Finite Group Databasebeta

← C31×#8511 #21465270

Subgroups — C31×#8511 #21465270

Order 7936.

728 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×#3561C211yesmaximal, characteristic
2C31×#1986C211yesmaximal, characteristic, complemented
3C31×#1983C211yesmaximal, characteristic, complemented
4C31×#1552C211yesmaximal, characteristic, complemented
5C31×#1527C211yesmaximal, characteristic, complemented
6C31×#1509C211yesmaximal, characteristic, complemented
7C31×#1483C211yesmaximal, characteristic, complemented
8C31×#563C412yescomplemented
9C31×#562C412yescomplemented
10C31×#556—21no—
11C31×#551—22no—
12D4×Q8×C31—21no—
13C31×#548—21no—
14C31×#546—21no—
15C31×#543—21no—
16C31×#542—21no—
17C31×#542—22no—
18C31×#538—21no—
19C31×#536—21no—
20C31×#533C2211yescharacteristic
21C31×#529C2211yescharacteristic
22C2×C31×#124C2211yescharacteristic
23C31×#432C2211yescharacteristic
24C2×C31×C4⋊C8C2211yescharacteristic
25C2×C31×#103—21no—
26C2×C31×#102—21no—
27C4×M16×C31C2211yescharacteristic
28C4×C4⋊C4×C31C2211yescharacteristic
29C31×#339—22no—
30C31×#332—21no—
31C31×#332—21no—
32C31×#329—22no—
33C31×#328—21no—
34C31×#327—21no—
35C31×Q8⋊C8—21no—
36C31×D4⋊C8—21no—
37C2×#41×C31—21no—
38C22×Q8×C31—21no—
39C22×Q8×C31D411yescharacteristic
40C22×D4×C31D411yescharacteristic
41C2×M16×C31—21no—
42C22×C8×C31—21no—
43C31×C4⋊Q8—21no—
44C31×C4⋊Q8—22no—
45C31×C4⋊Q8D411yescharacteristic
46C31×C4⋊Q8D411yescharacteristic
47C31×#126—22no—
48C31×#126—22no—
49C31×#126—22no—
50C31×#126—21no—
51C31×#125C2×C411yescharacteristic
52C31×#125C2×C412yes—
53C31×#124—21no—
54C31×#124—22no—
55C31×#124—22no—
56C31×#124—21no—
57C31×#124—21no—
58C31×#124C2×C412yes—
59C31×#124C2×C411yescharacteristic
60C31×#124D412yes—
61C31×#123—22no—
62C31×#123—22no—
63C31×#123—21no—
64C31×#123—22no—
65C31×#123—22no—
66C31×C22⋊Q8—22no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—21no—
69C31×C22⋊Q8—22no—
70C31×C22⋊Q8—22no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—21no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—22no—
75C31×C22⋊Q8—22no—
76C31×C22⋊Q8D411yescharacteristic
77C31×#121—21no—
78C31×#121—22no—
79C31×#121—22no—
80C31×#121D411yescharacteristic
81C31×#120—22no—
82C4×Q8×C31—21no—
83C4×Q8×C31—21no—
84C4×Q8×C31—21no—
85C4×Q8×C31D411yescharacteristic
86C4×D4×C31—22no—
87C4×D4×C31—21no—
88C4×D4×C31—22no—
89C4×D4×C31—21no—
90C4×D4×C31D411yescharacteristic
91C31×#117—22no—
92C31×#117—21no—
93C31×#117D412yes—
94C2×C4⋊C4×C31—21no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C42×C31—21no—
99C2×C42×C31C2311yescharacteristic
100C31×C4⋊C8—21no—
101C31×C4⋊C8—21no—
102C31×C4⋊C8—21no—
103C31×#103—42no—
104C31×#102—42no—
105C31×C22⋊C8—21no—
106C31×#97—21no—
107C31×#97—21no—
108C4×C8×C31—21no—
109C31×#95—21no—
110C24×C31—21no—
111#41×C31—81no—
112#41×C31—81no—
113C2×Q8×C31—22no—
114C2×Q8×C31—21no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—41no—
118C2×Q8×C31—41no—
119C2×Q8×C31—81no—
120C2×Q8×C31—81no—
121C2×Q8×C31C22⋊C411yescharacteristic
122C2×Q8×C31C22⋊C412yes—
123C2×D4×C31—21no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31—21no—
127C2×D4×C31—41no—
128C2×D4×C31—41no—
129C2×D4×C31C22⋊C411yescharacteristic
130C2×D4×C31C22⋊C412yes—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—41no—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140M16×C31—22no—
141M16×C31—41no—
142C2×C8×C31—41no—
143C2×C8×C31—41no—
144C2×C8×C31—41no—
145C2×C8×C31—41no—
146C2×C8×C31—41no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—21no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—21no—
156C4⋊C4×C31—21no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—41no—
161C4⋊C4×C31C22⋊C412yes—
162C4⋊C4×C31C22⋊C412yes—
163C4⋊C4×C31C22⋊C411yescharacteristic
164C4⋊C4×C31C22⋊C411yescharacteristic
165C22⋊C4×C31—22no—
166C22⋊C4×C31—22no—
167C22⋊C4×C31—22no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—21no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—21no—
175C22⋊C4×C31—21no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—21no—
179C22⋊C4×C31—21no—
180C42×C31—22no—
181C42×C31—22no—
182C42×C31—21no—
183C42×C31—42no—
184C42×C31C22×C411yescharacteristic
185C42×C31C2×D411yescharacteristic
186C42×C31C2×D411yescharacteristic
187C42×C31C2×D411yescharacteristic
188#8511C3111yesmaximal, characteristic, direct product factor, Sylow
189C23×C31—22no—
190C23×C31—21no—
191C23×C31—41no—
192C23×C31—41no—
193C23×C31—81no—
194C23×C31—81no—
195C23×C31#12011yescharacteristic
196Q8×C31—42no—
197Q8×C31—41no—
198Q8×C31—41no—
199Q8×C31—41no—
200Q8×C31—41no—
201Q8×C31—41no—
202Q8×C31—81no—
203Q8×C31—82no—
204D4×C31—42no—
205D4×C31—41no—
206D4×C31—41no—
207D4×C31—41no—
208D4×C31—41no—
209D4×C31—41no—
210D4×C31—82no—
211D4×C31—82no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—21no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—22no—
218C2×C4×C31—21no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—21no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—42no—
226C2×C4×C31—81no—
227C2×C4×C31—82no—
228C2×C4×C31—82no—
229C2×C4×C31—82no—
230C2×C4×C31—81no—
231C2×C4×C31—81no—
232C2×C4×C31C2×C22⋊C411yescharacteristic
233C2×C4×C31C2×C22⋊C411yescharacteristic
234C2×C4×C31C2×C22⋊C411yescharacteristic
235C2×C4×C31#12011yescharacteristic
236C2×C4×C31#12011yescharacteristic
237C2×C4×C31#12011yescharacteristic
238C8×C31—42no—
239C8×C31—81no—
240C8×C31—81no—
241#3561C2×C3111yescharacteristic
242#1986C2×C3111yescharacteristic, complemented
243#1983C2×C3111yescharacteristic, complemented
244#1552C2×C3111yescharacteristic, complemented
245#1527C2×C3111yescharacteristic, complemented
246#1509C2×C3111yescharacteristic, complemented
247#1483C2×C3111yescharacteristic, complemented
248C22×C31—21no—
249C22×C31—21no—
250C22×C31—41no—
251C22×C31—42no—
252C22×C31—41no—
253C22×C31—42no—
254C22×C31—42no—
255C22×C31—41no—
256C22×C31—161no—
257C22×C31—161no—
258C22×C31#37911yescharacteristic
259C22×C31#45411yescharacteristic
260C22×C31#45611yescharacteristic
261C4×C31—21no—
262C4×C31—21no—
263C4×C31—21no—
264C4×C31—21no—
265C4×C31—21no—
266C4×C31—41no—
267C4×C31—42no—
268C4×C31—42no—
269C4×C31—41no—
270C4×C31—41no—
271C4×C31—42no—
272C4×C31—82no—
273C4×C31#45311yescharacteristic
274C4×C31#45511yescharacteristic
275#563C4×C3112yescomplemented
276#562C4×C3112yescomplemented
277#556—21no—
278#551—22no—
279D4×Q8—21no—
280#548—21no—
281#546—21no—
282#543—21no—
283#542—21no—
284#542—22no—
285#538—21no—
286#536—21no—
287#533C22×C3111yescharacteristic
288#529C22×C3111yescharacteristic
289C2×#124C22×C3111yescharacteristic
290#432C22×C3111yescharacteristic
291C2×C4⋊C8C22×C3111yescharacteristic
292C2×#103—21no—
293C2×#102—21no—
294C4×M16C22×C3111yescharacteristic
295C4×C4⋊C4C22×C3111yescharacteristic
296#339—22no—
297#332—21no—
298#332—21no—
299#329—22no—
300#328—21no—
301#327—21no—
302Q8⋊C8—21no—
303D4⋊C8—21no—
304C2×C31—21no—
305C2×C31—21no—
306C2×C31—81no—
307C2×C31—82no—
308C2×C31#179211yescharacteristic
309C2×C31#179311yescharacteristic
310C2×C31#179411yescharacteristic
311C2×#41—21no—
312C22×Q8—21no—
313C22×Q8D4×C3111yescharacteristic
314C22×D4D4×C3111yescharacteristic
315C2×M16—21no—
316C22×C8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—22no—
319C4⋊Q8D4×C3111yescharacteristic
320C4⋊Q8D4×C3111yescharacteristic
321#126—22no—
322#126—22no—
323#126—22no—
324#126—21no—
325#125C2×C4×C3111yescharacteristic
326#125C2×C4×C3112yes—
327#124—21no—
328#124—22no—
329#124—22no—
330#124—21no—
331#124—21no—
332#124C2×C4×C3112yes—
333#124C2×C4×C3111yescharacteristic
334#124D4×C3112yes—
335#123—22no—
336#123—22no—
337#123—21no—
338#123—22no—
339#123—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—21no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—21no—
347C22⋊Q8—21no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8D4×C3111yescharacteristic
351#121—21no—
352#121—22no—
353#121—22no—
354#121D4×C3111yescharacteristic
355#120—22no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8D4×C3111yescharacteristic
360C4×D4—22no—
361C4×D4—21no—
362C4×D4—22no—
363C4×D4—21no—
364C4×D4D4×C3111yescharacteristic
365#117—22no—
366#117—21no—
367#117D4×C3112yes—
368C2×C4⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C42—21no—
373C2×C42C23×C3111yescharacteristic
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377#103—42no—
378#102—42no—
379C22⋊C8—21no—
380#97—21no—
381#97—21no—
382C4×C8—21no—
383#95—21no—
384C31#851111yescharacteristic, direct product factor, Sylow
385C24—21no—
386#41—81no—
387#41—81no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8—81no—
395C2×Q8—81no—
396C2×Q8C22⋊C4×C3111yescharacteristic
397C2×Q8C22⋊C4×C3112yes—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4C22⋊C4×C3111yescharacteristic
405C2×D4C22⋊C4×C3112yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—41no—
412C22×C4C2×D4×C3111yescharacteristic
413C22×C4C2×D4×C3111yescharacteristic
414C22×C4C2×D4×C3111yescharacteristic
415M16—22no—
416M16—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4C22⋊C4×C3112yes—
437C4⋊C4C22⋊C4×C3112yes—
438C4⋊C4C22⋊C4×C3111yescharacteristic
439C4⋊C4C22⋊C4×C3111yescharacteristic
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—21no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C42—22no—
456C42—22no—
457C42—21no—
458C42—42no—
459C42C22×C4×C3111yescharacteristic
460C42C2×D4×C3111yescharacteristic
461C42C2×D4×C3111yescharacteristic
462C42C2×D4×C3111yescharacteristic
463C23—22no—
464C23—21no—
465C23—41no—
466C23—41no—
467C23—81no—
468C23—81no—
469C23C31×#12011yescharacteristic
470Q8—42no—
471Q8—41no—
472Q8—41no—
473Q8—41no—
474Q8—41no—
475Q8—41no—
476Q8—81no—
477Q8—82no—
478D4—42no—
479D4—41no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—82no—
485D4—82no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4C2×C22⋊C4×C3111yescharacteristic
507C2×C4C2×C22⋊C4×C3111yescharacteristic
508C2×C4C2×C22⋊C4×C3111yescharacteristic
509C2×C4C31×#12011yescharacteristic
510C2×C4C31×#12011yescharacteristic
511C2×C4C31×#12011yescharacteristic
512C8—42no—
513C8—81no—
514C8—81no—
515C22—21no—
516C22—21no—
517C22—41no—
518C22—42no—
519C22—41no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—161no—
524C22—161no—
525C22C31×#37911yescharacteristic
526C22C31×#45411yescharacteristic
527C22C31×#45611yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—41no—
534C4—42no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—42no—
539C4—82no—
540C4C31×#45311yescharacteristic
541C4C31×#45511yescharacteristic
542C2—21no—
543C2—21no—
544C2—82no—
545C2—81no—
546C2C31×#179211yescharacteristic
547C2C31×#179311yescharacteristic
548C2C31×#179411yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3561C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1986C2semidirect
3. C31×#1983C2semidirect
4. C31×#1552C2semidirect
5. C31×#1527C2semidirect
6. C31×#1509C2semidirect
7. C31×#1483C2semidirect
8. C31×#563C4semidirect
9. C31×#562C4semidirect
188. #8511C31direct (N × H)
242. #1986C2×C31semidirect
243. #1983C2×C31semidirect
244. #1552C2×C31semidirect
245. #1527C2×C31semidirect
246. #1509C2×C31semidirect
247. #1483C2×C31semidirect
275. #563C4×C31semidirect
276. #562C4×C31semidirect
384. C31#8511direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35612
C31×#19862
C31×#19832
C31×#15522
C31×#15272
C31×#15092
C31×#14832
#851131