Finite Group Databasebeta

← C31×#25615 #21482374

Subgroups — C31×#25615 #21482374

Order 7936.

1116 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3511C211yesmaximal, characteristic
2C31×#3152C211yesmaximal, characteristic, complemented
3C31×#3039C211yesmaximal, characteristic, complemented
4C31×#2862C211yesmaximal, characteristic, complemented
5C31×#2007C214yesmaximal, complemented
6C31×#1997C211yesmaximal, characteristic, complemented
7C31×#1962C211yesmaximal, characteristic, complemented
8C31×#1882C214yesmaximal, complemented
9C4×C31×#104C211yesmaximal, characteristic, complemented
10C2×C31×#137C2214yes—
11C2×C31×#137C2212yes—
12C31×#556—21no—
13C31×#555—21no—
14C31×#554—21no—
15C31×#553—22no—
16C31×#548—22no—
17C31×#547—21no—
18C31×#542—22no—
19C31×#537C2212yes—
20C31×#533—21no—
21C31×#532—21no—
22C31×#532C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C2×C31×C4⋊Q8C2211yescharacteristic
25C31×#520C2212yes—
26C4×#41×C31C2211yescharacteristic
27C31×#497—22no—
28C31×#494—21no—
29C31×#492—21no—
30C31×#462—21no—
31C31×#462—21no—
32C31×#461—21no—
33C31×#461—21no—
34C31×#423C2211yescharacteristic
35C2×C31×#104C2212yes—
36C2×C31×#104C2214yes—
37C31×#419C2212yes—
38C4×M16×C31C2211yescharacteristic
39C31×#391—22no—
40C31×#391C2214yes—
41C31×#382—22no—
42C43×C31C2211yescharacteristic
43C31×#343C2214yes—
44C31×#339C2212yes—
45C2×#41×C31—22no—
46C2×#41×C31C2311yescharacteristic
47C2×#41×C31C2312yes—
48C22×Q8×C31D412yes—
49C22×Q8×C31C2312yes—
50C31×#137—28no—
51C31×#137—44no—
52C31×#137—44no—
53C2×Q16×C31—22no—
54C2×Q16×C31—24no—
55C2×SD16×C31—22no—
56C2×SD16×C31—24no—
57C2×M16×C31C2312yes—
58C2×M16×C31C2312yes—
59C31×C4⋊Q8—21no—
60C31×C4⋊Q8—21no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8D411yescharacteristic
68C31×#127D411yescharacteristic
69C31×#126—24no—
70C31×#125—22no—
71C31×#124—22no—
72C31×#124—24no—
73C31×#124D412yes—
74C31×#123—24no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—22no—
78C31×C22⋊Q8—22no—
79C31×#121—22no—
80C31×#121—22no—
81C31×#121—22no—
82C31×#121—21no—
83C31×#121—21no—
84C31×#121—24no—
85C4×Q8×C31—22no—
86C4×Q8×C31—22no—
87C4×Q8×C31D411yescharacteristic
88C4×Q8×C31D411yescharacteristic
89C4×D4×C31—22no—
90C4×D4×C31—22no—
91C4×D4×C31—21no—
92C4×D4×C31—21no—
93C4×D4×C31—24no—
94C4×D4×C31—21no—
95C4×D4×C31—21no—
96C4×D4×C31D411yescharacteristic
97C4×D4×C31D411yescharacteristic
98C31×#117—22no—
99C31×#117—22no—
100C31×#117—21no—
101C31×#117C2311yescharacteristic
102C2×C4⋊C4×C31—22no—
103C2×C4⋊C4×C31—21no—
104C2×C4⋊C4×C31—21no—
105C2×C4⋊C4×C31D412yes—
106C2×C4⋊C4×C31C2312yes—
107C2×C42×C31—21no—
108C2×C42×C31—21no—
109C2×C42×C31—21no—
110C2×C42×C31C2312yes—
111C2×C42×C31C2311yescharacteristic
112C31×C4⋊C8—21no—
113C31×C4⋊C8—21no—
114C31×#104—24no—
115C31×#104—44no—
116C31×#103—22no—
117C31×#103—22no—
118C31×#97—21no—
119C4×C8×C31—21no—
120C31×#95—28no—
121#41×C31—22no—
122#41×C31—24no—
123#41×C31—24no—
124#41×C31—22no—
125#41×C31—84no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31—22no—
129C2×Q8×C31—22no—
130C2×Q8×C31—24no—
131C2×Q8×C31—24no—
132C2×Q8×C31—44no—
133C2×Q8×C31C2×D412yes—
134C2×Q8×C31C2×D411yescharacteristic
135C2×D4×C31—21no—
136C2×D4×C31—22no—
137C2×D4×C31—24no—
138C2×D4×C31—21no—
139C2×D4×C31—21no—
140C2×D4×C31C2×D412yes—
141C2×D4×C31C2×D411yescharacteristic
142C22×C4×C31—22no—
143C22×C4×C31—24no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31C2×D411yescharacteristic
150C22×C4×C31C2×D411yescharacteristic
151C22×C4×C31C2×D414yes—
152C22×C4×C31C2411yescharacteristic
153Q16×C31—48no—
154Q16×C31—44no—
155SD16×C31—48no—
156SD16×C31—44no—
157M16×C31—22no—
158M16×C31—24no—
159M16×C31#4114yescomplemented
160C2×C8×C31—22no—
161C2×C8×C31—22no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—22no—
171C4⋊C4×C31—22no—
172C4⋊C4×C31—41no—
173C4⋊C4×C31—41no—
174C4⋊C4×C31—41no—
175C4⋊C4×C31—41no—
176C4⋊C4×C31C2×D411yescharacteristic
177C4⋊C4×C31C2×D411yescharacteristic
178C22⋊C4×C31—24no—
179C22⋊C4×C31—24no—
180C22⋊C4×C31—24no—
181C22⋊C4×C31—21no—
182C22⋊C4×C31—21no—
183C22⋊C4×C31—21no—
184C22⋊C4×C31—22no—
185C22⋊C4×C31—22no—
186C42×C31—24no—
187C42×C31—24no—
188C42×C31—21no—
189C42×C31—21no—
190C42×C31—21no—
191C42×C31—22no—
192C42×C31—22no—
193C42×C31—42no—
194C42×C31C2×D411yescharacteristic
195C42×C31C2×D411yescharacteristic
196C42×C31C2×D411yescharacteristic
197C42×C31C2×D411yescharacteristic
198#25615C3111yesmaximal, characteristic, direct product factor, Sylow
199C23×C31—21no—
200C23×C31—22no—
201C23×C31C22×D411yescharacteristic
202Q8×C31—21no—
203Q8×C31—21no—
204Q8×C31—24no—
205Q8×C31—44no—
206Q8×C31—42no—
207Q8×C31—42no—
208Q8×C31—44no—
209Q8×C31—82no—
210D4×C31—21no—
211D4×C31—21no—
212D4×C31—24no—
213D4×C31—42no—
214D4×C31—44no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—84no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—24no—
221C2×C4×C31—21no—
222C2×C4×C31—22no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—24no—
226C2×C4×C31—42no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—42no—
232C2×C4×C31—42no—
233C2×C4×C31—44no—
234C2×C4×C31—42no—
235C2×C4×C31—44no—
236C2×C4×C31—84no—
237C2×C4×C31#12011yescharacteristic
238C2×C4×C31#12011yescharacteristic
239C2×C4×C31#12011yescharacteristic
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31C22×D411yescharacteristic
242C2×C4×C31C22×D411yescharacteristic
243C2×C4×C31C2×#4112yes—
244C2×C4×C31H(5,𝔽2)12yes—
245C8×C31—24no—
246C8×C31—42no—
247#3511C2×C3111yescharacteristic
248#3152C2×C3111yescharacteristic, complemented
249#3039C2×C3111yescharacteristic, complemented
250#2862C2×C3111yescharacteristic, complemented
251#2007C2×C3114yescomplemented
252#1997C2×C3111yescharacteristic, complemented
253#1962C2×C3111yescharacteristic, complemented
254#1882C2×C3114yescomplemented
255C4×#104C2×C3111yescharacteristic, complemented
256C22×C31—21no—
257C22×C31—21no—
258C22×C31—42no—
259C22×C31—41no—
260C22×C31—41no—
261C22×C31—42no—
262C22×C31—41no—
263C22×C31—42no—
264C22×C31C2×#12011yescharacteristic
265C22×C31#55012yes—
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—44no—
269C4×C31—42no—
270C4×C31—41no—
271C4×C31—41no—
272C4×C31—44no—
273C4×C31—41no—
274C4×C31—82no—
275C4×C31#45512yes—
276C4×C31#45612yes—
277C4×C31#53211yescharacteristic
278C4×C31#53511yescharacteristic
279C4×C31#54612yes—
280C2×#137C22×C3114yes—
281C2×#137C22×C3112yes—
282#556—21no—
283#555—21no—
284#554—21no—
285#553—22no—
286#548—22no—
287#547—21no—
288#542—22no—
289#537C22×C3112yes—
290#533—21no—
291#532—21no—
292#532C22×C3111yescharacteristic
293C2×C4⋊Q8C22×C3111yescharacteristic
294C2×C4⋊Q8C22×C3111yescharacteristic
295#520C22×C3112yes—
296C4×#41C22×C3111yescharacteristic
297#497—22no—
298#494—21no—
299#492—21no—
300#462—21no—
301#462—21no—
302#461—21no—
303#461—21no—
304#423C22×C3111yescharacteristic
305C2×#104C22×C3112yes—
306C2×#104C22×C3114yes—
307#419C22×C3112yes—
308C4×M16C22×C3111yescharacteristic
309#391—22no—
310#391C22×C3114yes—
311#382—22no—
312C43C22×C3111yescharacteristic
313#343C22×C3114yes—
314#339C22×C3112yes—
315C2×C31—22no—
316C2×C31—81no—
317C2×C31—82no—
318C2×C31#243111yescharacteristic
319C2×C31C2×#45511yescharacteristic
320C2×C31C2×#45611yescharacteristic
321C2×#41—22no—
322C2×#41C23×C3111yescharacteristic
323C2×#41C23×C3112yes—
324C22×Q8D4×C3112yes—
325C22×Q8C23×C3112yes—
326#137—28no—
327#137—44no—
328#137—44no—
329C2×Q16—22no—
330C2×Q16—24no—
331C2×SD16—22no—
332C2×SD16—24no—
333C2×M16C23×C3112yes—
334C2×M16C23×C3112yes—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8—21no—
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8D4×C3111yescharacteristic
344#127D4×C3111yescharacteristic
345#126—24no—
346#125—22no—
347#124—22no—
348#124—24no—
349#124D4×C3112yes—
350#123—24no—
351C22⋊Q8—22no—
352C22⋊Q8—24no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121—21no—
359#121—21no—
360#121—24no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8D4×C3111yescharacteristic
364C4×Q8D4×C3111yescharacteristic
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—24no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4D4×C3111yescharacteristic
373C4×D4D4×C3111yescharacteristic
374#117—22no—
375#117—22no—
376#117—21no—
377#117C23×C3111yescharacteristic
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4D4×C3112yes—
382C2×C4⋊C4C23×C3112yes—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42C23×C3112yes—
387C2×C42C23×C3111yescharacteristic
388C4⋊C8—21no—
389C4⋊C8—21no—
390#104—24no—
391#104—44no—
392#103—22no—
393#103—22no—
394#97—21no—
395C4×C8—21no—
396#95—28no—
397C31#2561511yescharacteristic, direct product factor, Sylow
398#41—22no—
399#41—24no—
400#41—24no—
401#41—22no—
402#41—84no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—44no—
410C2×Q8C2×D4×C3112yes—
411C2×Q8C2×D4×C3111yescharacteristic
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4C2×D4×C3112yes—
418C2×D4C2×D4×C3111yescharacteristic
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4C2×D4×C3111yescharacteristic
427C22×C4C2×D4×C3111yescharacteristic
428C22×C4C2×D4×C3114yes—
429C22×C4C24×C3111yescharacteristic
430Q16—48no—
431Q16—44no—
432SD16—48no—
433SD16—44no—
434M16—22no—
435M16—24no—
436M16#41×C3114yescomplemented
437C2×C8—22no—
438C2×C8—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4C2×D4×C3111yescharacteristic
454C4⋊C4C2×D4×C3111yescharacteristic
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C42—24no—
464C42—24no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—22no—
469C42—22no—
470C42—42no—
471C42C2×D4×C3111yescharacteristic
472C42C2×D4×C3111yescharacteristic
473C42C2×D4×C3111yescharacteristic
474C42C2×D4×C3111yescharacteristic
475C23—21no—
476C23—22no—
477C23C22×D4×C3111yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—24no—
481Q8—44no—
482Q8—42no—
483Q8—42no—
484Q8—44no—
485Q8—82no—
486D4—21no—
487D4—21no—
488D4—24no—
489D4—42no—
490D4—44no—
491D4—42no—
492D4—42no—
493D4—84no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—84no—
513C2×C4C31×#12011yescharacteristic
514C2×C4C31×#12011yescharacteristic
515C2×C4C31×#12011yescharacteristic
516C2×C4C31×#12011yescharacteristic
517C2×C4C22×D4×C3111yescharacteristic
518C2×C4C22×D4×C3111yescharacteristic
519C2×C4C2×#41×C3112yes—
520C2×C4C31×H(5,𝔽2)12yes—
521C8—24no—
522C8—42no—
523C22—21no—
524C22—21no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—41no—
530C22—42no—
531C22C2×C31×#12011yescharacteristic
532C22C31×#55012yes—
533C4—21no—
534C4—21no—
535C4—44no—
536C4—42no—
537C4—41no—
538C4—41no—
539C4—44no—
540C4—41no—
541C4—82no—
542C4C31×#45512yes—
543C4C31×#45612yes—
544C4C31×#53211yescharacteristic
545C4C31×#53511yescharacteristic
546C4C31×#54612yes—
547C2—22no—
548C2—81no—
549C2—82no—
550C2C31×#243111yescharacteristic
551C2C2×C31×#45511yescharacteristic
552C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3511C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3152C2semidirect
3. C31×#3039C2semidirect
4. C31×#2862C2semidirect
5. C31×#2007C2semidirect
6. C31×#1997C2semidirect
7. C31×#1962C2semidirect
8. C31×#1882C2semidirect
9. C4×C31×#104C2semidirect
159. M16×C31#41semidirect
198. #25615C31direct (N × H)
248. #3152C2×C31semidirect
249. #3039C2×C31semidirect
250. #2862C2×C31semidirect
251. #2007C2×C31semidirect
252. #1997C2×C31semidirect
253. #1962C2×C31semidirect
254. #1882C2×C31semidirect
255. C4×#104C2×C31semidirect
397. C31#25615direct (N × H)
436. M16#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35112
C31×#31522
C31×#30392
C31×#28622
C31×#20072
C31×#19972
C31×#19622
C31×#18822
C4×C31×#1042
#2561531