Finite Group Databasebeta

← #31968595

Subgroups — #31968595

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#22597360C211yesmaximal, characteristic
2#22597289C211yesmaximal, characteristic, complemented
3#22587243C211yesmaximal, characteristic, complemented
4#22587231C211yesmaximal, characteristic, complemented
5#22587219C211yesmaximal, characteristic, complemented
6#22587212C211yesmaximal, characteristic, complemented
7#22587191C211yesmaximal, characteristic, complemented
8#22587184C211yesmaximal, characteristic, complemented
9#22587159C211yesmaximal, characteristic, complemented
10Q16×C71⋊C8C211yesmaximal, characteristic, complemented
11#22587125C211yesmaximal, characteristic, complemented
12#22586813C211yesmaximal, characteristic, complemented
13#22586727C211yesmaximal, characteristic
14#22586635C211yesmaximal, characteristic, complemented
15C71×#3107C211yesmaximal, characteristic
16#17986130—21no—
17#17986130—21no—
18#17986130C2211yescharacteristic
19#17986130C2211yescharacteristic
20Q8×C71⋊C8C2211yescharacteristic
21Q8×C71⋊C8C2211yescharacteristic
22Q8×C71⋊C8C2211yescharacteristic
23Q8×C71⋊C8C2211yescharacteristic
24#17986077—21no—
25#17986077—21no—
26#17986077C2211yescharacteristic
27#17986077C2211yescharacteristic
28#17986056—21no—
29#17986056C2211yescharacteristic
30#17986052C2211yescharacteristic
31#17986026—21no—
32C2×C4⋊(C71⋊C8)C2211yescharacteristic
33#17985955—21no—
34C4×#11765849—21no—
35C4×#11765849C2211yescharacteristic
36#17985752C2211yescharacteristic
37#17985752C2211yescharacteristic
38#17985750C2211yescharacteristic
39#17985750C2211yescharacteristic
40Q8⋊(C71⋊C8)C2211yescharacteristic
41Q8⋊(C71⋊C8)C2211yescharacteristic
42Q8⋊(C71⋊C8)C2211yescharacteristic
43Q8⋊(C71⋊C8)C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46D4⋊(C71⋊C8)C2211yescharacteristic
47D4⋊(C71⋊C8)C2211yescharacteristic
48#17985721C2211yescharacteristic
49#17985721C2211yescharacteristic
50#17985720C2211yescharacteristic
51C8×C71⋊C8C2211yescharacteristic
52#533×C71C2211yescharacteristic
53C4×#41×C71C2211yescharacteristic
54#485×C71C411yescharacteristic
55#485×C71C411yescharacteristic
56#475×C71C411yescharacteristic
57#474×C71C411yescharacteristic
58#462×C71C411yescharacteristic
59#461×C71C411yescharacteristic
60#452×C71C411yescharacteristic
61#452×C71C411yescharacteristic
62C4×Q16×C71C2211yescharacteristic
63C4×Q16×C71C2211yescharacteristic
64C4×SD16×C71C2211yescharacteristic
65C4×SD16×C71C2211yescharacteristic
66#431×C71C2211yescharacteristic
67C2×#11765849—21no—
68C2×#11765849—21no—
69C2×#11765849D411yescharacteristic
70C2×#11765849D411yescharacteristic
71C22×C71⋊C8—21no—
72C4⋊(C71⋊C8)—21no—
73C4⋊(C71⋊C8)—21no—
74C4⋊(C71⋊C8)—21no—
75C4⋊(C71⋊C8)—21no—
76C4⋊(C71⋊C8)—21no—
77C4⋊(C71⋊C8)—21no—
78C4⋊(C71⋊C8)—21no—
79C4⋊(C71⋊C8)—21no—
80C4⋊(C71⋊C8)C2311yescharacteristic
81C4⋊(C71⋊C8)C2311yescharacteristic
82C4⋊(C71⋊C8)C2311yescharacteristic
83C4⋊(C71⋊C8)C2311yescharacteristic
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#14556039D411yescharacteristic
92#14556039D411yescharacteristic
93C4×C71⋊C8—21no—
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×Q16×C71C2×C411yescharacteristic
104C2×SD16×C71C2×C411yescharacteristic
105C2×SD16×C71C2×C411yescharacteristic
106C4⋊Q8×C71C2×C411yescharacteristic
107C4⋊Q8×C71C2×C411yescharacteristic
108#125×C71—21no—
109C22⋊Q8×C71—21no—
110C22⋊Q8×C71C2×C411yescharacteristic
111C22⋊Q8×C71C2×C411yescharacteristic
112C4×Q8×C71—21no—
113C4×Q8×C71C2×C411yescharacteristic
114C4×Q8×C71C2311yescharacteristic
115C4×Q8×C71C2311yescharacteristic
116C4×Q8×C71C2311yescharacteristic
117C4×D4×C71—21no—
118C4×D4×C71—21no—
119C4×D4×C71C2×C411yescharacteristic
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#103×C71C2×C411yescharacteristic
137#103×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—22no—
145#11765849—41no—
146#11765849—42no—
147#11765849—42no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—42no—
155C2×C71⋊C8C2×D411yescharacteristic
156C2×C71⋊C8C2×D411yescharacteristic
157C2×C71⋊C8C2×D411yescharacteristic
158C2×C71⋊C8C2×D411yescharacteristic
159#41×C71—22no—
160#41×C71—41no—
161C2×Q8×C71—21no—
162C2×Q8×C71C22×C411yescharacteristic
163C2×Q8×C71C22×C411yescharacteristic
164C2×Q8×C71C22×C411yescharacteristic
165C2×D4×C71—21no—
166C2×D4×C71C22×C411yescharacteristic
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71C22×C411yescharacteristic
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71#4111yescharacteristic
173Q16×C71—22no—
174Q16×C71—22no—
175SD16×C71—41no—
176SD16×C71—41no—
177C2×C8×C71C22×C411yescharacteristic
178C2×C8×C71C22×C411yescharacteristic
179C2×C8×C71C22×C411yescharacteristic
180C2×C8×C71C22×C411yescharacteristic
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
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—
199C42×C71—21no—
200C42×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—21no—
212Q8×C71—21no—
213Q8×C71—21no—
214Q8×C71—21no—
215Q8×C71—81no—
216Q8×C71#13112yes—
217D4×C71—21no—
218D4×C71—21no—
219D4×C71—42no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
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—41no—
232C2×C4×C71—42no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—81no—
236C2×C4×C71C4×D411yescharacteristic
237C2×C4×C71C4×D411yescharacteristic
238C2×C4×C71C4×D411yescharacteristic
239C2×C4×C71C23×C411yescharacteristic
240C2×C4×C71C22×D411yescharacteristic
241C2×C4×C71C2×#4111yescharacteristic
242C8×C71—21no—
243C8×C71—21no—
244C8×C71—21no—
245C8×C71—21no—
246C71⋊C8—21no—
247C71⋊C8—21no—
248C71⋊C8—21no—
249C71⋊C8—41no—
250C71⋊C8—41no—
251C71⋊C8—41no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C71⋊C8#13712yes—
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×C71—41no—
271C4×C71—41no—
272C4×C71—41no—
273C4×C71—41no—
274C4×C71C2×#13111yescharacteristic
275C4×C71#56811yescharacteristic
276C4×C71C2×#13711yescharacteristic
277C4×C71#57511yescharacteristic
278#18013—711nomaximal, Sylow
279C2×C71—41no—
280C2×C71—81no—
281C2×C71#293311yescharacteristic
282C2×C71C4×#13711yescharacteristic
283C2×C71#295711yescharacteristic
284#3107D7111yescharacteristic
285#2971—711no—
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#533C2×D7111yescharacteristic
301C4×#41C2×D7111yescharacteristic
302#485Dic7111yescharacteristic
303#485Dic7111yescharacteristic
304#475Dic7111yescharacteristic
305#474Dic7111yescharacteristic
306#462Dic7111yescharacteristic
307#461Dic7111yescharacteristic
308#452Dic7111yescharacteristic
309#452Dic7111yescharacteristic
310#446—711no—
311#446—711no—
312#446—1421no—
313#446—1421no—
314C8×Q8—711no—
315C8×Q8—711no—
316C8×Q8—711no—
317C8×Q8—711no—
318C4×Q16C2×D7111yescharacteristic
319C4×Q16C2×D7111yescharacteristic
320C4×SD16C2×D7111yescharacteristic
321C4×SD16C2×D7111yescharacteristic
322#436—711no—
323#436—711no—
324#436—1421no—
325#436—1421no—
326#432—711no—
327#432—1421no—
328#431—711no—
329#431C2×D7111yescharacteristic
330#424—1421no—
331C2×C4⋊C8—711no—
332#405—1421no—
333C4×M16—711no—
334C4×M16—1421no—
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×#41C2×Dic7111yescharacteristic
352C2×Q16C2×Dic7111yescharacteristic
353C2×Q16C2×Dic7111yescharacteristic
354C2×SD16C2×Dic7111yescharacteristic
355C2×SD16C2×Dic7111yescharacteristic
356C2×M16—711no—
357C2×M16—711no—
358C2×M16—1421no—
359C2×M16—1421no—
360C22×C8—1421no—
361C4⋊Q8C2×Dic7111yescharacteristic
362C4⋊Q8C2×Dic7111yescharacteristic
363#125—21no—
364C22⋊Q8—21no—
365C22⋊Q8C2×Dic7111yescharacteristic
366C22⋊Q8C2×Dic7111yescharacteristic
367C4×Q8—21no—
368C4×Q8C2×Dic7111yescharacteristic
369C4×Q8C22×D7111yescharacteristic
370C4×Q8C22×D7111yescharacteristic
371C4×Q8C22×D7111yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C2×Dic7111yescharacteristic
375C4×D4C22×D7111yescharacteristic
376#117—21no—
377#117—21no—
378#117C2×Dic7111yescharacteristic
379C2×C42—21no—
380C2×C42C22×D7111yescharacteristic
381#107C2×Dic7111yescharacteristic
382#107C2×Dic7111yescharacteristic
383#106C2×Dic7111yescharacteristic
384#106C2×Dic7111yescharacteristic
385C4⋊C8—711no—
386C4⋊C8—711no—
387C4⋊C8—711no—
388C4⋊C8—711no—
389C4⋊C8—1421no—
390C4⋊C8—1421no—
391C4⋊C8—1421no—
392C4⋊C8—1421no—
393C4⋊C8—1421no—
394C4⋊C8—1421no—
395C4⋊C8—1421no—
396C4⋊C8—1421no—
397C4⋊C8C2×Dic7111yescharacteristic
398C4⋊C8C2×Dic7111yescharacteristic
399#103C2×Dic7111yescharacteristic
400#103C2×Dic7111yescharacteristic
401#103C2×Dic7111yescharacteristic
402#103C2×Dic7111yescharacteristic
403#103C2×Dic7111yescharacteristic
404#103C2×Dic7111yescharacteristic
405#102C2×Dic7111yescharacteristic
406#102C2×Dic7111yescharacteristic
407C22⋊C8—1421no—
408C22⋊C8—1421no—
409C22⋊C8—1421no—
410C22⋊C8—1421no—
411C22⋊C8C2×Dic7111yescharacteristic
412C22⋊C8C2×Dic7111yescharacteristic
413#97—711no—
414#97—711no—
415#97—1421no—
416#97—1421no—
417#97—1421no—
418C4×C8—711no—
419C4×C8—711no—
420C4×C8—711no—
421C4×C8—711no—
422C4×C8—1421no—
423C4×C8—1421no—
424C4×C8—1421no—
425C4×C8—1421no—
426C4×C8C22×D7111yescharacteristic
427C4×C8C22×D7111yescharacteristic
428#41—22no—
429#41—41no—
430C2×Q8—21no—
431C2×Q8C22×Dic7111yescharacteristic
432C2×Q8C22×Dic7111yescharacteristic
433C2×Q8C22×Dic7111yescharacteristic
434C2×D4—21no—
435C2×D4C22×Dic7111yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4C22×Dic7111yescharacteristic
440C22×C4D4×D7111yescharacteristic
441C22×C4#1176586411yescharacteristic
442Q16—22no—
443Q16—22no—
444SD16—41no—
445SD16—41no—
446M16—2842no—
447M16—2842no—
448M16—2841no—
449M16—1422no—
450C2×C8—2842no—
451C2×C8—711no—
452C2×C8—711no—
453C2×C8—711no—
454C2×C8—711no—
455C2×C8—1421no—
456C2×C8—1421no—
457C2×C8—1421no—
458C2×C8—1421no—
459C2×C8—1421no—
460C2×C8—1421no—
461C2×C8C22×Dic7111yescharacteristic
462C2×C8C22×Dic7111yescharacteristic
463C2×C8C22×Dic7111yescharacteristic
464C2×C8C22×Dic7111yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4C22×Dic7111yescharacteristic
475C4⋊C4C22×Dic7111yescharacteristic
476C4⋊C4C22×Dic7111yescharacteristic
477C4⋊C4C22×Dic7111yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42C22×Dic7111yescharacteristic
489C42D4×D7111yescharacteristic
490C42#1176586411yescharacteristic
491C42C23×D7111yescharacteristic
492C23—21no—
493C23D4×Dic7111yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—81no—
500Q8#1455613512yes—
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—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—81no—
520C2×C4D4×Dic7111yescharacteristic
521C2×C4D4×Dic7111yescharacteristic
522C2×C4D4×Dic7111yescharacteristic
523C2×C4C23×Dic7111yescharacteristic
524C2×C4C2×D4×D7111yescharacteristic
525C2×C4C2×#1176586411yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—2841no—
531C8—2841no—
532C8—2841no—
533C8—2841no—
534C8—2841no—
535C8—2841no—
536C8—712no—
537C8—1421no—
538C8—1421no—
539C8—1421no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×D4×Dic7111yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4C2×#1455613511yescharacteristic
559C4#1798676611yescharacteristic
560C4#137×D7111yescharacteristic
561C4#1798679811yescharacteristic
562C2—41no—
563C2—81no—
564C2#2259706711yescharacteristic
565C2#137×Dic7111yescharacteristic
566C2#2259725711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#3107C2
socleC22×C71C2×C4×D4
prime radicalC2×#41×C71C2×C4
cosocleC42×C71C24
hypercenter#3107D71
hypocenterC71#18013
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #22597289C2semidirect
3. #22587243C2semidirect
4. #22587231C2semidirect
5. #22587219C2semidirect
6. #22587212C2semidirect
7. #22587191C2semidirect
8. #22587184C2semidirect
9. #22587159C2semidirect
10. Q16×C71⋊C8C2semidirect
11. #22587125C2semidirect
12. #22586813C2semidirect
14. #22586635C2semidirect
299. C71#18013semidirect

Maximal subgroups

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

subgroupindex
#225973602
#225972892
#225872432
#225872312
#225872192
#225872122
#225871912
#225871842
#225871592
Q16×C71⋊C82
#225871252
#225868132
#225867272
#225866352
C71×#31072
#1801371