Finite Group Databasebeta

← C31×#21262 #21478021

Subgroups — C31×#21262 #21478021

Order 7936.

1534 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#536C211yesmaximal, characteristic
2C2×C31×#424C211yesmaximal, characteristic, complemented
3C2×C31×#417C211yesmaximal, characteristic
4C2×C31×#419C211yesmaximal, characteristic, complemented
5C2×C31×#416C211yesmaximal, characteristic
6C22×C31×#103C211yesmaximal, characteristic, complemented
7C31×#2274C211yesmaximal, characteristic, complemented
8C31×#1983C214yesmaximal, complemented
9C31×#1981C214yesmaximal, complemented
10C22×#41×C31C2211yescharacteristic
11C23×Q8×C31C2211yescharacteristic
12C22×M16×C31C2211yescharacteristic
13C23×C8×C31C2211yescharacteristic
14C31×#536—24no—
15C31×#536C418yescomplemented
16C2×C31×C4⋊Q8C2212yes—
17C2×C31×#124C2212yes—
18C2×C31×#123—22no—
19C2×C31×C22⋊Q8—22no—
20C2×C31×#117C2211yescharacteristic
21C2×C31×#117C2211yescharacteristic
22C22×C4⋊C4×C31C2211yescharacteristic
23C31×#424—24no—
24C2×C31×C4⋊C8C2214yes—
25C31×#419—24no—
26C31×#417—24no—
27C31×#416—24no—
28C2×C31×#103—24no—
29C2×C31×#103C2214yes—
30C2×C31×#103C2212yes—
31C2×C31×#103C2214yes—
32C2×C31×#103C2212yes—
33C2×C31×#102C2212yes—
34C2×C31×#102C2212yes—
35C4×C4⋊C4×C31—22no—
36C4×C4⋊C4×C31C2214yes—
37C4×C22⋊C4×C31—22no—
38C2×#41×C31—22no—
39C2×#41×C31—22no—
40C2×#41×C31—22no—
41C2×#41×C31—22no—
42C2×#41×C31—22no—
43C2×#41×C31C2×C414yes—
44C22×Q8×C31—21no—
45C22×Q8×C31—22no—
46C22×Q8×C31—22no—
47C22×Q8×C31—22no—
48C22×Q8×C31—22no—
49C22×Q8×C31—22no—
50C22×Q8×C31C2×C414yes—
51C22×Q8×C31C2312yes—
52C22×Q8×C31C2311yescharacteristic
53C22×D4×C31—21no—
54C22×D4×C31C2311yescharacteristic
55C23×C4×C31—21no—
56C23×C4×C31—21no—
57C23×C4×C31—21no—
58C23×C4×C31C2311yescharacteristic
59C2×M16×C31—22no—
60C2×M16×C31—22no—
61C2×M16×C31—22no—
62C22×C8×C31—22no—
63C22×C8×C31—22no—
64C22×C8×C31—22no—
65C22×C8×C31C2312yes—
66C22×C8×C31C2312yes—
67C31×C4⋊Q8—24no—
68C31×C4⋊Q8C2×C418yescomplemented
69C31×#124—24no—
70C31×#124C2×C418yescomplemented
71C31×#123—216no—
72C31×C22⋊Q8—216no—
73C31×#117—22no—
74C31×#117C2×C414yes—
75C31×#117D414yes—
76C31×#117D414yes—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—24no—
79C2×C4⋊C4×C31—22no—
80C2×C4⋊C4×C31D414yes—
81C2×C4⋊C4×C31D412yes—
82C2×C4⋊C4×C31C2312yes—
83C2×C4⋊C4×C31C2312yes—
84C2×C22⋊C4×C31—21no—
85C2×C22⋊C4×C31—24no—
86C2×C22⋊C4×C31D412yes—
87C2×C42×C31—21no—
88C2×C42×C31—24no—
89C2×C42×C31C2312yes—
90C31×C4⋊C8—28no—
91C31×#103—28no—
92C31×#103—24no—
93C31×#103—24no—
94C31×#103—28no—
95C31×#102—24no—
96C31×#102—24no—
97C31×#95—24no—
98C24×C31—21no—
99C24×C31C2×D411yescharacteristic
100#41×C31—28no—
101#41×C31—28no—
102#41×C31—44no—
103#41×C31—44no—
104C2×Q8×C31—24no—
105C2×Q8×C31—22no—
106C2×Q8×C31—21no—
107C2×Q8×C31—28no—
108C2×Q8×C31—28no—
109C2×Q8×C31—22no—
110C2×Q8×C31—22no—
111C2×Q8×C31—21no—
112C2×Q8×C31—21no—
113C2×Q8×C31—22no—
114C2×Q8×C31—22no—
115C2×Q8×C31—21no—
116C2×Q8×C31—21no—
117C2×Q8×C31—42no—
118C2×Q8×C31—42no—
119C2×Q8×C31—44no—
120C2×Q8×C31—44no—
121C2×Q8×C31C22×C414yes—
122C2×Q8×C31C22×C412yes—
123C2×D4×C31—24no—
124C2×D4×C31—21no—
125C2×D4×C31—21no—
126C2×D4×C31—21no—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—42no—
130C2×D4×C31—42no—
131C2×D4×C31C22×C412yes—
132C22×C4×C31—24no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—24no—
138C22×C4×C31—24no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31—42no—
142C22×C4×C31—42no—
143C22×C4×C31—82no—
144C22×C4×C31C22×C412yes—
145C22×C4×C31C2×D414yes—
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×D412yes—
148C22×C4×C31C2×D412yes—
149C22×C4×C31C2×D412yes—
150C22×C4×C31#4112yes—
151C22×C4×C31#4112yes—
152C22×C4×C31C2411yescharacteristic
153M16×C31—44no—
154C2×C8×C31—22no—
155C2×C8×C31—22no—
156C2×C8×C31—22no—
157C2×C8×C31—22no—
158C2×C8×C31—22no—
159C2×C8×C31—22no—
160C2×C8×C31—44no—
161C4⋊C4×C31—216no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—22no—
164C4⋊C4×C31—22no—
165C4⋊C4×C31C22⋊C418yes—
166C4⋊C4×C31C2×D414yes—
167C4⋊C4×C31C2×D414yes—
168C22⋊C4×C31—24no—
169C22⋊C4×C31—216no—
170C22⋊C4×C31C22⋊C418yes—
171C42×C31—24no—
172C42×C31—84no—
173C42×C31C22×C414yes—
174C42×C31C2×D414yes—
175#21262C3111yesmaximal, characteristic, direct product factor, Sylow
176C23×C31—22no—
177C23×C31—22no—
178C23×C31—22no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31C2×C22⋊C412yes—
184C23×C31#12112yes—
185C23×C31#12112yes—
186C23×C31C22×D411yescharacteristic
187Q8×C31—24no—
188Q8×C31—24no—
189Q8×C31—22no—
190Q8×C31—22no—
191Q8×C31—42no—
192Q8×C31—41no—
193Q8×C31—42no—
194Q8×C31—41no—
195Q8×C31—48no—
196D4×C31—22no—
197D4×C31—22no—
198D4×C31—48no—
199D4×C31—41no—
200D4×C31—41no—
201C2×C4×C31—24no—
202C2×C4×C31—24no—
203C2×C4×C31—24no—
204C2×C4×C31—22no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—41no—
208C2×C4×C31—41no—
209C2×C4×C31—42no—
210C2×C4×C31—42no—
211C2×C4×C31—42no—
212C2×C4×C31—44no—
213C2×C4×C31—42no—
214C2×C4×C31—48no—
215C2×C4×C31—48no—
216C2×C4×C31—84no—
217C2×C4×C31C2×C22⋊C418yes—
218C2×C4×C31C2×C22⋊C412yes—
219C2×C4×C31C4×D414yes—
220C2×C4×C31C4×D414yes—
221C2×C4×C31#12112yes—
222C2×C4×C31#12112yes—
223C2×C4×C31#12414yes—
224C2×C4×C31#12714yes—
225C2×C4×C31C23×C411yescharacteristic
226C2×C4×C31C22×D411yescharacteristic
227C2×C4×C31C22×D411yescharacteristic
228C2×C4×C31C22×D411yescharacteristic
229C2×C4×C31C2×#4111yescharacteristic
230C2×C4×C31C2×#4111yescharacteristic
231C8×C31—42no—
232C8×C31—42no—
233C2×#536C2×C3111yescharacteristic
234C2×#424C2×C3111yescharacteristic, complemented
235C2×#417C2×C3111yescharacteristic
236C2×#419C2×C3111yescharacteristic, complemented
237C2×#416C2×C3111yescharacteristic
238C22×#103C2×C3111yescharacteristic, complemented
239#2274C2×C3111yescharacteristic, complemented
240#1983C2×C3114yescomplemented
241#1981C2×C3114yescomplemented
242C22×C31—24no—
243C22×C31—22no—
244C22×C31—22no—
245C22×C31—24no—
246C22×C31—44no—
247C22×C31—42no—
248C22×C31—81no—
249C22×C31—81no—
250C22×C31—82no—
251C22×C31#39014yes—
252C22×C31C22×C22⋊C411yescharacteristic
253C22×C31C2×#12111yescharacteristic
254C22×C31C2×#12111yescharacteristic
255C22×C31#57712yes—
256C22×C31#57812yes—
257C4×C31—28no—
258C4×C31—44no—
259C4×C31—44no—
260C4×C31—42no—
261C4×C31—44no—
262C4×C31#39014yes—
263C4×C31C2×C4×D411yescharacteristic
264C4×C31C2×C4×D411yescharacteristic
265C4×C31C2×#12411yescharacteristic
266C4×C31C2×#12711yescharacteristic
267C22×#41C22×C3111yescharacteristic
268C23×Q8C22×C3111yescharacteristic
269C22×M16C22×C3111yescharacteristic
270C23×C8C22×C3111yescharacteristic
271#536—24no—
272#536C4×C3118yescomplemented
273C2×C4⋊Q8C22×C3112yes—
274C2×#124C22×C3112yes—
275C2×#123—22no—
276C2×C22⋊Q8—22no—
277C2×#117C22×C3111yescharacteristic
278C2×#117C22×C3111yescharacteristic
279C22×C4⋊C4C22×C3111yescharacteristic
280#424—24no—
281C2×C4⋊C8C22×C3114yes—
282#419—24no—
283#417—24no—
284#416—24no—
285C2×#103—24no—
286C2×#103C22×C3114yes—
287C2×#103C22×C3112yes—
288C2×#103C22×C3114yes—
289C2×#103C22×C3112yes—
290C2×#102C22×C3112yes—
291C2×#102C22×C3112yes—
292C4×C4⋊C4—22no—
293C4×C4⋊C4C22×C3114yes—
294C4×C22⋊C4—22no—
295C2×C31—24no—
296C2×C31—82no—
297C2×C31C2×#39011yescharacteristic
298C2×C31#289812yes—
299C2×C31#304211yescharacteristic
300C2×C31#304411yescharacteristic
301C2×C31#304612yes—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41—22no—
307C2×#41C2×C4×C3114yes—
308C22×Q8—21no—
309C22×Q8—22no—
310C22×Q8—22no—
311C22×Q8—22no—
312C22×Q8—22no—
313C22×Q8—22no—
314C22×Q8C2×C4×C3114yes—
315C22×Q8C23×C3112yes—
316C22×Q8C23×C3111yescharacteristic
317C22×D4—21no—
318C22×D4C23×C3111yescharacteristic
319C23×C4—21no—
320C23×C4—21no—
321C23×C4—21no—
322C23×C4C23×C3111yescharacteristic
323C2×M16—22no—
324C2×M16—22no—
325C2×M16—22no—
326C22×C8—22no—
327C22×C8—22no—
328C22×C8—22no—
329C22×C8C23×C3112yes—
330C22×C8C23×C3112yes—
331C4⋊Q8—24no—
332C4⋊Q8C2×C4×C3118yescomplemented
333#124—24no—
334#124C2×C4×C3118yescomplemented
335#123—216no—
336C22⋊Q8—216no—
337#117—22no—
338#117C2×C4×C3114yes—
339#117D4×C3114yes—
340#117D4×C3114yes—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4D4×C3114yes—
345C2×C4⋊C4D4×C3112yes—
346C2×C4⋊C4C23×C3112yes—
347C2×C4⋊C4C23×C3112yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4D4×C3112yes—
351C2×C42—21no—
352C2×C42—24no—
353C2×C42C23×C3112yes—
354C4⋊C8—28no—
355#103—28no—
356#103—24no—
357#103—24no—
358#103—28no—
359#102—24no—
360#102—24no—
361#95—24no—
362C31#2126211yescharacteristic, direct product factor, Sylow
363C24—21no—
364C24C2×D4×C3111yescharacteristic
365#41—28no—
366#41—28no—
367#41—44no—
368#41—44no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8C22×C4×C3114yes—
387C2×Q8C22×C4×C3112yes—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4C22×C4×C3112yes—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—82no—
409C22×C4C22×C4×C3112yes—
410C22×C4C2×D4×C3114yes—
411C22×C4C2×D4×C3111yescharacteristic
412C22×C4C2×D4×C3112yes—
413C22×C4C2×D4×C3112yes—
414C22×C4C2×D4×C3112yes—
415C22×C4#41×C3112yes—
416C22×C4#41×C3112yes—
417C22×C4C24×C3111yescharacteristic
418M16—44no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—44no—
426C4⋊C4—216no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4C22⋊C4×C3118yes—
431C4⋊C4C2×D4×C3114yes—
432C4⋊C4C2×D4×C3114yes—
433C22⋊C4—24no—
434C22⋊C4—216no—
435C22⋊C4C22⋊C4×C3118yes—
436C42—24no—
437C42—84no—
438C42C22×C4×C3114yes—
439C42C2×D4×C3114yes—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23C2×C22⋊C4×C3112yes—
448C23C31×#12112yes—
449C23C31×#12112yes—
450C23C22×D4×C3111yescharacteristic
451Q8—24no—
452Q8—24no—
453Q8—22no—
454Q8—22no—
455Q8—42no—
456Q8—41no—
457Q8—42no—
458Q8—41no—
459Q8—48no—
460D4—22no—
461D4—22no—
462D4—48no—
463D4—41no—
464D4—41no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—84no—
481C2×C4C2×C22⋊C4×C3118yes—
482C2×C4C2×C22⋊C4×C3112yes—
483C2×C4C4×D4×C3114yes—
484C2×C4C4×D4×C3114yes—
485C2×C4C31×#12112yes—
486C2×C4C31×#12112yes—
487C2×C4C31×#12414yes—
488C2×C4C31×#12714yes—
489C2×C4C23×C4×C3111yescharacteristic
490C2×C4C22×D4×C3111yescharacteristic
491C2×C4C22×D4×C3111yescharacteristic
492C2×C4C22×D4×C3111yescharacteristic
493C2×C4C2×#41×C3111yescharacteristic
494C2×C4C2×#41×C3111yescharacteristic
495C8—42no—
496C8—42no—
497C22—24no—
498C22—22no—
499C22—22no—
500C22—24no—
501C22—44no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—82no—
506C22C31×#39014yes—
507C22C22×C22⋊C4×C3111yescharacteristic
508C22C2×C31×#12111yescharacteristic
509C22C2×C31×#12111yescharacteristic
510C22C31×#57712yes—
511C22C31×#57812yes—
512C4—28no—
513C4—44no—
514C4—44no—
515C4—42no—
516C4—44no—
517C4C31×#39014yes—
518C4C2×C4×D4×C3111yescharacteristic
519C4C2×C4×D4×C3111yescharacteristic
520C4C2×C31×#12411yescharacteristic
521C4C2×C31×#12711yescharacteristic
522C2—24no—
523C2—82no—
524C2C2×C31×#39011yescharacteristic
525C2C31×#289812yes—
526C2C31×#304211yescharacteristic
527C2C31×#304411yescharacteristic
528C2C31×#304612yes—
subgroupG/H
center, socleC23×C31C22×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#424C2semidirect
4. C2×C31×#419C2semidirect
6. C22×C31×#103C2semidirect
7. C31×#2274C2semidirect
8. C31×#1983C2semidirect
9. C31×#1981C2semidirect
15. C31×#536C4semidirect
68. C31×C4⋊Q8C2×C4semidirect
70. C31×#124C2×C4semidirect
175. #21262C31direct (N × H)
234. C2×#424C2×C31semidirect
236. C2×#419C2×C31semidirect
238. C22×#103C2×C31semidirect
239. #2274C2×C31semidirect
240. #1983C2×C31semidirect
241. #1981C2×C31semidirect
272. #536C4×C31semidirect
332. C4⋊Q8C2×C4×C31semidirect
334. #124C2×C4×C31semidirect
362. C31#21262direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5362
C2×C31×#4242
C2×C31×#4172
C2×C31×#4192
C2×C31×#4162
C22×C31×#1032
C31×#22742
C31×#19832
C31×#19812
#2126231