Finite Group Databasebeta

← #31968578

Subgroups — #31968578

Order 18176.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598420C211yesmaximal, characteristic
2#22597344C211yesmaximal, characteristic
3#22587247C211yesmaximal, characteristic, complemented
4#22587230C211yesmaximal, characteristic, complemented
5#22587211C211yesmaximal, characteristic, complemented
6#22587195C211yesmaximal, characteristic, complemented
7#22587188C211yesmaximal, characteristic, complemented
8#22587183C211yesmaximal, characteristic, complemented
9#22587150C211yesmaximal, characteristic, complemented
10C8×#11765855C211yesmaximal, characteristic, complemented
11#22587131C211yesmaximal, characteristic, complemented
12#22586803C211yesmaximal, characteristic
13#22586735C211yesmaximal, characteristic, complemented
14#22586637C211yesmaximal, characteristic, complemented
15Q8×M16×C71C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17#17986329C411yescharacteristic
18#17986312C411yescharacteristic
19#17986306C411yescharacteristic
20#17986306C411yescharacteristic
21#17986221C411yescharacteristic
22#17986221C411yescharacteristic
23#17986155C411yescharacteristic
24#17986138C411yescharacteristic
25#17986129C2211yescharacteristic
26C8×Dic142C2211yescharacteristic
27C4×#11765855C2211yescharacteristic
28C4×#11765855C2211yescharacteristic
29C4×#11765852C2211yescharacteristic
30C4×#11765852C2211yescharacteristic
31#17986078C2211yescharacteristic
32#17986075—21no—
33#17986068—21no—
34C8×D284C2211yescharacteristic
35#17986061—21no—
36#17986052C2211yescharacteristic
37#17986052C2211yescharacteristic
38#17986052C2211yescharacteristic
39#17985956—21no—
40#17985751C2211yescharacteristic
41#17985751C2211yescharacteristic
42#17985749C2211yescharacteristic
43#17985749C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46Q8⋊(C71⋊C8)C2211yescharacteristic
47Q8⋊(C71⋊C8)C2211yescharacteristic
48#17985729C2211yescharacteristic
49#17985729C2211yescharacteristic
50#17985726C2211yescharacteristic
51#17985726C2211yescharacteristic
52#17985721C2211yescharacteristic
53#17985721C2211yescharacteristic
54#17985720C2211yescharacteristic
55C8×C71⋊C8C2211yescharacteristic
56C2×C4×Q8×C71C2211yescharacteristic
57#446×C71—21no—
58#446×C71—21no—
59#446×C71C2211yescharacteristic
60#446×C71C2211yescharacteristic
61C8×Q8×C71C2211yescharacteristic
62C8×Q8×C71C2211yescharacteristic
63#423×C71—21no—
64#423×C71C2211yescharacteristic
65C4×M16×C71—21no—
66C4×M16×C71C2211yescharacteristic
67C2×#11765855C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765852C2×C411yescharacteristic
70C2×#11765852C2×C411yescharacteristic
71C2×#11765848—21no—
72C2×C8×D71—21no—
73#14556122—21no—
74#14556120C2×C411yescharacteristic
75#14556116C2×C411yescharacteristic
76#14556111—21no—
77#14556107C2×C411yescharacteristic
78#14556102C2×C411yescharacteristic
79C4×Dic142C2311yescharacteristic
80C4×D284C2311yescharacteristic
81C4×C22⋊D71—21no—
82#14556085—21no—
83#14556061C2×C411yescharacteristic
84#14556061C2×C411yescharacteristic
85#14556059C2×C411yescharacteristic
86#14556059C2×C411yescharacteristic
87C4⋊(C71⋊C8)—21no—
88C4⋊(C71⋊C8)C2×C411yescharacteristic
89C4⋊(C71⋊C8)C2×C411yescharacteristic
90C4⋊(C71⋊C8)C2311yescharacteristic
91C4⋊(C71⋊C8)C2311yescharacteristic
92Dic71⋊C8—21no—
93Dic71⋊C8—21no—
94#14556052C2×C411yescharacteristic
95#14556052C2×C411yescharacteristic
96#14556052C2×C411yescharacteristic
97#14556052C2×C411yescharacteristic
98#14556051C2×C411yescharacteristic
99#14556051C2×C411yescharacteristic
100#14556048C2×C411yescharacteristic
101#14556048C2×C411yescharacteristic
102C22⋊(C71⋊C8)—21no—
103C22⋊(C71⋊C8)—21no—
104C22⋊(C71⋊C8)C2×C411yescharacteristic
105C22⋊(C71⋊C8)C2×C411yescharacteristic
106(C2×D71)⋊C8—21no—
107(C2×D71)⋊C8—21no—
108#14556038—21no—
109C4×C71⋊C8—21no—
110C4×C71⋊C8C2311yescharacteristic
111C4×C71⋊C8C2311yescharacteristic
112C4×C71⋊C8C2311yescharacteristic
113C4×C71⋊C8C2311yescharacteristic
114C8×Dic71—21no—
115C22×Q8×C71C2×C411yescharacteristic
116C2×M16×C71—21no—
117C2×M16×C71D411yescharacteristic
118C2×M16×C71D411yescharacteristic
119C4×Q8×C71—21no—
120C4×Q8×C71—21no—
121C4×Q8×C71C2×C411yescharacteristic
122C4×Q8×C71C2×C411yescharacteristic
123C4×Q8×C71C2311yescharacteristic
124C4×Q8×C71C2311yescharacteristic
125C2×C4⋊C4×C71—21no—
126C2×C4⋊C4×C71C2×C411yescharacteristic
127C2×C42×C71—21no—
128C2×C42×C71C2311yescharacteristic
129C4⋊C8×C71—21no—
130C4⋊C8×C71—21no—
131C4⋊C8×C71—21no—
132C4⋊C8×C71—21no—
133C4⋊C8×C71—21no—
134C4⋊C8×C71C2311yescharacteristic
135C4⋊C8×C71C2311yescharacteristic
136#97×C71—21no—
137#97×C71—21no—
138#97×C71D411yescharacteristic
139#97×C71D411yescharacteristic
140C4×C8×C71—21no—
141C4×C8×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C2×Dic142C22×C411yescharacteristic
145C2×D284C22×C411yescharacteristic
146C2×C22⋊D71—21no—
147C2×C4×D71—21no—
148#11765855—22no—
149#11765855—41no—
150#11765852—22no—
151#11765852—41no—
152#11765848—42no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8C22×C411yescharacteristic
158C2×C71⋊C8C22×C411yescharacteristic
159C2×C71⋊C8C22×C411yescharacteristic
160C2×C71⋊C8C22×C411yescharacteristic
161C8×D71—42no—
162C4⋊Dic71C22×C411yescharacteristic
163C4⋊Dic71C22×C411yescharacteristic
164Dic71⋊C4—21no—
165Dic71⋊C4—21no—
166Dic71⋊C4—21no—
167Dic71⋊C4—21no—
168C22⋊Dic71—21no—
169(C2×D71)⋊C4—21no—
170(C2×D71)⋊C4—21no—
171(C2×D71)⋊C4—21no—
172(C2×D71)⋊C4—21no—
173C4×Dic71—21no—
174C2×Q8×C71—21no—
175C2×Q8×C71—22no—
176C2×Q8×C71—41no—
177C2×Q8×C71C22×C411yescharacteristic
178C2×Q8×C71C22×C411yescharacteristic
179C22×C4×C71—21no—
180C22×C4×C71—21no—
181C22×C4×C71C22×C411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71#4111yescharacteristic
184M16×C71—22no—
185M16×C71—41no—
186M16×C71—42no—
187C2×C8×C71—21no—
188C2×C8×C71—21no—
189C2×C8×C71C2×D411yescharacteristic
190C2×C8×C71C2×D411yescharacteristic
191C2×C8×C71C2×D411yescharacteristic
192C2×C8×C71C2×D411yescharacteristic
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71C22×C411yescharacteristic
199C4⋊C4×C71C22×C411yescharacteristic
200C42×C71—21no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71—21no—
204C42×C71C22×C411yescharacteristic
205C42×C71C2×D411yescharacteristic
206C42×C71#4111yescharacteristic
207C42×C71C2411yescharacteristic
208C23×C71C4×D411yescharacteristic
209C22×D71—21no—
210Q8×C71—21no—
211Q8×C71—21no—
212Q8×C71—21no—
213Q8×C71—42no—
214Q8×C71#13112yes—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—41no—
224C2×C4×C71—42no—
225C2×C4×C71—81no—
226C2×C4×C71C4×D411yescharacteristic
227C2×C4×C71C4×D411yescharacteristic
228C2×C4×C71C4×D411yescharacteristic
229C2×C4×C71C23×C411yescharacteristic
230C2×C4×C71C22×D411yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C22⋊D71—81no—
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236D284—21no—
237D284—21no—
238C4×D71—41no—
239C4×D71—41no—
240Dic142—21no—
241Dic142—21no—
242C8×C71—21no—
243C8×C71—21no—
244C8×C71—21no—
245C8×C71—41no—
246C8×C71—41no—
247C8×C71#13712yes—
248C71⋊C8—21no—
249C71⋊C8—21no—
250C71⋊C8—21no—
251C71⋊C8—21no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C71⋊C8—41no—
256C22×C71—21no—
257C22×C71—21no—
258C22×C71—21no—
259C22×C71C2×C4×D411yescharacteristic
260C2×D71—41no—
261C2×D71—41no—
262C2×D71—41no—
263C4×C71—21no—
264C4×C71—22no—
265C4×C71—21no—
266C4×C71—21no—
267C4×C71—21no—
268C4×C71—41no—
269C4×C71—41no—
270C4×C71—41no—
271C4×C71C2×#13111yescharacteristic
272C4×C71#56811yescharacteristic
273C4×C71C2×#13711yescharacteristic
274C4×C71#57511yescharacteristic
275Dic71—41no—
276Dic71—41no—
277Dic71—41no—
278#18012—711nomaximal, Sylow
279C2×C71—41no—
280C2×C71#293311yescharacteristic
281C2×C71C4×#13711yescharacteristic
282C2×C71#295711yescharacteristic
283D71—81no—
284#3108—711no—
285#2969—711no—
286Q8×M16D7111yescharacteristic, complemented
287#1593—711no—
288#1590—711no—
289#1587—711no—
290#1585—711no—
291#1584—711no—
292#1583—711no—
293SD16⋊C8—711no—
294C8×Q16—711no—
295#1574—711no—
296#1495—711no—
297#1479—711no—
298#1453—711no—
299C71#1801211yescharacteristic, complemented, Sylow
300#529—711no—
301C2×C4×Q8C2×D7111yescharacteristic
302#483—711no—
303#480—711no—
304#479—711no—
305#479—711no—
306#464—711no—
307#464—711no—
308#451—711no—
309#448—711no—
310#446—21no—
311#446—21no—
312#446—711no—
313#446C2×D7111yescharacteristic
314#446C2×D7111yescharacteristic
315C8×Q8—711no—
316C8×Q8C2×D7111yescharacteristic
317C8×Q8C2×D7111yescharacteristic
318C4×Q16—711no—
319C4×Q16—711no—
320C4×SD16—711no—
321C4×SD16—711no—
322#436—711no—
323#436—1421no—
324#435—1421no—
325C8×D4—711no—
326#433—1421no—
327#431—711no—
328#431—711no—
329#431—711no—
330#423—21no—
331#423C2×D7111yescharacteristic
332#405—1421no—
333C4×M16—21no—
334C4×M16C2×D7111yescharacteristic
335#335—711no—
336#335—711no—
337#334—711no—
338#334—711no—
339Q8⋊C8—711no—
340Q8⋊C8—711no—
341Q8⋊C8—711no—
342Q8⋊C8—711no—
343Q8⋊C8—711no—
344Q8⋊C8—711no—
345D4⋊C8—711no—
346D4⋊C8—711no—
347#322—711no—
348#322—711no—
349#322—711no—
350C82—711no—
351C22×Q8C4×D7111yescharacteristic
352C2×Q16—711no—
353C2×Q16—711no—
354C2×SD16—711no—
355C2×SD16—711no—
356C2×M16—21no—
357C2×M16—1421no—
358C2×M16C22⋊D7111yescharacteristic
359C2×M16C22⋊D7111yescharacteristic
360C22×C8—1421no—
361#126—1421no—
362#125—711no—
363#124—711no—
364#123—1421no—
365C22⋊Q8—711no—
366#121—711no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—711no—
370C4×Q8C4×D7111yescharacteristic
371C4×Q8C4×D7111yescharacteristic
372C4×Q8C22×D7111yescharacteristic
373C4×Q8C22×D7111yescharacteristic
374C4×D4—711no—
375C4×D4—1421no—
376#117—1421no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C4×D7111yescharacteristic
379C2×C42—21no—
380C2×C42C22×D7111yescharacteristic
381#107—711no—
382#107—711no—
383#106—711no—
384#106—711no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—711no—
391C4⋊C8—711no—
392C4⋊C8—711no—
393C4⋊C8—711no—
394C4⋊C8—1421no—
395C4⋊C8—1421no—
396C4⋊C8—1421no—
397C4⋊C8C22×D7111yescharacteristic
398C4⋊C8C22×D7111yescharacteristic
399#103—711no—
400#103—711no—
401#103—711no—
402#103—711no—
403#103—711no—
404#103—711no—
405#102—711no—
406#102—711no—
407C22⋊C8—711no—
408C22⋊C8—711no—
409C22⋊C8—1421no—
410C22⋊C8—1421no—
411C22⋊C8—1421no—
412C22⋊C8—1421no—
413#97—21no—
414#97—21no—
415#97—1421no—
416#97C22⋊D7111yescharacteristic
417#97C22⋊D7111yescharacteristic
418C4×C8—21no—
419C4×C8—21no—
420C4×C8—711no—
421C4×C8—711no—
422C4×C8—711no—
423C4×C8—711no—
424C4×C8—1421no—
425C4×C8—1421no—
426C4×C8C22×D7111yescharacteristic
427C4×C8C22×D7111yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—711no—
432C2×Q8C2×C4×D7111yescharacteristic
433C2×Q8C2×C4×D7111yescharacteristic
434C2×D4—711no—
435C2×D4—1421no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—1421no—
439C22×C4C2×C4×D7111yescharacteristic
440C22×C4C2×C22⋊D7111yescharacteristic
441C22×C4#1176586511yescharacteristic
442Q16—2841no—
443Q16—1422no—
444SD16—2841no—
445SD16—1422no—
446M16—22no—
447M16—41no—
448M16—42no—
449M16—2842no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—2842no—
453C2×C8—711no—
454C2×C8—711no—
455C2×C8—711no—
456C2×C8—711no—
457C2×C8—1421no—
458C2×C8—1421no—
459C2×C8—1421no—
460C2×C8—1421no—
461C2×C8C2×C22⋊D7111yescharacteristic
462C2×C8C2×C22⋊D7111yescharacteristic
463C2×C8C2×C22⋊D7111yescharacteristic
464C2×C8C2×C22⋊D7111yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—711no—
471C4⋊C4—711no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4C2×C4×D7111yescharacteristic
477C4⋊C4C2×C4×D7111yescharacteristic
478C22⋊C4—1421no—
479C22⋊C4—1421no—
480C22⋊C4—1421no—
481C22⋊C4—1421no—
482C22⋊C4—1421no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—1421no—
488C42C2×C4×D7111yescharacteristic
489C42C2×C22⋊D7111yescharacteristic
490C42#1176586511yescharacteristic
491C42C23×D7111yescharacteristic
492C23—1421no—
493C23C4×C22⋊D7111yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—1421no—
499Q8—1421no—
500Q8#1455613612yes—
501D4—5681no—
502D4—1421no—
503D4—1421no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—2841no—
516C2×C4—2841no—
517C2×C4—1421no—
518C2×C4—1421no—
519C2×C4—1421no—
520C2×C4C4×C22⋊D7111yescharacteristic
521C2×C4C4×C22⋊D7111yescharacteristic
522C2×C4C4×C22⋊D7111yescharacteristic
523C2×C4C22×C4×D7111yescharacteristic
524C2×C4C22×C22⋊D7111yescharacteristic
525C2×C4C2×#1176586511yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—2841no—
532C8—2841no—
533C8—2841no—
534C8—2841no—
535C8—1421no—
536C8—1421no—
537C8—1421no—
538C8—1421no—
539C8#1455615912yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—2841no—
544C22—2841no—
545C22—2841no—
546C22C2×C4×C22⋊D7111yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—2841no—
556C4—2841no—
557C4—2841no—
558C4C2×#1455613611yescharacteristic
559C4#1798676511yescharacteristic
560C4C2×#1455615911yescharacteristic
561C4#1798679711yescharacteristic
562C2—41no—
563C2—5681no—
564C2#2259706511yescharacteristic
565C2C4×#1455615911yescharacteristic
566C2#2259725511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingQ8×M16×C71C2
socleC22×C71C2×C4×D4
prime radical#14556102C2×C4
cosocleC42×C71C24
hypercenterQ8×M16D71
hypocenterC71#18012
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. #22587247C2semidirect
4. #22587230C2semidirect
5. #22587211C2semidirect
6. #22587195C2semidirect
7. #22587188C2semidirect
8. #22587183C2semidirect
9. #22587150C2semidirect
10. C8×#11765855C2semidirect
11. #22587131C2semidirect
13. #22586735C2semidirect
14. #22586637C2semidirect
15. Q8×M16×C71C2semidirect
286. Q8×M16D71semidirect
299. C71#18012semidirect

Maximal subgroups

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

subgroupindex
#225984202
#225973442
#225872472
#225872302
#225872112
#225871952
#225871882
#225871832
#225871502
C8×#117658552
#225871312
#225868032
#225867352
#225866372
Q8×M16×C712
#1801271