Finite Group Databasebeta

← C31×#25604 #21482363

Subgroups — C31×#25604 #21482363

Order 7936.

1178 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#529C211yesmaximal, characteristic
2C2×C31×#497C211yesmaximal, characteristic, complemented
3C2×C31×#491C211yesmaximal, characteristic, complemented
4C31×#2857C211yesmaximal, characteristic, complemented
5C31×#2001C212yesmaximal, complemented
6C31×#2000C212yesmaximal
7C31×#1992C211yesmaximal, characteristic, complemented
8C31×#1959C211yesmaximal, characteristic, complemented
9C31×#1878C212yesmaximal, complemented
10C31×#1877C212yesmaximal, complemented
11C31×#1749C211yesmaximal, characteristic, complemented
12C22×Q16×C31C2212yes—
13C22×SD16×C31C2212yes—
14C2×C31×C4⋊Q8C2211yescharacteristic
15C31×#529—24no—
16C31×#529—24no—
17C2×C31×#126—21no—
18C2×C31×#125C2211yescharacteristic
19C2×C31×#124C2211yescharacteristic
20C2×C31×#123—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×C22⋊Q8C2212yes—
23C2×C31×#121C2211yescharacteristic
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×D4×C31—21no—
26C2×C4×D4×C31C2211yescharacteristic
27C2×C31×#117—21no—
28C22×C42×C31C2211yescharacteristic
29C31×#497—24no—
30C31×#491—24no—
31C2×C31×#103C2212yes—
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×C22⋊C8C2212yes—
36C2×C31×#97C2211yescharacteristic
37C2×C31×#97C2211yescharacteristic
38C31×#391C2212yes—
39C31×#389—21no—
40C31×#389—21no—
41C31×#384C2211yescharacteristic
42C31×#380—21no—
43C31×#380—21no—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C22×Q8×C31C2312yes—
48C22×Q8×C31C2311yescharacteristic
49C22×D4×C31—21no—
50C22×D4×C31C2311yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31D411yescharacteristic
53C23×C4×C31D411yescharacteristic
54C23×C4×C31C2311yescharacteristic
55C2×Q16×C31—24no—
56C2×Q16×C31—24no—
57C2×Q16×C31—42no—
58C2×SD16×C31—24no—
59C2×SD16×C31—24no—
60C2×SD16×C31—42no—
61C22×C8×C31C2312yes—
62C22×C8×C31C2312yes—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8—22no—
65C31×#126—28no—
66C31×#125—22no—
67C31×#125D414yes—
68C31×#124—22no—
69C31×#124D414yes—
70C31×#123—28no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—42no—
74C31×C22⋊Q8—42no—
75C31×#121—22no—
76C31×#121—22no—
77C4×Q8×C31—22no—
78C4×Q8×C31—22no—
79C4×D4×C31—28no—
80C4×D4×C31—22no—
81C4×D4×C31—22no—
82C31×#117—28no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31C2312yes—
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—22no—
101C2×C42×C31—21no—
102C2×C42×C31—22no—
103C2×C42×C31—22no—
104C2×C42×C31D411yescharacteristic
105C2×C42×C31D411yescharacteristic
106C2×C42×C31C2311yescharacteristic
107C2×C42×C31C2311yescharacteristic
108C31×#103—24no—
109C31×#103—24no—
110C31×#103—42no—
111C31×#102—42no—
112C31×C22⋊C8—24no—
113C31×#97—22no—
114C31×#97—22no—
115C31×#95—22no—
116C31×#95—22no—
117C24×C31—21no—
118C24×C31C2×D411yescharacteristic
119C2×Q8×C31—21no—
120C2×Q8×C31—22no—
121C2×Q8×C31—22no—
122C2×Q8×C31—22no—
123C2×Q8×C31—22no—
124C2×Q8×C31—22no—
125C2×Q8×C31—22no—
126C2×Q8×C31—22no—
127C2×Q8×C31—22no—
128C2×Q8×C31C2×D412yes—
129C2×D4×C31—24no—
130C2×D4×C31—21no—
131C2×D4×C31—22no—
132C2×D4×C31—22no—
133C2×D4×C31—42no—
134C2×D4×C31—42no—
135C2×D4×C31C2×D412yes—
136C22×C4×C31—24no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—22no—
150C22×C4×C31—41no—
151C22×C4×C31—41no—
152C22×C4×C31—42no—
153C22×C4×C31—42no—
154C22×C4×C31C2×D411yescharacteristic
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2411yescharacteristic
160Q16×C31—48no—
161SD16×C31—48no—
162C2×C8×C31—22no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31#4114yes—
168C4⋊C4×C31—24no—
169C4⋊C4×C31—24no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—42no—
177C4⋊C4×C31—42no—
178C4⋊C4×C31—44no—
179C4⋊C4×C31—42no—
180C4⋊C4×C31—42no—
181C4⋊C4×C31C2×D412yes—
182C4⋊C4×C31C2×D412yes—
183C22⋊C4×C31—24no—
184C22⋊C4×C31—24no—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—44no—
189C42×C31—24no—
190C42×C31—22no—
191C42×C31—22no—
192C42×C31—21no—
193C42×C31—21no—
194C42×C31C2×D412yes—
195C42×C31C2×D412yes—
196#25604C3111yesmaximal, characteristic, direct product factor, Sylow
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—21no—
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—21no—
203C23×C31—22no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31—41no—
208C23×C31—41no—
209C23×C31—41no—
210C23×C31#13712yes—
211C23×C31C22×D411yescharacteristic
212Q8×C31—24no—
213Q8×C31—42no—
214Q8×C31—42no—
215Q8×C31—42no—
216Q8×C31—42no—
217Q8×C31—42no—
218D4×C31—24no—
219D4×C31—48no—
220D4×C31—42no—
221C2×C4×C31—21no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—21no—
228C2×C4×C31—21no—
229C2×C4×C31—22no—
230C2×C4×C31—22no—
231C2×C4×C31—22no—
232C2×C4×C31—21no—
233C2×C4×C31—21no—
234C2×C4×C31—21no—
235C2×C4×C31—21no—
236C2×C4×C31—44no—
237C2×C4×C31—44no—
238C2×C4×C31—48no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31—41no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31#12012yes—
254C2×C4×C31#12012yes—
255C2×C4×C31#13712yes—
256C2×C4×C31C22×D411yescharacteristic
257C2×C4×C31C22×D411yescharacteristic
258C2×C4×C31C2×#4112yes—
259C2×C4×C31H(5,𝔽2)12yes—
260C8×C31—24no—
261C8×C31—42no—
262C2×#529C2×C3111yescharacteristic
263C2×#497C2×C3111yescharacteristic, complemented
264C2×#491C2×C3111yescharacteristic, complemented
265#2857C2×C3111yescharacteristic, complemented
266#2001C2×C3112yescomplemented
267#2000C2×C3112yes—
268#1992C2×C3111yescharacteristic, complemented
269#1959C2×C3111yescharacteristic, complemented
270#1878C2×C3112yescomplemented
271#1877C2×C3112yescomplemented
272#1749C2×C3111yescharacteristic, complemented
273C22×C31—22no—
274C22×C31—22no—
275C22×C31—22no—
276C22×C31—41no—
277C22×C31—41no—
278C22×C31—41no—
279C22×C31—41no—
280C22×C31—42no—
281C22×C31—42no—
282C22×C31—42no—
283C22×C31—81no—
284C22×C31—81no—
285C22×C31—81no—
286C22×C31—81no—
287C22×C31C2×#12011yescharacteristic
288C22×C31#53211yescharacteristic
289C22×C31#53511yescharacteristic
290C22×C31C2×#13711yescharacteristic
291C22×C31C2×#13711yescharacteristic
292C22×C31#57511yescharacteristic
293C22×C31#57511yescharacteristic
294C4×C31—22no—
295C4×C31—22no—
296C4×C31—22no—
297C4×C31—41no—
298C4×C31—42no—
299C4×C31—41no—
300C4×C31—42no—
301C4×C31—42no—
302C4×C31—82no—
303C4×C31—82no—
304C4×C31#54612yes—
305C4×C31#55012yes—
306C22×Q16C22×C3112yes—
307C22×SD16C22×C3112yes—
308C2×C4⋊Q8C22×C3111yescharacteristic
309#529—24no—
310#529—24no—
311C2×#126—21no—
312C2×#125C22×C3111yescharacteristic
313C2×#124C22×C3111yescharacteristic
314C2×#123—21no—
315C2×C22⋊Q8C22×C3111yescharacteristic
316C2×C22⋊Q8C22×C3112yes—
317C2×#121C22×C3111yescharacteristic
318C2×C4×Q8C22×C3111yescharacteristic
319C2×C4×D4—21no—
320C2×C4×D4C22×C3111yescharacteristic
321C2×#117—21no—
322C22×C42C22×C3111yescharacteristic
323#497—24no—
324#491—24no—
325C2×#103C22×C3112yes—
326C2×#103C22×C3112yes—
327C2×#103C22×C3112yes—
328C2×#102C22×C3112yes—
329C2×C22⋊C8C22×C3112yes—
330C2×#97C22×C3111yescharacteristic
331C2×#97C22×C3111yescharacteristic
332#391C22×C3112yes—
333#389—21no—
334#389—21no—
335#384C22×C3111yescharacteristic
336#380—21no—
337#380—21no—
338#340C22×C3112yes—
339#340C22×C3112yes—
340#336C22×C3112yes—
341C2×C31—42no—
342C2×C31—82no—
343C2×C31#243111yescharacteristic
344C2×C31#300711yescharacteristic
345C2×C31#300711yescharacteristic
346C2×C31#317211yescharacteristic
347C2×C31#318111yescharacteristic
348C2×C31#321811yescharacteristic
349C2×C31#321911yescharacteristic
350C22×Q8C23×C3112yes—
351C22×Q8C23×C3111yescharacteristic
352C22×D4—21no—
353C22×D4C23×C3111yescharacteristic
354C23×C4—21no—
355C23×C4D4×C3111yescharacteristic
356C23×C4D4×C3111yescharacteristic
357C23×C4C23×C3111yescharacteristic
358C2×Q16—24no—
359C2×Q16—24no—
360C2×Q16—42no—
361C2×SD16—24no—
362C2×SD16—24no—
363C2×SD16—42no—
364C22×C8C23×C3112yes—
365C22×C8C23×C3112yes—
366C4⋊Q8—22no—
367C4⋊Q8—22no—
368#126—28no—
369#125—22no—
370#125D4×C3114yes—
371#124—22no—
372#124D4×C3114yes—
373#123—28no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—42no—
377C22⋊Q8—42no—
378#121—22no—
379#121—22no—
380C4×Q8—22no—
381C4×Q8—22no—
382C4×D4—28no—
383C4×D4—22no—
384C4×D4—22no—
385#117—28no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4C23×C3112yes—
396C2×C4⋊C4C23×C3111yescharacteristic
397C2×C4⋊C4C23×C3111yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—22no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42D4×C3111yescharacteristic
408C2×C42D4×C3111yescharacteristic
409C2×C42C23×C3111yescharacteristic
410C2×C42C23×C3111yescharacteristic
411#103—24no—
412#103—24no—
413#103—42no—
414#102—42no—
415C22⋊C8—24no—
416#97—22no—
417#97—22no—
418#95—22no—
419#95—22no—
420C31#2560411yescharacteristic, direct product factor, Sylow
421C24—21no—
422C24C2×D4×C3111yescharacteristic
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×D4×C3112yes—
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4C2×D4×C3112yes—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4C2×D4×C3111yescharacteristic
459C22×C4C2×D4×C3111yescharacteristic
460C22×C4C2×D4×C3111yescharacteristic
461C22×C4C2×D4×C3111yescharacteristic
462C22×C4C2×D4×C3111yescharacteristic
463C22×C4C24×C3111yescharacteristic
464Q16—48no—
465SD16—48no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8#41×C3114yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4C2×D4×C3112yes—
486C4⋊C4C2×D4×C3112yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—44no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42C2×D4×C3112yes—
499C42C2×D4×C3112yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C31×#13712yes—
514C23C22×D4×C3111yescharacteristic
515Q8—24no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—42no—
521D4—24no—
522D4—48no—
523D4—42no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4C31×#12012yes—
557C2×C4C31×#12012yes—
558C2×C4C31×#13712yes—
559C2×C4C22×D4×C3111yescharacteristic
560C2×C4C22×D4×C3111yescharacteristic
561C2×C4C2×#41×C3112yes—
562C2×C4C31×H(5,𝔽2)12yes—
563C8—24no—
564C8—42no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C2×C31×#12011yescharacteristic
580C22C31×#53211yescharacteristic
581C22C31×#53511yescharacteristic
582C22C2×C31×#13711yescharacteristic
583C22C2×C31×#13711yescharacteristic
584C22C31×#57511yescharacteristic
585C22C31×#57511yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—82no—
595C4—82no—
596C4C31×#54612yes—
597C4C31×#55012yes—
598C2—42no—
599C2—82no—
600C2C31×#243111yescharacteristic
601C2C31×#300711yescharacteristic
602C2C31×#300711yescharacteristic
603C2C31×#317211yescharacteristic
604C2C31×#318111yescharacteristic
605C2C31×#321811yescharacteristic
606C2C31×#321911yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C22×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#497C2semidirect
3. C2×C31×#491C2semidirect
4. C31×#2857C2semidirect
5. C31×#2001C2semidirect
7. C31×#1992C2semidirect
8. C31×#1959C2semidirect
9. C31×#1878C2semidirect
10. C31×#1877C2semidirect
11. C31×#1749C2semidirect
196. #25604C31direct (N × H)
263. C2×#497C2×C31semidirect
264. C2×#491C2×C31semidirect
265. #2857C2×C31semidirect
266. #2001C2×C31semidirect
268. #1992C2×C31semidirect
269. #1959C2×C31semidirect
270. #1878C2×C31semidirect
271. #1877C2×C31semidirect
272. #1749C2×C31semidirect
420. C31#25604direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5292
C2×C31×#4972
C2×C31×#4912
C31×#28572
C31×#20012
C31×#20002
C31×#19922
C31×#19592
C31×#18782
C31×#18772
C31×#17492
#2560431