Finite Group Databasebeta

← C31×#25923 #21482682

Subgroups — C31×#25923 #21482682

Order 7936.

1014 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3503C211yesmaximal, characteristic
2C31×#3139C211yesmaximal, characteristic, complemented
3C31×#3039C211yesmaximal, characteristic
4C31×#2862C211yesmaximal, characteristic, complemented
5C31×#2028C214yesmaximal, complemented
6C31×#1997C211yesmaximal, characteristic, complemented
7C31×#1994C211yesmaximal, characteristic, complemented
8C31×#1882C214yesmaximal
9C4×C31×#104C211yesmaximal, characteristic
10C2×C31×#137C2212yes—
11Q82×C31—21no—
12C31×#557—21no—
13C31×#554—21no—
14C31×#554—22no—
15D4×Q8×C31—22no—
16D4×Q8×C31—21no—
17C31×#544—22no—
18C31×#544C2212yes—
19C31×#533—21no—
20C31×#533C2211yescharacteristic
21C2×C31×C4⋊Q8—21no—
22C2×C31×C4⋊Q8C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C31×#520C2212yes—
25C4×#41×C31C2211yescharacteristic
26C31×#491—22no—
27C31×#487—21no—
28C31×#487—21no—
29C31×#462—21no—
30C31×#462—21no—
31C31×#461—21no—
32C31×#461—21no—
33C31×#423C2211yescharacteristic
34C2×C31×#104C2212yes—
35C31×#419C2214yes—
36C31×#419C2212yes—
37C4×M16×C31C2211yescharacteristic
38C31×#391—22no—
39C31×#391C2214yes—
40C31×#382—22no—
41C43×C31C2211yescharacteristic
42C31×#343C2214yes—
43C31×#339C2214yes—
44C31×#339C2212yes—
45C2×#41×C31C2311yescharacteristic
46C22×Q8×C31—22no—
47C22×Q8×C31D412yes—
48C22×Q8×C31C2312yes—
49C31×#137—28no—
50C2×Q16×C31—22no—
51C2×SD16×C31—22no—
52C2×M16×C31C2312yes—
53C2×M16×C31C2312yes—
54C31×C4⋊Q8—22no—
55C31×C4⋊Q8—22no—
56C31×C4⋊Q8—24no—
57C31×C4⋊Q8—22no—
58C31×C4⋊Q8—22no—
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×C4⋊Q8D411yescharacteristic
69C31×#125—24no—
70C31×#125—22no—
71C31×#125—22no—
72C31×#125D412yes—
73C31×C22⋊Q8—22no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—24no—
76C31×C22⋊Q8—24no—
77C31×C22⋊Q8—21no—
78C31×C22⋊Q8—21no—
79C31×C22⋊Q8—24no—
80C4×Q8×C31—22no—
81C4×Q8×C31—22no—
82C4×Q8×C31—22no—
83C4×Q8×C31—21no—
84C4×Q8×C31—21no—
85C4×Q8×C31D411yescharacteristic
86C4×Q8×C31D411yescharacteristic
87C4×D4×C31—22no—
88C4×D4×C31—21no—
89C4×D4×C31—21no—
90C4×D4×C31—24no—
91C4×D4×C31D411yescharacteristic
92C4×D4×C31D411yescharacteristic
93C31×#117—22no—
94C31×#117—21no—
95C31×#117C2311yescharacteristic
96C31×#117C2312yes—
97C2×C4⋊C4×C31—22no—
98C2×C4⋊C4×C31—22no—
99C2×C4⋊C4×C31—21no—
100C2×C4⋊C4×C31—21no—
101C2×C4⋊C4×C31D412yes—
102C2×C4⋊C4×C31C2312yes—
103C2×C42×C31—21no—
104C2×C42×C31—21no—
105C2×C42×C31—21no—
106C2×C42×C31C2312yes—
107C2×C42×C31C2311yescharacteristic
108C31×C4⋊C8—21no—
109C31×C4⋊C8—21no—
110C31×#104—24no—
111C31×#103—22no—
112C31×#103—22no—
113C31×#103—24no—
114C31×#103—24no—
115C31×#97—21no—
116C4×C8×C31—21no—
117C31×#95—28no—
118#41×C31—22no—
119#41×C31#4114yes—
120C2×Q8×C31—24no—
121C2×Q8×C31—21no—
122C2×Q8×C31—22no—
123C2×Q8×C31—21no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—24no—
128C2×Q8×C31—24no—
129C2×Q8×C31—44no—
130C2×Q8×C31—84no—
131C2×Q8×C31C2×D411yescharacteristic
132C2×D4×C31—21no—
133C2×D4×C31C2×D411yescharacteristic
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—24no—
137C22×C4×C31—21no—
138C22×C4×C31—21no—
139C22×C4×C31—22no—
140C22×C4×C31C2×D411yescharacteristic
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D414yes—
143C22×C4×C31C2411yescharacteristic
144Q16×C31—44no—
145SD16×C31—44no—
146M16×C31—22no—
147M16×C31—24no—
148M16×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—22no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31—41no—
166C4⋊C4×C31—41no—
167C4⋊C4×C31—41no—
168C4⋊C4×C31—41no—
169C4⋊C4×C31C2×D412yes—
170C4⋊C4×C31C2×D412yes—
171C4⋊C4×C31C2×D411yescharacteristic
172C4⋊C4×C31C2×D411yescharacteristic
173C22⋊C4×C31—24no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—21no—
176C22⋊C4×C31—21no—
177C22⋊C4×C31—21no—
178C42×C31—24no—
179C42×C31—24no—
180C42×C31—21no—
181C42×C31—21no—
182C42×C31—21no—
183C42×C31—22no—
184C42×C31—22no—
185C42×C31—22no—
186C42×C31—42no—
187C42×C31C2×D411yescharacteristic
188C42×C31C2×D411yescharacteristic
189C42×C31C2×D411yescharacteristic
190C42×C31C2×D411yescharacteristic
191#25923C3111yesmaximal, characteristic, direct product factor, Sylow
192C23×C31—21no—
193C23×C31C22×D411yescharacteristic
194Q8×C31—21no—
195Q8×C31—44no—
196Q8×C31—42no—
197Q8×C31—42no—
198Q8×C31—42no—
199Q8×C31—42no—
200Q8×C31—44no—
201Q8×C31—82no—
202Q8×C31—84no—
203Q8×C31#12412yes—
204D4×C31—24no—
205D4×C31—21no—
206D4×C31#12412yes—
207C2×C4×C31—21no—
208C2×C4×C31—24no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—24no—
212C2×C4×C31—22no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—24no—
216C2×C4×C31—22no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—44no—
222C2×C4×C31—42no—
223C2×C4×C31—44no—
224C2×C4×C31—84no—
225C2×C4×C31#12011yescharacteristic
226C2×C4×C31#12011yescharacteristic
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31C22×D411yescharacteristic
230C2×C4×C31C22×D411yescharacteristic
231C2×C4×C31C2×#4112yes—
232C2×C4×C312−1+412yes—
233C8×C31—42no—
234C8×C31—42no—
235#3503C2×C3111yescharacteristic
236#3139C2×C3111yescharacteristic, complemented
237#3039C2×C3111yescharacteristic
238#2862C2×C3111yescharacteristic, complemented
239#2028C2×C3114yescomplemented
240#1997C2×C3111yescharacteristic, complemented
241#1994C2×C3111yescharacteristic, complemented
242#1882C2×C3114yes—
243C4×#104C2×C3111yescharacteristic
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—22no—
247C22×C31—41no—
248C22×C31—41no—
249C22×C31C2×#12011yescharacteristic
250C22×C31#54812yes—
251C4×C31—22no—
252C4×C31—21no—
253C4×C31—21no—
254C4×C31—44no—
255C4×C31—41no—
256C4×C31—44no—
257C4×C31—42no—
258C4×C31—41no—
259C4×C31—42no—
260C4×C31—82no—
261C4×C31#45512yes—
262C4×C31#45612yes—
263C4×C31C2×#12411yescharacteristic
264C4×C31#53611yescharacteristic
265C4×C31#54812yes—
266C2×#137C22×C3112yes—
267Q82—21no—
268#557—21no—
269#554—21no—
270#554—22no—
271D4×Q8—22no—
272D4×Q8—21no—
273#544—22no—
274#544C22×C3112yes—
275#533—21no—
276#533C22×C3111yescharacteristic
277C2×C4⋊Q8—21no—
278C2×C4⋊Q8C22×C3111yescharacteristic
279C2×C4⋊Q8C22×C3111yescharacteristic
280#520C22×C3112yes—
281C4×#41C22×C3111yescharacteristic
282#491—22no—
283#487—21no—
284#487—21no—
285#462—21no—
286#462—21no—
287#461—21no—
288#461—21no—
289#423C22×C3111yescharacteristic
290C2×#104C22×C3112yes—
291#419C22×C3114yes—
292#419C22×C3112yes—
293C4×M16C22×C3111yescharacteristic
294#391—22no—
295#391C22×C3114yes—
296#382—22no—
297C43C22×C3111yescharacteristic
298#343C22×C3114yes—
299#339C22×C3114yes—
300#339C22×C3112yes—
301C2×C31—22no—
302C2×C31—42no—
303C2×C31#243611yescharacteristic
304C2×C31C2×#45511yescharacteristic
305C2×C31C2×#45611yescharacteristic
306C2×#41C23×C3111yescharacteristic
307C22×Q8—22no—
308C22×Q8D4×C3112yes—
309C22×Q8C23×C3112yes—
310#137—28no—
311C2×Q16—22no—
312C2×SD16—22no—
313C2×M16C23×C3112yes—
314C2×M16C23×C3112yes—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—24no—
318C4⋊Q8—22no—
319C4⋊Q8—22no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8D4×C3111yescharacteristic
329C4⋊Q8D4×C3111yescharacteristic
330#125—24no—
331#125—22no—
332#125—22no—
333#125D4×C3112yes—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—24no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—21no—
345C4×Q8—21no—
346C4×Q8D4×C3111yescharacteristic
347C4×Q8D4×C3111yescharacteristic
348C4×D4—22no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—24no—
352C4×D4D4×C3111yescharacteristic
353C4×D4D4×C3111yescharacteristic
354#117—22no—
355#117—21no—
356#117C23×C3111yescharacteristic
357#117C23×C3112yes—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4D4×C3112yes—
363C2×C4⋊C4C23×C3112yes—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42C23×C3112yes—
368C2×C42C23×C3111yescharacteristic
369C4⋊C8—21no—
370C4⋊C8—21no—
371#104—24no—
372#103—22no—
373#103—22no—
374#103—24no—
375#103—24no—
376#97—21no—
377C4×C8—21no—
378#95—28no—
379C31#2592311yescharacteristic, direct product factor, Sylow
380#41—22no—
381#41#41×C3114yes—
382C2×Q8—24no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—44no—
392C2×Q8—84no—
393C2×Q8C2×D4×C3111yescharacteristic
394C2×D4—21no—
395C2×D4C2×D4×C3111yescharacteristic
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4C2×D4×C3111yescharacteristic
403C22×C4C2×D4×C3111yescharacteristic
404C22×C4C2×D4×C3114yes—
405C22×C4C24×C3111yescharacteristic
406Q16—44no—
407SD16—44no—
408M16—22no—
409M16—24no—
410M16—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4C2×D4×C3112yes—
432C4⋊C4C2×D4×C3112yes—
433C4⋊C4C2×D4×C3111yescharacteristic
434C4⋊C4C2×D4×C3111yescharacteristic
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C42—24no—
441C42—24no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—42no—
449C42C2×D4×C3111yescharacteristic
450C42C2×D4×C3111yescharacteristic
451C42C2×D4×C3111yescharacteristic
452C42C2×D4×C3111yescharacteristic
453C23—21no—
454C23C22×D4×C3111yescharacteristic
455Q8—21no—
456Q8—44no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460Q8—42no—
461Q8—44no—
462Q8—82no—
463Q8—84no—
464Q8C31×#12412yes—
465D4—24no—
466D4—21no—
467D4C31×#12412yes—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—84no—
486C2×C4C31×#12011yescharacteristic
487C2×C4C31×#12011yescharacteristic
488C2×C4C31×#12011yescharacteristic
489C2×C4C31×#12011yescharacteristic
490C2×C4C22×D4×C3111yescharacteristic
491C2×C4C22×D4×C3111yescharacteristic
492C2×C4C2×#41×C3112yes—
493C2×C4C31×2−1+412yes—
494C8—42no—
495C8—42no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22C2×C31×#12011yescharacteristic
502C22C31×#54812yes—
503C4—22no—
504C4—21no—
505C4—21no—
506C4—44no—
507C4—41no—
508C4—44no—
509C4—42no—
510C4—41no—
511C4—42no—
512C4—82no—
513C4C31×#45512yes—
514C4C31×#45612yes—
515C4C2×C31×#12411yescharacteristic
516C4C31×#53611yescharacteristic
517C4C31×#54812yes—
518C2—22no—
519C2—42no—
520C2C31×#243611yescharacteristic
521C2C2×C31×#45511yescharacteristic
522C2C2×C31×#45611yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×#41×C31C23
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×#3139C2semidirect
4. C31×#2862C2semidirect
5. C31×#2028C2semidirect
6. C31×#1997C2semidirect
7. C31×#1994C2semidirect
191. #25923C31direct (N × H)
236. #3139C2×C31semidirect
238. #2862C2×C31semidirect
239. #2028C2×C31semidirect
240. #1997C2×C31semidirect
241. #1994C2×C31semidirect
379. C31#25923direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35032
C31×#31392
C31×#30392
C31×#28622
C31×#20282
C31×#19972
C31×#19942
C31×#18822
C4×C31×#1042
#2592331