Finite Group Databasebeta

← C31×#25924 #21482683

Subgroups — C31×#25924 #21482683

Order 7936.

1138 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic, complemented
2C31×#3138C211yesmaximal, characteristic, complemented
3C31×#3038C211yesmaximal, characteristic
4C31×#2858C211yesmaximal, characteristic, complemented
5C31×#2027C214yesmaximal, complemented
6C31×#1997C211yesmaximal, characteristic, complemented
7C31×#1994C211yesmaximal, characteristic, complemented
8C31×#1883C214yesmaximal, complemented
9C4×C31×#104C211yesmaximal, characteristic, complemented
10C2×C31×AGL(1,ℤ/8)C2212yes—
11Q82×C31—21no—
12C31×#557—21no—
13C31×#554—21no—
14C31×#554—22no—
15D4×Q8×C31—22no—
16D4×Q8×C31—21no—
17C31×#544—22no—
18C31×#544C2212yes—
19C31×#533—21no—
20C31×#533C2211yescharacteristic
21C2×C31×C4⋊Q8—21no—
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×#127C2211yescharacteristic
24C2×C31×#124—22no—
25C31×#520C2212yes—
26C4×#41×C31C2211yescharacteristic
27C31×#490—22no—
28C31×#486—21no—
29C31×#486—21no—
30C31×#460—21no—
31C31×#460—21no—
32C31×#459—21no—
33C31×#459—21no—
34C31×#423C2211yescharacteristic
35C2×C31×#104C2212yes—
36C31×#418C2214yes—
37C31×#418C2212yes—
38C4×M16×C31C2211yescharacteristic
39C31×#390—22no—
40C31×#390C2214yes—
41C43×C31C2211yescharacteristic
42C31×#343C2214yes—
43C31×#339C2214yes—
44C31×#339C2212yes—
45C2×#41×C31C2311yescharacteristic
46C22×Q8×C31—22no—
47C22×Q8×C31D412yes—
48C22×D4×C31—21no—
49C22×D4×C31—21no—
50C22×D4×C31C2312yes—
51C31×AGL(1,ℤ/8)—28no—
52C2×SD16×C31—22no—
53C2×D8×C31—22no—
54C2×M16×C31C2312yes—
55C2×M16×C31C2312yes—
56C31×C4⋊Q8—22no—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—24no—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8D411yescharacteristic
66C31×C4⋊Q8D411yescharacteristic
67C31×#127—21no—
68C31×#127—21no—
69C31×#127—21no—
70C31×#127—21no—
71C31×#125—24no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#125D412yes—
75C31×#124—48no—
76C31×C22⋊Q8—22no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—24no—
79C31×C22⋊Q8—24no—
80C31×C22⋊Q8—21no—
81C31×C22⋊Q8—21no—
82C31×C22⋊Q8—24no—
83C4×Q8×C31—22no—
84C4×Q8×C31—22no—
85C4×Q8×C31—22no—
86C4×Q8×C31—21no—
87C4×Q8×C31—21no—
88C4×Q8×C31D411yescharacteristic
89C4×Q8×C31D411yescharacteristic
90C4×D4×C31—22no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31—24no—
94C4×D4×C31D411yescharacteristic
95C4×D4×C31D411yescharacteristic
96C31×#117—22no—
97C31×#117—21no—
98C31×#117C2311yescharacteristic
99C31×#117C2312yes—
100C2×C4⋊C4×C31—22no—
101C2×C4⋊C4×C31—22no—
102C2×C4⋊C4×C31D412yes—
103C2×C4⋊C4×C31C2312yes—
104C2×C22⋊C4×C31—28no—
105C2×C42×C31—21no—
106C2×C42×C31—21no—
107C2×C42×C31—21no—
108C2×C42×C31C2312yes—
109C2×C42×C31C2311yescharacteristic
110C31×C4⋊C8—21no—
111C31×C4⋊C8—21no—
112C31×#104—24no—
113C31×#102—22no—
114C31×#102—22no—
115C31×#102—24no—
116C31×#102—24no—
117C31×#97—21no—
118C4×C8×C31—21no—
119C24×C31—22no—
120#41×C31—22no—
121#41×C31#4114yes—
122C2×Q8×C31—24no—
123C2×Q8×C31—21no—
124C2×Q8×C31—22no—
125C2×Q8×C31—21no—
126C2×Q8×C31—22no—
127C2×Q8×C31—44no—
128C2×Q8×C31—84no—
129C2×Q8×C31C2×D411yescharacteristic
130C2×D4×C31—21no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—24no—
134C2×D4×C31—24no—
135C2×D4×C31—44no—
136C2×D4×C31—44no—
137C2×D4×C31—41no—
138C2×D4×C31—41no—
139C2×D4×C31—41no—
140C2×D4×C31—41no—
141C2×D4×C31C2×D411yescharacteristic
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—24no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D414yes—
150C22×C4×C31C2411yescharacteristic
151SD16×C31—44no—
152D8×C31—44no—
153M16×C31—22no—
154M16×C31—24no—
155M16×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—24no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31—22no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31C2×D412yes—
173C4⋊C4×C31C2×D412yes—
174C4⋊C4×C31C2×D411yescharacteristic
175C4⋊C4×C31C2×D411yescharacteristic
176C22⋊C4×C31—24no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—21no—
179C22⋊C4×C31—21no—
180C22⋊C4×C31—21no—
181C22⋊C4×C31—416no—
182C42×C31—24no—
183C42×C31—24no—
184C42×C31—21no—
185C42×C31—21no—
186C42×C31—21no—
187C42×C31—22no—
188C42×C31—22no—
189C42×C31—22no—
190C42×C31—42no—
191C42×C31C2×D411yescharacteristic
192C42×C31C2×D411yescharacteristic
193C42×C31C2×D411yescharacteristic
194C42×C31C2×D411yescharacteristic
195#25924C3111yesmaximal, characteristic, direct product factor, Sylow
196C23×C31—21no—
197C23×C31—44no—
198C23×C31—44no—
199C23×C31—42no—
200C23×C31—44no—
201C23×C31C22×D411yescharacteristic
202Q8×C31—21no—
203Q8×C31—44no—
204Q8×C31—42no—
205Q8×C31—42no—
206Q8×C31—82no—
207Q8×C31—84no—
208Q8×C31#12412yescomplemented
209D4×C31—24no—
210D4×C31—21no—
211D4×C31—42no—
212D4×C31—42no—
213D4×C31—44no—
214D4×C31—82no—
215D4×C31—82no—
216D4×C31—82no—
217D4×C31—82no—
218D4×C31#12412yescomplemented
219C2×C4×C31—21no—
220C2×C4×C31—24no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—24no—
224C2×C4×C31—22no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—24no—
228C2×C4×C31—22no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—44no—
234C2×C4×C31—84no—
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31#12011yescharacteristic
237C2×C4×C31#12011yescharacteristic
238C2×C4×C31#12011yescharacteristic
239C2×C4×C31C22×D411yescharacteristic
240C2×C4×C31C22×D411yescharacteristic
241C2×C4×C31C2×#4112yes—
242C2×C4×C312−1+412yes—
243C8×C31—42no—
244C8×C31—42no—
245#3503C2×C3111yescharacteristic, complemented
246#3138C2×C3111yescharacteristic, complemented
247#3038C2×C3111yescharacteristic
248#2858C2×C3111yescharacteristic, complemented
249#2027C2×C3114yescomplemented
250#1997C2×C3111yescharacteristic, complemented
251#1994C2×C3111yescharacteristic, complemented
252#1883C2×C3114yescomplemented
253C4×#104C2×C3111yescharacteristic, complemented
254C22×C31—21no—
255C22×C31—21no—
256C22×C31—22no—
257C22×C31—41no—
258C22×C31—41no—
259C22×C31—88no—
260C22×C31—82no—
261C22×C31—82no—
262C22×C31—82no—
263C22×C31C2×#12011yescharacteristic
264C22×C31#54812yes—
265C4×C31—22no—
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—44no—
269C4×C31—41no—
270C4×C31—44no—
271C4×C31—42no—
272C4×C31—41no—
273C4×C31—42no—
274C4×C31#45312yes—
275C4×C31#45412yes—
276C4×C31C2×#12411yescharacteristic
277C4×C31#53611yescharacteristic
278C4×C31#54812yes—
279C2×AGL(1,ℤ/8)C22×C3112yes—
280Q82—21no—
281#557—21no—
282#554—21no—
283#554—22no—
284D4×Q8—22no—
285D4×Q8—21no—
286#544—22no—
287#544C22×C3112yes—
288#533—21no—
289#533C22×C3111yescharacteristic
290C2×C4⋊Q8—21no—
291C2×C4⋊Q8C22×C3111yescharacteristic
292C2×#127C22×C3111yescharacteristic
293C2×#124—22no—
294#520C22×C3112yes—
295C4×#41C22×C3111yescharacteristic
296#490—22no—
297#486—21no—
298#486—21no—
299#460—21no—
300#460—21no—
301#459—21no—
302#459—21no—
303#423C22×C3111yescharacteristic
304C2×#104C22×C3112yes—
305#418C22×C3114yes—
306#418C22×C3112yes—
307C4×M16C22×C3111yescharacteristic
308#390—22no—
309#390C22×C3114yes—
310C43C22×C3111yescharacteristic
311#343C22×C3114yes—
312#339C22×C3114yes—
313#339C22×C3112yes—
314C2×C31—22no—
315C2×C31—42no—
316C2×C31—162no—
317C2×C31#243611yescharacteristic
318C2×C31C2×#45311yescharacteristic
319C2×C31C2×#45411yescharacteristic
320C2×#41C23×C3111yescharacteristic
321C22×Q8—22no—
322C22×Q8D4×C3112yes—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C23×C3112yes—
326AGL(1,ℤ/8)—28no—
327C2×SD16—22no—
328C2×D8—22no—
329C2×M16C23×C3112yes—
330C2×M16C23×C3112yes—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—24no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8D4×C3111yescharacteristic
341C4⋊Q8D4×C3111yescharacteristic
342#127—21no—
343#127—21no—
344#127—21no—
345#127—21no—
346#125—24no—
347#125—22no—
348#125—22no—
349#125D4×C3112yes—
350#124—48no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—24no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—24no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8D4×C3111yescharacteristic
364C4×Q8D4×C3111yescharacteristic
365C4×D4—22no—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—24no—
369C4×D4D4×C3111yescharacteristic
370C4×D4D4×C3111yescharacteristic
371#117—22no—
372#117—21no—
373#117C23×C3111yescharacteristic
374#117C23×C3112yes—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4D4×C3112yes—
378C2×C4⋊C4C23×C3112yes—
379C2×C22⋊C4—28no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42C23×C3112yes—
384C2×C42C23×C3111yescharacteristic
385C4⋊C8—21no—
386C4⋊C8—21no—
387#104—24no—
388#102—22no—
389#102—22no—
390#102—24no—
391#102—24no—
392#97—21no—
393C4×C8—21no—
394C31#2592411yescharacteristic, direct product factor, Sylow
395C24—22no—
396#41—22no—
397#41#41×C3114yes—
398C2×Q8—24no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—84no—
405C2×Q8C2×D4×C3111yescharacteristic
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4C2×D4×C3111yescharacteristic
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4C2×D4×C3111yescharacteristic
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4C2×D4×C3114yes—
426C22×C4C24×C3111yescharacteristic
427SD16—44no—
428D8—44no—
429M16—22no—
430M16—24no—
431M16—22no—
432C2×C8—22no—
433C2×C8—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4C2×D4×C3112yes—
449C4⋊C4C2×D4×C3112yes—
450C4⋊C4C2×D4×C3111yescharacteristic
451C4⋊C4C2×D4×C3111yescharacteristic
452C22⋊C4—24no—
453C22⋊C4—22no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—416no—
458C42—24no—
459C42—24no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—42no—
467C42C2×D4×C3111yescharacteristic
468C42C2×D4×C3111yescharacteristic
469C42C2×D4×C3111yescharacteristic
470C42C2×D4×C3111yescharacteristic
471C23—21no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—44no—
476C23C22×D4×C3111yescharacteristic
477Q8—21no—
478Q8—44no—
479Q8—42no—
480Q8—42no—
481Q8—82no—
482Q8—84no—
483Q8C31×#12412yescomplemented
484D4—24no—
485D4—21no—
486D4—42no—
487D4—42no—
488D4—44no—
489D4—82no—
490D4—82no—
491D4—82no—
492D4—82no—
493D4C31×#12412yescomplemented
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—84no—
510C2×C4C31×#12011yescharacteristic
511C2×C4C31×#12011yescharacteristic
512C2×C4C31×#12011yescharacteristic
513C2×C4C31×#12011yescharacteristic
514C2×C4C22×D4×C3111yescharacteristic
515C2×C4C22×D4×C3111yescharacteristic
516C2×C4C2×#41×C3112yes—
517C2×C4C31×2−1+412yes—
518C8—42no—
519C8—42no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—88no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22C2×C31×#12011yescharacteristic
530C22C31×#54812yes—
531C4—22no—
532C4—21no—
533C4—21no—
534C4—44no—
535C4—41no—
536C4—44no—
537C4—42no—
538C4—41no—
539C4—42no—
540C4C31×#45312yes—
541C4C31×#45412yes—
542C4C2×C31×#12411yescharacteristic
543C4C31×#53611yescharacteristic
544C4C31×#54812yes—
545C2—22no—
546C2—42no—
547C2—162no—
548C2C31×#243611yescharacteristic
549C2C2×C31×#45311yescharacteristic
550C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3038C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3503C2semidirect
2. C31×#3138C2semidirect
4. C31×#2858C2semidirect
5. C31×#2027C2semidirect
6. C31×#1997C2semidirect
7. C31×#1994C2semidirect
8. C31×#1883C2semidirect
9. C4×C31×#104C2semidirect
195. #25924C31direct (N × H)
208. Q8×C31#124semidirect
218. D4×C31#124semidirect
245. #3503C2×C31semidirect
246. #3138C2×C31semidirect
248. #2858C2×C31semidirect
249. #2027C2×C31semidirect
250. #1997C2×C31semidirect
251. #1994C2×C31semidirect
252. #1883C2×C31semidirect
253. C4×#104C2×C31semidirect
394. C31#25924direct (N × H)
483. Q8C31×#124semidirect
493. D4C31×#124semidirect

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#31382
C31×#30382
C31×#28582
C31×#20272
C31×#19972
C31×#19942
C31×#18832
C4×C31×#1042
#2592431