Finite Group Databasebeta

← C31×#18538 #21475297

Subgroups — C31×#18538 #21475297

Order 7936.

922 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3482C211yesmaximal, characteristic
2C31×#3139C211yesmaximal, characteristic, complemented
3C31×#3079C211yesmaximal, characteristic, complemented
4C31×#3079C211yesmaximal, characteristic, complemented
5C31×#1638C212yesmaximal, complemented
6C31×#1637C212yesmaximal, complemented
7C31×#1636C212yesmaximal, complemented
8C31×#1635C212yesmaximal, complemented
9C31×#1561C211yesmaximal, characteristic, complemented
10C31×#1489C211yesmaximal, characteristic, complemented
11C31×#1489C211yesmaximal, characteristic, complemented
12C31×#557C2211yescharacteristic
13C31×#556—21no—
14C31×#556C2211yescharacteristic
15C31×#552—22no—
16C31×#552—21no—
17D4×Q8×C31C2211yescharacteristic
18C31×#548—21no—
19C31×#547—21no—
20C31×#547C2211yescharacteristic
21C31×#533—21no—
22C31×#533C2211yescharacteristic
23C2×C31×C4⋊Q8C2211yescharacteristic
24C31×#529—21no—
25C31×#529—22no—
26C4×#41×C31—21no—
27C4×#41×C31C2211yescharacteristic
28C4×#41×C31C2211yescharacteristic
29C2×C4×Q8×C31—21no—
30C31×#491—22no—
31C31×#487C2212yes—
32C31×#487C2212yes—
33C31×#477C2212yes—
34C31×#477C2212yes—
35C31×#476C2212yes—
36C31×#476C2212yes—
37C31×#475—21no—
38C31×#475—21no—
39C31×#474—21no—
40C31×#474—21no—
41C31×#428—21no—
42C31×#428—21no—
43C31×#423C2211yescharacteristic
44C31×#423C2211yescharacteristic
45C31×#419—22no—
46C31×#417—21no—
47C31×#417—21no—
48C4×M16×C31C2211yescharacteristic
49C31×#332C2212yes—
50C31×#332C2212yes—
51C31×Q8⋊C8C2212yes—
52C31×Q8⋊C8C2212yes—
53C31×D4⋊C8C2212yes—
54C31×D4⋊C8C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31D411yescharacteristic
57C2×#41×C31D411yescharacteristic
58C22×Q8×C31—21no—
59C22×Q8×C31—21no—
60C2×M16×C31—21no—
61C2×M16×C31—21no—
62C2×M16×C31—21no—
63C31×C4⋊Q8—21no—
64C31×C4⋊Q8—21no—
65C31×C4⋊Q8—21no—
66C31×C4⋊Q8—21no—
67C31×C4⋊Q8—21no—
68C31×C4⋊Q8C2312yes—
69C31×C4⋊Q8C2311yescharacteristic
70C31×C4⋊Q8C2311yescharacteristic
71C31×#126—22no—
72C31×#126—22no—
73C31×#126—22no—
74C31×#125—22no—
75C31×#125—22no—
76C31×#125—21no—
77C31×#125D412yes—
78C31×#124—22no—
79C31×#124—21no—
80C31×#123—22no—
81C31×#123—22no—
82C31×#123—22no—
83C31×C22⋊Q8—22no—
84C31×C22⋊Q8—21no—
85C31×C22⋊Q8—21no—
86C31×C22⋊Q8—21no—
87C31×C22⋊Q8—21no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—21no—
90C31×#121—22no—
91C31×#121—21no—
92C4×Q8×C31—22no—
93C4×Q8×C31—22no—
94C4×Q8×C31—21no—
95C4×Q8×C31—21no—
96C4×Q8×C31—21no—
97C4×Q8×C31—21no—
98C4×Q8×C31—21no—
99C4×Q8×C31D411yescharacteristic
100C4×Q8×C31D411yescharacteristic
101C4×Q8×C31C2311yescharacteristic
102C4×Q8×C31C2311yescharacteristic
103C4×D4×C31—22no—
104C4×D4×C31—22no—
105C4×D4×C31—21no—
106C4×D4×C31—21no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31—21no—
110C4×D4×C31D411yescharacteristic
111C4×D4×C31D411yescharacteristic
112C4×D4×C31C2311yescharacteristic
113C4×D4×C31C2311yescharacteristic
114C31×#117—22no—
115C31×#117—22no—
116C31×#117—21no—
117C31×#117—21no—
118C31×#117—21no—
119C31×#117D412yes—
120C31×#117D411yescharacteristic
121C31×#117D411yescharacteristic
122C2×C4⋊C4×C31—22no—
123C2×C4⋊C4×C31—21no—
124C2×C4⋊C4×C31—21no—
125C2×C4⋊C4×C31—21no—
126C2×C42×C31—22no—
127C2×C42×C31—21no—
128C2×C42×C31—21no—
129C2×C42×C31C2311yescharacteristic
130C31×#107—22no—
131C31×#107—22no—
132C31×#106—22no—
133C31×#106—22no—
134C31×C4⋊C8—21no—
135C31×C4⋊C8—21no—
136C31×C4⋊C8C2312yes—
137C31×C4⋊C8C2312yes—
138C31×#103—22no—
139C31×#103—22no—
140C31×#103—24no—
141C31×#103—24no—
142C31×#102—22no—
143C31×#102—22no—
144C31×#97—21no—
145C4×C8×C31C2312yes—
146#41×C31—22no—
147#41×C31—22no—
148#41×C31—41no—
149#41×C31—41no—
150#41×C31—42no—
151#41×C31—42no—
152C2×Q8×C31—22no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—22no—
156C2×Q8×C31—44no—
157C2×Q8×C31—42no—
158C2×Q8×C31—42no—
159C2×Q8×C31—41no—
160C2×Q8×C31C2×D411yescharacteristic
161C2×Q8×C31C2×D411yescharacteristic
162C2×D4×C31—22no—
163C2×D4×C31—21no—
164C2×D4×C31—21no—
165C2×D4×C31C2×D411yescharacteristic
166C2×D4×C31C2×D411yescharacteristic
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31—21no—
170C22×C4×C31—21no—
171C22×C4×C31—21no—
172C22×C4×C31—21no—
173C22×C4×C31—21no—
174C22×C4×C31—21no—
175C22×C4×C31—41no—
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D411yescharacteristic
179M16×C31—42no—
180M16×C31—42no—
181M16×C31—42no—
182C2×C8×C31—22no—
183C2×C8×C31—22no—
184C2×C8×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—21no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—22no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—41no—
202C4⋊C4×C31—41no—
203C4⋊C4×C31C2×D412yes—
204C4⋊C4×C31C2×D412yes—
205C4⋊C4×C31C2×D411yescharacteristic
206C4⋊C4×C31C2×D411yescharacteristic
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D411yescharacteristic
209C22⋊C4×C31—22no—
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—21no—
214C22⋊C4×C31—21no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C42×C31—22no—
221C42×C31—22no—
222C42×C31—21no—
223C42×C31—21no—
224C42×C31—21no—
225C42×C31—22no—
226C42×C31—21no—
227C42×C31—21no—
228C42×C31—21no—
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2411yescharacteristic
233#18538C3111yesmaximal, characteristic, direct product factor, Sylow
234C23×C31—21no—
235C23×C31—21no—
236C23×C31#12011yescharacteristic
237Q8×C31—21no—
238Q8×C31—21no—
239Q8×C31—21no—
240Q8×C31—44no—
241Q8×C31—44no—
242Q8×C31—41no—
243Q8×C31—41no—
244Q8×C31—41no—
245Q8×C31—41no—
246Q8×C31—82no—
247Q8×C31#13712yes—
248D4×C31—21no—
249D4×C31—21no—
250D4×C31—21no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—44no—
254D4×C31#13712yes—
255C2×C4×C31—21no—
256C2×C4×C31—21no—
257C2×C4×C31—21no—
258C2×C4×C31—21no—
259C2×C4×C31—21no—
260C2×C4×C31—22no—
261C2×C4×C31—21no—
262C2×C4×C31—21no—
263C2×C4×C31—21no—
264C2×C4×C31—22no—
265C2×C4×C31—21no—
266C2×C4×C31—21no—
267C2×C4×C31—22no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—44no—
271C2×C4×C31—44no—
272C2×C4×C31—41no—
273C2×C4×C31—41no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×C4×C31—42no—
279C2×C4×C31—82no—
280C2×C4×C31#12011yescharacteristic
281C2×C4×C31#12011yescharacteristic
282C2×C4×C31#12011yescharacteristic
283C2×C4×C31C22×D411yescharacteristic
284C2×C4×C31C22×D411yescharacteristic
285C2×C4×C31C22×D411yescharacteristic
286C8×C31—42no—
287C8×C31—42no—
288C8×C31—42no—
289#3482C2×C3111yescharacteristic
290#3139C2×C3111yescharacteristic, complemented
291#3079C2×C3111yescharacteristic, complemented
292#3079C2×C3111yescharacteristic, complemented
293#1638C2×C3112yescomplemented
294#1637C2×C3112yescomplemented
295#1636C2×C3112yescomplemented
296#1635C2×C3112yescomplemented
297#1561C2×C3111yescharacteristic, complemented
298#1489C2×C3111yescharacteristic, complemented
299#1489C2×C3111yescharacteristic, complemented
300C22×C31—21no—
301C22×C31—21no—
302C22×C31—22no—
303C22×C31—21no—
304C22×C31—41no—
305C22×C31—41no—
306C22×C31—41no—
307C22×C31—41no—
308C22×C31—41no—
309C22×C31C2×#12011yescharacteristic
310C4×C31—21no—
311C4×C31—21no—
312C4×C31—22no—
313C4×C31—21no—
314C4×C31—41no—
315C4×C31—41no—
316C4×C31—42no—
317C4×C31—42no—
318C4×C31—41no—
319C4×C31—41no—
320C4×C31—41no—
321C4×C31—82no—
322C4×C31#45412yes—
323C4×C31C2×#13711yescharacteristic
324C4×C31C2×#13711yescharacteristic
325C4×C31#57511yescharacteristic
326C4×C31#57511yescharacteristic
327#557C22×C3111yescharacteristic
328#556—21no—
329#556C22×C3111yescharacteristic
330#552—22no—
331#552—21no—
332D4×Q8C22×C3111yescharacteristic
333#548—21no—
334#547—21no—
335#547C22×C3111yescharacteristic
336#533—21no—
337#533C22×C3111yescharacteristic
338C2×C4⋊Q8C22×C3111yescharacteristic
339#529—21no—
340#529—22no—
341C4×#41—21no—
342C4×#41C22×C3111yescharacteristic
343C4×#41C22×C3111yescharacteristic
344C2×C4×Q8—21no—
345#491—22no—
346#487C22×C3112yes—
347#487C22×C3112yes—
348#477C22×C3112yes—
349#477C22×C3112yes—
350#476C22×C3112yes—
351#476C22×C3112yes—
352#475—21no—
353#475—21no—
354#474—21no—
355#474—21no—
356#428—21no—
357#428—21no—
358#423C22×C3111yescharacteristic
359#423C22×C3111yescharacteristic
360#419—22no—
361#417—21no—
362#417—21no—
363C4×M16C22×C3111yescharacteristic
364#332C22×C3112yes—
365#332C22×C3112yes—
366Q8⋊C8C22×C3112yes—
367Q8⋊C8C22×C3112yes—
368D4⋊C8C22×C3112yes—
369D4⋊C8C22×C3112yes—
370C2×C31—41no—
371C2×C31—42no—
372C2×C31—81no—
373C2×C31#300711yescharacteristic
374C2×C31#300711yescharacteristic
375C2×C31C2×#45411yescharacteristic
376C2×#41—21no—
377C2×#41D4×C3111yescharacteristic
378C2×#41D4×C3111yescharacteristic
379C22×Q8—21no—
380C22×Q8—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C4⋊Q8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8C23×C3112yes—
390C4⋊Q8C23×C3111yescharacteristic
391C4⋊Q8C23×C3111yescharacteristic
392#126—22no—
393#126—22no—
394#126—22no—
395#125—22no—
396#125—22no—
397#125—21no—
398#125D4×C3112yes—
399#124—22no—
400#124—21no—
401#123—22no—
402#123—22no—
403#123—22no—
404C22⋊Q8—22no—
405C22⋊Q8—21no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8—21no—
411#121—22no—
412#121—21no—
413C4×Q8—22no—
414C4×Q8—22no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8D4×C3111yescharacteristic
421C4×Q8D4×C3111yescharacteristic
422C4×Q8C23×C3111yescharacteristic
423C4×Q8C23×C3111yescharacteristic
424C4×D4—22no—
425C4×D4—22no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4D4×C3111yescharacteristic
432C4×D4D4×C3111yescharacteristic
433C4×D4C23×C3111yescharacteristic
434C4×D4C23×C3111yescharacteristic
435#117—22no—
436#117—22no—
437#117—21no—
438#117—21no—
439#117—21no—
440#117D4×C3111yescharacteristic
441#117D4×C3111yescharacteristic
442#117D4×C3112yes—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C42—22no—
448C2×C42—21no—
449C2×C42—21no—
450C2×C42C23×C3111yescharacteristic
451#107—22no—
452#107—22no—
453#106—22no—
454#106—22no—
455C4⋊C8—21no—
456C4⋊C8—21no—
457C4⋊C8C23×C3112yes—
458C4⋊C8C23×C3112yes—
459#103—22no—
460#103—22no—
461#103—24no—
462#103—24no—
463#102—22no—
464#102—22no—
465#97—21no—
466C4×C8C23×C3112yes—
467C31#1853811yescharacteristic, direct product factor, Sylow
468#41—22no—
469#41—22no—
470#41—41no—
471#41—41no—
472#41—42no—
473#41—42no—
474C2×Q8—22no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—22no—
478C2×Q8—44no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—41no—
482C2×Q8C2×D4×C3111yescharacteristic
483C2×Q8C2×D4×C3111yescharacteristic
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4C2×D4×C3111yescharacteristic
488C2×D4C2×D4×C3111yescharacteristic
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—41no—
498C22×C4C2×D4×C3111yescharacteristic
499C22×C4C2×D4×C3111yescharacteristic
500C22×C4C2×D4×C3111yescharacteristic
501M16—42no—
502M16—42no—
503M16—42no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4C2×D4×C3112yes—
526C4⋊C4C2×D4×C3112yes—
527C4⋊C4C2×D4×C3111yescharacteristic
528C4⋊C4C2×D4×C3111yescharacteristic
529C4⋊C4C2×D4×C3111yescharacteristic
530C4⋊C4C2×D4×C3111yescharacteristic
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—21no—
537C22⋊C4—22no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—22no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42C2×D4×C3111yescharacteristic
552C42C2×D4×C3111yescharacteristic
553C42C2×D4×C3111yescharacteristic
554C42C24×C3111yescharacteristic
555C23—21no—
556C23—21no—
557C23C31×#12011yescharacteristic
558Q8—21no—
559Q8—21no—
560Q8—21no—
561Q8—44no—
562Q8—44no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—82no—
568Q8C31×#13712yes—
569D4—21no—
570D4—21no—
571D4—21no—
572D4—42no—
573D4—42no—
574D4—44no—
575D4C31×#13712yes—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4C31×#12011yescharacteristic
602C2×C4C31×#12011yescharacteristic
603C2×C4C31×#12011yescharacteristic
604C2×C4C22×D4×C3111yescharacteristic
605C2×C4C22×D4×C3111yescharacteristic
606C2×C4C22×D4×C3111yescharacteristic
607C8—42no—
608C8—42no—
609C8—42no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—21no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22C2×C31×#12011yescharacteristic
620C4—21no—
621C4—21no—
622C4—22no—
623C4—21no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—82no—
632C4C31×#45412yes—
633C4C2×C31×#13711yescharacteristic
634C4C2×C31×#13711yescharacteristic
635C4C31×#57511yescharacteristic
636C4C31×#57511yescharacteristic
637C2—41no—
638C2—42no—
639C2—81no—
640C2C31×#300711yescharacteristic
641C2C31×#300711yescharacteristic
642C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3482C2
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×#3139C2semidirect
3. C31×#3079C2semidirect
4. C31×#3079C2semidirect
5. C31×#1638C2semidirect
6. C31×#1637C2semidirect
7. C31×#1636C2semidirect
8. C31×#1635C2semidirect
9. C31×#1561C2semidirect
10. C31×#1489C2semidirect
11. C31×#1489C2semidirect
233. #18538C31direct (N × H)
290. #3139C2×C31semidirect
291. #3079C2×C31semidirect
292. #3079C2×C31semidirect
293. #1638C2×C31semidirect
294. #1637C2×C31semidirect
295. #1636C2×C31semidirect
296. #1635C2×C31semidirect
297. #1561C2×C31semidirect
298. #1489C2×C31semidirect
299. #1489C2×C31semidirect
467. C31#18538direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34822
C31×#31392
C31×#30792
C31×#16382
C31×#16372
C31×#16362
C31×#16352
C31×#15612
C31×#14892
#1853831