Finite Group Databasebeta

← #31968438

Subgroups — #31968438

Order 18176.

610 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#22597354C211yesmaximal, characteristic, complemented
2#22597345C211yesmaximal, characteristic, complemented
3#22587249C211yesmaximal, characteristic, complemented
4#22587243C211yesmaximal, characteristic, complemented
5#22587212C211yesmaximal, characteristic, complemented
6#22587205C211yesmaximal, characteristic, complemented
7#22587198C211yesmaximal, characteristic, complemented
8#22587198C211yesmaximal, characteristic, complemented
9#22587164C211yesmaximal, characteristic, complemented
10SD16⋊(C71⋊C8)C211yesmaximal, characteristic, complemented
11#22587120C211yesmaximal, characteristic, complemented
12#22586800C211yesmaximal, characteristic, complemented
13#22586722C211yesmaximal, characteristic, complemented
14#22586630C211yesmaximal, characteristic, complemented
15C71×#3102C211yesmaximal, characteristic
16#17986130C2211yescharacteristic
17#17986130C2211yescharacteristic
18#17986130C2211yescharacteristic
19Q8×C71⋊C8C2211yescharacteristic
20#17986077—21no—
21#17986077C2211yescharacteristic
22#17986071—21no—
23#17986071—21no—
24#17986071—21no—
25D4×C71⋊C8C2211yescharacteristic
26D4×C71⋊C8C2211yescharacteristic
27D4×C71⋊C8C2211yescharacteristic
28#17986060—21no—
29#17986056—21no—
30#17986056C2211yescharacteristic
31#17986052C2211yescharacteristic
32#17985955—21no—
33C4×#11765849C2211yescharacteristic
34C2×#14556039—21no—
35C2×C4×C71⋊C8C2211yescharacteristic
36#17985752C2211yescharacteristic
37#17985752C2211yescharacteristic
38#17985750C2211yescharacteristic
39#17985750C2211yescharacteristic
40Q8⋊(C71⋊C8)C2211yescharacteristic
41Q8⋊(C71⋊C8)C2211yescharacteristic
42Q8⋊(C71⋊C8)C2211yescharacteristic
43Q8⋊(C71⋊C8)C2211yescharacteristic
44D4⋊(C71⋊C8)C2211yescharacteristic
45D4⋊(C71⋊C8)C2211yescharacteristic
46D4⋊(C71⋊C8)C2211yescharacteristic
47D4⋊(C71⋊C8)C2211yescharacteristic
48#17985721C2211yescharacteristic
49#17985721C2211yescharacteristic
50#17985720C2211yescharacteristic
51#17985720C2211yescharacteristic
52#529×C71C2211yescharacteristic
53C4×#41×C71C2211yescharacteristic
54#485×C71C411yescharacteristic
55#482×C71C411yescharacteristic
56#479×C71C411yescharacteristic
57#478×C71C411yescharacteristic
58#464×C71C411yescharacteristic
59#463×C71C411yescharacteristic
60#452×C71C411yescharacteristic
61#449×C71C411yescharacteristic
62C4×Q16×C71C2211yescharacteristic
63C4×SD16×C71C2211yescharacteristic
64C4×SD16×C71C2211yescharacteristic
65C4×D8×C71C2211yescharacteristic
66#431×C71C2211yescharacteristic
67C2×#11765849—21no—
68C2×#11765849D411yescharacteristic
69C2×#11765849D411yescharacteristic
70C22×C71⋊C8—21no—
71C22×C71⋊C8—21no—
72C22×C71⋊C8—21no—
73C22×C71⋊C8—21no—
74C4⋊(C71⋊C8)—21no—
75C4⋊(C71⋊C8)—21no—
76C4⋊(C71⋊C8)—21no—
77C4⋊(C71⋊C8)—21no—
78C4⋊(C71⋊C8)C2311yescharacteristic
79C4⋊(C71⋊C8)C2311yescharacteristic
80C4⋊(C71⋊C8)C2311yescharacteristic
81C4⋊(C71⋊C8)C2311yescharacteristic
82C22⋊(C71⋊C8)—21no—
83C22⋊(C71⋊C8)—21no—
84C22⋊(C71⋊C8)—21no—
85C22⋊(C71⋊C8)—21no—
86C22⋊(C71⋊C8)—21no—
87C22⋊(C71⋊C8)—21no—
88#14556039—21no—
89#14556039—21no—
90#14556039—21no—
91#14556039—21no—
92#14556039D411yescharacteristic
93#14556039D411yescharacteristic
94C4×C71⋊C8—21no—
95C4×C71⋊C8—21no—
96C4×C71⋊C8—21no—
97C4×C71⋊C8C2311yescharacteristic
98C4×C71⋊C8C2311yescharacteristic
99C4×C71⋊C8C2311yescharacteristic
100C4×C71⋊C8C2311yescharacteristic
101C2×#41×C71C2×C411yescharacteristic
102C2×Q16×C71C2×C411yescharacteristic
103C2×SD16×C71C2×C411yescharacteristic
104C2×SD16×C71C2×C411yescharacteristic
105C2×D8×C71C2×C411yescharacteristic
106#126×C71—21no—
107#125×C71C2×C411yescharacteristic
108#124×C71C2×C411yescharacteristic
109#123×C71—21no—
110C22⋊Q8×C71C2×C411yescharacteristic
111#121×C71C2×C411yescharacteristic
112C4×Q8×C71C2×C411yescharacteristic
113C4×Q8×C71C2311yescharacteristic
114C4×Q8×C71C2311yescharacteristic
115C4×D4×C71—21no—
116C4×D4×C71—21no—
117C4×D4×C71—21no—
118C4×D4×C71C2×C411yescharacteristic
119C4×D4×C71C2311yescharacteristic
120C4×D4×C71C2311yescharacteristic
121#117×C71—21no—
122#117×C71—21no—
123#117×C71C2×C411yescharacteristic
124C2×C42×C71—21no—
125C2×C42×C71C2311yescharacteristic
126#107×C71C2×C411yescharacteristic
127#107×C71C2×C411yescharacteristic
128#106×C71C2×C411yescharacteristic
129#106×C71C2×C411yescharacteristic
130C4⋊C8×C71C2×C411yescharacteristic
131C4⋊C8×C71C2×C411yescharacteristic
132#103×C71C2×C411yescharacteristic
133#103×C71C2×C411yescharacteristic
134#103×C71C2×C411yescharacteristic
135#103×C71C2×C411yescharacteristic
136#102×C71C2×C411yescharacteristic
137#102×C71C2×C411yescharacteristic
138#102×C71C2×C411yescharacteristic
139#102×C71C2×C411yescharacteristic
140C22⋊C8×C71C2×C411yescharacteristic
141C22⋊C8×C71C2×C411yescharacteristic
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144#11765849—41no—
145#11765849—41no—
146#11765849—42no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—42no—
154C2×C71⋊C8—42no—
155C2×C71⋊C8—42no—
156C2×C71⋊C8—42no—
157C2×C71⋊C8C2×D411yescharacteristic
158C2×C71⋊C8C2×D411yescharacteristic
159C2×C71⋊C8C2×D411yescharacteristic
160C2×C71⋊C8C2×D411yescharacteristic
161#41×C71—22no—
162#41×C71—41no—
163C2×Q8×C71C22×C411yescharacteristic
164C2×Q8×C71C22×C411yescharacteristic
165C2×D4×C71—21no—
166C2×D4×C71—21no—
167C2×D4×C71C22×C411yescharacteristic
168C2×D4×C71C22×C411yescharacteristic
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71C22×C411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71#4111yescharacteristic
176Q16×C71—41no—
177SD16×C71—22no—
178SD16×C71—41no—
179D8×C71—22no—
180C2×C8×C71C22×C411yescharacteristic
181C2×C8×C71C22×C411yescharacteristic
182C2×C8×C71C22×C411yescharacteristic
183C2×C8×C71C22×C411yescharacteristic
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71C22×C411yescharacteristic
191C4⋊C4×C71C22×C411yescharacteristic
192C4⋊C4×C71C22×C411yescharacteristic
193C4⋊C4×C71C22×C411yescharacteristic
194C22⋊C4×C71—21no—
195C22⋊C4×C71—21no—
196C22⋊C4×C71—21no—
197C22⋊C4×C71—21no—
198C22⋊C4×C71—21no—
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C22⋊C4×C71—21no—
202C42×C71—21no—
203C42×C71—21no—
204C42×C71—21no—
205C42×C71—21no—
206C42×C71C22×C411yescharacteristic
207C42×C71C2×D411yescharacteristic
208C42×C71#4111yescharacteristic
209C42×C71C2411yescharacteristic
210C23×C71—21no—
211C23×C71—21no—
212C23×C71C4×D411yescharacteristic
213Q8×C71—21no—
214Q8×C71—21no—
215Q8×C71—21no—
216Q8×C71—21no—
217D4×C71—21no—
218D4×C71—21no—
219D4×C71—21no—
220D4×C71—42no—
221D4×C71—81no—
222D4×C71#13112yes—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—41no—
234C2×C4×C71—42no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—81no—
240C2×C4×C71C4×D411yescharacteristic
241C2×C4×C71C4×D411yescharacteristic
242C2×C4×C71C4×D411yescharacteristic
243C2×C4×C71C23×C411yescharacteristic
244C2×C4×C71C22×D411yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C8×C71—21no—
247C8×C71—21no—
248C8×C71—21no—
249C8×C71—21no—
250C71⋊C8—22no—
251C71⋊C8—21no—
252C71⋊C8—21no—
253C71⋊C8—21no—
254C71⋊C8—21no—
255C71⋊C8—41no—
256C71⋊C8—41no—
257C71⋊C8—41no—
258C71⋊C8—41no—
259C71⋊C8—41no—
260C22×C71—21no—
261C22×C71—22no—
262C22×C71—21no—
263C22×C71—21no—
264C22×C71—41no—
265C22×C71—41no—
266C22×C71—41no—
267C22×C71—41no—
268C22×C71—41no—
269C22×C71C2×C4×D411yescharacteristic
270C4×C71—21no—
271C4×C71—21no—
272C4×C71—21no—
273C4×C71—21no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71—41no—
277C4×C71—41no—
278C4×C71—41no—
279C4×C71—41no—
280C4×C71C2×#13111yescharacteristic
281C4×C71#56811yescharacteristic
282C4×C71#57511yescharacteristic
283C4×C71#57511yescharacteristic
284#18002—711nomaximal, Sylow
285C2×C71—41no—
286C2×C71—42no—
287C2×C71—81no—
288C2×C71#293311yescharacteristic
289C2×C71#294311yescharacteristic
290C2×C71#295611yescharacteristic
291#3102D7111yescharacteristic
292#2970—711no—
293#2969—711no—
294#1593—711no—
295#1592—711no—
296#1587—711no—
297#1586—711no—
298#1585—711no—
299#1585—711no—
300#1580—711no—
301SD16⋊C8—711no—
302#1572—711no—
303#1494—711no—
304#1476—711no—
305#1451—711no—
306C71#1800211yescharacteristic, complemented, Sylow
307#529C2×D7111yescharacteristic
308C4×#41C2×D7111yescharacteristic
309#485Dic7111yescharacteristic
310#482Dic7111yescharacteristic
311#479Dic7111yescharacteristic
312#478Dic7111yescharacteristic
313#464Dic7111yescharacteristic
314#463Dic7111yescharacteristic
315#452Dic7111yescharacteristic
316#449Dic7111yescharacteristic
317#446—711no—
318#446—711no—
319#446—711no—
320C8×Q8—711no—
321C4×Q16C2×D7111yescharacteristic
322C4×SD16C2×D7111yescharacteristic
323C4×SD16C2×D7111yescharacteristic
324C4×D8C2×D7111yescharacteristic
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×M16—711no—
340C2×#97—1421no—
341C2×C4×C8—711no—
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#322—711no—
355#322—711no—
356#322—711no—
357#322—711no—
358C2×#41C2×Dic7111yescharacteristic
359C2×Q16C2×Dic7111yescharacteristic
360C2×SD16C2×Dic7111yescharacteristic
361C2×SD16C2×Dic7111yescharacteristic
362C2×D8C2×Dic7111yescharacteristic
363C2×M16—711no—
364C2×M16—711no—
365C2×M16—1421no—
366C22×C8—1421no—
367C22×C8—1421no—
368C22×C8—1421no—
369C22×C8—1421no—
370#126—21no—
371#125C2×Dic7111yescharacteristic
372#124C2×Dic7111yescharacteristic
373#123—21no—
374C22⋊Q8C2×Dic7111yescharacteristic
375#121C2×Dic7111yescharacteristic
376C4×Q8C2×Dic7111yescharacteristic
377C4×Q8C22×D7111yescharacteristic
378C4×Q8C22×D7111yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C2×Dic7111yescharacteristic
383C4×D4C22×D7111yescharacteristic
384C4×D4C22×D7111yescharacteristic
385#117—21no—
386#117—21no—
387#117C2×Dic7111yescharacteristic
388C2×C42—21no—
389C2×C42C22×D7111yescharacteristic
390#107C2×Dic7111yescharacteristic
391#107C2×Dic7111yescharacteristic
392#106C2×Dic7111yescharacteristic
393#106C2×Dic7111yescharacteristic
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⋊C8C2×Dic7111yescharacteristic
403C4⋊C8C2×Dic7111yescharacteristic
404#103C2×Dic7111yescharacteristic
405#103C2×Dic7111yescharacteristic
406#103C2×Dic7111yescharacteristic
407#103C2×Dic7111yescharacteristic
408#102C2×Dic7111yescharacteristic
409#102C2×Dic7111yescharacteristic
410#102C2×Dic7111yescharacteristic
411#102C2×Dic7111yescharacteristic
412C22⋊C8—1421no—
413C22⋊C8—1421no—
414C22⋊C8—1421no—
415C22⋊C8—1421no—
416C22⋊C8—1421no—
417C22⋊C8—1421no—
418C22⋊C8C2×Dic7111yescharacteristic
419C22⋊C8C2×Dic7111yescharacteristic
420#97—711no—
421#97—711no—
422#97—1421no—
423#97—1421no—
424#97—1421no—
425#97—1421no—
426C4×C8—711no—
427C4×C8—711no—
428C4×C8—711no—
429C4×C8—711no—
430C4×C8—1421no—
431C4×C8—1421no—
432C4×C8—1421no—
433C4×C8C22×D7111yescharacteristic
434C4×C8C22×D7111yescharacteristic
435#41—22no—
436#41—41no—
437C2×Q8C22×Dic7111yescharacteristic
438C2×Q8C22×Dic7111yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4C22×Dic7111yescharacteristic
442C2×D4C22×Dic7111yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4C22×Dic7111yescharacteristic
448C22×C4D4×D7111yescharacteristic
449C22×C4#1176586411yescharacteristic
450Q16—41no—
451SD16—22no—
452SD16—41no—
453D8—22no—
454M16—2842no—
455M16—2841no—
456M16—2841no—
457C2×C8—2842no—
458C2×C8—2842no—
459C2×C8—2842no—
460C2×C8—2842no—
461C2×C8—711no—
462C2×C8—711no—
463C2×C8—711no—
464C2×C8—711no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1421no—
469C2×C8—1421no—
470C2×C8—1421no—
471C2×C8C22×Dic7111yescharacteristic
472C2×C8C22×Dic7111yescharacteristic
473C2×C8C22×Dic7111yescharacteristic
474C2×C8C22×Dic7111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4C22×Dic7111yescharacteristic
482C4⋊C4C22×Dic7111yescharacteristic
483C4⋊C4C22×Dic7111yescharacteristic
484C4⋊C4C22×Dic7111yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42C22×Dic7111yescharacteristic
498C42D4×D7111yescharacteristic
499C42#1176586411yescharacteristic
500C42C23×D7111yescharacteristic
501C23—21no—
502C23—21no—
503C23D4×Dic7111yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8—21no—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—42no—
512D4—81no—
513D4#1455613512yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4D4×Dic7111yescharacteristic
532C2×C4D4×Dic7111yescharacteristic
533C2×C4D4×Dic7111yescharacteristic
534C2×C4C23×Dic7111yescharacteristic
535C2×C4C2×D4×D7111yescharacteristic
536C2×C4C2×#1176586411yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—2841no—
542C8—2841no—
543C8—2841no—
544C8—2841no—
545C8—2841no—
546C8—1422no—
547C8—1421no—
548C8—1421no—
549C8—1421no—
550C8—1421no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22C2×D4×Dic7111yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4C2×#1455613511yescharacteristic
572C4#1798676611yescharacteristic
573C4#1798679611yescharacteristic
574C4#1798679611yescharacteristic
575C2—41no—
576C2—42no—
577C2—81no—
578C2#2259706711yescharacteristic
579C2#2259714111yescharacteristic
580C2#2259725011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#3102C2
socleC22×C71C2×C4×D4
prime radical#449×C71C4
cosocleC42×C71C24
hypercenter#3102D71
hypocenterC71#18002
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3102

Complements

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

NH ≅ G/Ntype
1. #22597354C2semidirect
2. #22597345C2semidirect
3. #22587249C2semidirect
4. #22587243C2semidirect
5. #22587212C2semidirect
6. #22587205C2semidirect
7. #22587198C2semidirect
8. #22587198C2semidirect
9. #22587164C2semidirect
10. SD16⋊(C71⋊C8)C2semidirect
11. #22587120C2semidirect
12. #22586800C2semidirect
13. #22586722C2semidirect
14. #22586630C2semidirect
306. C71#18002semidirect

Maximal subgroups

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

subgroupindex
#225973542
#225973452
#225872492
#225872432
#225872122
#225872052
#225871982
#225871642
SD16⋊(C71⋊C8)2
#225871202
#225868002
#225867222
#225866302
C71×#31022
#1800271