Finite Group Databasebeta

← #31968366

Subgroups — #31968366

Order 18176.

618 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598360C211yesmaximal, characteristic
2#22597355C211yesmaximal, characteristic, complemented
3#22597344C211yesmaximal, characteristic, complemented
4#22587238C211yesmaximal, characteristic, complemented
5#22587232C211yesmaximal, characteristic, complemented
6#22587192C211yesmaximal, characteristic, complemented
7#22587178C211yesmaximal, characteristic, complemented
8#22587178C211yesmaximal, characteristic, complemented
9#22587171C211yesmaximal, characteristic, complemented
10C8×#11765854C211yesmaximal, characteristic, complemented
11C8×D568C211yesmaximal, characteristic, complemented
12#22587120C211yesmaximal, characteristic, complemented
13#22586801C211yesmaximal, characteristic, complemented
14#22586723C211yesmaximal, characteristic, complemented
15C8×M16×C71C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17C4×#11765865C2211yescharacteristic
18#17986341C411yescharacteristic
19#17986324C411yescharacteristic
20#17986307C411yescharacteristic
21#17986300C411yescharacteristic
22#17986222C411yescharacteristic
23#17986215C411yescharacteristic
24#17986163C411yescharacteristic
25#17986146C411yescharacteristic
26#17986129C2211yescharacteristic
27#17986129C2211yescharacteristic
28#17986129C2211yescharacteristic
29C8×Dic142C2211yescharacteristic
30C4×Dic284C2211yescharacteristic
31C4×#11765854C2211yescharacteristic
32C4×#11765854C2211yescharacteristic
33C4×D568C2211yescharacteristic
34#17986078C2211yescharacteristic
35#17986075—21no—
36#17986068—21no—
37#17986068—21no—
38#17986068—21no—
39C8×D284C2211yescharacteristic
40C8×D284C2211yescharacteristic
41C8×D284C2211yescharacteristic
42#17986061—21no—
43#17986056C2211yescharacteristic
44#17986054—21no—
45#17986052C2211yescharacteristic
46#17985956—21no—
47#97×D71—21no—
48#17985752C2211yescharacteristic
49#17985752C2211yescharacteristic
50#17985750C2211yescharacteristic
51#17985750C2211yescharacteristic
52#17985731C2211yescharacteristic
53#17985731C2211yescharacteristic
54#17985731C2211yescharacteristic
55#17985731C2211yescharacteristic
56#17985728C2211yescharacteristic
57#17985728C2211yescharacteristic
58#17985728C2211yescharacteristic
59#17985728C2211yescharacteristic
60#431×C71C2211yescharacteristic
61C4×M16×C71C2211yescharacteristic
62C2×C4×C8×C71C2211yescharacteristic
63#322×C71C2211yescharacteristic
64#322×C71C2211yescharacteristic
65C82×C71C2211yescharacteristic
66C82×C71C2211yescharacteristic
67C2×#11765865C2×C411yescharacteristic
68C2×Dic284C2×C411yescharacteristic
69C2×#11765854C2×C411yescharacteristic
70C2×#11765854C2×C411yescharacteristic
71C2×D568C2×C411yescharacteristic
72C2×#11765848—21no—
73C2×C8×D71—21no—
74C2×C8×D71—21no—
75C2×C8×D71—21no—
76#14556122—21no—
77#14556120C2×C411yescharacteristic
78#14556116C2×C411yescharacteristic
79#14556111—21no—
80#14556107C2×C411yescharacteristic
81#14556102C2×C411yescharacteristic
82C4×Dic142C2×C411yescharacteristic
83C4×Dic142C2311yescharacteristic
84C4×Dic142C2311yescharacteristic
85C4×D284C2×C411yescharacteristic
86C4×D284C2311yescharacteristic
87C4×D284C2311yescharacteristic
88C4×C22⋊D71—21no—
89C4×C22⋊D71—21no—
90C4×C22⋊D71—21no—
91#14556087C2×C411yescharacteristic
92#14556085—21no—
93#14556085—21no—
94C42×D71—21no—
95#14556062C2×C411yescharacteristic
96#14556062C2×C411yescharacteristic
97#14556060C2×C411yescharacteristic
98#14556060C2×C411yescharacteristic
99C4⋊(C71⋊C8)C2311yescharacteristic
100C4⋊(C71⋊C8)C2311yescharacteristic
101C4⋊(C71⋊C8)C2311yescharacteristic
102C4⋊(C71⋊C8)C2311yescharacteristic
103Dic71⋊C8—21no—
104Dic71⋊C8—21no—
105Dic71⋊C8—21no—
106Dic71⋊C8—21no—
107#14556053C2×C411yescharacteristic
108#14556053C2×C411yescharacteristic
109#14556053C2×C411yescharacteristic
110#14556053C2×C411yescharacteristic
111#14556050C2×C411yescharacteristic
112#14556050C2×C411yescharacteristic
113#14556050C2×C411yescharacteristic
114#14556050C2×C411yescharacteristic
115C22⋊(C71⋊C8)—21no—
116C22⋊(C71⋊C8)—21no—
117(C2×D71)⋊C8—21no—
118(C2×D71)⋊C8—21no—
119(C2×D71)⋊C8—21no—
120(C2×D71)⋊C8—21no—
121#14556039—21no—
122#14556038—21no—
123#14556038—21no—
124#14556038—21no—
125C4×C71⋊C8—21no—
126C8×Dic71—21no—
127C2×M16×C71D411yescharacteristic
128C2×M16×C71D411yescharacteristic
129C22×C8×C71—21no—
130C2×C42×C71C2311yescharacteristic
131C4⋊C8×C71C2×C411yescharacteristic
132C4⋊C8×C71C2×C411yescharacteristic
133C22⋊C8×C71C2×C411yescharacteristic
134C22⋊C8×C71C2×C411yescharacteristic
135#97×C71D411yescharacteristic
136#97×C71D411yescharacteristic
137C4×C8×C71—21no—
138C4×C8×C71C2311yescharacteristic
139C4×C8×C71C2311yescharacteristic
140C4×C8×C71C2311yescharacteristic
141C4×C8×C71C2311yescharacteristic
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144#11765865—22no—
145#11765865—41no—
146C2×Dic142C22×C411yescharacteristic
147C2×Dic142C22×C411yescharacteristic
148C2×D284C22×C411yescharacteristic
149C2×D284C22×C411yescharacteristic
150C2×C22⋊D71—21no—
151C2×C22⋊D71—21no—
152C2×C4×D71—21no—
153C2×C4×D71—21no—
154C2×C4×D71—21no—
155C2×C4×D71—21no—
156Dic284—41no—
157#11765854—22no—
158#11765854—41no—
159D568—22no—
160#11765848—42no—
161C2×C71⋊C8—21no—
162C2×C71⋊C8—21no—
163C2×C71⋊C8—21no—
164C2×C71⋊C8—21no—
165C8×D71—42no—
166C8×D71—42no—
167C8×D71—42no—
168C4⋊Dic71C22×C411yescharacteristic
169C4⋊Dic71C22×C411yescharacteristic
170C4⋊Dic71C22×C411yescharacteristic
171C4⋊Dic71C22×C411yescharacteristic
172Dic71⋊C4—21no—
173Dic71⋊C4—21no—
174Dic71⋊C4—21no—
175Dic71⋊C4—21no—
176Dic71⋊C4—21no—
177Dic71⋊C4—21no—
178C22⋊Dic71—21no—
179C22⋊Dic71—21no—
180(C2×D71)⋊C4—21no—
181(C2×D71)⋊C4—21no—
182(C2×D71)⋊C4—21no—
183(C2×D71)⋊C4—21no—
184(C2×D71)⋊C4—21no—
185(C2×D71)⋊C4—21no—
186C4×Dic71—21no—
187C4×Dic71—21no—
188C4×Dic71—21no—
189C4×Dic71—21no—
190C22×C4×C71C22×C411yescharacteristic
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71#4111yescharacteristic
193M16×C71—22no—
194M16×C71—22no—
195C2×C8×C71—21no—
196C2×C8×C71—21no—
197C2×C8×C71—42no—
198C2×C8×C71C22×C411yescharacteristic
199C2×C8×C71C22×C411yescharacteristic
200C2×C8×C71C22×C411yescharacteristic
201C2×C8×C71C22×C411yescharacteristic
202C2×C8×C71C2×D411yescharacteristic
203C2×C8×C71C2×D411yescharacteristic
204C2×C8×C71C2×D411yescharacteristic
205C2×C8×C71C2×D411yescharacteristic
206C42×C71C22×C411yescharacteristic
207C42×C71C2×D411yescharacteristic
208C42×C71#4111yescharacteristic
209C42×C71C2411yescharacteristic
210C23×C71C4×D411yescharacteristic
211C22×D71—21no—
212C22×D71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—41no—
218C2×C4×C71C4×D411yescharacteristic
219C2×C4×C71C4×D411yescharacteristic
220C2×C4×C71C4×D411yescharacteristic
221C2×C4×C71C23×C411yescharacteristic
222C2×C4×C71C22×D411yescharacteristic
223C2×C4×C71C2×#4111yescharacteristic
224C22⋊D71—42no—
225C22⋊D71—81no—
226C2×Dic71—21no—
227C2×Dic71—21no—
228C2×Dic71—21no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232D284—21no—
233D284—21no—
234D284—21no—
235D284#13112yes—
236C4×D71—42no—
237C4×D71—41no—
238C4×D71—41no—
239C4×D71—41no—
240C4×D71—41no—
241C4×D71—81no—
242Dic142—21no—
243Dic142—21no—
244Dic142—21no—
245Dic142—21no—
246C8×C71—22no—
247C8×C71—21no—
248C8×C71—21no—
249C8×C71—21no—
250C8×C71—21no—
251C8×C71—21no—
252C8×C71—21no—
253C8×C71—41no—
254C8×C71#13512yes—
255C8×C71#13512yes—
256C71⋊C8—41no—
257C71⋊C8—41no—
258C71⋊C8—41no—
259C71⋊C8—41no—
260C22×C71—21no—
261C22×C71—21no—
262C22×C71—21no—
263C22×C71C2×C4×D411yescharacteristic
264C2×D71—22no—
265C2×D71—41no—
266C2×D71—41no—
267C2×D71—41no—
268C2×D71—41no—
269C2×D71—41no—
270C4×C71—21no—
271C4×C71—21no—
272C4×C71—21no—
273C4×C71—21no—
274C4×C71C2×#13111yescharacteristic
275C4×C71#56811yescharacteristic
276C4×C71C2×#13511yescharacteristic
277C4×C71C2×#13511yescharacteristic
278Dic71—41no—
279Dic71—41no—
280Dic71—41no—
281Dic71—41no—
282Dic71—41no—
283Dic71—41no—
284#17996—711nomaximal, Sylow
285C2×C71—41no—
286C2×C71#293311yescharacteristic
287C2×C71C4×#13511yescharacteristic
288C2×C71#295611yescharacteristic
289D71—42no—
290D71—81no—
291#3102—711no—
292#2970—711no—
293#2969—711no—
294#1591—711no—
295#1590—711no—
296#1584—711no—
297#1582—711no—
298#1582—711no—
299#1581—711no—
300C8×SD16—711no—
301C8×D8—711no—
302#1572—711no—
303#1494—711no—
304#1476—711no—
305C8×M16D7111yescharacteristic, complemented
306C71#1799611yescharacteristic, complemented, Sylow
307#529—711no—
308C4×#41—711no—
309#485—711no—
310#482—711no—
311#479—711no—
312#478—711no—
313#464—711no—
314#463—711no—
315#452—711no—
316#449—711no—
317#446—711no—
318#446—711no—
319#446—711no—
320C8×Q8—711no—
321C4×Q16—711no—
322C4×SD16—711no—
323C4×SD16—711no—
324C4×D8—711no—
325#436—711no—
326#436—1421no—
327#435—1421no—
328#435—1421no—
329#435—1421no—
330C8×D4—711no—
331C8×D4—711no—
332C8×D4—711no—
333#433—1421no—
334#432—711no—
335#432—1421no—
336#431—711no—
337#431C2×D7111yescharacteristic
338#405—1421no—
339C4×M16C2×D7111yescharacteristic
340C2×#97—1421no—
341C2×C4×C8C2×D7111yescharacteristic
342#335—711no—
343#335—711no—
344#334—711no—
345#334—711no—
346Q8⋊C8—711no—
347Q8⋊C8—711no—
348Q8⋊C8—711no—
349Q8⋊C8—711no—
350D4⋊C8—711no—
351D4⋊C8—711no—
352D4⋊C8—711no—
353D4⋊C8—711no—
354#322C2×D7111yescharacteristic
355#322C2×D7111yescharacteristic
356C82C2×D7111yescharacteristic
357C82C2×D7111yescharacteristic
358C2×#41—711no—
359C2×Q16—711no—
360C2×SD16—711no—
361C2×SD16—711no—
362C2×D8—711no—
363C2×M16—1421no—
364C2×M16D28411yescharacteristic
365C2×M16D28411yescharacteristic
366C22×C8—21no—
367C22×C8—1421no—
368C22×C8—1421no—
369C22×C8—1421no—
370#126—1421no—
371#125—711no—
372#124—711no—
373#123—1421no—
374C22⋊Q8—711no—
375#121—711no—
376C4×Q8—711no—
377C4×Q8—711no—
378C4×Q8—711no—
379C4×D4—711no—
380C4×D4—711no—
381C4×D4—711no—
382C4×D4—1421no—
383C4×D4—1421no—
384C4×D4—1421no—
385#117—711no—
386#117—1421no—
387#117—1421no—
388C2×C42—1421no—
389C2×C42C22×D7111yescharacteristic
390#107—711no—
391#107—711no—
392#106—711no—
393#106—711no—
394C4⋊C8—711no—
395C4⋊C8—711no—
396C4⋊C8—711no—
397C4⋊C8—711no—
398C4⋊C8—1421no—
399C4⋊C8—1421no—
400C4⋊C8—1421no—
401C4⋊C8—1421no—
402C4⋊C8C4×D7111yescharacteristic
403C4⋊C8C4×D7111yescharacteristic
404#103—711no—
405#103—711no—
406#103—711no—
407#103—711no—
408#102—711no—
409#102—711no—
410#102—711no—
411#102—711no—
412C22⋊C8—1421no—
413C22⋊C8—1421no—
414C22⋊C8—1421no—
415C22⋊C8—1421no—
416C22⋊C8—1421no—
417C22⋊C8—1421no—
418C22⋊C8C4×D7111yescharacteristic
419C22⋊C8C4×D7111yescharacteristic
420#97—1421no—
421#97—1421no—
422#97—1421no—
423#97—1421no—
424#97D28411yescharacteristic
425#97D28411yescharacteristic
426C4×C8—21no—
427C4×C8—1421no—
428C4×C8—1421no—
429C4×C8C22×D7111yescharacteristic
430C4×C8C22×D7111yescharacteristic
431C4×C8C22×D7111yescharacteristic
432C4×C8C22×D7111yescharacteristic
433C4×C8C22×D7111yescharacteristic
434C4×C8C22×D7111yescharacteristic
435#41—2841no—
436#41—1422no—
437C2×Q8—711no—
438C2×Q8—711no—
439C2×D4—711no—
440C2×D4—711no—
441C2×D4—1421no—
442C2×D4—1421no—
443C22×C4—1421no—
444C22×C4—1421no—
445C22×C4—1421no—
446C22×C4—1421no—
447C22×C4C2×C4×D7111yescharacteristic
448C22×C4C2×D28411yescharacteristic
449C22×C4#1176586511yescharacteristic
450Q16—2841no—
451SD16—2841no—
452SD16—1422no—
453D8—1422no—
454M16—22no—
455M16—22no—
456M16—2842no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—42no—
460C2×C8—2842no—
461C2×C8—2842no—
462C2×C8—2842no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8C2×C4×D7111yescharacteristic
468C2×C8C2×C4×D7111yescharacteristic
469C2×C8C2×C4×D7111yescharacteristic
470C2×C8C2×C4×D7111yescharacteristic
471C2×C8C2×D28411yescharacteristic
472C2×C8C2×D28411yescharacteristic
473C2×C8C2×D28411yescharacteristic
474C2×C8C2×D28411yescharacteristic
475C4⋊C4—711no—
476C4⋊C4—711no—
477C4⋊C4—711no—
478C4⋊C4—711no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C4⋊C4—1421no—
483C4⋊C4—1421no—
484C4⋊C4—1421no—
485C22⋊C4—1421no—
486C22⋊C4—1421no—
487C22⋊C4—1421no—
488C22⋊C4—1421no—
489C22⋊C4—1421no—
490C22⋊C4—1421no—
491C22⋊C4—1421no—
492C22⋊C4—1421no—
493C42—1421no—
494C42—1421no—
495C42—1421no—
496C42—1421no—
497C42C2×C4×D7111yescharacteristic
498C42C2×D28411yescharacteristic
499C42#1176586511yescharacteristic
500C42C23×D7111yescharacteristic
501C23—1421no—
502C23—1421no—
503C23C4×D28411yescharacteristic
504Q8—1421no—
505Q8—1421no—
506Q8—1421no—
507Q8—1421no—
508D4—2842no—
509D4—5681no—
510D4—712no—
511D4—1421no—
512D4—1421no—
513D4—1421no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—2842no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—2841no—
524C2×C4—5681no—
525C2×C4—1421no—
526C2×C4—1421no—
527C2×C4—1421no—
528C2×C4—1421no—
529C2×C4—1421no—
530C2×C4—1421no—
531C2×C4C4×D28411yescharacteristic
532C2×C4C4×D28411yescharacteristic
533C2×C4C4×D28411yescharacteristic
534C2×C4C22×C4×D7111yescharacteristic
535C2×C4C22×D28411yescharacteristic
536C2×C4C2×#1176586511yescharacteristic
537C8—22no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—41no—
545C8—2841no—
546C8—2841no—
547C8—2841no—
548C8—2841no—
549C8#1455615012yes—
550C8#1455615012yes—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—2841no—
555C22—2841no—
556C22—2841no—
557C22—2841no—
558C22—2841no—
559C22—1422no—
560C22C2×C4×D28411yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—2841no—
566C4—2841no—
567C4—2841no—
568C4—2841no—
569C4—2841no—
570C4—2841no—
571C4#131×D7111yescharacteristic
572C4#1798676411yescharacteristic
573C4C2×#1455615011yescharacteristic
574C4C2×#1455615011yescharacteristic
575C2—41no—
576C2—2842no—
577C2—5681no—
578C2#2259706811yescharacteristic
579C2C4×#1455615011yescharacteristic
580C2#2259725111yescharacteristic
subgroupG/H
centerC2×C4C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC8×M16×C71C2
socleC22×C71C2×C4×D4
prime radical#17986146C4
cosocleC42×C71C24
hypercenterC8×M16D71
hypocenterC71#17996
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #22597355C2semidirect
3. #22597344C2semidirect
4. #22587238C2semidirect
5. #22587232C2semidirect
6. #22587192C2semidirect
7. #22587178C2semidirect
8. #22587178C2semidirect
9. #22587171C2semidirect
10. C8×#11765854C2semidirect
11. C8×D568C2semidirect
12. #22587120C2semidirect
13. #22586801C2semidirect
14. #22586723C2semidirect
15. C8×M16×C71C2semidirect
305. C8×M16D71semidirect
306. C71#17996semidirect

Maximal subgroups

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

subgroupindex
#225983602
#225973552
#225973442
#225872382
#225872322
#225871922
#225871782
#225871712
C8×#117658542
C8×D5682
#225871202
#225868012
#225867232
C8×M16×C712
#1799671