Finite Group Databasebeta

← C31×#24071 #21480830

Subgroups — C31×#24071 #21480830

Order 7936.

1002 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#541C211yesmaximal, characteristic
2C2×C31×#484C211yesmaximal, characteristic
3C2×C31×#483C211yesmaximal, characteristic
4C31×#2317C211yesmaximal, characteristic
5C31×#2059C212yesmaximal, complemented
6C31×#2043C212yesmaximal, complemented
7C31×#1873C211yesmaximal, characteristic
8C31×#1873C211yesmaximal, characteristic
9C31×#1865C212yesmaximal, complemented
10C31×#1863C212yesmaximal, complemented
11C31×#1818C211yesmaximal, characteristic
12C31×#541—24no—
13C31×#541—24no—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C2×C31×#126—21no—
16C2×C31×#126—21no—
17C2×C31×#125—21no—
18C2×C31×#125—21no—
19C2×C31×#125C2211yescharacteristic
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8×C31C2211yescharacteristic
23C2×C4×Q8×C31C2211yescharacteristic
24C2×C31×#117C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#484—24no—
28C31×#483—24no—
29C2×C31×#107C2212yes—
30C2×C31×#106C2212yes—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#103C2212yes—
35C2×C31×C22⋊C8C2211yescharacteristic
36C2×C31×C22⋊C8C2211yescharacteristic
37C31×#389C2212yes—
38C31×#389C2212yes—
39C31×#385—21no—
40C31×#385—21no—
41C31×#384—21no—
42C31×#382—21no—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31C2311yescharacteristic
48C22×Q8×C31C2311yescharacteristic
49C23×C4×C31—21no—
50C23×C4×C31C2311yescharacteristic
51C22×C8×C31C2312yes—
52C22×C8×C31C2312yes—
53C31×C4⋊Q8—22no—
54C31×C4⋊Q8—22no—
55C31×#126—28no—
56C31×#126—28no—
57C31×#125—28no—
58C31×#125—28no—
59C31×#125—22no—
60C31×#125—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8D414yes—
64C31×C22⋊Q8D414yes—
65C4×Q8×C31—22no—
66C4×Q8×C31—22no—
67C4×Q8×C31—22no—
68C4×Q8×C31—22no—
69C31×#117—22no—
70C31×#117—22no—
71C31×#117—24no—
72C2×C4⋊C4×C31—21no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—22no—
81C2×C4⋊C4×C31—22no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31D411yescharacteristic
86C2×C4⋊C4×C31D411yescharacteristic
87C2×C4⋊C4×C31C2312yes—
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2312yes—
91C2×C22⋊C4×C31—21no—
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31D411yescharacteristic
95C2×C22⋊C4×C31D411yescharacteristic
96C2×C42×C31—21no—
97C2×C42×C31—21no—
98C2×C42×C31—21no—
99C2×C42×C31C2311yescharacteristic
100C2×C42×C31C2311yescharacteristic
101C31×#107—24no—
102C31×#106—24no—
103C31×#103—24no—
104C31×#103—24no—
105C31×#103—24no—
106C31×#103—24no—
107C31×C22⋊C8—22no—
108C31×C22⋊C8—22no—
109C31×#95—22no—
110C31×#95—22no—
111C24×C31C2×D411yescharacteristic
112C2×Q8×C31—21no—
113C2×Q8×C31—21no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C2×Q8×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31C2×D412yes—
119C2×Q8×C31C2×D412yes—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—22no—
128C22×C4×C31—42no—
129C22×C4×C31—82no—
130C22×C4×C31C2×D411yescharacteristic
131C22×C4×C31C2×D411yescharacteristic
132C22×C4×C31C2×D411yescharacteristic
133C22×C4×C31C2×D411yescharacteristic
134C22×C4×C31C2×D411yescharacteristic
135C22×C4×C31C2×D412yes—
136C22×C4×C31C2×D412yes—
137C22×C4×C31C2411yescharacteristic
138C2×C8×C31—22no—
139C2×C8×C31—22no—
140C2×C8×C31—22no—
141C2×C8×C31—22no—
142C2×C8×C31—22no—
143C2×C8×C31—22no—
144C4⋊C4×C31—24no—
145C4⋊C4×C31—24no—
146C4⋊C4×C31—24no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—24no—
149C4⋊C4×C31—24no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—42no—
160C4⋊C4×C31—82no—
161C4⋊C4×C31—82no—
162C4⋊C4×C31C2×D412yes—
163C4⋊C4×C31C2×D412yes—
164C4⋊C4×C31#4114yes—
165C22⋊C4×C31—24no—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C42×C31—24no—
171C42×C31—24no—
172C42×C31—24no—
173C42×C31—21no—
174C42×C31—21no—
175C42×C31—21no—
176C42×C31—21no—
177#24071C3111yesmaximal, characteristic, direct product factor, Sylow
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—22no—
181C23×C31—22no—
182C23×C31#12012yes—
183C23×C31C22×D411yescharacteristic
184Q8×C31—24no—
185Q8×C31—24no—
186Q8×C31—42no—
187Q8×C31—42no—
188C2×C4×C31—21no—
189C2×C4×C31—21no—
190C2×C4×C31—24no—
191C2×C4×C31—24no—
192C2×C4×C31—21no—
193C2×C4×C31—21no—
194C2×C4×C31—21no—
195C2×C4×C31—21no—
196C2×C4×C31—22no—
197C2×C4×C31—22no—
198C2×C4×C31—22no—
199C2×C4×C31—22no—
200C2×C4×C31—24no—
201C2×C4×C31—22no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—41no—
206C2×C4×C31—41no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—42no—
213C2×C4×C31—42no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—84no—
217C2×C4×C31#12012yes—
218C2×C4×C31C22×D411yescharacteristic
219C2×C4×C31C22×D411yescharacteristic
220C2×C4×C31C2×#4112yes—
221C2×C4×C312−1+412yes—
222C8×C31—42no—
223C8×C31—42no—
224C2×#541C2×C3111yescharacteristic
225C2×#484C2×C3111yescharacteristic
226C2×#483C2×C3111yescharacteristic
227#2317C2×C3111yescharacteristic
228#2059C2×C3112yescomplemented
229#2043C2×C3112yescomplemented
230#1873C2×C3111yescharacteristic
231#1873C2×C3111yescharacteristic
232#1865C2×C3112yescomplemented
233#1863C2×C3112yescomplemented
234#1818C2×C3111yescharacteristic
235C22×C31—22no—
236C22×C31—22no—
237C22×C31—22no—
238C22×C31—42no—
239C22×C31—42no—
240C22×C31C2×#12011yescharacteristic
241C22×C31#52511yescharacteristic
242C22×C31#53611yescharacteristic
243C22×C31#57712yes—
244C22×C31#57812yes—
245C4×C31—22no—
246C4×C31—44no—
247C4×C31—42no—
248C4×C31—42no—
249C4×C31—41no—
250C4×C31—41no—
251C4×C31—42no—
252C4×C31—41no—
253C4×C31—41no—
254C4×C31—42no—
255C4×C31—82no—
256C4×C31#54712yes—
257C4×C31#54712yes—
258#541—24no—
259#541—24no—
260C2×C4⋊Q8C22×C3111yescharacteristic
261C2×#126—21no—
262C2×#126—21no—
263C2×#125—21no—
264C2×#125—21no—
265C2×#125C22×C3111yescharacteristic
266C2×C22⋊Q8C22×C3111yescharacteristic
267C2×C22⋊Q8C22×C3111yescharacteristic
268C2×C4×Q8C22×C3111yescharacteristic
269C2×C4×Q8C22×C3111yescharacteristic
270C2×#117C22×C3111yescharacteristic
271C2×#117C22×C3111yescharacteristic
272C22×C4⋊C4C22×C3111yescharacteristic
273#484—24no—
274#483—24no—
275C2×#107C22×C3112yes—
276C2×#106C22×C3112yes—
277C2×#103C22×C3112yes—
278C2×#103C22×C3112yes—
279C2×#103C22×C3112yes—
280C2×#103C22×C3112yes—
281C2×C22⋊C8C22×C3111yescharacteristic
282C2×C22⋊C8C22×C3111yescharacteristic
283#389C22×C3112yes—
284#389C22×C3112yes—
285#385—21no—
286#385—21no—
287#384—21no—
288#382—21no—
289#340C22×C3112yes—
290#340C22×C3112yes—
291#336C22×C3112yes—
292#336C22×C3112yes—
293C2×C31—42no—
294C2×C31#240611yescharacteristic
295C2×C31#301011yescharacteristic
296C2×C31#301111yescharacteristic
297C2×C31#312012yes—
298C2×C31#324212yes—
299C22×Q8C23×C3111yescharacteristic
300C22×Q8C23×C3111yescharacteristic
301C23×C4—21no—
302C23×C4C23×C3111yescharacteristic
303C22×C8C23×C3112yes—
304C22×C8C23×C3112yes—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307#126—28no—
308#126—28no—
309#125—28no—
310#125—28no—
311#125—22no—
312#125—22no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C22⋊Q8D4×C3114yes—
316C22⋊Q8D4×C3114yes—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321#117—22no—
322#117—22no—
323#117—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4D4×C3111yescharacteristic
338C2×C4⋊C4D4×C3111yescharacteristic
339C2×C4⋊C4C23×C3112yes—
340C2×C4⋊C4C23×C3111yescharacteristic
341C2×C4⋊C4C23×C3111yescharacteristic
342C2×C4⋊C4C23×C3112yes—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4D4×C3111yescharacteristic
347C2×C22⋊C4D4×C3111yescharacteristic
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42C23×C3111yescharacteristic
352C2×C42C23×C3111yescharacteristic
353#107—24no—
354#106—24no—
355#103—24no—
356#103—24no—
357#103—24no—
358#103—24no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361#95—22no—
362#95—22no—
363C31#2407111yescharacteristic, direct product factor, Sylow
364C24C2×D4×C3111yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8C2×D4×C3112yes—
372C2×Q8C2×D4×C3112yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—42no—
382C22×C4—82no—
383C22×C4C2×D4×C3111yescharacteristic
384C22×C4C2×D4×C3111yescharacteristic
385C22×C4C2×D4×C3111yescharacteristic
386C22×C4C2×D4×C3111yescharacteristic
387C22×C4C2×D4×C3111yescharacteristic
388C22×C4C2×D4×C3112yes—
389C22×C4C2×D4×C3112yes—
390C22×C4C24×C3111yescharacteristic
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4C2×D4×C3112yes—
416C4⋊C4C2×D4×C3112yes—
417C4⋊C4#41×C3114yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C42—24no—
424C42—24no—
425C42—24no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23C31×#12012yes—
435C23C22×D4×C3111yescharacteristic
436Q8—24no—
437Q8—24no—
438Q8—42no—
439Q8—42no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4C31×#12012yes—
470C2×C4C22×D4×C3111yescharacteristic
471C2×C4C22×D4×C3111yescharacteristic
472C2×C4C2×#41×C3112yes—
473C2×C4C31×2−1+412yes—
474C8—42no—
475C8—42no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—42no—
480C22—42no—
481C22C2×C31×#12011yescharacteristic
482C22C31×#52511yescharacteristic
483C22C31×#53611yescharacteristic
484C22C31×#57712yes—
485C22C31×#57812yes—
486C4—22no—
487C4—44no—
488C4—42no—
489C4—42no—
490C4—41no—
491C4—41no—
492C4—42no—
493C4—41no—
494C4—41no—
495C4—42no—
496C4—82no—
497C4C31×#54712yes—
498C4C31×#54712yes—
499C2—42no—
500C2C31×#240611yescharacteristic
501C2C31×#301011yescharacteristic
502C2C31×#301111yescharacteristic
503C2C31×#312012yes—
504C2C31×#324212yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
5. C31×#2059C2semidirect
6. C31×#2043C2semidirect
9. C31×#1865C2semidirect
10. C31×#1863C2semidirect
177. #24071C31direct (N × H)
228. #2059C2×C31semidirect
229. #2043C2×C31semidirect
232. #1865C2×C31semidirect
233. #1863C2×C31semidirect
363. C31#24071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5412
C2×C31×#4842
C2×C31×#4832
C31×#23172
C31×#20592
C31×#20432
C31×#18732
C31×#18652
C31×#18632
C31×#18182
#2407131