Finite Group Databasebeta

← C31×#25899 #21482658

Subgroups — C31×#25899 #21482658

Order 7936.

1574 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3496C211yesmaximal, characteristic
2C31×#2436C212yesmaximal, complemented
3C31×#2409C211yesmaximal, characteristic, complemented
4C31×#2044C214yesmaximal, complemented
5C31×#1964C212yesmaximal, complemented
6C31×#1790C214yesmaximal, complemented
7C4×C31×#99C211yesmaximal, characteristic, complemented
8C2×C31×H(5,𝔽2)C2212yes—
9C23×Q8×C31C2211yescharacteristic, complemented
10C31×#547—22no—
11C31×#546—24no—
12D42×C31—22no—
13C31×#537—21no—
14C31×#534—21no—
15C31×#534C2212yes—
16C31×#532—21no—
17C2×C31×#124—22no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×#121—21no—
20C2×C31×#121C2212yes—
21C2×C4×D4×C31C2211yescharacteristic
22C2×C31×#99C2212yes—
23C2×C31×#99C2218yescomplemented
24C31×#395—21no—
25C31×#393—21no—
26C31×#393—24no—
27C31×#391—22no—
28C31×#391—21no—
29C31×#386C2212yes—
30C31×#386C2218yescomplemented
31C31×#385C2212yescomplemented
32C31×#380C2211yescharacteristic, complemented
33C4×C22⋊C4×C31C2212yescomplemented
34C31×#342C2212yescomplemented
35C31×H(5,𝔽2)—28no—
36C31×H(5,𝔽2)—44no—
37C2×#41×C31—22no—
38C2×#41×C31—24no—
39C22×Q8×C31—21no—
40C22×Q8×C31—22no—
41C22×Q8×C31—21no—
42C22×Q8×C31—22no—
43C22×Q8×C31—44no—
44C22×D4×C31—22no—
45C22×D4×C31—24no—
46C22×D4×C31—21no—
47C22×D4×C31D412yes—
48C22×D4×C31C2314yes—
49C22×D4×C31C2311yescharacteristic
50C23×C4×C31D411yescharacteristic
51C23×C4×C31C2312yes—
52C23×C4×C31C2311yescharacteristic
53C31×C4⋊Q8—21no—
54C31×#127—21no—
55C31×#124—22no—
56C31×#124—48no—
57C31×#123—24no—
58C31×#123—24no—
59C31×C22⋊Q8—24no—
60C31×C22⋊Q8—44no—
61C31×#121—22no—
62C31×#121—22no—
63C31×#121—24no—
64C31×#121—22no—
65C31×#121—22no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121—22no—
69C31×#121—22no—
70C31×#120—24no—
71C31×#120—24no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31—22no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31D411yescharacteristic
80C2×C4⋊C4×C31D412yes—
81C2×C22⋊C4×C31—22no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—28no—
84C2×C22⋊C4×C31D411yescharacteristic
85C2×C22⋊C4×C31D414yes—
86C2×C22⋊C4×C31C2314yes—
87C2×C22⋊C4×C31C2312yes—
88C2×C22⋊C4×C31C2311yescharacteristic
89C2×C42×C31—22no—
90C2×C42×C31D411yescharacteristic
91C31×#99—24no—
92C31×#99—48no—
93C31×#95—21no—
94C31×#95—22no—
95C31×#95—22no—
96C31×#95—22no—
97C31×#95—28no—
98C24×C31—22no—
99C24×C31C2×D414yes—
100C24×C31C2×D411yescharacteristic
101C24×C31C2411yescharacteristic
102#41×C31—416no—
103#41×C31—44no—
104#41×C31—44no—
105C2×Q8×C31—22no—
106C2×Q8×C31—24no—
107C2×Q8×C31—28no—
108C2×Q8×C31—44no—
109C2×Q8×C31—42no—
110C2×Q8×C31—48no—
111C2×Q8×C31—44no—
112C2×Q8×C31—41no—
113C2×Q8×C31—42no—
114C2×Q8×C31—41no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—24no—
118C2×D4×C31—22no—
119C2×D4×C31—24no—
120C2×D4×C31—24no—
121C2×D4×C31—28no—
122C2×D4×C31—28no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—22no—
127C2×D4×C31—44no—
128C2×D4×C31—416no—
129C2×D4×C31—44no—
130C2×D4×C31—44no—
131C2×D4×C31—82no—
132C2×D4×C31#4114yes—
133C22×C4×C31—22no—
134C22×C4×C31—24no—
135C22×C4×C31—24no—
136C22×C4×C31—22no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—24no—
140C22×C4×C31—22no—
141C22×C4×C31—44no—
142C22×C4×C31—42no—
143C22×C4×C31—42no—
144C22×C4×C31C2×D414yes—
145C22×C4×C31C2×D412yes—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D411yescharacteristic
148C22×C4×C31C2×D411yescharacteristic
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D412yes—
151C22×C4×C31C2×D411yescharacteristic
152C4⋊C4×C31—22no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—42no—
158C4⋊C4×C31—44no—
159C22⋊C4×C31—24no—
160C22⋊C4×C31—24no—
161C22⋊C4×C31—22no—
162C22⋊C4×C31—24no—
163C22⋊C4×C31—22no—
164C22⋊C4×C31—24no—
165C22⋊C4×C31—24no—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—44no—
169C22⋊C4×C31—416no—
170C42×C31—21no—
171C42×C31—21no—
172C42×C31—44no—
173#25899C3111yesmaximal, characteristic, direct product factor, Sylow
174C23×C31—21no—
175C23×C31—21no—
176C23×C31—24no—
177C23×C31—28no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—44no—
183C23×C31—44no—
184C23×C31—44no—
185C23×C31—82no—
186C23×C31—82no—
187C23×C31#12412yescomplemented
188C23×C31C22×D411yescharacteristic
189C23×C31C22×D412yes—
190C23×C31C2×#4112yes—
191C23×C31H(5,𝔽2)12yes—
192Q8×C31—44no—
193Q8×C31—48no—
194Q8×C31—44no—
195Q8×C31—44no—
196D4×C31—28no—
197D4×C31—44no—
198D4×C31—48no—
199D4×C31—44no—
200D4×C31—44no—
201D4×C31—48no—
202D4×C31—44no—
203D4×C31—82no—
204D4×C31—82no—
205D4×C31—82no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—24no—
209C2×C4×C31—22no—
210C2×C4×C31—28no—
211C2×C4×C31—24no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—24no—
215C2×C4×C31—28no—
216C2×C4×C31—42no—
217C2×C4×C31—41no—
218C2×C4×C31—48no—
219C2×C4×C31—42no—
220C2×C4×C31—44no—
221C2×C4×C31—44no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—44no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—44no—
230C2×C4×C31—42no—
231C2×C4×C31—44no—
232C2×C4×C31—81no—
233C2×C4×C31#12011yescharacteristic
234C2×C4×C31#12012yes—
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31#12412yescomplemented
237#3496C2×C3111yescharacteristic
238#2436C2×C3112yescomplemented
239#2409C2×C3111yescharacteristic, complemented
240#2044C2×C3114yescomplemented
241#1964C2×C3112yescomplemented
242#1790C2×C3114yescomplemented
243C4×#99C2×C3111yescharacteristic, complemented
244C22×C31—21no—
245C22×C31—22no—
246C22×C31—21no—
247C22×C31—22no—
248C22×C31—22no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—41no—
253C22×C31—41no—
254C22×C31—44no—
255C22×C31—44no—
256C22×C31—44no—
257C22×C31—81no—
258C22×C31—88no—
259C22×C31—164no—
260C22×C31C2×#12011yescharacteristic
261C22×C31C2×#12411yescharacteristic
262C22×C31#53511yescharacteristic
263C22×C31#54614yes—
264C4×C31—22no—
265C4×C31—21no—
266C4×C31—22no—
267C4×C31—44no—
268C4×C31—44no—
269C4×C31—41no—
270C4×C31—41no—
271C4×C31—82no—
272C4×C31—84no—
273C4×C31B(4,𝔽2)12yes—
274C2×H(5,𝔽2)C22×C3112yes—
275C23×Q8C22×C3111yescharacteristic, complemented
276#547—22no—
277#546—24no—
278D42—22no—
279#537—21no—
280#534—21no—
281#534C22×C3112yes—
282#532—21no—
283C2×#124—22no—
284C2×C22⋊Q8—21no—
285C2×#121—21no—
286C2×#121C22×C3112yes—
287C2×C4×D4C22×C3111yescharacteristic
288C2×#99C22×C3112yes—
289C2×#99C22×C3118yescomplemented
290#395—21no—
291#393—21no—
292#393—24no—
293#391—22no—
294#391—21no—
295#386C22×C3112yes—
296#386C22×C3118yescomplemented
297#385C22×C3112yescomplemented
298#380C22×C3111yescharacteristic, complemented
299C4×C22⋊C4C22×C3112yescomplemented
300#342C22×C3112yescomplemented
301C2×C31—21no—
302C2×C31—24no—
303C2×C31—21no—
304C2×C31—42no—
305C2×C31—84no—
306C2×C31#243511yescharacteristic
307C2×C31C2×B(4,𝔽2)11yescharacteristic
308C2×C31#302811yescharacteristic
309H(5,𝔽2)—28no—
310H(5,𝔽2)—44no—
311C2×#41—22no—
312C2×#41—24no—
313C22×Q8—21no—
314C22×Q8—22no—
315C22×Q8—22no—
316C22×Q8—21no—
317C22×Q8—44no—
318C22×D4—22no—
319C22×D4—24no—
320C22×D4—21no—
321C22×D4D4×C3112yes—
322C22×D4C23×C3114yes—
323C22×D4C23×C3111yescharacteristic
324C23×C4D4×C3111yescharacteristic
325C23×C4C23×C3112yes—
326C23×C4C23×C3111yescharacteristic
327C4⋊Q8—21no—
328#127—21no—
329#124—22no—
330#124—48no—
331#123—24no—
332#123—24no—
333C22⋊Q8—24no—
334C22⋊Q8—44no—
335#121—22no—
336#121—22no—
337#121—24no—
338#121—22no—
339#121—22no—
340#121—22no—
341#121—22no—
342#121—22no—
343#121—22no—
344#120—24no—
345#120—24no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4D4×C3111yescharacteristic
354C2×C4⋊C4D4×C3112yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—28no—
358C2×C22⋊C4D4×C3111yescharacteristic
359C2×C22⋊C4D4×C3114yes—
360C2×C22⋊C4C23×C3114yes—
361C2×C22⋊C4C23×C3112yes—
362C2×C22⋊C4C23×C3111yescharacteristic
363C2×C42—22no—
364C2×C42D4×C3111yescharacteristic
365#99—24no—
366#99—48no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—28no—
372C31#2589911yescharacteristic, direct product factor, Sylow
373C24—22no—
374C24C2×D4×C3114yes—
375C24C2×D4×C3111yescharacteristic
376C24C24×C3111yescharacteristic
377#41—416no—
378#41—44no—
379#41—44no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—28no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8—44no—
387C2×Q8—41no—
388C2×Q8—42no—
389C2×Q8—41no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—44no—
403C2×D4—416no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—82no—
407C2×D4#41×C3114yes—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4C2×D4×C3114yes—
420C22×C4C2×D4×C3112yes—
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4C2×D4×C3111yescharacteristic
423C22×C4C2×D4×C3111yescharacteristic
424C22×C4C2×D4×C3111yescharacteristic
425C22×C4C2×D4×C3112yes—
426C22×C4C2×D4×C3111yescharacteristic
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—44no—
444C22⋊C4—416no—
445C42—21no—
446C42—21no—
447C42—44no—
448C23—21no—
449C23—21no—
450C23—24no—
451C23—28no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—82no—
460C23—82no—
461C23C31×#12412yescomplemented
462C23C22×D4×C3111yescharacteristic
463C23C22×D4×C3112yes—
464C23C2×#41×C3112yes—
465C23C31×H(5,𝔽2)12yes—
466Q8—44no—
467Q8—48no—
468Q8—44no—
469Q8—44no—
470D4—28no—
471D4—44no—
472D4—48no—
473D4—44no—
474D4—44no—
475D4—48no—
476D4—44no—
477D4—82no—
478D4—82no—
479D4—82no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—81no—
507C2×C4C31×#12011yescharacteristic
508C2×C4C31×#12012yes—
509C2×C4C31×#12011yescharacteristic
510C2×C4C31×#12412yescomplemented
511C22—21no—
512C22—22no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—41no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—81no—
525C22—88no—
526C22—164no—
527C22C2×C31×#12011yescharacteristic
528C22C2×C31×#12411yescharacteristic
529C22C31×#53511yescharacteristic
530C22C31×#54614yes—
531C4—22no—
532C4—21no—
533C4—22no—
534C4—44no—
535C4—44no—
536C4—41no—
537C4—41no—
538C4—82no—
539C4—84no—
540C4C31×B(4,𝔽2)12yes—
541C2—21no—
542C2—24no—
543C2—21no—
544C2—42no—
545C2—84no—
546C2C31×#243511yescharacteristic
547C2C2×C31×B(4,𝔽2)11yescharacteristic
548C2C31×#302811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC24C24×C31
prime radicalC31×#3496C2
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×#2436C2semidirect
3. C31×#2409C2semidirect
4. C31×#2044C2semidirect
5. C31×#1964C2semidirect
6. C31×#1790C2semidirect
7. C4×C31×#99C2semidirect
9. C23×Q8×C31C22semidirect
23. C2×C31×#99C22semidirect
30. C31×#386C22semidirect
31. C31×#385C22semidirect
32. C31×#380C22semidirect
33. C4×C22⋊C4×C31C22semidirect
34. C31×#342C22semidirect
173. #25899C31direct (N × H)
187. C23×C31#124semidirect
236. C2×C4×C31#124semidirect
238. #2436C2×C31semidirect
239. #2409C2×C31semidirect
240. #2044C2×C31semidirect
241. #1964C2×C31semidirect
242. #1790C2×C31semidirect
243. C4×#99C2×C31semidirect
275. C23×Q8C22×C31semidirect
289. C2×#99C22×C31semidirect
296. #386C22×C31semidirect
297. #385C22×C31semidirect
298. #380C22×C31semidirect
299. C4×C22⋊C4C22×C31semidirect
300. #342C22×C31semidirect
372. C31#25899direct (N × H)
461. C23C31×#124semidirect
510. C2×C4C31×#124semidirect

Maximal subgroups

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

subgroupindex
C31×#34962
C31×#24362
C31×#24092
C31×#20442
C31×#19642
C31×#17902
C4×C31×#992
#2589931