Finite Group Databasebeta

← C31×#20911 #21477670

Subgroups — C31×#20911 #21477670

Order 7936.

1934 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×C22⋊Q8C211yesmaximal, characteristic
2C22×C31×#117C211yesmaximal, characteristic
3C2×C31×#419C212yesmaximal, complemented
4C2×C31×#407C211yesmaximal, characteristic
5C31×#1873C214yesmaximal, complemented
6C31×#1872C214yesmaximal, complemented
7C31×#1736C212yesmaximal, complemented
8C23×Q8×C31C2211yescharacteristic
9C24×C4×C31C2211yescharacteristic
10C22×M16×C31C2212yes—
11C2×C31×C22⋊Q8—24no—
12C2×C31×C22⋊Q8—21no—
13C2×C31×C22⋊Q8—21no—
14C2×C31×C22⋊Q8C418yescomplemented
15C2×C31×C22⋊Q8C2212yes—
16C2×C31×C22⋊Q8C2212yes—
17C2×C31×#117—21no—
18C2×C31×#117—21no—
19C2×C31×#117—28no—
20C2×C31×#117C2212yes—
21C2×C31×#117C2212yes—
22C22×C4⋊C4×C31—21no—
23C22×C4⋊C4×C31C2212yes—
24C22×C4⋊C4×C31C2211yescharacteristic
25C22×C22⋊C4×C31—21no—
26C22×C22⋊C4×C31—21no—
27C22×C42×C31—21no—
28C31×#419—28no—
29C2×C31×#103C2218yescomplemented
30C31×#407—24no—
31C2×C31×C22⋊C8C2212yes—
32C2×C31×C22⋊C8C2212yes—
33C31×#340C2218yescomplemented
34C25×C31D411yescharacteristic
35C22×Q8×C31—22no—
36C22×Q8×C31—21no—
37C22×Q8×C31—24no—
38C22×Q8×C31—24no—
39C22×Q8×C31C2×C414yescomplemented
40C22×Q8×C31C2312yes—
41C23×C4×C31—24no—
42C23×C4×C31—24no—
43C23×C4×C31—22no—
44C23×C4×C31—21no—
45C23×C4×C31—21no—
46C23×C4×C31C2×C414yes—
47C23×C4×C31D411yescharacteristic
48C23×C4×C31D411yescharacteristic
49C23×C4×C31D411yescharacteristic
50C23×C4×C31D414yes—
51C23×C4×C31C2311yescharacteristic
52C23×C4×C31C2311yescharacteristic
53C23×C4×C31C2311yescharacteristic
54C2×M16×C31—24no—
55C2×M16×C31—24no—
56C2×M16×C31—24no—
57C22×C8×C31C2314yes—
58C31×C22⋊Q8—28no—
59C31×C22⋊Q8—28no—
60C31×C22⋊Q8—24no—
61C31×C22⋊Q8—24no—
62C31×C22⋊Q8C2×C418yes—
63C31×C22⋊Q8C2×C418yescomplemented
64C31×#117—28no—
65C31×#117—28no—
66C31×#117—24no—
67C31×#117—28no—
68C31×#117D418yes—
69C2×C4⋊C4×C31—28no—
70C2×C4⋊C4×C31—22no—
71C2×C4⋊C4×C31—22no—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—22no—
75C2×C4⋊C4×C31—28no—
76C2×C4⋊C4×C31C2×C414yescomplemented
77C2×C4⋊C4×C31C2312yes—
78C2×C4⋊C4×C31C2314yes—
79C2×C22⋊C4×C31—28no—
80C2×C22⋊C4×C31—28no—
81C2×C22⋊C4×C31—22no—
82C2×C22⋊C4×C31—22no—
83C2×C22⋊C4×C31—22no—
84C2×C22⋊C4×C31—22no—
85C2×C42×C31—28no—
86C2×C42×C31—22no—
87C2×C42×C31—22no—
88C31×#103—216no—
89C31×C22⋊C8—24no—
90C31×C22⋊C8—24no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—24no—
94C24×C31—42no—
95C24×C31C22⋊C414yes—
96C24×C31C2×D411yescharacteristic
97C24×C31C2×D411yescharacteristic
98C24×C31C2×D411yescharacteristic
99C2×Q8×C31—24no—
100C2×Q8×C31—22no—
101C2×Q8×C31—216no—
102C2×Q8×C31—24no—
103C2×Q8×C31—24no—
104C2×Q8×C31—42no—
105C2×Q8×C31—48no—
106C2×Q8×C31—42no—
107C2×Q8×C31C22×C414yes—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—216no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—216no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—28no—
121C22×C4×C31—24no—
122C22×C4×C31—24no—
123C22×C4×C31—24no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—44no—
127C22×C4×C31—44no—
128C22×C4×C31—42no—
129C22×C4×C31—42no—
130C22×C4×C31—44no—
131C22×C4×C31C22⋊C414yes—
132C22×C4×C31C22⋊C414yes—
133C22×C4×C31C22⋊C414yes—
134C22×C4×C31C22×C412yes—
135C22×C4×C31C22×C412yes—
136C22×C4×C31C22×C412yes—
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D412yes—
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D414yes—
144C22×C4×C31#4114yes—
145C22×C4×C31C2411yescharacteristic
146M16×C31—48no—
147C2×C8×C31—24no—
148C2×C8×C31—24no—
149C2×C8×C31—24no—
150C4⋊C4×C31—28no—
151C4⋊C4×C31—28no—
152C4⋊C4×C31—28no—
153C4⋊C4×C31—24no—
154C4⋊C4×C31—24no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31C22×C414yes—
157C4⋊C4×C31C2×D418yes—
158C22⋊C4×C31—28no—
159C22⋊C4×C31—28no—
160C22⋊C4×C31—28no—
161C22⋊C4×C31—28no—
162C42×C31—28no—
163C42×C31—28no—
164#20911C3111yesmaximal, characteristic, direct product factor, Sylow
165C23×C31—21no—
166C23×C31—21no—
167C23×C31—22no—
168C23×C31—22no—
169C23×C31—22no—
170C23×C31—21no—
171C23×C31—21no—
172C23×C31—22no—
173C23×C31—22no—
174C23×C31—28no—
175C23×C31—28no—
176C23×C31—44no—
177C23×C31—48no—
178C23×C31—48no—
179C23×C31C2×C22⋊C412yes—
180C23×C31C2×C22⋊C412yes—
181C23×C31C2×C22⋊C412yes—
182C23×C31#12012yes—
183C23×C31#12312yes—
184C23×C31#13714yes—
185C23×C31C22×D411yescharacteristic
186Q8×C31—28no—
187Q8×C31—44no—
188Q8×C31—48no—
189C2×C4×C31—21no—
190C2×C4×C31—21no—
191C2×C4×C31—21no—
192C2×C4×C31—21no—
193C2×C4×C31—28no—
194C2×C4×C31—24no—
195C2×C4×C31—24no—
196C2×C4×C31—24no—
197C2×C4×C31—24no—
198C2×C4×C31—28no—
199C2×C4×C31—28no—
200C2×C4×C31—24no—
201C2×C4×C31—24no—
202C2×C4×C31—28no—
203C2×C4×C31—48no—
204C2×C4×C31—416no—
205C2×C4×C31—42no—
206C2×C4×C31—42no—
207C2×C4×C31—44no—
208C2×C4×C31—44no—
209C2×C4×C31—44no—
210C2×C4×C31—48no—
211C2×C4×C31C2×C22⋊C412yes—
212C2×C4×C31C2×C22⋊C412yes—
213C2×C4×C31C2×C22⋊C412yes—
214C2×C4×C31C4×D418yes—
215C2×C4×C31#12012yes—
216C2×C4×C31#12118yes—
217C2×C4×C31#12312yes—
218C2×C4×C31C23×C411yescharacteristic
219C2×C4×C31C22×D412yes—
220C2×C4×C31C22×D411yescharacteristic
221C2×C4×C31C2×#4112yes—
222C8×C31—44no—
223C22×C22⋊Q8C2×C3111yescharacteristic
224C22×#117C2×C3111yescharacteristic
225C2×#419C2×C3112yescomplemented
226C2×#407C2×C3111yescharacteristic
227#1873C2×C3114yescomplemented
228#1872C2×C3114yescomplemented
229#1736C2×C3112yescomplemented
230C22×C31—24no—
231C22×C31—24no—
232C22×C31—22no—
233C22×C31—22no—
234C22×C31—22no—
235C22×C31—24no—
236C22×C31—24no—
237C22×C31—22no—
238C22×C31—22no—
239C22×C31—22no—
240C22×C31—22no—
241C22×C31—416no—
242C22×C31—42no—
243C22×C31—42no—
244C22×C31#38614yes—
245C22×C31#41914yescomplemented
246C22×C31C22×C22⋊C411yescharacteristic
247C22×C31C2×#12011yescharacteristic
248C22×C31C2×#12311yescharacteristic
249C22×C31C2×#13712yes—
250C22×C31#57512yes—
251C4×C31—22no—
252C4×C31—22no—
253C4×C31—44no—
254C4×C31—48no—
255C4×C31—44no—
256C4×C31#38614yes—
257C4×C31C2×C4×D412yes—
258C4×C31C2×#12112yes—
259C23×Q8C22×C3111yescharacteristic
260C24×C4C22×C3111yescharacteristic
261C22×M16C22×C3112yes—
262C2×C22⋊Q8—24no—
263C2×C22⋊Q8—21no—
264C2×C22⋊Q8—21no—
265C2×C22⋊Q8C4×C3118yescomplemented
266C2×C22⋊Q8C22×C3112yes—
267C2×C22⋊Q8C22×C3112yes—
268C2×#117—21no—
269C2×#117—21no—
270C2×#117—28no—
271C2×#117C22×C3112yes—
272C2×#117C22×C3112yes—
273C22×C4⋊C4—21no—
274C22×C4⋊C4C22×C3112yes—
275C22×C4⋊C4C22×C3111yescharacteristic
276C22×C22⋊C4—21no—
277C22×C22⋊C4—21no—
278C22×C42—21no—
279#419—28no—
280C2×#103C22×C3118yescomplemented
281#407—24no—
282C2×C22⋊C8C22×C3112yes—
283C2×C22⋊C8C22×C3112yes—
284#340C22×C3118yescomplemented
285C2×C31—24no—
286C2×C31—24no—
287C2×C31—42no—
288C2×C31C2×#38611yescharacteristic
289C2×C31C2×#41911yescharacteristic
290C2×C31#289611yescharacteristic
291C2×C31#300712yes—
292C2×C31#309512yes—
293C25D4×C3111yescharacteristic
294C22×Q8—22no—
295C22×Q8—21no—
296C22×Q8—24no—
297C22×Q8—24no—
298C22×Q8C2×C4×C3114yescomplemented
299C22×Q8C23×C3112yes—
300C23×C4—24no—
301C23×C4—24no—
302C23×C4—21no—
303C23×C4—21no—
304C23×C4—22no—
305C23×C4C2×C4×C3114yes—
306C23×C4D4×C3111yescharacteristic
307C23×C4D4×C3111yescharacteristic
308C23×C4D4×C3111yescharacteristic
309C23×C4D4×C3114yes—
310C23×C4C23×C3111yescharacteristic
311C23×C4C23×C3111yescharacteristic
312C23×C4C23×C3111yescharacteristic
313C2×M16—24no—
314C2×M16—24no—
315C2×M16—24no—
316C22×C8C23×C3114yes—
317C22⋊Q8—28no—
318C22⋊Q8—28no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8C2×C4×C3118yes—
322C22⋊Q8C2×C4×C3118yescomplemented
323#117—28no—
324#117—28no—
325#117—24no—
326#117—28no—
327#117D4×C3118yes—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4C2×C4×C3114yescomplemented
336C2×C4⋊C4C23×C3112yes—
337C2×C4⋊C4C23×C3114yes—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—28no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C42—28no—
345C2×C42—22no—
346C2×C42—22no—
347#103—216no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350C31#2091111yescharacteristic, direct product factor, Sylow
351C24—22no—
352C24—22no—
353C24—24no—
354C24—42no—
355C24C22⋊C4×C3114yes—
356C24C2×D4×C3111yescharacteristic
357C24C2×D4×C3111yescharacteristic
358C24C2×D4×C3111yescharacteristic
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—216no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—42no—
365C2×Q8—48no—
366C2×Q8—42no—
367C2×Q8C22×C4×C3114yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—216no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4C22⋊C4×C3114yes—
392C22×C4C22⋊C4×C3114yes—
393C22×C4C22⋊C4×C3114yes—
394C22×C4C22×C4×C3112yes—
395C22×C4C22×C4×C3112yes—
396C22×C4C22×C4×C3112yes—
397C22×C4C2×D4×C3111yescharacteristic
398C22×C4C2×D4×C3111yescharacteristic
399C22×C4C2×D4×C3112yes—
400C22×C4C2×D4×C3112yes—
401C22×C4C2×D4×C3111yescharacteristic
402C22×C4C2×D4×C3112yes—
403C22×C4C2×D4×C3114yes—
404C22×C4#41×C3114yes—
405C22×C4C24×C3111yescharacteristic
406M16—48no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4C22×C4×C3114yes—
417C4⋊C4C2×D4×C3118yes—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C42—28no—
423C42—28no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—28no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23—28no—
435C23—44no—
436C23—48no—
437C23—48no—
438C23C2×C22⋊C4×C3112yes—
439C23C2×C22⋊C4×C3112yes—
440C23C2×C22⋊C4×C3112yes—
441C23C31×#12012yes—
442C23C31×#12312yes—
443C23C31×#13714yes—
444C23C22×D4×C3111yescharacteristic
445Q8—28no—
446Q8—44no—
447Q8—48no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—48no—
463C2×C4—416no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4C2×C22⋊C4×C3112yes—
471C2×C4C2×C22⋊C4×C3112yes—
472C2×C4C2×C22⋊C4×C3112yes—
473C2×C4C4×D4×C3118yes—
474C2×C4C31×#12012yes—
475C2×C4C31×#12118yes—
476C2×C4C31×#12312yes—
477C2×C4C23×C4×C3111yescharacteristic
478C2×C4C22×D4×C3112yes—
479C2×C4C22×D4×C3111yescharacteristic
480C2×C4C2×#41×C3112yes—
481C8—44no—
482C22—24no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—24no—
488C22—24no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—416no—
494C22—42no—
495C22—42no—
496C22C31×#38614yes—
497C22C31×#41914yescomplemented
498C22C22×C22⋊C4×C3111yescharacteristic
499C22C2×C31×#12011yescharacteristic
500C22C2×C31×#12311yescharacteristic
501C22C2×C31×#13712yes—
502C22C31×#57512yes—
503C4—22no—
504C4—22no—
505C4—44no—
506C4—48no—
507C4—44no—
508C4C31×#38614yes—
509C4C2×C4×D4×C3112yes—
510C4C2×C31×#12112yes—
511C2—24no—
512C2—24no—
513C2—42no—
514C2C2×C31×#38611yescharacteristic
515C2C2×C31×#41911yescharacteristic
516C2C31×#289611yescharacteristic
517C2C31×#300712yes—
518C2C31×#309512yes—
subgroupG/H
center, socleC23×C31C22×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC25×C31D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C31×#419C2semidirect
5. C31×#1873C2semidirect
6. C31×#1872C2semidirect
7. C31×#1736C2semidirect
14. C2×C31×C22⋊Q8C4semidirect
29. C2×C31×#103C22semidirect
33. C31×#340C22semidirect
39. C22×Q8×C31C2×C4semidirect
63. C31×C22⋊Q8C2×C4semidirect
76. C2×C4⋊C4×C31C2×C4semidirect
164. #20911C31direct (N × H)
225. C2×#419C2×C31semidirect
227. #1873C2×C31semidirect
228. #1872C2×C31semidirect
229. #1736C2×C31semidirect
245. C22×C31#419semidirect
265. C2×C22⋊Q8C4×C31semidirect
280. C2×#103C22×C31semidirect
284. #340C22×C31semidirect
298. C22×Q8C2×C4×C31semidirect
322. C22⋊Q8C2×C4×C31semidirect
335. C2×C4⋊C4C2×C4×C31semidirect
350. C31#20911direct (N × H)
497. C22C31×#419semidirect

Maximal subgroups

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

subgroupindex
C22×C31×C22⋊Q82
C22×C31×#1172
C2×C31×#4192
C2×C31×#4072
C31×#18732
C31×#18722
C31×#17362
#2091131