Finite Group Databasebeta

← C31×#18277 #21475036

Subgroups — C31×#18277 #21475036

Order 7936.

934 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3559C211yesmaximal, characteristic
2C31×#3112C211yesmaximal, characteristic, complemented
3C31×#3108C211yesmaximal, characteristic, complemented
4C31×#2584C211yesmaximal, characteristic, complemented
5C31×#1610C212yesmaximal, complemented
6C31×#1609C212yesmaximal, complemented
7C31×#1604C212yesmaximal, complemented
8C31×#1603C212yesmaximal, complemented
9C31×#1509C211yesmaximal, characteristic, complemented
10C31×#1502C211yesmaximal, characteristic, complemented
11C31×#1470C211yesmaximal, characteristic, complemented
12C31×#563—21no—
13C31×#562—21no—
14C31×#554—21no—
15C31×#553—21no—
16C31×#553—21no—
17C31×#547—22no—
18C31×#547—21no—
19C31×#544—21no—
20C31×#540—21no—
21C31×#539—22no—
22C31×#539C2212yes—
23C31×#537C2212yes—
24C31×#532C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C2×C4×Q8×C31C2211yescharacteristic
28C31×#483C2211yescharacteristic
29C31×#480C2211yescharacteristic
30C31×#480C2212yes—
31C31×#479—21no—
32C31×#477—21no—
33C31×#464—21no—
34C31×#460—21no—
35C31×#451C2211yescharacteristic
36C31×#448C2211yescharacteristic
37C31×#448C2212yes—
38C31×#440—22no—
39C4×Q16×C31—21no—
40C4×SD16×C31—21no—
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#398C2212yes—
44C31×#395C2212yes—
45C31×#389—21no—
46C31×#389—22no—
47C31×#389—21no—
48C4×C4⋊C4×C31C2211yescharacteristic
49C31×#328C2212yes—
50C31×#327C2212yes—
51C31×#327C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31—22no—
57C22×Q8×C31C2311yescharacteristic
58C2×Q16×C31—21no—
59C2×SD16×C31—21no—
60C2×SD16×C31—22no—
61C31×C4⋊Q8—21no—
62C31×C4⋊Q8—22no—
63C31×C4⋊Q8D411yescharacteristic
64C31×#127D411yescharacteristic
65C31×#126—22no—
66C31×#126—22no—
67C31×#126—21no—
68C31×#125—21no—
69C31×#125—22no—
70C31×#125—21no—
71C31×#125—21no—
72C31×#125—21no—
73C31×#125—21no—
74C31×#125D411yescharacteristic
75C31×#124—21no—
76C31×#124D411yescharacteristic
77C31×#123—22no—
78C31×#123—22no—
79C31×#123—22no—
80C31×#123—22no—
81C31×#123—22no—
82C31×#123—21no—
83C31×#123—22no—
84C31×C22⋊Q8—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8C2311yescharacteristic
91C31×#121—21no—
92C31×#121—22no—
93C31×#121—21no—
94C31×#121—22no—
95C31×#121—22no—
96C31×#121—21no—
97C31×#121—22no—
98C31×#121C2311yescharacteristic
99C31×#121C2312yes—
100C4×Q8×C31—21no—
101C4×Q8×C31—21no—
102C4×Q8×C31—21no—
103C4×Q8×C31—21no—
104C4×Q8×C31D411yescharacteristic
105C4×D4×C31—22no—
106C4×D4×C31—21no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31—22no—
110C4×D4×C31—21no—
111C4×D4×C31D411yescharacteristic
112C4×D4×C31D412yes—
113C31×#117—21no—
114C31×#117—21no—
115C2×C4⋊C4×C31—22no—
116C2×C4⋊C4×C31—21no—
117C2×C4⋊C4×C31—21no—
118C2×C4⋊C4×C31—22no—
119C2×C4⋊C4×C31—21no—
120C2×C4⋊C4×C31—22no—
121C2×C4⋊C4×C31D411yescharacteristic
122C2×C4⋊C4×C31D412yes—
123C2×C4⋊C4×C31D411yescharacteristic
124C2×C4⋊C4×C31C2311yescharacteristic
125C2×C4⋊C4×C31C2312yes—
126C2×C42×C31—21no—
127C2×C42×C31—21no—
128C2×C42×C31C2311yescharacteristic
129C31×#107—21no—
130C31×#107—22no—
131C31×#106—21no—
132C31×C4⋊C8—21no—
133C31×C4⋊C8—21no—
134C31×#103—21no—
135C31×#103—21no—
136C31×#103—21no—
137C31×#103—22no—
138C31×#102—21no—
139C31×#102—22no—
140C31×C22⋊C8C2311yescharacteristic
141C31×C22⋊C8C2311yescharacteristic
142C31×C22⋊C8C2312yes—
143C31×#97—21no—
144C4×C8×C31—21no—
145C31×#95—22no—
146C31×#95—22no—
147C31×#95C2312yes—
148#41×C31—81no—
149#41×C31—81no—
150#41×C31—82no—
151#41×C31—82no—
152C2×Q8×C31—22no—
153C2×Q8×C31—21no—
154C2×Q8×C31—21no—
155C2×Q8×C31—21no—
156C2×Q8×C31—42no—
157C2×Q8×C31C2×D411yescharacteristic
158C2×D4×C31—21no—
159C2×D4×C31—22no—
160C2×D4×C31—22no—
161C2×D4×C31—22no—
162C2×D4×C31—21no—
163C2×D4×C31—21no—
164C2×D4×C31C2×D411yescharacteristic
165C2×D4×C31C2×D412yes—
166C22×C4×C31—22no—
167C22×C4×C31—21no—
168C22×C4×C31—22no—
169C22×C4×C31—22no—
170C22×C4×C31—22no—
171C22×C4×C31—21no—
172C22×C4×C31—22no—
173C22×C4×C31—21no—
174C22×C4×C31C2×D412yes—
175C22×C4×C31C2×D411yescharacteristic
176C22×C4×C31C2×D411yescharacteristic
177C22×C4×C31C2×D412yes—
178C22×C4×C31C2411yescharacteristic
179Q16×C31—42no—
180SD16×C31—44no—
181SD16×C31—42no—
182C2×C8×C31—21no—
183C2×C8×C31—21no—
184C2×C8×C31—22no—
185C4⋊C4×C31—24no—
186C4⋊C4×C31—22no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—21no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—22no—
193C4⋊C4×C31—21no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—22no—
196C4⋊C4×C31—21no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—22no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—21no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—44no—
206C4⋊C4×C31—41no—
207C4⋊C4×C31—41no—
208C4⋊C4×C31—82no—
209C4⋊C4×C31C2×D411yescharacteristic
210C4⋊C4×C31C2×D412yes—
211C4⋊C4×C31C2×D411yescharacteristic
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—22no—
216C22⋊C4×C31—22no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—21no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—21no—
223C22⋊C4×C31—22no—
224C42×C31—22no—
225C42×C31—21no—
226C42×C31—41no—
227C42×C31—41no—
228C42×C31—81no—
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232C42×C31C2×D411yescharacteristic
233#18277C3111yesmaximal, characteristic, direct product factor, Sylow
234C23×C31—21no—
235C23×C31—22no—
236C23×C31C22×D411yescharacteristic
237Q8×C31—22no—
238Q8×C31—22no—
239Q8×C31—41no—
240Q8×C31—41no—
241Q8×C31—41no—
242Q8×C31—81no—
243Q8×C31—82no—
244D4×C31—42no—
245D4×C31—41no—
246D4×C31—42no—
247D4×C31—41no—
248D4×C31—42no—
249D4×C31—41no—
250D4×C31—42no—
251D4×C31—81no—
252D4×C31—82no—
253D4×C31—82no—
254C2×C4×C31—21no—
255C2×C4×C31—21no—
256C2×C4×C31—22no—
257C2×C4×C31—24no—
258C2×C4×C31—22no—
259C2×C4×C31—21no—
260C2×C4×C31—21no—
261C2×C4×C31—22no—
262C2×C4×C31—22no—
263C2×C4×C31—21no—
264C2×C4×C31—22no—
265C2×C4×C31—22no—
266C2×C4×C31—22no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—41no—
270C2×C4×C31—41no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—41no—
275C2×C4×C31—41no—
276C2×C4×C31—41no—
277C2×C4×C31—82no—
278C2×C4×C31—82no—
279C2×C4×C31—82no—
280C2×C4×C31—82no—
281C2×C4×C31#12011yescharacteristic
282C2×C4×C31#12011yescharacteristic
283C2×C4×C31#12011yescharacteristic
284C2×C4×C31#12011yescharacteristic
285C2×C4×C31C22×D411yescharacteristic
286C2×C4×C31C22×D411yescharacteristic
287C8×C31—42no—
288C8×C31—41no—
289C8×C31—41no—
290#3559C2×C3111yescharacteristic
291#3112C2×C3111yescharacteristic, complemented
292#3108C2×C3111yescharacteristic, complemented
293#2584C2×C3111yescharacteristic, complemented
294#1610C2×C3112yescomplemented
295#1609C2×C3112yescomplemented
296#1604C2×C3112yescomplemented
297#1603C2×C3112yescomplemented
298#1509C2×C3111yescharacteristic, complemented
299#1502C2×C3111yescharacteristic, complemented
300#1470C2×C3111yescharacteristic, complemented
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—42no—
304C22×C31—41no—
305C22×C31—41no—
306C22×C31—42no—
307C22×C31—42no—
308C22×C31—41no—
309C22×C31C2×#12011yescharacteristic
310C22×C31#57611yescharacteristic
311C22×C31#57711yescharacteristic
312C4×C31—21no—
313C4×C31—21no—
314C4×C31—21no—
315C4×C31—21no—
316C4×C31—21no—
317C4×C31—44no—
318C4×C31—42no—
319C4×C31—42no—
320C4×C31—41no—
321C4×C31—41no—
322C4×C31—42no—
323C4×C31—42no—
324C4×C31—41no—
325C4×C31—81no—
326C4×C31#45612yes—
327#563—21no—
328#562—21no—
329#554—21no—
330#553—21no—
331#553—21no—
332#547—22no—
333#547—21no—
334#544—21no—
335#540—21no—
336#539—22no—
337#539C22×C3112yes—
338#537C22×C3112yes—
339#532C22×C3111yescharacteristic
340#529C22×C3111yescharacteristic
341C2×#125C22×C3111yescharacteristic
342C2×C4×Q8C22×C3111yescharacteristic
343#483C22×C3111yescharacteristic
344#480C22×C3111yescharacteristic
345#480C22×C3112yes—
346#479—21no—
347#477—21no—
348#464—21no—
349#460—21no—
350#451C22×C3111yescharacteristic
351#448C22×C3111yescharacteristic
352#448C22×C3112yes—
353#440—22no—
354C4×Q16—21no—
355C4×SD16—21no—
356#432C22×C3111yescharacteristic
357#431C22×C3111yescharacteristic
358#398C22×C3112yes—
359#395C22×C3112yes—
360#389—21no—
361#389—22no—
362#389—21no—
363C4×C4⋊C4C22×C3111yescharacteristic
364#328C22×C3112yes—
365#327C22×C3112yes—
366#327C22×C3112yes—
367#327C22×C3112yes—
368#324C22×C3112yes—
369#324C22×C3112yes—
370C2×C31—21no—
371C2×C31—21no—
372C2×C31—81no—
373C2×C31—82no—
374C2×C31#301211yescharacteristic
375C2×C31C2×#45611yescharacteristic
376C2×C31#302511yescharacteristic
377C2×#41—21no—
378C2×#41—22no—
379C22×Q8C23×C3111yescharacteristic
380C2×Q16—21no—
381C2×SD16—21no—
382C2×SD16—22no—
383C4⋊Q8—21no—
384C4⋊Q8—22no—
385C4⋊Q8D4×C3111yescharacteristic
386#127D4×C3111yescharacteristic
387#126—22no—
388#126—22no—
389#126—21no—
390#125—21no—
391#125—22no—
392#125—21no—
393#125—21no—
394#125—21no—
395#125—21no—
396#125D4×C3111yescharacteristic
397#124—21no—
398#124D4×C3111yescharacteristic
399#123—22no—
400#123—22no—
401#123—22no—
402#123—22no—
403#123—22no—
404#123—21no—
405#123—22no—
406C22⋊Q8—22no—
407C22⋊Q8—22no—
408C22⋊Q8—22no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8C23×C3111yescharacteristic
413#121—21no—
414#121—22no—
415#121—21no—
416#121—22no—
417#121—22no—
418#121—21no—
419#121—22no—
420#121C23×C3111yescharacteristic
421#121C23×C3112yes—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8D4×C3111yescharacteristic
427C4×D4—22no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—22no—
432C4×D4—21no—
433C4×D4D4×C3111yescharacteristic
434C4×D4D4×C3112yes—
435#117—21no—
436#117—21no—
437C2×C4⋊C4—22no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—22no—
442C2×C4⋊C4—22no—
443C2×C4⋊C4D4×C3111yescharacteristic
444C2×C4⋊C4D4×C3112yes—
445C2×C4⋊C4D4×C3111yescharacteristic
446C2×C4⋊C4C23×C3111yescharacteristic
447C2×C4⋊C4C23×C3112yes—
448C2×C42—21no—
449C2×C42—21no—
450C2×C42C23×C3111yescharacteristic
451#107—21no—
452#107—22no—
453#106—21no—
454C4⋊C8—21no—
455C4⋊C8—21no—
456#103—21no—
457#103—21no—
458#103—21no—
459#103—22no—
460#102—21no—
461#102—22no—
462C22⋊C8C23×C3111yescharacteristic
463C22⋊C8C23×C3111yescharacteristic
464C22⋊C8C23×C3112yes—
465#97—21no—
466C4×C8—21no—
467#95—22no—
468#95—22no—
469#95C23×C3112yes—
470C31#1827711yescharacteristic, direct product factor, Sylow
471#41—81no—
472#41—81no—
473#41—82no—
474#41—82no—
475C2×Q8—22no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—42no—
480C2×Q8C2×D4×C3111yescharacteristic
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4C2×D4×C3111yescharacteristic
488C2×D4C2×D4×C3112yes—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4C2×D4×C3112yes—
498C22×C4C2×D4×C3111yescharacteristic
499C22×C4C2×D4×C3111yescharacteristic
500C22×C4C2×D4×C3112yes—
501C22×C4C24×C3111yescharacteristic
502Q16—42no—
503SD16—44no—
504SD16—42no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—44no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—82no—
532C4⋊C4C2×D4×C3111yescharacteristic
533C4⋊C4C2×D4×C3112yes—
534C4⋊C4C2×D4×C3111yescharacteristic
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—22no—
547C42—22no—
548C42—21no—
549C42—41no—
550C42—41no—
551C42—81no—
552C42C2×D4×C3111yescharacteristic
553C42C2×D4×C3111yescharacteristic
554C42C2×D4×C3111yescharacteristic
555C42C2×D4×C3111yescharacteristic
556C23—21no—
557C23—22no—
558C23C22×D4×C3111yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—81no—
565Q8—82no—
566D4—42no—
567D4—41no—
568D4—42no—
569D4—41no—
570D4—42no—
571D4—41no—
572D4—42no—
573D4—81no—
574D4—82no—
575D4—82no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4C31×#12011yescharacteristic
604C2×C4C31×#12011yescharacteristic
605C2×C4C31×#12011yescharacteristic
606C2×C4C31×#12011yescharacteristic
607C2×C4C22×D4×C3111yescharacteristic
608C2×C4C22×D4×C3111yescharacteristic
609C8—42no—
610C8—41no—
611C8—41no—
612C22—21no—
613C22—21no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22C2×C31×#12011yescharacteristic
621C22C31×#57611yescharacteristic
622C22C31×#57711yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—81no—
637C4C31×#45612yes—
638C2—21no—
639C2—21no—
640C2—81no—
641C2—82no—
642C2C31×#301211yescharacteristic
643C2C2×C31×#45611yescharacteristic
644C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3559C2
LCSG ▷ C22×C4 ▷ 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×#3112C2semidirect
3. C31×#3108C2semidirect
4. C31×#2584C2semidirect
5. C31×#1610C2semidirect
6. C31×#1609C2semidirect
7. C31×#1604C2semidirect
8. C31×#1603C2semidirect
9. C31×#1509C2semidirect
10. C31×#1502C2semidirect
11. C31×#1470C2semidirect
233. #18277C31direct (N × H)
291. #3112C2×C31semidirect
292. #3108C2×C31semidirect
293. #2584C2×C31semidirect
294. #1610C2×C31semidirect
295. #1609C2×C31semidirect
296. #1604C2×C31semidirect
297. #1603C2×C31semidirect
298. #1509C2×C31semidirect
299. #1502C2×C31semidirect
300. #1470C2×C31semidirect
470. C31#18277direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35592
C31×#31122
C31×#31082
C31×#25842
C31×#16102
C31×#16092
C31×#16042
C31×#16032
C31×#15092
C31×#15022
C31×#14702
#1827731