Finite Group Databasebeta

← C31×#25944 #21482703

Subgroups — C31×#25944 #21482703

Order 7936.

1100 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#557C211yesmaximal, characteristic, complemented
2C2×C31×#490C211yesmaximal, characteristic
3C2×C31×#460C211yesmaximal, characteristic
4C31×#2548C211yesmaximal, characteristic
5C31×#2025C212yesmaximal, complemented
6C31×#2023C212yesmaximal, complemented
7C31×#1989C211yesmaximal, characteristic, complemented
8C31×#1985C211yesmaximal, characteristic, complemented
9C31×#1880C212yesmaximal, complemented
10C31×#1880C212yesmaximal, complemented
11C31×#1763C211yesmaximal, characteristic, complemented
12C22×SD16×C31C2212yes—
13C31×#557—24no—
14C31×#557—24no—
15C2×C31×C4⋊Q8—21no—
16C2×C31×C4⋊Q8C2211yescharacteristic
17C2×C31×#127C2211yescharacteristic
18C2×C31×#125—21no—
19C2×C31×#125—21no—
20C2×C31×#125C2211yescharacteristic
21C2×C31×#125C2211yescharacteristic
22C2×C31×#124—21no—
23C2×C4×Q8×C31—21no—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×Q8×C31C2211yescharacteristic
26C2×C4×Q8×C31C2211yescharacteristic
27C2×C4×Q8×C31C2211yescharacteristic
28C31×#490—24no—
29C31×#460—24no—
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×#103C2212yes—
32C2×C31×#102C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#97C2211yescharacteristic
36C31×#394—21no—
37C31×#394—21no—
38C31×#390—21no—
39C31×#390C2212yes—
40C31×#390C2212yes—
41C4×C4⋊C4×C31C2211yescharacteristic
42C31×#340C2212yes—
43C31×#340C2212yes—
44C31×#340C2212yes—
45C31×#336C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31—21no—
48C22×Q8×C31D411yescharacteristic
49C22×Q8×C31C2311yescharacteristic
50C22×D4×C31—21no—
51C22×D4×C31—21no—
52C22×D4×C31C2312yes—
53C2×SD16×C31—24no—
54C2×SD16×C31—24no—
55C2×SD16×C31—24no—
56C22×C8×C31C2312yes—
57C22×C8×C31C2312yes—
58C31×C4⋊Q8—28no—
59C31×C4⋊Q8—22no—
60C31×C4⋊Q8—22no—
61C31×#127—22no—
62C31×#127—22no—
63C31×#125—28no—
64C31×#125—28no—
65C31×#125—22no—
66C31×#125—22no—
67C31×#125—22no—
68C31×#125D414yes—
69C31×#124—44no—
70C4×Q8×C31—28no—
71C4×Q8×C31—22no—
72C4×Q8×C31—22no—
73C4×Q8×C31—22no—
74C4×Q8×C31—22no—
75C4×Q8×C31—22no—
76C4×Q8×C31—22no—
77C4×Q8×C31—22no—
78C4×Q8×C31D414yes—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31C2312yes—
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31—22no—
98C2×C42×C31—21no—
99C2×C42×C31—21no—
100C2×C42×C31—21no—
101C2×C42×C31—21no—
102C2×C42×C31C2311yescharacteristic
103C2×C42×C31C2311yescharacteristic
104C2×C42×C31C2311yescharacteristic
105C31×C4⋊C8—22no—
106C31×#103—24no—
107C31×#102—24no—
108C31×#102—24no—
109C31×#102—24no—
110C31×#97—22no—
111C31×#95—21no—
112C24×C31—22no—
113C2×Q8×C31—24no—
114C2×Q8×C31—22no—
115C2×Q8×C31—22no—
116C2×Q8×C31—21no—
117C2×Q8×C31—22no—
118C2×Q8×C31—41no—
119C2×Q8×C31—42no—
120C2×Q8×C31—42no—
121C2×Q8×C31C2×D412yes—
122C2×Q8×C31#4114yes—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—42no—
130C2×D4×C31—42no—
131C2×D4×C31—42no—
132C2×D4×C31—42no—
133C2×D4×C31—82no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—21no—
138C22×C4×C31—22no—
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2411yescharacteristic
146SD16×C31—48no—
147C2×C8×C31—22no—
148C2×C8×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C2×C8×C31—22no—
152C2×C8×C31—22no—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—24no—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—24no—
158C4⋊C4×C31—24no—
159C4⋊C4×C31—24no—
160C4⋊C4×C31—24no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31—22no—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D412yes—
171C4⋊C4×C31C2×D412yes—
172C22⋊C4×C31—44no—
173C22⋊C4×C31—44no—
174C22⋊C4×C31—82no—
175C22⋊C4×C31—82no—
176C42×C31—24no—
177C42×C31—24no—
178C42×C31—24no—
179C42×C31—21no—
180C42×C31—21no—
181C42×C31—21no—
182C42×C31—21no—
183C42×C31—42no—
184C42×C31C2×D412yes—
185C42×C31C2×D412yes—
186#25944C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31C22×D411yescharacteristic
195Q8×C31—22no—
196Q8×C31—24no—
197Q8×C31—44no—
198Q8×C31—48no—
199Q8×C31#12414yescomplemented
200D4×C31—42no—
201D4×C31—42no—
202D4×C31—42no—
203D4×C31—42no—
204D4×C31—84no—
205D4×C31—84no—
206C2×C4×C31—22no—
207C2×C4×C31—21no—
208C2×C4×C31—24no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—22no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—22no—
216C2×C4×C31—21no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31#12012yes—
232C2×C4×C31#12012yes—
233C2×C4×C31AGL(1,ℤ/8)12yes—
234C2×C4×C31C22×D411yescharacteristic
235C2×C4×C31C22×D411yescharacteristic
236C2×C4×C31C2×#4112yes—
237C2×C4×C312−1+412yes—
238C8×C31—42no—
239C8×C31—42no—
240C2×#557C2×C3111yescharacteristic, complemented
241C2×#490C2×C3111yescharacteristic
242C2×#460C2×C3111yescharacteristic
243#2548C2×C3111yescharacteristic
244#2025C2×C3112yescomplemented
245#2023C2×C3112yescomplemented
246#1989C2×C3111yescharacteristic, complemented
247#1985C2×C3111yescharacteristic, complemented
248#1880C2×C3112yescomplemented
249#1880C2×C3112yescomplemented
250#1763C2×C3111yescharacteristic, complemented
251C22×C31—82no—
252C22×C31—82no—
253C22×C31—82no—
254C22×C31—82no—
255C22×C31—82no—
256C22×C31—82no—
257C22×C31—82no—
258C22×C31C2×#12011yescharacteristic
259C22×C31#54812yes—
260C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C31#57511yescharacteristic
262C22×C31#57712yes—
263C4×C31—24no—
264C4×C31—22no—
265C4×C31—22no—
266C4×C31—22no—
267C4×C31—41no—
268C4×C31—42no—
269C4×C31—42no—
270C4×C31—41no—
271C4×C31—41no—
272C4×C31—42no—
273C4×C31—82no—
274C4×C31C2×#12411yescharacteristic
275C4×C31#53611yescharacteristic
276C4×C31#54812yes—
277C22×SD16C22×C3112yes—
278#557—24no—
279#557—24no—
280C2×C4⋊Q8—21no—
281C2×C4⋊Q8C22×C3111yescharacteristic
282C2×#127C22×C3111yescharacteristic
283C2×#125—21no—
284C2×#125—21no—
285C2×#125C22×C3111yescharacteristic
286C2×#125C22×C3111yescharacteristic
287C2×#124—21no—
288C2×C4×Q8—21no—
289C2×C4×Q8C22×C3111yescharacteristic
290C2×C4×Q8C22×C3111yescharacteristic
291C2×C4×Q8C22×C3111yescharacteristic
292C2×C4×Q8C22×C3111yescharacteristic
293#490—24no—
294#460—24no—
295C2×C4⋊C8C22×C3111yescharacteristic
296C2×#103C22×C3112yes—
297C2×#102C22×C3112yes—
298C2×#102C22×C3112yes—
299C2×#102C22×C3112yes—
300C2×#97C22×C3111yescharacteristic
301#394—21no—
302#394—21no—
303#390—21no—
304#390C22×C3112yes—
305#390C22×C3112yes—
306C4×C4⋊C4C22×C3111yescharacteristic
307#340C22×C3112yes—
308#340C22×C3112yes—
309#340C22×C3112yes—
310#336C22×C3112yes—
311#336C22×C3112yes—
312C2×C31—162no—
313C2×C31#243611yescharacteristic
314C2×C31#300611yescharacteristic
315C2×C31#301011yescharacteristic
316C2×C31#336812yes—
317C2×C31#337112yes—
318C22×Q8—21no—
319C22×Q8D4×C3111yescharacteristic
320C22×Q8C23×C3111yescharacteristic
321C22×D4—21no—
322C22×D4—21no—
323C22×D4C23×C3112yes—
324C2×SD16—24no—
325C2×SD16—24no—
326C2×SD16—24no—
327C22×C8C23×C3112yes—
328C22×C8C23×C3112yes—
329C4⋊Q8—28no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332#127—22no—
333#127—22no—
334#125—28no—
335#125—28no—
336#125—22no—
337#125—22no—
338#125—22no—
339#125D4×C3114yes—
340#124—44no—
341C4×Q8—28no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8D4×C3114yes—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4D4×C3111yescharacteristic
359C2×C4⋊C4D4×C3111yescharacteristic
360C2×C4⋊C4D4×C3111yescharacteristic
361C2×C4⋊C4C23×C3112yes—
362C2×C4⋊C4C23×C3111yescharacteristic
363C2×C4⋊C4C23×C3111yescharacteristic
364C2×C4⋊C4C23×C3111yescharacteristic
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42C23×C3111yescharacteristic
374C2×C42C23×C3111yescharacteristic
375C2×C42C23×C3111yescharacteristic
376C4⋊C8—22no—
377#103—24no—
378#102—24no—
379#102—24no—
380#102—24no—
381#97—22no—
382#95—21no—
383C31#2594411yescharacteristic, direct product factor, Sylow
384C24—22no—
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—41no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C2×D4×C3112yes—
394C2×Q8#41×C3114yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—82no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415C22×C4C2×D4×C3111yescharacteristic
416C22×C4C2×D4×C3111yescharacteristic
417C22×C4C24×C3111yescharacteristic
418SD16—48no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4C2×D4×C3112yes—
442C4⋊C4C2×D4×C3112yes—
443C4⋊C4C2×D4×C3112yes—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C42—24no—
449C42—24no—
450C42—24no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—42no—
456C42C2×D4×C3112yes—
457C42C2×D4×C3112yes—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C22×D4×C3111yescharacteristic
466Q8—22no—
467Q8—24no—
468Q8—44no—
469Q8—48no—
470Q8C31×#12414yescomplemented
471D4—42no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—84no—
476D4—84no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4C31×#12012yes—
503C2×C4C31×#12012yes—
504C2×C4C31×AGL(1,ℤ/8)12yes—
505C2×C4C22×D4×C3111yescharacteristic
506C2×C4C22×D4×C3111yescharacteristic
507C2×C4C2×#41×C3112yes—
508C2×C4C31×2−1+412yes—
509C8—42no—
510C8—42no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22C2×C31×#12011yescharacteristic
519C22C31×#54812yes—
520C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
521C22C31×#57511yescharacteristic
522C22C31×#57712yes—
523C4—24no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—82no—
534C4C2×C31×#12411yescharacteristic
535C4C31×#53611yescharacteristic
536C4C31×#54812yes—
537C2—162no—
538C2C31×#243611yescharacteristic
539C2C31×#300611yescharacteristic
540C2C31×#301011yescharacteristic
541C2C31×#336812yes—
542C2C31×#337112yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#557C2semidirect
5. C31×#2025C2semidirect
6. C31×#2023C2semidirect
7. C31×#1989C2semidirect
8. C31×#1985C2semidirect
9. C31×#1880C2semidirect
10. C31×#1880C2semidirect
11. C31×#1763C2semidirect
186. #25944C31direct (N × H)
199. Q8×C31#124semidirect
240. C2×#557C2×C31semidirect
244. #2025C2×C31semidirect
245. #2023C2×C31semidirect
246. #1989C2×C31semidirect
247. #1985C2×C31semidirect
248. #1880C2×C31semidirect
249. #1880C2×C31semidirect
250. #1763C2×C31semidirect
383. C31#25944direct (N × H)
470. Q8C31×#124semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5572
C2×C31×#4902
C2×C31×#4602
C31×#25482
C31×#20252
C31×#20232
C31×#19892
C31×#19852
C31×#18802
C31×#17632
#2594431