Finite Group Databasebeta

← C31×#18238 #21474997

Subgroups — C31×#18238 #21474997

Order 7936.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3538C211yesmaximal, characteristic
2C31×#3095C211yesmaximal, characteristic, complemented
3C31×#3091C211yesmaximal, characteristic
4C31×#2625C211yesmaximal, characteristic, complemented
5C31×#1617C212yesmaximal, complemented
6C31×#1615C212yesmaximal, complemented
7C31×#1611C212yesmaximal, complemented
8C31×#1609C212yesmaximal, complemented
9C31×#1520C211yesmaximal, characteristic, complemented
10C31×#1513C211yesmaximal, characteristic
11C31×#1472C211yesmaximal, characteristic, complemented
12C31×#563—22no—
13C31×#552—21no—
14C31×#552—21no—
15C31×#551—22no—
16D4×Q8×C31—21no—
17D4×Q8×C31—21no—
18C31×#544—21no—
19C31×#544C2212yes—
20C31×#541—21no—
21C31×#539—21no—
22C31×#539C2212yes—
23C31×#536—21no—
24C31×#525C2211yescharacteristic
25C2×C31×C22⋊Q8C2211yescharacteristic
26C2×C31×C22⋊Q8C2211yescharacteristic
27C2×C31×#117C2211yescharacteristic
28C31×#485—21no—
29C31×#485—22no—
30C31×#484C2211yescharacteristic
31C31×#484C2212yes—
32C31×#483C2211yescharacteristic
33C31×#483C2212yes—
34C31×#482—21no—
35C31×#481C2211yescharacteristic
36C31×#480C2211yescharacteristic
37C31×#428—21no—
38C31×#427—21no—
39C31×#419—21no—
40C31×#416—21no—
41C31×#407C2211yescharacteristic
42C2×C31×C22⋊C8C2211yescharacteristic
43C31×#400C2212yes—
44C31×#398C2212yes—
45C31×#397—22no—
46C31×#391—21no—
47C31×#389—21no—
48C31×#380C2211yescharacteristic
49C31×#328C2212yes—
50C31×#328C2212yes—
51C31×#328C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C2×#41×C31D411yescharacteristic
56C22×Q8×C31—21no—
57C22×Q8×C31D411yescharacteristic
58C22×Q8×C31D411yescharacteristic
59C23×C4×C31C2311yescharacteristic
60C2×M16×C31—21no—
61C22×C8×C31—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8—22no—
66C31×#126—22no—
67C31×#126—22no—
68C31×#126—22no—
69C31×#126—22no—
70C31×#125—21no—
71C31×#125—22no—
72C31×#125—22no—
73C31×#125—22no—
74C31×#125—22no—
75C31×#124—22no—
76C31×#123—22no—
77C31×#123—22no—
78C31×#123—22no—
79C31×#123—21no—
80C31×#123—22no—
81C31×C22⋊Q8—21no—
82C31×C22⋊Q8—21no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—21no—
89C31×C22⋊Q8—21no—
90C31×C22⋊Q8—21no—
91C31×C22⋊Q8—21no—
92C31×C22⋊Q8—22no—
93C31×C22⋊Q8D412yes—
94C31×C22⋊Q8D411yescharacteristic
95C31×C22⋊Q8C2312yes—
96C31×C22⋊Q8C2311yescharacteristic
97C31×#121D411yescharacteristic
98C31×#121C2311yescharacteristic
99C31×#120—21no—
100C4×Q8×C31—21no—
101C4×Q8×C31—21no—
102C4×D4×C31—22no—
103C4×D4×C31—21no—
104C4×D4×C31—21no—
105C4×D4×C31—22no—
106C31×#117—21no—
107C31×#117—22no—
108C31×#117—22no—
109C31×#117—21no—
110C31×#117—21no—
111C31×#117—21no—
112C31×#117—21no—
113C31×#117D411yescharacteristic
114C2×C4⋊C4×C31—21no—
115C2×C4⋊C4×C31—21no—
116C2×C4⋊C4×C31—22no—
117C2×C4⋊C4×C31D411yescharacteristic
118C2×C4⋊C4×C31D411yescharacteristic
119C2×C4⋊C4×C31C2311yescharacteristic
120C2×C4⋊C4×C31C2311yescharacteristic
121C2×C4⋊C4×C31C2312yes—
122C2×C22⋊C4×C31—21no—
123C2×C22⋊C4×C31—21no—
124C2×C22⋊C4×C31—21no—
125C2×C22⋊C4×C31D412yes—
126C2×C42×C31—21no—
127C31×#107—21no—
128C31×#107—21no—
129C31×#107—22no—
130C31×#106—21no—
131C31×#106—21no—
132C31×#106—22no—
133C31×#103—21no—
134C31×#103—21no—
135C31×#103—22no—
136C31×#103—22no—
137C31×#102—21no—
138C31×#102—21no—
139C31×C22⋊C8—21no—
140C31×C22⋊C8—21no—
141C31×C22⋊C8C2311yescharacteristic
142C31×C22⋊C8C2311yescharacteristic
143C31×C22⋊C8C2312yes—
144C31×#95—22no—
145C31×#95—22no—
146C31×#95—22no—
147C31×#95C2312yes—
148C24×C31C2×D411yescharacteristic
149#41×C31—42no—
150C2×Q8×C31—22no—
151C2×Q8×C31—21no—
152C2×Q8×C31—21no—
153C2×Q8×C31—21no—
154C2×Q8×C31—42no—
155C2×Q8×C31—42no—
156C2×Q8×C31—82no—
157C2×Q8×C31C2×D412yes—
158C2×Q8×C31C2×D411yescharacteristic
159C2×D4×C31—21no—
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31C2×D411yescharacteristic
163C22×C4×C31—21no—
164C22×C4×C31—22no—
165C22×C4×C31—21no—
166C22×C4×C31—21no—
167C22×C4×C31—21no—
168C22×C4×C31—22no—
169C22×C4×C31—21no—
170C22×C4×C31—42no—
171C22×C4×C31C2×D412yes—
172C22×C4×C31C2×D411yescharacteristic
173C22×C4×C31C2×D411yescharacteristic
174C22×C4×C31C2×D411yescharacteristic
175C22×C4×C31C2×D412yes—
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2411yescharacteristic
179M16×C31—42no—
180C2×C8×C31—21no—
181C2×C8×C31—21no—
182C2×C8×C31—22no—
183C2×C8×C31—42no—
184C4⋊C4×C31—22no—
185C4⋊C4×C31—22no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—22no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—22no—
198C4⋊C4×C31—22no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—21no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—42no—
206C4⋊C4×C31—82no—
207C4⋊C4×C31C2×D412yes—
208C4⋊C4×C31C2×D411yescharacteristic
209C4⋊C4×C31C2×D411yescharacteristic
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—22no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—21no—
222C22⋊C4×C31—41no—
223C22⋊C4×C31—41no—
224C22⋊C4×C31—81no—
225C42×C31—22no—
226C42×C31—22no—
227C42×C31—22no—
228C42×C31—21no—
229C42×C31—41no—
230C42×C31—41no—
231#18238C3111yesmaximal, characteristic, direct product factor, Sylow
232C23×C31—21no—
233C23×C31—21no—
234C23×C31—21no—
235C23×C31—41no—
236C23×C31—41no—
237C23×C31#12011yescharacteristic
238C23×C31#12011yescharacteristic
239C23×C31C22×D411yescharacteristic
240Q8×C31—42no—
241Q8×C31—41no—
242Q8×C31—42no—
243Q8×C31—42no—
244Q8×C31—81no—
245Q8×C31—82no—
246D4×C31—41no—
247D4×C31—42no—
248D4×C31—41no—
249D4×C31—41no—
250D4×C31—81no—
251C2×C4×C31—22no—
252C2×C4×C31—22no—
253C2×C4×C31—22no—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—22no—
257C2×C4×C31—22no—
258C2×C4×C31—21no—
259C2×C4×C31—22no—
260C2×C4×C31—22no—
261C2×C4×C31—22no—
262C2×C4×C31—22no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—42no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—81no—
278C2×C4×C31—82no—
279C2×C4×C31#12011yescharacteristic
280C2×C4×C31#12011yescharacteristic
281C2×C4×C31C22×D411yescharacteristic
282C2×C4×C31C22×D411yescharacteristic
283C8×C31—41no—
284C8×C31—41no—
285C8×C31—42no—
286#3538C2×C3111yescharacteristic
287#3095C2×C3111yescharacteristic, complemented
288#3091C2×C3111yescharacteristic
289#2625C2×C3111yescharacteristic, complemented
290#1617C2×C3112yescomplemented
291#1615C2×C3112yescomplemented
292#1611C2×C3112yescomplemented
293#1609C2×C3112yescomplemented
294#1520C2×C3111yescharacteristic, complemented
295#1513C2×C3111yescharacteristic
296#1472C2×C3111yescharacteristic, complemented
297C22×C31—21no—
298C22×C31—21no—
299C22×C31—21no—
300C22×C31—21no—
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—21no—
304C22×C31—21no—
305C22×C31—41no—
306C22×C31—41no—
307C22×C31—41no—
308C22×C31—81no—
309C22×C31—81no—
310C22×C31C2×#12011yescharacteristic
311C22×C31#57711yescharacteristic
312C22×C31#57811yescharacteristic
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—21no—
316C4×C31—42no—
317C4×C31—41no—
318C4×C31—42no—
319C4×C31—41no—
320C4×C31—41no—
321C4×C31—42no—
322C4×C31—42no—
323C4×C31—82no—
324C4×C31—81no—
325C4×C31—81no—
326C4×C31#45812yes—
327#563—22no—
328#552—21no—
329#552—21no—
330#551—22no—
331D4×Q8—21no—
332D4×Q8—21no—
333#544—21no—
334#544C22×C3112yes—
335#541—21no—
336#539—21no—
337#539C22×C3112yes—
338#536—21no—
339#525C22×C3111yescharacteristic
340C2×C22⋊Q8C22×C3111yescharacteristic
341C2×C22⋊Q8C22×C3111yescharacteristic
342C2×#117C22×C3111yescharacteristic
343#485—21no—
344#485—22no—
345#484C22×C3111yescharacteristic
346#484C22×C3112yes—
347#483C22×C3111yescharacteristic
348#483C22×C3112yes—
349#482—21no—
350#481C22×C3111yescharacteristic
351#480C22×C3111yescharacteristic
352#428—21no—
353#427—21no—
354#419—21no—
355#416—21no—
356#407C22×C3111yescharacteristic
357C2×C22⋊C8C22×C3111yescharacteristic
358#400C22×C3112yes—
359#398C22×C3112yes—
360#397—22no—
361#391—21no—
362#389—21no—
363#380C22×C3111yescharacteristic
364#328C22×C3112yes—
365#328C22×C3112yes—
366#328C22×C3112yes—
367#327C22×C3112yes—
368#324C22×C3112yes—
369#324C22×C3112yes—
370C2×C31—21no—
371C2×C31—21no—
372C2×C31—41no—
373C2×C31—41no—
374C2×C31—81no—
375C2×C31#301311yescharacteristic
376C2×C31#302011yescharacteristic
377C2×C31C2×#45811yescharacteristic
378C2×#41D4×C3111yescharacteristic
379C22×Q8—21no—
380C22×Q8D4×C3111yescharacteristic
381C22×Q8D4×C3111yescharacteristic
382C23×C4C23×C3111yescharacteristic
383C2×M16—21no—
384C22×C8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—22no—
387C4⋊Q8—22no—
388C4⋊Q8—22no—
389#126—22no—
390#126—22no—
391#126—22no—
392#126—22no—
393#125—21no—
394#125—22no—
395#125—22no—
396#125—22no—
397#125—22no—
398#124—22no—
399#123—22no—
400#123—22no—
401#123—22no—
402#123—21no—
403#123—22no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—21no—
410C22⋊Q8—21no—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—22no—
416C22⋊Q8D4×C3112yes—
417C22⋊Q8D4×C3111yescharacteristic
418C22⋊Q8C23×C3112yes—
419C22⋊Q8C23×C3111yescharacteristic
420#121D4×C3111yescharacteristic
421#121C23×C3111yescharacteristic
422#120—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×D4—22no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—22no—
429#117—21no—
430#117—22no—
431#117—22no—
432#117—21no—
433#117—21no—
434#117—21no—
435#117—21no—
436#117D4×C3111yescharacteristic
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—22no—
440C2×C4⋊C4D4×C3111yescharacteristic
441C2×C4⋊C4D4×C3111yescharacteristic
442C2×C4⋊C4C23×C3111yescharacteristic
443C2×C4⋊C4C23×C3111yescharacteristic
444C2×C4⋊C4C23×C3112yes—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4D4×C3112yes—
449C2×C42—21no—
450#107—21no—
451#107—21no—
452#107—22no—
453#106—21no—
454#106—21no—
455#106—22no—
456#103—21no—
457#103—21no—
458#103—22no—
459#103—22no—
460#102—21no—
461#102—21no—
462C22⋊C8—21no—
463C22⋊C8—21no—
464C22⋊C8C23×C3112yes—
465C22⋊C8C23×C3111yescharacteristic
466C22⋊C8C23×C3111yescharacteristic
467#95—22no—
468#95—22no—
469#95—22no—
470#95C23×C3112yes—
471C31#1823811yescharacteristic, direct product factor, Sylow
472C24C2×D4×C3111yescharacteristic
473#41—42no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—82no—
481C2×Q8C2×D4×C3112yes—
482C2×Q8C2×D4×C3111yescharacteristic
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×D4×C3111yescharacteristic
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—42no—
495C22×C4C2×D4×C3112yes—
496C22×C4C2×D4×C3111yescharacteristic
497C22×C4C2×D4×C3111yescharacteristic
498C22×C4C2×D4×C3111yescharacteristic
499C22×C4C2×D4×C3112yes—
500C22×C4C2×D4×C3111yescharacteristic
501C22×C4C2×D4×C3111yescharacteristic
502C22×C4C24×C3111yescharacteristic
503M16—42no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—22no—
507C2×C8—42no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—82no—
531C4⋊C4C2×D4×C3112yes—
532C4⋊C4C2×D4×C3111yescharacteristic
533C4⋊C4C2×D4×C3111yescharacteristic
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—21no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—81no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—21no—
553C42—41no—
554C42—41no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23C31×#12011yescharacteristic
561C23C31×#12011yescharacteristic
562C23C22×D4×C3111yescharacteristic
563Q8—42no—
564Q8—41no—
565Q8—42no—
566Q8—42no—
567Q8—81no—
568Q8—82no—
569D4—41no—
570D4—42no—
571D4—41no—
572D4—41no—
573D4—81no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4C31×#12011yescharacteristic
603C2×C4C31×#12011yescharacteristic
604C2×C4C22×D4×C3111yescharacteristic
605C2×C4C22×D4×C3111yescharacteristic
606C8—41no—
607C8—41no—
608C8—42no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—81no—
621C22—81no—
622C22C2×C31×#12011yescharacteristic
623C22C31×#57711yescharacteristic
624C22C31×#57811yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—81no—
638C4C31×#45812yes—
639C2—21no—
640C2—21no—
641C2—41no—
642C2—41no—
643C2—81no—
644C2C31×#301311yescharacteristic
645C2C31×#302011yescharacteristic
646C2C2×C31×#45811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3095C2semidirect
4. C31×#2625C2semidirect
5. C31×#1617C2semidirect
6. C31×#1615C2semidirect
7. C31×#1611C2semidirect
8. C31×#1609C2semidirect
9. C31×#1520C2semidirect
11. C31×#1472C2semidirect
231. #18238C31direct (N × H)
287. #3095C2×C31semidirect
289. #2625C2×C31semidirect
290. #1617C2×C31semidirect
291. #1615C2×C31semidirect
292. #1611C2×C31semidirect
293. #1609C2×C31semidirect
294. #1520C2×C31semidirect
296. #1472C2×C31semidirect
471. C31#18238direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35382
C31×#30952
C31×#30912
C31×#26252
C31×#16172
C31×#16152
C31×#16112
C31×#16092
C31×#15202
C31×#15132
C31×#14722
#1823831