Finite Group Databasebeta

← C31×#23926 #21480685

Subgroups — C31×#23926 #21480685

Order 7936.

1888 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#536C211yesmaximal, characteristic, complemented
2C2×C31×#535C211yesmaximal, characteristic
3C2×C31×#448C212yesmaximal, complemented
4C2×C31×#450C212yesmaximal, complemented
5C31×#2000C214yesmaximal, complemented
6C31×#1875C212yesmaximal, complemented
7C31×#1873C212yesmaximal, complemented
8C31×#1780C211yesmaximal, characteristic, complemented
9C22×#41×C31C2211yescharacteristic
10C23×Q8×C31C2211yescharacteristic, complemented
11C23×D4×C31C2211yescharacteristic
12C22×SD16×C31C2214yes—
13C31×#536—28no—
14C31×#535—28no—
15C2×C31×C4⋊Q8—21no—
16C2×C31×#127—21no—
17C2×C31×#124—21no—
18C2×C31×#124—21no—
19C2×C31×#123—21no—
20C2×C31×#123—21no—
21C2×C31×C22⋊Q8C2212yes—
22C2×C31×C22⋊Q8C2212yescomplemented
23C2×C31×#121C2212yes—
24C2×C31×#121C2212yes—
25C2×C31×#120—21no—
26C2×C31×#120—21no—
27C2×C31×#117C2211yescharacteristic
28C2×C31×#117C2211yescharacteristic, complemented
29C31×#450—28no—
30C31×#448—28no—
31C2×C31×#103C2214yescomplemented
32C2×C31×#102C2214yes—
33C2×C31×C22⋊C8C2212yes—
34C2×C31×C22⋊C8C2212yescomplemented
35C2×C31×C22⋊C8C2212yescomplemented
36C31×#340C2214yescomplemented
37C25×C31—21no—
38C2×#41×C31—24no—
39C2×#41×C31—24no—
40C2×#41×C31—42no—
41C22×Q8×C31—24no—
42C22×Q8×C31—21no—
43C22×Q8×C31—24no—
44C22×Q8×C31—42no—
45C22×Q8×C31D411yescharacteristic
46C22×Q8×C31C2312yes—
47C22×D4×C31—22no—
48C22×D4×C31—22no—
49C22×D4×C31—24no—
50C22×D4×C31—21no—
51C22×D4×C31—24no—
52C22×D4×C31—42no—
53C22×D4×C31D411yescharacteristic
54C22×D4×C31D411yescharacteristic
55C22×D4×C31D411yescharacteristic
56C22×D4×C31C2312yes—
57C23×C4×C31—21no—
58C23×C4×C31C2311yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C23×C4×C31C2311yescharacteristic
61C2×SD16×C31—28no—
62C2×SD16×C31—28no—
63C2×SD16×C31—44no—
64C22×C8×C31C2314yes—
65C31×C4⋊Q8—28no—
66C31×#127—28no—
67C31×#124—28no—
68C31×#124—28no—
69C31×#123—28no—
70C31×#123—28no—
71C31×C22⋊Q8—24no—
72C31×C22⋊Q8—24no—
73C31×C22⋊Q8—24no—
74C31×C22⋊Q8D418yescomplemented
75C31×#121—24no—
76C31×#121—24no—
77C31×#121—24no—
78C31×#121D418yes—
79C31×#120—28no—
80C31×#120—28no—
81C31×#117—24no—
82C31×#117—24no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31C2312yes—
86C2×C4⋊C4×C31C2312yes—
87C2×C22⋊C4×C31—21no—
88C2×C22⋊C4×C31—21no—
89C2×C22⋊C4×C31—22no—
90C2×C22⋊C4×C31—22no—
91C2×C22⋊C4×C31—22no—
92C2×C22⋊C4×C31—22no—
93C2×C42×C31—21no—
94C2×C42×C31—21no—
95C31×#103—28no—
96C31×#102—28no—
97C31×C22⋊C8—24no—
98C31×C22⋊C8—24no—
99C31×C22⋊C8—42no—
100C24×C31—24no—
101C24×C31—22no—
102C24×C31—42no—
103C24×C31—42no—
104C24×C31—84no—
105C24×C31C2×D411yescharacteristic
106C24×C31C2×D411yescharacteristic
107C24×C31C2×D411yescharacteristic
108#41×C31—416no—
109C2×Q8×C31—24no—
110C2×Q8×C31—22no—
111C2×Q8×C31—22no—
112C2×Q8×C31—22no—
113C2×Q8×C31—24no—
114C2×Q8×C31—24no—
115C2×Q8×C31—41no—
116C2×Q8×C31—416no—
117C2×Q8×C31—42no—
118C2×Q8×C31—42no—
119C2×Q8×C31C2×D414yes—
120C2×D4×C31—24no—
121C2×D4×C31—22no—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—28no—
126C2×D4×C31—28no—
127C2×D4×C31—22no—
128C2×D4×C31—22no—
129C2×D4×C31—24no—
130C2×D4×C31—22no—
131C2×D4×C31—24no—
132C2×D4×C31—416no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—41no—
136C2×D4×C31—42no—
137C2×D4×C31—42no—
138C2×D4×C31—84no—
139C2×D4×C31—84no—
140C2×D4×C31C2×D414yes—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—22no—
145C22×C4×C31—24no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—42no—
153C22×C4×C31—42no—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D414yes—
158C22×C4×C31C2×D414yes—
159C22×C4×C31C2411yescharacteristic
160SD16×C31—416no—
161C2×C8×C31—24no—
162C2×C8×C31—24no—
163C2×C8×C31—24no—
164C4⋊C4×C31—28no—
165C4⋊C4×C31—28no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31C2×D414yes—
169C4⋊C4×C31C2×D414yes—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—28no—
173C22⋊C4×C31—28no—
174C22⋊C4×C31—28no—
175C22⋊C4×C31—28no—
176C42×C31—24no—
177C42×C31—24no—
178#23926C3111yesmaximal, characteristic, direct product factor, Sylow
179C23×C31—21no—
180C23×C31—21no—
181C23×C31—21no—
182C23×C31—21no—
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—21no—
186C23×C31—24no—
187C23×C31—21no—
188C23×C31—21no—
189C23×C31—24no—
190C23×C31—48no—
191C23×C31—42no—
192C23×C31—42no—
193C23×C31—44no—
194C23×C31—44no—
195C23×C31—88no—
196C23×C31—88no—
197C23×C31—88no—
198C23×C31#12012yes—
199C23×C31#12012yes—
200C23×C31C22×D411yescharacteristic
201Q8×C31—28no—
202Q8×C31—44no—
203Q8×C31—48no—
204Q8×C31—44no—
205D4×C31—28no—
206D4×C31—44no—
207D4×C31—44no—
208D4×C31—48no—
209D4×C31—44no—
210D4×C31—44no—
211D4×C31—88no—
212D4×C31—88no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—28no—
218C2×C4×C31—28no—
219C2×C4×C31—24no—
220C2×C4×C31—24no—
221C2×C4×C31—21no—
222C2×C4×C31—24no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—48no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—44no—
229C2×C4×C31—44no—
230C2×C4×C31—44no—
231C2×C4×C31#12012yes—
232C2×C4×C31#12012yes—
233C2×C4×C31C22×D412yes—
234C2×C4×C31C22×D412yes—
235C2×C4×C31H(5,𝔽2)12yes—
236C8×C31—44no—
237C2×#536C2×C3111yescharacteristic, complemented
238C2×#535C2×C3111yescharacteristic
239C2×#448C2×C3112yescomplemented
240C2×#450C2×C3112yescomplemented
241#2000C2×C3114yescomplemented
242#1875C2×C3112yescomplemented
243#1873C2×C3112yescomplemented
244#1780C2×C3111yescharacteristic, complemented
245C22×C31—22no—
246C22×C31—22no—
247C22×C31—24no—
248C22×C31—24no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—42no—
253C22×C31—41no—
254C22×C31—44no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—44no—
258C22×C31—44no—
259C22×C31—42no—
260C22×C31—816no—
261C22×C31—84no—
262C22×C31—84no—
263C22×C31C2×#12011yescharacteristic
264C22×C31C2×#12011yescharacteristic
265C22×C31#53411yescharacteristic
266C22×C31#57714yes—
267C4×C31—22no—
268C4×C31—22no—
269C4×C31—41no—
270C4×C31—44no—
271C4×C31—44no—
272C4×C31—44no—
273C4×C31D4214yes—
274C22×#41C22×C3111yescharacteristic
275C23×Q8C22×C3111yescharacteristic, complemented
276C23×D4C22×C3111yescharacteristic
277C22×SD16C22×C3114yes—
278#536—28no—
279#535—28no—
280C2×C4⋊Q8—21no—
281C2×#127—21no—
282C2×#124—21no—
283C2×#124—21no—
284C2×#123—21no—
285C2×#123—21no—
286C2×C22⋊Q8C22×C3112yes—
287C2×C22⋊Q8C22×C3112yescomplemented
288C2×#121C22×C3112yes—
289C2×#121C22×C3112yes—
290C2×#120—21no—
291C2×#120—21no—
292C2×#117C22×C3111yescharacteristic
293C2×#117C22×C3111yescharacteristic, complemented
294#450—28no—
295#448—28no—
296C2×#103C22×C3114yescomplemented
297C2×#102C22×C3114yes—
298C2×C22⋊C8C22×C3112yes—
299C2×C22⋊C8C22×C3112yescomplemented
300C2×C22⋊C8C22×C3112yescomplemented
301#340C22×C3114yescomplemented
302C2×C31—42no—
303C2×C31—42no—
304C2×C31—81no—
305C2×C31—84no—
306C2×C31#240311yescharacteristic
307C2×C31#300912yes—
308C2×C31#301012yes—
309C2×C31#320012yes—
310C25—21no—
311C2×#41—24no—
312C2×#41—24no—
313C2×#41—42no—
314C22×Q8—24no—
315C22×Q8—21no—
316C22×Q8—24no—
317C22×Q8—42no—
318C22×Q8D4×C3111yescharacteristic
319C22×Q8C23×C3112yes—
320C22×D4—22no—
321C22×D4—22no—
322C22×D4—24no—
323C22×D4—21no—
324C22×D4—24no—
325C22×D4—42no—
326C22×D4D4×C3111yescharacteristic
327C22×D4D4×C3111yescharacteristic
328C22×D4D4×C3111yescharacteristic
329C22×D4C23×C3112yes—
330C23×C4—21no—
331C23×C4C23×C3111yescharacteristic
332C23×C4C23×C3111yescharacteristic
333C23×C4C23×C3111yescharacteristic
334C2×SD16—28no—
335C2×SD16—28no—
336C2×SD16—44no—
337C22×C8C23×C3114yes—
338C4⋊Q8—28no—
339#127—28no—
340#124—28no—
341#124—28no—
342#123—28no—
343#123—28no—
344C22⋊Q8—24no—
345C22⋊Q8—24no—
346C22⋊Q8—24no—
347C22⋊Q8D4×C3118yescomplemented
348#121—24no—
349#121—24no—
350#121—24no—
351#121D4×C3118yes—
352#120—28no—
353#120—28no—
354#117—24no—
355#117—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4C23×C3112yes—
359C2×C4⋊C4C23×C3112yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C42—21no—
367C2×C42—21no—
368#103—28no—
369#102—28no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—42no—
373C31#2392611yescharacteristic, direct product factor, Sylow
374C24—24no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—84no—
379C24C2×D4×C3111yescharacteristic
380C24C2×D4×C3111yescharacteristic
381C24C2×D4×C3111yescharacteristic
382#41—416no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—41no—
390C2×Q8—416no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8C2×D4×C3114yes—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—416no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4C2×D4×C3114yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4C2×D4×C3111yescharacteristic
429C22×C4C2×D4×C3111yescharacteristic
430C22×C4C2×D4×C3111yescharacteristic
431C22×C4C2×D4×C3114yes—
432C22×C4C2×D4×C3114yes—
433C22×C4C24×C3111yescharacteristic
434SD16—416no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4C2×D4×C3114yes—
443C4⋊C4C2×D4×C3114yes—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C42—24no—
451C42—24no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—24no—
457C23—24no—
458C23—21no—
459C23—24no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—48no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—88no—
469C23—88no—
470C23—88no—
471C23C31×#12012yes—
472C23C31×#12012yes—
473C23C22×D4×C3111yescharacteristic
474Q8—28no—
475Q8—44no—
476Q8—48no—
477Q8—44no—
478D4—28no—
479D4—44no—
480D4—44no—
481D4—48no—
482D4—44no—
483D4—44no—
484D4—88no—
485D4—88no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4C31×#12012yes—
505C2×C4C31×#12012yes—
506C2×C4C22×D4×C3112yes—
507C2×C4C22×D4×C3112yes—
508C2×C4C31×H(5,𝔽2)12yes—
509C8—44no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—24no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—41no—
519C22—44no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—816no—
526C22—84no—
527C22—84no—
528C22C2×C31×#12011yescharacteristic
529C22C2×C31×#12011yescharacteristic
530C22C31×#53411yescharacteristic
531C22C31×#57714yes—
532C4—22no—
533C4—22no—
534C4—41no—
535C4—44no—
536C4—44no—
537C4—44no—
538C4D42×C3114yes—
539C2—42no—
540C2—42no—
541C2—81no—
542C2—84no—
543C2C31×#240311yescharacteristic
544C2C31×#300912yes—
545C2C31×#301012yes—
546C2C31×#320012yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#536C2semidirect
3. C2×C31×#448C2semidirect
4. C2×C31×#450C2semidirect
5. C31×#2000C2semidirect
6. C31×#1875C2semidirect
7. C31×#1873C2semidirect
8. C31×#1780C2semidirect
10. C23×Q8×C31C22semidirect
22. C2×C31×C22⋊Q8C22semidirect
28. C2×C31×#117C22semidirect
31. C2×C31×#103C22semidirect
34. C2×C31×C22⋊C8C22semidirect
35. C2×C31×C22⋊C8C22semidirect
36. C31×#340C22semidirect
74. C31×C22⋊Q8D4semidirect
178. #23926C31direct (N × H)
237. C2×#536C2×C31semidirect
239. C2×#448C2×C31semidirect
240. C2×#450C2×C31semidirect
241. #2000C2×C31semidirect
242. #1875C2×C31semidirect
243. #1873C2×C31semidirect
244. #1780C2×C31semidirect
275. C23×Q8C22×C31semidirect
287. C2×C22⋊Q8C22×C31semidirect
293. C2×#117C22×C31semidirect
296. C2×#103C22×C31semidirect
299. C2×C22⋊C8C22×C31semidirect
300. C2×C22⋊C8C22×C31semidirect
301. #340C22×C31semidirect
347. C22⋊Q8D4×C31semidirect
373. C31#23926direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5362
C2×C31×#5352
C2×C31×#4482
C2×C31×#4502
C31×#20002
C31×#18752
C31×#18732
C31×#17802
#2392631