Finite Group Databasebeta

← #31969235

Subgroups — #31969235

Order 18176.

600 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598904C211yesmaximal, characteristic
2#22597344C211yesmaximal, characteristic, complemented
3#22587235C211yesmaximal, characteristic, complemented
4#22587223C211yesmaximal, characteristic, complemented
5#22587183C211yesmaximal, characteristic, complemented
6#22587181C211yesmaximal, characteristic, complemented
7#22587176C211yesmaximal, characteristic, complemented
8#22587174C211yesmaximal, characteristic, complemented
9C8×#11765855C211yesmaximal, characteristic, complemented
10C8×#11765850C211yesmaximal, characteristic, complemented
11#22587086C211yesmaximal, characteristic, complemented
12#22586748C211yesmaximal, characteristic, complemented
13#22586746C211yesmaximal, characteristic, complemented
14M16×C71⋊C8C211yesmaximal, characteristic, complemented
15C71×#2969C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17#17986398C411yescharacteristic
18#17986381C411yescharacteristic
19#17986356C411yescharacteristic
20#17986353C411yescharacteristic
21#17986238C411yescharacteristic
22#17986233C411yescharacteristic
23#17986230C411yescharacteristic
24#17986226C411yescharacteristic
25#17986129C2211yescharacteristic
26C8×Dic142C2211yescharacteristic
27C4×#11765855C2211yescharacteristic
28C4×#11765852C2211yescharacteristic
29C4×#11765853C2211yescharacteristic
30C4×#11765850C2211yescharacteristic
31#17986078C2211yescharacteristic
32#17986075—21no—
33#17986068—21no—
34C8×D284C2211yescharacteristic
35#17986061—21no—
36#17986052C2211yescharacteristic
37#17986052C2211yescharacteristic
38#17985956—21no—
39C2×C4×C71⋊C8C2211yescharacteristic
40#17985751C2211yescharacteristic
41#17985751C2211yescharacteristic
42#17985749C2211yescharacteristic
43#17985749C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46#17985729C2211yescharacteristic
47#17985729C2211yescharacteristic
48D4⋊(C71⋊C8)C2211yescharacteristic
49D4⋊(C71⋊C8)C2211yescharacteristic
50#17985726C2211yescharacteristic
51#17985726C2211yescharacteristic
52#17985720C2211yescharacteristic
53#17985720C2211yescharacteristic
54C8×C71⋊C8C2211yescharacteristic
55C8×C71⋊C8C2211yescharacteristic
56#529×C71C2211yescharacteristic
57#446×C71C2211yescharacteristic
58C8×Q8×C71C2211yescharacteristic
59#436×C71—21no—
60#436×C71C2211yescharacteristic
61#435×C71—21no—
62C8×D4×C71C2211yescharacteristic
63#433×C71—21no—
64#431×C71C2211yescharacteristic
65#405×C71—21no—
66C4×M16×C71C2211yescharacteristic
67C2×#11765855C2×C411yescharacteristic
68C2×#11765853C2×C411yescharacteristic
69C2×#11765852C2×C411yescharacteristic
70C2×#11765850C2×C411yescharacteristic
71C2×#11765848—21no—
72C22×C71⋊C8C2×C411yescharacteristic
73C22×C71⋊C8C2×C411yescharacteristic
74C2×C8×D71—21no—
75#14556122—21no—
76#14556120C2×C411yescharacteristic
77#14556116C2×C411yescharacteristic
78#14556111—21no—
79#14556107C2×C411yescharacteristic
80#14556102C2×C411yescharacteristic
81C4×Dic142C2311yescharacteristic
82C4×D284C2311yescharacteristic
83C4×C22⋊D71—21no—
84#14556085—21no—
85#14556061C2×C411yescharacteristic
86#14556061C2×C411yescharacteristic
87#14556059C2×C411yescharacteristic
88#14556059C2×C411yescharacteristic
89C4⋊(C71⋊C8)—21no—
90C4⋊(C71⋊C8)C2311yescharacteristic
91C4⋊(C71⋊C8)C2311yescharacteristic
92Dic71⋊C8—21no—
93Dic71⋊C8—21no—
94#14556052C2×C411yescharacteristic
95#14556052C2×C411yescharacteristic
96#14556051C2×C411yescharacteristic
97#14556051C2×C411yescharacteristic
98#14556049C2×C411yescharacteristic
99#14556049C2×C411yescharacteristic
100#14556048C2×C411yescharacteristic
101#14556048C2×C411yescharacteristic
102C22⋊(C71⋊C8)—21no—
103C22⋊(C71⋊C8)—21no—
104(C2×D71)⋊C8—21no—
105(C2×D71)⋊C8—21no—
106#14556038—21no—
107C4×C71⋊C8—21no—
108C4×C71⋊C8C2×C411yescharacteristic
109C4×C71⋊C8C2×C411yescharacteristic
110C4×C71⋊C8C2311yescharacteristic
111C4×C71⋊C8C2311yescharacteristic
112C4×C71⋊C8C2311yescharacteristic
113C4×C71⋊C8C2311yescharacteristic
114C8×Dic71—21no—
115C2×M16×C71—21no—
116C2×M16×C71D411yescharacteristic
117C2×M16×C71D411yescharacteristic
118C22×C8×C71—21no—
119#126×C71—21no—
120#125×C71C2×C411yescharacteristic
121#124×C71C2×C411yescharacteristic
122#123×C71—21no—
123C22⋊Q8×C71C2×C411yescharacteristic
124#121×C71C2×C411yescharacteristic
125C4×Q8×C71C2311yescharacteristic
126C4×D4×C71—21no—
127C4×D4×C71C2311yescharacteristic
128#117×C71—21no—
129C2×C42×C71C2311yescharacteristic
130C4⋊C8×C71—21no—
131C4⋊C8×C71—21no—
132C4⋊C8×C71C2311yescharacteristic
133C4⋊C8×C71C2311yescharacteristic
134C22⋊C8×C71—21no—
135C22⋊C8×C71—21no—
136C22⋊C8×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#11765853—41no—
150#11765852—41no—
151#11765850—22no—
152#11765848—42no—
153C2×C71⋊C8—22no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—41no—
159C2×C71⋊C8C22×C411yescharacteristic
160C2×C71⋊C8C22×C411yescharacteristic
161C2×C71⋊C8C22×C411yescharacteristic
162C2×C71⋊C8C22×C411yescharacteristic
163C8×D71—42no—
164C4⋊Dic71C22×C411yescharacteristic
165C4⋊Dic71C22×C411yescharacteristic
166Dic71⋊C4—21no—
167Dic71⋊C4—21no—
168Dic71⋊C4—21no—
169Dic71⋊C4—21no—
170C22⋊Dic71—21no—
171(C2×D71)⋊C4—21no—
172(C2×D71)⋊C4—21no—
173(C2×D71)⋊C4—21no—
174(C2×D71)⋊C4—21no—
175C4×Dic71—21no—
176C2×Q8×C71C22×C411yescharacteristic
177C2×D4×C71—21no—
178C2×D4×C71C22×C411yescharacteristic
179C22×C4×C71—21no—
180C22×C4×C71C22×C411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71#4111yescharacteristic
183M16×C71—22no—
184M16×C71—22no—
185M16×C71—42no—
186C2×C8×C71—21no—
187C2×C8×C71—21no—
188C2×C8×C71—42no—
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×C71C22×C411yescharacteristic
198C4⋊C4×C71C22×C411yescharacteristic
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C22⋊C4×C71—21no—
202C22⋊C4×C71—21no—
203C22⋊C4×C71—21no—
204C42×C71—21no—
205C42×C71C22×C411yescharacteristic
206C42×C71C2×D411yescharacteristic
207C42×C71#4111yescharacteristic
208C42×C71C2411yescharacteristic
209C23×C71—21no—
210C23×C71C4×D411yescharacteristic
211C22×D71—21no—
212Q8×C71—21no—
213Q8×C71—21no—
214D4×C71—21no—
215D4×C71—21no—
216D4×C71—81no—
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—21no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71C4×D411yescharacteristic
228C2×C4×C71C4×D411yescharacteristic
229C2×C4×C71C4×D411yescharacteristic
230C2×C4×C71C23×C411yescharacteristic
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C22⋊D71—81no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237D284—21no—
238D284—21no—
239C4×D71—41no—
240C4×D71—41no—
241Dic142—21no—
242Dic142—21no—
243C8×C71—21no—
244C8×C71—21no—
245C8×C71—41no—
246C8×C71—41no—
247C8×C71#13512yes—
248C8×C71#13512yes—
249C71⋊C8—21no—
250C71⋊C8—21no—
251C71⋊C8—21no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C71⋊C8—41no—
256C71⋊C8#13112yes—
257C22×C71—21no—
258C22×C71—21no—
259C22×C71—21no—
260C22×C71—41no—
261C22×C71—41no—
262C22×C71—41no—
263C22×C71C2×C4×D411yescharacteristic
264C2×D71—41no—
265C2×D71—41no—
266C2×D71—41no—
267C4×C71—21no—
268C4×C71—21no—
269C4×C71—21no—
270C4×C71—21no—
271C4×C71—41no—
272C4×C71—41no—
273C4×C71—41no—
274C4×C71C2×#13111yescharacteristic
275C4×C71#56811yescharacteristic
276C4×C71C2×#13511yescharacteristic
277C4×C71C2×#13511yescharacteristic
278Dic71—41no—
279Dic71—41no—
280Dic71—41no—
281#18069—711nomaximal, Sylow
282C2×C71—41no—
283C2×C71—81no—
284C2×C71#293511yescharacteristic
285C2×C71C4×#13511yescharacteristic
286C2×C71#295711yescharacteristic
287D71—81no—
288#3168—711no—
289#2969—711no—
290#2969D7111yescharacteristic, complemented
291#1591—711no—
292#1589—711no—
293#1583—711no—
294#1583—711no—
295#1582—711no—
296#1582—711no—
297C8×Q16—711no—
298C8×D8—711no—
299#1564—711no—
300#1482—711no—
301#1482—711no—
302C8×M16—711no—
303C71#1806911yescharacteristic, complemented, Sylow
304#529—711no—
305#529C2×D7111yescharacteristic
306#499—711no—
307#495—711no—
308#488—711no—
309#488—711no—
310#467—711no—
311#466—711no—
312#465—711no—
313#465—711no—
314#446—711no—
315#446C2×D7111yescharacteristic
316C8×Q8—711no—
317C8×Q8C2×D7111yescharacteristic
318C4×Q16—711no—
319C4×SD16—711no—
320C4×SD16—711no—
321C4×D8—711no—
322#436—21no—
323#436—711no—
324#436—1421no—
325#436C2×D7111yescharacteristic
326#435—21no—
327#435—1421no—
328C8×D4—711no—
329C8×D4C2×D7111yescharacteristic
330#433—21no—
331#433—1421no—
332#431—711no—
333#431—711no—
334#431C2×D7111yescharacteristic
335#405—21no—
336#405—1421no—
337C4×M16C2×D7111yescharacteristic
338C2×C4×C8—711no—
339#335—711no—
340#335—711no—
341#334—711no—
342#334—711no—
343Q8⋊C8—711no—
344Q8⋊C8—711no—
345Q8⋊C8—711no—
346Q8⋊C8—711no—
347D4⋊C8—711no—
348D4⋊C8—711no—
349D4⋊C8—711no—
350D4⋊C8—711no—
351#322—711no—
352#322—711no—
353C82—711no—
354C82—711no—
355C2×Q16—711no—
356C2×SD16—711no—
357C2×SD16—711no—
358C2×D8—711no—
359C2×M16—21no—
360C2×M16—1421no—
361C2×M16C22⋊D7111yescharacteristic
362C2×M16C22⋊D7111yescharacteristic
363C22×C8—21no—
364C22×C8—711no—
365C22×C8—711no—
366C22×C8—1421no—
367#126—21no—
368#126—1421no—
369#125—711no—
370#125C4×D7111yescharacteristic
371#124—711no—
372#124C4×D7111yescharacteristic
373#123—21no—
374#123—1421no—
375C22⋊Q8—711no—
376C22⋊Q8C4×D7111yescharacteristic
377#121—711no—
378#121C4×D7111yescharacteristic
379C4×Q8—711no—
380C4×Q8C22×D7111yescharacteristic
381C4×D4—21no—
382C4×D4—711no—
383C4×D4—1421no—
384C4×D4C22×D7111yescharacteristic
385#117—21no—
386#117—1421no—
387C2×C42C22×D7111yescharacteristic
388#107—711no—
389#107—711no—
390#106—711no—
391#106—711no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—711no—
395C4⋊C8—711no—
396C4⋊C8—1421no—
397C4⋊C8—1421no—
398C4⋊C8—1421no—
399C4⋊C8C22×D7111yescharacteristic
400C4⋊C8C22×D7111yescharacteristic
401#103—711no—
402#103—711no—
403#103—711no—
404#103—711no—
405#102—711no—
406#102—711no—
407#102—711no—
408#102—711no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—1421no—
413C22⋊C8—1421no—
414C22⋊C8—1421no—
415C22⋊C8—1421no—
416#97—21no—
417#97—1421no—
418#97C22⋊D7111yescharacteristic
419#97C22⋊D7111yescharacteristic
420C4×C8—21no—
421C4×C8—21no—
422C4×C8—711no—
423C4×C8—711no—
424C4×C8—711no—
425C4×C8—711no—
426C4×C8—711no—
427C4×C8—711no—
428C4×C8—1421no—
429C4×C8—1421no—
430C4×C8C22×D7111yescharacteristic
431C4×C8C22×D7111yescharacteristic
432C2×Q8—711no—
433C2×Q8C2×C4×D7111yescharacteristic
434C2×D4—21no—
435C2×D4—711no—
436C2×D4—1421no—
437C2×D4C2×C4×D7111yescharacteristic
438C22×C4—21no—
439C22×C4—1421no—
440C22×C4C2×C4×D7111yescharacteristic
441C22×C4C2×C22⋊D7111yescharacteristic
442C22×C4#1176586511yescharacteristic
443Q16—1422no—
444SD16—2841no—
445SD16—2841no—
446D8—1422no—
447M16—22no—
448M16—22no—
449M16—42no—
450M16—2842no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—2842no—
455C2×C8—2841no—
456C2×C8—711no—
457C2×C8—711no—
458C2×C8—711no—
459C2×C8—711no—
460C2×C8—1421no—
461C2×C8—1421no—
462C2×C8—1421no—
463C2×C8—1421no—
464C2×C8—1422no—
465C2×C8C2×C22⋊D7111yescharacteristic
466C2×C8C2×C22⋊D7111yescharacteristic
467C2×C8C2×C22⋊D7111yescharacteristic
468C2×C8C2×C22⋊D7111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—711no—
474C4⋊C4—711no—
475C4⋊C4—1421no—
476C4⋊C4—1421no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C4⋊C4C2×C4×D7111yescharacteristic
480C4⋊C4C2×C4×D7111yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—1421no—
487C22⋊C4—1421no—
488C22⋊C4—1421no—
489C22⋊C4—1421no—
490C22⋊C4—1421no—
491C42—21no—
492C42—1421no—
493C42C2×C4×D7111yescharacteristic
494C42C2×C22⋊D7111yescharacteristic
495C42#1176586511yescharacteristic
496C42C23×D7111yescharacteristic
497C23—21no—
498C23—1421no—
499C23C4×C22⋊D7111yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—1421no—
503Q8—1421no—
504D4—21no—
505D4—21no—
506D4—81no—
507D4—5681no—
508D4—1421no—
509D4—1421no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—1421no—
523C2×C4—1421no—
524C2×C4—1421no—
525C2×C4C4×C22⋊D7111yescharacteristic
526C2×C4C4×C22⋊D7111yescharacteristic
527C2×C4C4×C22⋊D7111yescharacteristic
528C2×C4C22×C4×D7111yescharacteristic
529C2×C4C22×C22⋊D7111yescharacteristic
530C2×C4C2×#1176586511yescharacteristic
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—2841no—
536C8—2841no—
537C8—2841no—
538C8—2841no—
539C8—712no—
540C8—1421no—
541C8—1421no—
542C8—1421no—
543C8#1455614812yes—
544C8#1455614812yes—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—2841no—
552C22—2841no—
553C22—2841no—
554C22C2×C4×C22⋊D7111yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—2841no—
563C4—2841no—
564C4—2841no—
565C4#131×D7111yescharacteristic
566C4#1798676411yescharacteristic
567C4C2×#1455614811yescharacteristic
568C4C2×#1455614811yescharacteristic
569C2—41no—
570C2—81no—
571C2—5681no—
572C2#2259708211yescharacteristic
573C2C4×#1455614811yescharacteristic
574C2#2259725511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#2969C2
socleC22×C71C2×C4×D4
prime radical#17986233C4
cosocleC42×C71C24
hypercenter#2969D71
hypocenterC71#18069
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. #22597344C2semidirect
3. #22587235C2semidirect
4. #22587223C2semidirect
5. #22587183C2semidirect
6. #22587181C2semidirect
7. #22587176C2semidirect
8. #22587174C2semidirect
9. C8×#11765855C2semidirect
10. C8×#11765850C2semidirect
11. #22587086C2semidirect
12. #22586748C2semidirect
13. #22586746C2semidirect
14. M16×C71⋊C8C2semidirect
15. C71×#2969C2semidirect
290. #2969D71semidirect
303. C71#18069semidirect

Maximal subgroups

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

subgroupindex
#225989042
#225973442
#225872352
#225872232
#225871832
#225871812
#225871762
#225871742
C8×#117658552
C8×#117658502
#225870862
#225867482
#225867462
M16×C71⋊C82
C71×#29692
#1806971