Finite Group Databasebeta

← C31×#24351 #21481110

Subgroups — C31×#24351 #21481110

Order 7936.

1122 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C211yesmaximal, characteristic
2C31×#2855C212yesmaximal, complemented
3C4×C31×#123C211yesmaximal, characteristic
4C31×#2062C212yesmaximal, complemented
5C31×#2030C212yesmaximal, complemented
6C31×#1996C212yesmaximal, complemented
7C31×#1890C212yesmaximal
8C31×#1867C212yesmaximal
9C31×#1799C211yesmaximal, characteristic
10C31×#554—24no—
11C31×#554—22no—
12D4×Q8×C31—22no—
13C31×#544—21no—
14C31×#543—21no—
15C31×#543C2212yes—
16C31×#533—21no—
17C2×C31×#123C2211yescharacteristic
18C2×C31×C22⋊Q8—21no—
19C2×C31×C22⋊Q8C2212yes—
20C31×#518C2212yes—
21C2×C4×D4×C31C2211yescharacteristic
22C22×C42×C31C2211yescharacteristic
23C2×C31×#99C2212yes—
24C31×#391—22no—
25C31×#389—22no—
26C31×#388—21no—
27C31×#387—24no—
28C31×#387—21no—
29C31×#386C2211yescharacteristic
30C31×#385C2211yescharacteristic
31C31×#385C2214yes—
32C31×#381C2214yes—
33C31×#381C2211yescharacteristic
34C31×#380C2212yes—
35C4×C4⋊C4×C31—21no—
36C4×C22⋊C4×C31—21no—
37C4×C22⋊C4×C31C2211yescharacteristic
38C31×#342C2212yes—
39C31×#342C2214yes—
40C31×#342C2214yes—
41C22×Q8×C31—21no—
42C22×Q8×C31D412yes—
43C22×D4×C31C2311yescharacteristic
44C23×C4×C31D411yescharacteristic
45C23×C4×C31C2311yescharacteristic
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C31×C4⋊Q8—21no—
49C31×C4⋊Q8—21no—
50C31×C4⋊Q8—24no—
51C31×#125—22no—
52C31×#125—24no—
53C31×#123—22no—
54C31×#123Q814yescomplemented
55C31×C22⋊Q8—22no—
56C31×C22⋊Q8—22no—
57C31×C22⋊Q8—24no—
58C31×C22⋊Q8—24no—
59C31×C22⋊Q8—22no—
60C31×C22⋊Q8—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—22no—
64C31×C22⋊Q8—22no—
65C31×C22⋊Q8—24no—
66C31×C22⋊Q8—24no—
67C4×Q8×C31—22no—
68C4×Q8×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31—22no—
71C4×D4×C31—24no—
72C4×D4×C31—24no—
73C31×#117—22no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—22no—
79C2×C4⋊C4×C31—24no—
80C2×C4⋊C4×C31D411yescharacteristic
81C2×C4⋊C4×C31D414yes—
82C2×C4⋊C4×C31D412yes—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—21no—
86C2×C22⋊C4×C31D411yescharacteristic
87C2×C22⋊C4×C31C2314yes—
88C2×C22⋊C4×C31C2311yescharacteristic
89C2×C22⋊C4×C31C2311yescharacteristic
90C2×C22⋊C4×C31C2314yes—
91C2×C22⋊C4×C31C2311yescharacteristic
92C2×C42×C31—21no—
93C2×C42×C31—22no—
94C2×C42×C31—21no—
95C2×C42×C31—22no—
96C2×C42×C31—21no—
97C2×C42×C31D411yescharacteristic
98C31×#99—24no—
99C31×#95—21no—
100C31×#95—21no—
101C31×#95—21no—
102C31×#95—21no—
103C31×#95—21no—
104C31×#95—22no—
105C31×#95—24no—
106C31×#95—24no—
107C31×#95—24no—
108C24×C31C2×D411yescharacteristic
109C24×C31C2411yescharacteristic
110C2×Q8×C31—22no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—22no—
114C2×Q8×C31—44no—
115C2×Q8×C31—82no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31C2×Q812yes—
119C2×D4×C31#4112yes—
120C22×C4×C31—22no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—22no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—24no—
133C22×C4×C31—42no—
134C22×C4×C31—42no—
135C22×C4×C31—42no—
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D412yes—
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D414yes—
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D414yes—
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D412yes—
145C22×C4×C31C2×Q812yescomplemented
146C4⋊C4×C31—22no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—44no—
159C4⋊C4×C31—42no—
160C4⋊C4×C31—48no—
161C22⋊C4×C31—24no—
162C22⋊C4×C31—24no—
163C22⋊C4×C31—24no—
164C22⋊C4×C31—22no—
165C22⋊C4×C31—24no—
166C22⋊C4×C31—21no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—42no—
172C22⋊C4×C31C2×Q812yescomplemented
173C42×C31—24no—
174C42×C31—21no—
175C42×C31—21no—
176C42×C31—42no—
177C42×C31—41no—
178C42×C31—42no—
179C42×C31—41no—
180#24351C3111yesmaximal, characteristic, direct product factor, Sylow
181C23×C31—21no—
182C23×C31—21no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—41no—
187C23×C31C22⋊Q812yes—
188C23×C31C22×D411yescharacteristic
189C23×C31C22×D412yes—
190C23×C31C22×Q811yescharacteristic
191C23×C31C2×#4111yescharacteristic
192C23×C312−1+412yes—
193Q8×C31—44no—
194Q8×C31—82no—
195Q8×C31—82no—
196Q8×C31—82no—
197D4×C31—24no—
198D4×C31—24no—
199C2×C4×C31—21no—
200C2×C4×C31—21no—
201C2×C4×C31—22no—
202C2×C4×C31—24no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—21no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—24no—
212C2×C4×C31—42no—
213C2×C4×C31—41no—
214C2×C4×C31—41no—
215C2×C4×C31—41no—
216C2×C4×C31—42no—
217C2×C4×C31—41no—
218C2×C4×C31—41no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—44no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—44no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—44no—
230C2×C4×C31—44no—
231C2×C4×C31—41no—
232C2×C4×C31—44no—
233C2×C4×C31—81no—
234C2×C4×C31—82no—
235C2×C4×C31—82no—
236C2×C4×C31#12011yescharacteristic
237C2×C4×C31#12012yes—
238C2×C4×C31#12011yescharacteristic
239C2×C4×C31C22⋊Q812yes—
240#3501C2×C3111yescharacteristic
241#2855C2×C3112yescomplemented
242C4×#123C2×C3111yescharacteristic
243#2062C2×C3112yescomplemented
244#2030C2×C3112yescomplemented
245#1996C2×C3112yescomplemented
246#1890C2×C3112yes—
247#1867C2×C3112yes—
248#1799C2×C3111yescharacteristic
249C22×C31—22no—
250C22×C31—21no—
251C22×C31—22no—
252C22×C31—21no—
253C22×C31—22no—
254C22×C31—41no—
255C22×C31—41no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—41no—
259C22×C31—41no—
260C22×C31—81no—
261C22×C31C2×#12011yescharacteristic
262C22×C31C2×C22⋊Q811yescharacteristic
263C22×C31#53611yescharacteristic
264C22×C31#54812yes—
265C22×C31D4×Q812yes—
266C4×C31—22no—
267C4×C31—21no—
268C4×C31—22no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—41no—
272C4×C31—44no—
273C4×C31—42no—
274C4×C31—44no—
275C4×C31—41no—
276C4×C31—84no—
277C4×C31—81no—
278C4×C31#45812yes—
279#554—24no—
280#554—22no—
281D4×Q8—22no—
282#544—21no—
283#543—21no—
284#543C22×C3112yes—
285#533—21no—
286C2×#123C22×C3111yescharacteristic
287C2×C22⋊Q8—21no—
288C2×C22⋊Q8C22×C3112yes—
289#518C22×C3112yes—
290C2×C4×D4C22×C3111yescharacteristic
291C22×C42C22×C3111yescharacteristic
292C2×#99C22×C3112yes—
293#391—22no—
294#389—22no—
295#388—21no—
296#387—24no—
297#387—21no—
298#386C22×C3111yescharacteristic
299#385C22×C3111yescharacteristic
300#385C22×C3114yes—
301#381C22×C3114yes—
302#381C22×C3111yescharacteristic
303#380C22×C3112yes—
304C4×C4⋊C4—21no—
305C4×C22⋊C4—21no—
306C4×C22⋊C4C22×C3111yescharacteristic
307#342C22×C3112yes—
308#342C22×C3114yes—
309#342C22×C3114yes—
310C2×C31—21no—
311C2×C31—22no—
312C2×C31—21no—
313C2×C31—22no—
314C2×C31—42no—
315C2×C31#240911yescharacteristic
316C2×C31C2×#45811yescharacteristic
317C2×C31#302511yescharacteristic
318C22×Q8—21no—
319C22×Q8D4×C3112yes—
320C22×D4C23×C3111yescharacteristic
321C23×C4D4×C3111yescharacteristic
322C23×C4C23×C3111yescharacteristic
323C23×C4C23×C3111yescharacteristic
324C23×C4C23×C3111yescharacteristic
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—24no—
328#125—22no—
329#125—24no—
330#123—22no—
331#123Q8×C3114yescomplemented
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—24no—
343C22⋊Q8—24no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—24no—
349C4×D4—24no—
350#117—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4D4×C3111yescharacteristic
358C2×C4⋊C4D4×C3114yes—
359C2×C4⋊C4D4×C3112yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4D4×C3111yescharacteristic
364C2×C22⋊C4C23×C3111yescharacteristic
365C2×C22⋊C4C23×C3114yes—
366C2×C22⋊C4C23×C3111yescharacteristic
367C2×C22⋊C4C23×C3114yes—
368C2×C22⋊C4C23×C3111yescharacteristic
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—22no—
374C2×C42D4×C3111yescharacteristic
375#99—24no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385C31#2435111yescharacteristic, direct product factor, Sylow
386C24C2×D4×C3111yescharacteristic
387C24C24×C3111yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4C2×Q8×C3112yes—
397C2×D4#41×C3112yes—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3112yes—
416C22×C4C2×D4×C3111yescharacteristic
417C22×C4C2×D4×C3114yes—
418C22×C4C2×D4×C3111yescharacteristic
419C22×C4C2×D4×C3111yescharacteristic
420C22×C4C2×D4×C3114yes—
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C2×D4×C3112yes—
423C22×C4C2×Q8×C3112yescomplemented
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—48no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—24no—
444C22⋊C4—21no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4C2×Q8×C3112yescomplemented
451C42—24no—
452C42—21no—
453C42—21no—
454C42—42no—
455C42—41no—
456C42—42no—
457C42—41no—
458C23—21no—
459C23—21no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23C31×C22⋊Q812yes—
465C23C22×D4×C3111yescharacteristic
466C23C22×D4×C3112yes—
467C23C22×Q8×C3111yescharacteristic
468C23C2×#41×C3111yescharacteristic
469C23C31×2−1+412yes—
470Q8—44no—
471Q8—82no—
472Q8—82no—
473Q8—82no—
474D4—24no—
475D4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4C31×#12011yescharacteristic
514C2×C4C31×#12012yes—
515C2×C4C31×#12011yescharacteristic
516C2×C4C31×C22⋊Q812yes—
517C22—22no—
518C22—21no—
519C22—22no—
520C22—21no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22C2×C31×#12011yescharacteristic
530C22C2×C31×C22⋊Q811yescharacteristic
531C22C31×#53611yescharacteristic
532C22C31×#54812yes—
533C22D4×Q8×C3112yes—
534C4—22no—
535C4—21no—
536C4—22no—
537C4—42no—
538C4—42no—
539C4—41no—
540C4—44no—
541C4—42no—
542C4—44no—
543C4—41no—
544C4—84no—
545C4—81no—
546C4C31×#45812yes—
547C2—21no—
548C2—22no—
549C2—21no—
550C2—22no—
551C2—42no—
552C2C31×#240911yescharacteristic
553C2C2×C31×#45811yescharacteristic
554C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC22×D4×C31C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2855C2semidirect
4. C31×#2062C2semidirect
5. C31×#2030C2semidirect
6. C31×#1996C2semidirect
54. C31×#123Q8semidirect
145. C22×C4×C31C2×Q8semidirect
172. C22⋊C4×C31C2×Q8semidirect
180. #24351C31direct (N × H)
241. #2855C2×C31semidirect
243. #2062C2×C31semidirect
244. #2030C2×C31semidirect
245. #1996C2×C31semidirect
331. #123Q8×C31semidirect
385. C31#24351direct (N × H)
423. C22×C4C2×Q8×C31semidirect
450. C22⋊C4C2×Q8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#28552
C4×C31×#1232
C31×#20622
C31×#20302
C31×#19962
C31×#18902
C31×#18672
C31×#17992
#2435131