—
410#95—713no—
411#95C22×D7113yes—
412#95C22×D7113yes—
413C24—713no—
414#41—14232no—
415C2×Q8C2×C4×D7114yes—
416C2×Q8#1176586414yes—
417C2×Q8#1176586614yes—
418C2×D4—7112no—
419C2×D4—1426no—
420C2×D4—1426no—
421C22×C4—2846no—
422C22×C4—2846no—
423C22×C4—7124no—
424C22×C4—711no—
425C22×C4—711no—
426C22×C4—713no—
427C22×C4—711no—
428C22×C4—7112no—
429C22×C4—711no—
430C22×C4—713no—
431C22×C4—713no—
432C22×C4—1426no—
433C22×C4—1426no—
434C22×C4C23×D7111yescharacteristic
435C22×C4C23×D7113yes—
436C22×C4C23×D7113yes—
437C22×C4C23×D7111yescharacteristic
438C22×C4C23×D7113yes—
439C22×C4C23×D7111yescharacteristic
440C22×C4C23×D7113yes—
441C4⋊C4—26no—
442C4⋊C4—26no—
443C4⋊C4—22no—
444C4⋊C4—43no—
445C4⋊C4—46no—
446C4⋊C4—2843no—
447C4⋊C4—7112no—
448C4⋊C4—714no—
449C4⋊C4—7112no—
450C4⋊C4—1426no—
451C4⋊C4—1426no—
452C4⋊C4—1426no—
453C4⋊C4C2×C4×D71112yes—
454C22⋊C4—2843no—
455C22⋊C4—2846no—
456C22⋊C4—2843no—
457C22⋊C4—2846no—
458C22⋊C4—7112no—
459C22⋊C4—7112no—
460C22⋊C4—1426no—
461C22⋊C4—1426no—
462C42—26no—
463C42—26no—
464C42—2843no—
465C42—2843no—
466C42—2846no—
467C42—7112no—
468C42—1426no—
469C42—1426no—
470C42—1426no—
471C42—1422no—
472C42C2×C4×D71112yes—
473C23—716no—
474C23—1423no—
475C23—1423no—
476C23—1423no—
477C23—1423no—
478C23—1423no—
479C23—1423no—
480C23C24×D7111yescharacteristic
481Q8#14556086116yescomplemented
482D4—14224no—
483C2×C4—21no—
484C2×C4—23no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—23no—
489C2×C4—21no—
490C2×C4—23no—
491C2×C4—23no—
492C2×C4—23no—
493C2×C4—23no—
494C2×C4—21no—
495C2×C4—23no—
496C2×C4—23no—
497C2×C4—21no—
498C2×C4—28424no—
499C2×C4—712no—
500C2×C4—716no—
501C2×C4—712no—
502C2×C4—712no—
503C2×C4—1421no—
504C2×C4—1423no—
505C2×C4—1423no—
506C2×C4—1423no—
507C2×C4—1423no—
508C2×C4—1423no—
509C2×C4—1421no—
510C2×C4—1423no—
511C2×C4—1421no—
512C2×C4—1421no—
513C2×C4—1421no—
514C2×C4—1421no—
515C2×C4—1423no—
516C2×C4—1421no—
517C2×C4—1421no—
518C2×C4—1421no—
519C2×C4—1423no—
520C2×C4—14224no—
521C2×C4C22×C4×D7112yes—
522C2×C4C22×C4×D7116yes—
523C2×C4C22×C4×D7116yes—
524C2×C4C2×#1176586416yes—
525C2×C4C2×#1176586616yes—
526C2×C4#1455617612yes—
527C22—2843no—
528C22—2843no—
529C22—2843no—
530C22—2843no—
531C22—1426no—
532C22—1426no—
533C22—1426no—
534C22C23×C4×D7111yescharacteristic
535C22C22×#1176586411yescharacteristic
536C22C22×#1176586611yescharacteristic
537C22H(5,𝔽2)×D7111yescharacteristic
538C22C2×#1455617611yescharacteristic
539C22#1798683511yescharacteristic
540C22#1798683611yescharacteristic
541C4—26no—
542C4—22no—
543C4—22no—
544C4—41no—
545C4—43no—
546C4—43no—
547C4—2843no—
548C4—2841no—
549C4—2843no—
550C4—1422no—
551C4—1426no—
552C4—1422no—
553C4—1422no—
554C4C2×#14556086112yes—
555C2—2846no—
556C2C22×#1455608611yescharacteristic
557C2#2260154611yescharacteristic
558C2C4×#1455617611yescharacteristic
559C2#2260233411yescharacteristic
560C2#2260233711yescharacteristic
561C2#2260242611yescharacteristic
562C2#2260245211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingQ8×C4⋊C4×C71C2
socle, cosocleC23×C71C25
prime radicalC22×#11765866C22
hypercenterQ8×C4⋊C4D71
hypocenterC71#34263
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22591311C2semidirect
3. #22591291C2semidirect
4. #22591261C2semidirect
5. #22591216C2semidirect
6. #22591208C2semidirect
9. #22590979C2semidirect
10. #22590948C2semidirect
11. #22590833C2semidirect
12. #22590798C2semidirect
13. #22590705C2semidirect
14. C4×Q8×Dic71C2semidirect
15. Q8×C4⋊C4×C71C2semidirect
17. #17986481C4semidirect
199. Q8×C71#117semidirect
286. Q8×C4⋊C4D71semidirect
297. C71#34263semidirect
481. Q8#14556086semidirect

Maximal subgroups

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

subgroupindex
C2×#179864812
#225913112
#225912912
#225912612
#225912162
#225912082
#225911782
#225909792
#225909482
#225908332
#225907982
#225907052
C4×Q8×Dic712
Q8×C4⋊C4×C712
#3426371