Finite Group Databasebeta

← #31968593

Subgroups — #31968593

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#22598411C211yesmaximal, characteristic
2#22597287C211yesmaximal, characteristic, complemented
3#22587242C211yesmaximal, characteristic, complemented
4#22587228C211yesmaximal, characteristic, complemented
5#22587216C211yesmaximal, characteristic, complemented
6#22587211C211yesmaximal, characteristic, complemented
7#22587190C211yesmaximal, characteristic, complemented
8#22587181C211yesmaximal, characteristic, complemented
9#22587157C211yesmaximal, characteristic, complemented
10C8×#11765855C211yesmaximal, characteristic, complemented
11#22587126C211yesmaximal, characteristic, complemented
12#22586812C211yesmaximal, characteristic, complemented
13#22586727C211yesmaximal, characteristic
14#22586637C211yesmaximal, characteristic, complemented
15C71×#2971C211yesmaximal, characteristic
16#17986556C2211yescharacteristic
17#17986340C411yescharacteristic
18#17986340C411yescharacteristic
19#17986284C411yescharacteristic
20#17986279C411yescharacteristic
21#17986209C411yescharacteristic
22#17986204C411yescharacteristic
23#17986161C411yescharacteristic
24#17986161C411yescharacteristic
25#17986129C2211yescharacteristic
26#17986129C2211yescharacteristic
27#17986127—21no—
28#17986127—21no—
29C8×Dic142C2211yescharacteristic
30C8×Dic142C2211yescharacteristic
31C4×#11765855C2211yescharacteristic
32C4×#11765855C2211yescharacteristic
33C4×#11765853C2211yescharacteristic
34C4×#11765853C2211yescharacteristic
35#17986052C2211yescharacteristic
36#17986052C2211yescharacteristic
37#17986027—21no—
38C2×C4⋊(C71⋊C8)C2211yescharacteristic
39#17985957—21no—
40#17985751C2211yescharacteristic
41#17985751C2211yescharacteristic
42#17985749C2211yescharacteristic
43#17985749C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46#17985729C2211yescharacteristic
47#17985729C2211yescharacteristic
48#17985729C2211yescharacteristic
49#17985729C2211yescharacteristic
50D4⋊(C71⋊C8)C2211yescharacteristic
51D4⋊(C71⋊C8)C2211yescharacteristic
52#17985721C2211yescharacteristic
53#17985721C2211yescharacteristic
54#17985720C2211yescharacteristic
55C8×C71⋊C8C2211yescharacteristic
56C4×#41×C71C2211yescharacteristic
57C8×Q8×C71C2211yescharacteristic
58C8×Q8×C71C2211yescharacteristic
59#436×C71—21no—
60#436×C71—21no—
61#436×C71C2211yescharacteristic
62#436×C71C2211yescharacteristic
63#432×C71—21no—
64#432×C71C2211yescharacteristic
65C4×M16×C71—21no—
66C4×M16×C71C2211yescharacteristic
67C2×#11765855C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765853C2×C411yescharacteristic
70C2×#11765853C2×C411yescharacteristic
71C22×C71⋊C8—21no—
72#14556128C2×C411yescharacteristic
73#14556128C2×C411yescharacteristic
74#14556120—21no—
75#14556107—21no—
76#14556107C2×C411yescharacteristic
77#14556107C2×C411yescharacteristic
78C4×Dic142—21no—
79C4×Dic142C2311yescharacteristic
80C4×Dic142C2311yescharacteristic
81#14556087—21no—
82#14556061C2×C411yescharacteristic
83#14556061C2×C411yescharacteristic
84#14556059C2×C411yescharacteristic
85#14556059C2×C411yescharacteristic
86C4⋊(C71⋊C8)—21no—
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—
94Dic71⋊C8—21no—
95Dic71⋊C8—21no—
96#14556052C2×C411yescharacteristic
97#14556052C2×C411yescharacteristic
98#14556051C2×C411yescharacteristic
99#14556051C2×C411yescharacteristic
100#14556051C2×C411yescharacteristic
101#14556051C2×C411yescharacteristic
102#14556049C2×C411yescharacteristic
103#14556049C2×C411yescharacteristic
104C22⋊(C71⋊C8)—21no—
105C22⋊(C71⋊C8)C2×C411yescharacteristic
106C22⋊(C71⋊C8)C2×C411yescharacteristic
107#14556038—21no—
108#14556038—21no—
109C4×C71⋊C8C2311yescharacteristic
110C4×C71⋊C8C2311yescharacteristic
111C4×C71⋊C8C2311yescharacteristic
112C4×C71⋊C8C2311yescharacteristic
113C8×Dic71—21no—
114C8×Dic71—21no—
115C2×#41×C71C2×C411yescharacteristic
116C2×M16×C71—21no—
117C2×M16×C71—21no—
118C2×M16×C71D411yescharacteristic
119C2×M16×C71D411yescharacteristic
120C4×Q8×C71C2×C411yescharacteristic
121C4×Q8×C71C2311yescharacteristic
122C4×D4×C71—21no—
123C4×D4×C71—21no—
124C4×D4×C71C2×C411yescharacteristic
125C4×D4×C71C2311yescharacteristic
126#117×C71—21no—
127#117×C71C2×C411yescharacteristic
128C2×C42×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×Dic142—21no—
145C2×Dic142C22×C411yescharacteristic
146C2×Dic142C22×C411yescharacteristic
147#11765855—22no—
148#11765855—22no—
149#11765853—41no—
150#11765853—41no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8—42no—
156C2×C71⋊C8C22×C411yescharacteristic
157C2×C71⋊C8C22×C411yescharacteristic
158C2×C71⋊C8C22×C411yescharacteristic
159C2×C71⋊C8C22×C411yescharacteristic
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71—21no—
163C4⋊Dic71C22×C411yescharacteristic
164C4⋊Dic71C22×C411yescharacteristic
165Dic71⋊C4—21no—
166Dic71⋊C4—21no—
167Dic71⋊C4—21no—
168Dic71⋊C4—21no—
169C22⋊Dic71—21no—
170C22⋊Dic71—21no—
171C4×Dic71—21no—
172C4×Dic71—21no—
173#41×C71—22no—
174#41×C71—41no—
175C2×Q8×C71C22×C411yescharacteristic
176C2×D4×C71—21no—
177C2×D4×C71C22×C411yescharacteristic
178C22×C4×C71—21no—
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—
187M16×C71—42no—
188C2×C8×C71—21no—
189C2×C8×C71—21no—
190C2×C8×C71C2×D411yescharacteristic
191C2×C8×C71C2×D411yescharacteristic
192C2×C8×C71C2×D411yescharacteristic
193C2×C8×C71C2×D411yescharacteristic
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71C22×C411yescharacteristic
197C4⋊C4×C71C22×C411yescharacteristic
198C22⋊C4×C71—21no—
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71—21no—
204C42×C71C22×C411yescharacteristic
205C42×C71C2×D411yescharacteristic
206C42×C71#4111yescharacteristic
207C42×C71C2411yescharacteristic
208C23×C71—21no—
209C23×C71C4×D411yescharacteristic
210Q8×C71—21no—
211Q8×C71#13112yes—
212D4×C71—21no—
213D4×C71—21no—
214D4×C71—42no—
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—41no—
223C2×C4×C71—42no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—81no—
227C2×C4×C71C4×D411yescharacteristic
228C2×C4×C71C4×D411yescharacteristic
229C2×C4×C71C4×D411yescharacteristic
230C2×C4×C71C23×C411yescharacteristic
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237Dic142—21no—
238Dic142—21no—
239Dic142—21no—
240Dic142—21no—
241Dic142—81no—
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×C71—41no—
260C22×C71—41no—
261C22×C71—41no—
262C22×C71C2×C4×D411yescharacteristic
263C4×C71—21no—
264C4×C71—22no—
265C4×C71—21no—
266C4×C71—21no—
267C4×C71—21no—
268C4×C71—41no—
269C4×C71—41no—
270C4×C71C2×#13111yescharacteristic
271C4×C71#56811yescharacteristic
272C4×C71C2×#13711yescharacteristic
273C4×C71#57511yescharacteristic
274Dic71—41no—
275Dic71—41no—
276Dic71—41no—
277Dic71—41no—
278#18013—711nomaximal, Sylow
279C2×C71—41no—
280C2×C71—81no—
281C2×C71#293311yescharacteristic
282C2×C71C4×#13711yescharacteristic
283C2×C71#295711yescharacteristic
284#3107—711no—
285#2971D7111yescharacteristic
286#2962—711no—
287#1592—711no—
288#1590—711no—
289#1588—711no—
290#1587—711no—
291#1584—711no—
292#1583—711no—
293#1579—711no—
294C8×Q16—711no—
295#1573—711no—
296#1497—711no—
297#1477—711no—
298#1453—711no—
299C71#1801311yescharacteristic, complemented, Sylow
300#533—711no—
301C4×#41C2×D7111yescharacteristic
302#485—711no—
303#485—711no—
304#475—711no—
305#474—711no—
306#462—711no—
307#461—711no—
308#452—711no—
309#452—711no—
310#446—711no—
311#446—711no—
312#446—1421no—
313#446—1421no—
314C8×Q8—711no—
315C8×Q8—711no—
316C8×Q8C2×D7111yescharacteristic
317C8×Q8C2×D7111yescharacteristic
318C4×Q16—711no—
319C4×Q16—711no—
320C4×SD16—711no—
321C4×SD16—711no—
322#436—21no—
323#436—21no—
324#436C2×D7111yescharacteristic
325#436C2×D7111yescharacteristic
326#432—21no—
327#432C2×D7111yescharacteristic
328#431—711no—
329#431—711no—
330#424—1421no—
331C2×C4⋊C8—711no—
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—
351C2×#41C4×D7111yescharacteristic
352C2×Q16—711no—
353C2×Q16—711no—
354C2×SD16—711no—
355C2×SD16—711no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16C22⋊D7111yescharacteristic
359C2×M16C22⋊D7111yescharacteristic
360C22×C8—1421no—
361C4⋊Q8—711no—
362C4⋊Q8—711no—
363#125—1421no—
364C22⋊Q8—711no—
365C22⋊Q8—711no—
366C22⋊Q8—1421no—
367C4×Q8—711no—
368C4×Q8—711no—
369C4×Q8—1421no—
370C4×Q8C4×D7111yescharacteristic
371C4×Q8C22×D7111yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C4×D7111yescharacteristic
375C4×D4C22×D7111yescharacteristic
376#117—21no—
377#117—1421no—
378#117C4×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—711no—
388C4⋊C8—711no—
389C4⋊C8—711no—
390C4⋊C8—711no—
391C4⋊C8—1421no—
392C4⋊C8—1421no—
393C4⋊C8—1421no—
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—21no—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—711no—
411C22⋊C8—711no—
412C22⋊C8—1421no—
413#97—21no—
414#97—1421no—
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
428#41—22no—
429#41—41no—
430C2×Q8—711no—
431C2×Q8—711no—
432C2×Q8—1421no—
433C2×Q8C2×C4×D7111yescharacteristic
434C2×D4—21no—
435C2×D4C2×C4×D7111yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4C2×C4×D7111yescharacteristic
440C22×C4C2×C22⋊D7111yescharacteristic
441C22×C4#1176586511yescharacteristic
442Q16—1422no—
443Q16—1422no—
444SD16—2841no—
445SD16—2841no—
446M16—22no—
447M16—41no—
448M16—42no—
449M16—42no—
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—711no—
468C4⋊C4—711no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4C2×C4×D7111yescharacteristic
477C4⋊C4C2×C4×D7111yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—1421no—
482C22⋊C4—1421no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—1421no—
487C42—1421no—
488C42C2×C4×D7111yescharacteristic
489C42C2×C22⋊D7111yescharacteristic
490C42#1176586511yescharacteristic
491C42C23×D7111yescharacteristic
492C23—21no—
493C23C4×C22⋊D7111yescharacteristic
494Q8—21no—
495Q8—5681no—
496Q8—1421no—
497Q8—1421no—
498Q8—1421no—
499Q8—1421no—
500Q8#1455613612yes—
501D4—21no—
502D4—21no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4—1421no—
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#1455615812yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×C4×C22⋊D7111yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—2841no—
555C4—2841no—
556C4—2841no—
557C4—2841no—
558C4C2×#1455613611yescharacteristic
559C4#1798676511yescharacteristic
560C4C2×#1455615811yescharacteristic
561C4#1798679711yescharacteristic
562C2—41no—
563C2—81no—
564C2#2259706511yescharacteristic
565C2C4×#1455615811yescharacteristic
566C2#2259725511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#2971C2
socleC22×C71C2×C4×D4
prime radicalC2×#41×C71C2×C4
cosocleC42×C71C24
hypercenter#2971D71
hypocenterC71#18013
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. #22597287C2semidirect
3. #22587242C2semidirect
4. #22587228C2semidirect
5. #22587216C2semidirect
6. #22587211C2semidirect
7. #22587190C2semidirect
8. #22587181C2semidirect
9. #22587157C2semidirect
10. C8×#11765855C2semidirect
11. #22587126C2semidirect
12. #22586812C2semidirect
14. #22586637C2semidirect
299. C71#18013semidirect

Maximal subgroups

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

subgroupindex
#225984112
#225972872
#225872422
#225872282
#225872162
#225872112
#225871902
#225871812
#225871572
C8×#117658552
#225871262
#225868122
#225867272
#225866372
C71×#29712
#1801371