Finite Group Databasebeta

← C31×#25915 #21482674

Subgroups — C31×#25915 #21482674

Order 7936.

1110 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#544C211yesmaximal, characteristic, complemented
2C2×C31×#490C211yesmaximal, characteristic, complemented
3C2×C31×#486C211yesmaximal, characteristic, complemented
4C31×#2619C211yesmaximal, characteristic
5C31×#2025C212yesmaximal, complemented
6C31×#2023C212yesmaximal, complemented
7C31×#1993C211yesmaximal, characteristic, complemented
8C31×#1991C211yesmaximal, characteristic, complemented
9C31×#1875C212yesmaximal, complemented
10C31×#1874C212yesmaximal, complemented
11C31×#1750C211yesmaximal, characteristic, complemented
12C31×#544—24no—
13C31×#544—24no—
14C2×C31×C4⋊Q8—21no—
15C2×C31×C4⋊Q8C2211yescharacteristic
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#127C2211yescharacteristic
18C2×C31×#125—21no—
19C2×C31×#125C2211yescharacteristic
20C2×C31×#125C2211yescharacteristic
21C2×C31×C22⋊Q8—21no—
22C2×C31×C22⋊Q8—21no—
23C2×C31×#121C2212yes—
24C2×C31×#117C2211yescharacteristic
25C2×C31×#117C2211yescharacteristic
26C22×C4⋊C4×C31C2211yescharacteristic
27C31×#490—24no—
28C31×#486—24no—
29C2×C31×#102C2212yes—
30C2×C31×#102C2212yes—
31C2×C31×#102C2212yes—
32C2×C31×#102C2212yes—
33C2×C31×C22⋊C8C2212yes—
34C2×C31×#97C2211yescharacteristic
35C2×C4×C8×C31C2211yescharacteristic
36C31×#394—21no—
37C31×#394—21no—
38C31×#392—21no—
39C31×#392—21no—
40C31×#390C2212yes—
41C31×#382C2211yescharacteristic
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31—21no—
49C22×D4×C31—22no—
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C22×D4×C31C2312yes—
53C23×C4×C31—21no—
54C23×C4×C31C2311yescharacteristic
55C22×C8×C31C2312yes—
56C22×C8×C31C2312yes—
57C31×C4⋊Q8—28no—
58C31×C4⋊Q8—22no—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×C4⋊Q8D414yescomplemented
62C31×#127—22no—
63C31×#127—22no—
64C31×#125—28no—
65C31×#125—22no—
66C31×#125—22no—
67C31×#125—22no—
68C31×#125D414yescomplemented
69C31×C22⋊Q8—28no—
70C31×C22⋊Q8—28no—
71C31×#121—28no—
72C31×#117—22no—
73C31×#117—22no—
74C31×#117—22no—
75C31×#117—22no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
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×C31C2311yescharacteristic
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C42×C31—21no—
99C2×C42×C31C2311yescharacteristic
100C2×C42×C31C2311yescharacteristic
101C31×#102—24no—
102C31×#102—24no—
103C31×#102—24no—
104C31×#102—24no—
105C31×C22⋊C8—24no—
106C31×#97—22no—
107C4×C8×C31—22no—
108C31×#95—21no—
109C31×#95—21no—
110C24×C31—22no—
111C24×C31C2×D411yescharacteristic
112C2×Q8×C31—24no—
113C2×Q8×C31—24no—
114C2×Q8×C31—82no—
115C2×Q8×C31—82no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31—24no—
119C2×D4×C31—24no—
120C2×D4×C31—44no—
121C2×D4×C31—42no—
122C2×D4×C31—42no—
123C2×D4×C31—82no—
124C2×D4×C31—82no—
125C2×D4×C31—82no—
126C2×D4×C31—82no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—24no—
135C22×C4×C31—22no—
136C22×C4×C31—82no—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31#4114yes—
143C22×C4×C31C2411yescharacteristic
144C2×C8×C31—22no—
145C2×C8×C31—22no—
146C2×C8×C31—22no—
147C2×C8×C31—22no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C4⋊C4×C31—24no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31C2×D412yes—
165C4⋊C4×C31C2×D412yes—
166C4⋊C4×C31C2×D412yes—
167C4⋊C4×C31C2×D412yes—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—44no—
172C22⋊C4×C31—82no—
173C22⋊C4×C31—82no—
174C42×C31—24no—
175C42×C31—21no—
176C42×C31—21no—
177C42×C31C2×D412yes—
178C42×C31C2×D412yes—
179#25915C3111yesmaximal, characteristic, direct product factor, Sylow
180C23×C31—21no—
181C23×C31—21no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31#12412yes—
192C23×C31C22×D411yescharacteristic
193Q8×C31—84no—
194Q8×C31—84no—
195D4×C31—44no—
196D4×C31—44no—
197D4×C31—84no—
198D4×C31—84no—
199D4×C31—88no—
200C2×C4×C31—21no—
201C2×C4×C31—21no—
202C2×C4×C31—22no—
203C2×C4×C31—22no—
204C2×C4×C31—24no—
205C2×C4×C31—24no—
206C2×C4×C31—21no—
207C2×C4×C31—22no—
208C2×C4×C31—21no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—42no—
215C2×C4×C31—42no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—84no—
222C2×C4×C31#12012yes—
223C2×C4×C31#12012yes—
224C2×C4×C31#12412yes—
225C2×C4×C31C22×D411yescharacteristic
226C2×C4×C31C22×D411yescharacteristic
227C2×C4×C31C2×#4112yes—
228C2×C4×C312−1+412yes—
229C8×C31—42no—
230C8×C31—42no—
231C2×#544C2×C3111yescharacteristic, complemented
232C2×#490C2×C3111yescharacteristic, complemented
233C2×#486C2×C3111yescharacteristic, complemented
234#2619C2×C3111yescharacteristic
235#2025C2×C3112yescomplemented
236#2023C2×C3112yescomplemented
237#1993C2×C3111yescharacteristic, complemented
238#1991C2×C3111yescharacteristic, complemented
239#1875C2×C3112yescomplemented
240#1874C2×C3112yescomplemented
241#1750C2×C3111yescharacteristic, complemented
242C22×C31—24no—
243C22×C31—22no—
244C22×C31—42no—
245C22×C31—41no—
246C22×C31—41no—
247C22×C31—82no—
248C22×C31—82no—
249C22×C31—82no—
250C22×C31—82no—
251C22×C31—82no—
252C22×C31—82no—
253C22×C31—82no—
254C22×C31C2×#12011yescharacteristic
255C22×C31C2×#12411yescharacteristic
256C22×C31#53611yescharacteristic
257C22×C31#57612yes—
258C22×C31#57712yes—
259C4×C31—22no—
260C4×C31—22no—
261C4×C31—22no—
262C4×C31—41no—
263C4×C31—42no—
264C4×C31—42no—
265C4×C31—42no—
266C4×C31—41no—
267C4×C31—42no—
268C4×C31—82no—
269C4×C31#54812yes—
270C4×C31#54812yes—
271#544—24no—
272#544—24no—
273C2×C4⋊Q8—21no—
274C2×C4⋊Q8C22×C3111yescharacteristic
275C2×C4⋊Q8C22×C3111yescharacteristic
276C2×#127C22×C3111yescharacteristic
277C2×#125—21no—
278C2×#125C22×C3111yescharacteristic
279C2×#125C22×C3111yescharacteristic
280C2×C22⋊Q8—21no—
281C2×C22⋊Q8—21no—
282C2×#121C22×C3112yes—
283C2×#117C22×C3111yescharacteristic
284C2×#117C22×C3111yescharacteristic
285C22×C4⋊C4C22×C3111yescharacteristic
286#490—24no—
287#486—24no—
288C2×#102C22×C3112yes—
289C2×#102C22×C3112yes—
290C2×#102C22×C3112yes—
291C2×#102C22×C3112yes—
292C2×C22⋊C8C22×C3112yes—
293C2×#97C22×C3111yescharacteristic
294C2×C4×C8C22×C3111yescharacteristic
295#394—21no—
296#394—21no—
297#392—21no—
298#392—21no—
299#390C22×C3112yes—
300#382C22×C3111yescharacteristic
301#340C22×C3112yes—
302#340C22×C3112yes—
303#340C22×C3112yes—
304#340C22×C3112yes—
305#336C22×C3112yes—
306C2×C31—42no—
307C2×C31—162no—
308C2×C31#243611yescharacteristic
309C2×C31#300811yescharacteristic
310C2×C31#301011yescharacteristic
311C2×C31#313612yes—
312C2×C31#324112yes—
313C22×Q8—21no—
314C22×Q8—21no—
315C22×D4—22no—
316C22×D4—21no—
317C22×D4—21no—
318C22×D4C23×C3112yes—
319C23×C4—21no—
320C23×C4C23×C3111yescharacteristic
321C22×C8C23×C3112yes—
322C22×C8C23×C3112yes—
323C4⋊Q8—28no—
324C4⋊Q8—22no—
325C4⋊Q8—22no—
326C4⋊Q8—22no—
327C4⋊Q8D4×C3114yescomplemented
328#127—22no—
329#127—22no—
330#125—28no—
331#125—22no—
332#125—22no—
333#125—22no—
334#125D4×C3114yescomplemented
335C22⋊Q8—28no—
336C22⋊Q8—28no—
337#121—28no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4D4×C3111yescharacteristic
352C2×C4⋊C4D4×C3111yescharacteristic
353C2×C4⋊C4C23×C3112yes—
354C2×C4⋊C4C23×C3111yescharacteristic
355C2×C4⋊C4C23×C3111yescharacteristic
356C2×C4⋊C4C23×C3111yescharacteristic
357C2×C4⋊C4C23×C3111yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4D4×C3111yescharacteristic
363C2×C22⋊C4D4×C3111yescharacteristic
364C2×C42—21no—
365C2×C42C23×C3111yescharacteristic
366C2×C42C23×C3111yescharacteristic
367#102—24no—
368#102—24no—
369#102—24no—
370#102—24no—
371C22⋊C8—24no—
372#97—22no—
373C4×C8—22no—
374#95—21no—
375#95—21no—
376C31#2591511yescharacteristic, direct product factor, Sylow
377C24—22no—
378C24C2×D4×C3111yescharacteristic
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—82no—
404C22×C4C2×D4×C3111yescharacteristic
405C22×C4C2×D4×C3111yescharacteristic
406C22×C4C2×D4×C3111yescharacteristic
407C22×C4C2×D4×C3111yescharacteristic
408C22×C4C2×D4×C3111yescharacteristic
409C22×C4#41×C3114yes—
410C22×C4C24×C3111yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4C2×D4×C3112yes—
432C4⋊C4C2×D4×C3112yes—
433C4⋊C4C2×D4×C3112yes—
434C4⋊C4C2×D4×C3112yes—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—44no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42C2×D4×C3112yes—
445C42C2×D4×C3112yes—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23C31×#12412yes—
458C23C22×D4×C3111yescharacteristic
459Q8—84no—
460Q8—84no—
461D4—44no—
462D4—44no—
463D4—84no—
464D4—84no—
465D4—88no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4C31×#12012yes—
489C2×C4C31×#12012yes—
490C2×C4C31×#12412yes—
491C2×C4C22×D4×C3111yescharacteristic
492C2×C4C22×D4×C3111yescharacteristic
493C2×C4C2×#41×C3112yes—
494C2×C4C31×2−1+412yes—
495C8—42no—
496C8—42no—
497C22—24no—
498C22—22no—
499C22—42no—
500C22—41no—
501C22—41no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22C2×C31×#12011yescharacteristic
510C22C2×C31×#12411yescharacteristic
511C22C31×#53611yescharacteristic
512C22C31×#57612yes—
513C22C31×#57712yes—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—41no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—41no—
522C4—42no—
523C4—82no—
524C4C31×#54812yes—
525C4C31×#54812yes—
526C2—42no—
527C2—162no—
528C2C31×#243611yescharacteristic
529C2C31×#300811yescharacteristic
530C2C31×#301011yescharacteristic
531C2C31×#313612yes—
532C2C31×#324112yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#2619C2
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
1. C2×C31×#544C2semidirect
2. C2×C31×#490C2semidirect
3. C2×C31×#486C2semidirect
5. C31×#2025C2semidirect
6. C31×#2023C2semidirect
7. C31×#1993C2semidirect
8. C31×#1991C2semidirect
9. C31×#1875C2semidirect
10. C31×#1874C2semidirect
11. C31×#1750C2semidirect
61. C31×C4⋊Q8D4semidirect
68. C31×#125D4semidirect
179. #25915C31direct (N × H)
231. C2×#544C2×C31semidirect
232. C2×#490C2×C31semidirect
233. C2×#486C2×C31semidirect
235. #2025C2×C31semidirect
236. #2023C2×C31semidirect
237. #1993C2×C31semidirect
238. #1991C2×C31semidirect
239. #1875C2×C31semidirect
240. #1874C2×C31semidirect
241. #1750C2×C31semidirect
327. C4⋊Q8D4×C31semidirect
334. #125D4×C31semidirect
376. C31#25915direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5442
C2×C31×#4902
C2×C31×#4862
C31×#26192
C31×#20252
C31×#20232
C31×#19932
C31×#19912
C31×#18752
C31×#18742
C31×#17502
#2591531