Finite Group Databasebeta

← C31×#26012 #21482771

Subgroups — C31×#26012 #21482771

Order 7936.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2441C211yesmaximal, characteristic
2C31×#2344C211yesmaximal, characteristic
3C31×#2099C211yesmaximal, characteristic, complemented
4C31×#2088C211yesmaximal, characteristic, complemented
5C31×#2087C211yesmaximal, characteristic, complemented
6C31×#2059C211yesmaximal, characteristic, complemented
7C31×#1989C211yesmaximal, characteristic, complemented
8C31×#1947C211yesmaximal, characteristic, complemented
9C31×#1920C211yesmaximal, characteristic, complemented
10C31×#1919C211yesmaximal, characteristic, complemented
11C31×#1854C211yesmaximal, characteristic
12C31×#1851C211yesmaximal, characteristic
13C31×#1824C211yesmaximal, characteristic
14C31×#1809C211yesmaximal, characteristic
15C31×#1736C211yesmaximal, characteristic
16C22×M16×C31C2211yescharacteristic
17C2×C31×#125C2211yescharacteristic
18C2×C31×#117—21no—
19C2×C31×#117C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic
21C2×C31×#107C2211yescharacteristic
22C2×C31×#107C2211yescharacteristic
23C2×C31×#106C2211yescharacteristic
24C2×C31×#106C2211yescharacteristic
25C2×C31×C4⋊C8C2211yescharacteristic
26C2×C31×C4⋊C8C2211yescharacteristic
27C2×C31×C22⋊C8C2211yescharacteristic
28C31×#399—21no—
29C31×#396—21no—
30C31×#389—21no—
31C31×#389C2211yescharacteristic
32C31×#389C2211yescharacteristic
33C31×#389C2211yescharacteristic
34C31×#389C2211yescharacteristic
35C31×#387—21no—
36C31×#387—21no—
37C31×#387—21no—
38C31×#384C2211yescharacteristic
39C31×#381—21no—
40C31×#380C2211yescharacteristic
41C31×#380C2211yescharacteristic
42C4×C4⋊C4×C31C2211yescharacteristic
43C4×C4⋊C4×C31C2211yescharacteristic
44C4×C22⋊C4×C31C2211yescharacteristic
45C31×#340C2211yescharacteristic
46C31×#340C2211yescharacteristic
47C31×#340C2211yescharacteristic
48C31×#340C2211yescharacteristic
49C31×#340C2211yescharacteristic
50C31×#340C2211yescharacteristic
51C31×#340C2211yescharacteristic
52C31×#340C2211yescharacteristic
53C31×#340C2211yescharacteristic
54C31×#340C2211yescharacteristic
55C31×#340C2211yescharacteristic
56C31×#340C2211yescharacteristic
57C31×#336C2211yescharacteristic
58C31×#336C2211yescharacteristic
59C23×C4×C31—21no—
60C23×C4×C31C2311yescharacteristic
61C2×M16×C31—22no—
62C2×M16×C31—22no—
63C2×M16×C31—41no—
64C22×C8×C31C2311yescharacteristic
65C22×C8×C31C2311yescharacteristic
66C22×C8×C31C2311yescharacteristic
67C22×C8×C31C2311yescharacteristic
68C31×#125—22no—
69C31×#125—41no—
70C31×#117—22no—
71C31×#117—44no—
72C31×#117—41no—
73C2×C4⋊C4×C31—21no—
74C2×C4⋊C4×C31—22no—
75C2×C4⋊C4×C31—21no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31—21no—
79C2×C4⋊C4×C31—21no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31Q814yes—
84C2×C4⋊C4×C31C2311yescharacteristic
85C2×C4⋊C4×C31C2311yescharacteristic
86C2×C4⋊C4×C31C2311yescharacteristic
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31D411yescharacteristic
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C42×C31—21no—
98C2×C42×C31—21no—
99C2×C42×C31—21no—
100C2×C42×C31C2311yescharacteristic
101C2×C42×C31C2311yescharacteristic
102C31×#107—41no—
103C31×#107—41no—
104C31×#106—22no—
105C31×#106—22no—
106C31×C4⋊C8—22no—
107C31×C4⋊C8—41no—
108C31×C22⋊C8—22no—
109C31×#95—21no—
110C31×#95—21no—
111C31×#95—21no—
112C31×#95—21no—
113C31×#95—21no—
114C31×#95—21no—
115C31×#95—21no—
116C31×#95—21no—
117C31×#95D411yescharacteristic
118C31×#95D411yescharacteristic
119C24×C31C2×D411yescharacteristic
120C22×C4×C31—21no—
121C22×C4×C31—21no—
122C22×C4×C31—21no—
123C22×C4×C31—21no—
124C22×C4×C31—24no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—42no—
134C22×C4×C31—42no—
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×Q812yes—
141C22×C4×C31#4112yes—
142C22×C4×C31C2411yescharacteristic
143M16×C31—44no—
144C2×C8×C31—21no—
145C2×C8×C31—21no—
146C2×C8×C31—21no—
147C2×C8×C31—21no—
148C2×C8×C31—21no—
149C2×C8×C31—21no—
150C2×C8×C31—21no—
151C2×C8×C31—21no—
152C2×C8×C31—21no—
153C2×C8×C31—21no—
154C2×C8×C31—21no—
155C2×C8×C31#4112yes—
156C4⋊C4×C31—24no—
157C4⋊C4×C31—21no—
158C4⋊C4×C31—21no—
159C4⋊C4×C31—21no—
160C4⋊C4×C31—21no—
161C4⋊C4×C31—21no—
162C4⋊C4×C31—21no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—21no—
166C4⋊C4×C31—21no—
167C4⋊C4×C31—21no—
168C4⋊C4×C31—21no—
169C4⋊C4×C31—21no—
170C4⋊C4×C31—42no—
171C4⋊C4×C31—42no—
172C4⋊C4×C31—42no—
173C4⋊C4×C31—42no—
174C4⋊C4×C31—42no—
175C4⋊C4×C31—42no—
176C4⋊C4×C31—81no—
177C4⋊C4×C31—81no—
178C4⋊C4×C31C2×Q812yes—
179C4⋊C4×C31C2×Q812yes—
180C4⋊C4×C31#4112yes—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—41no—
183C22⋊C4×C31—42no—
184C22⋊C4×C31—42no—
185C22⋊C4×C31—81no—
186C42×C31—21no—
187C42×C31—21no—
188C42×C31—21no—
189C42×C31—21no—
190C42×C31—42no—
191C42×C31—42no—
192C42×C31—42no—
193#26012C3111yesmaximal, characteristic, direct product factor, Sylow
194C23×C31—21no—
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—21no—
200C23×C31C22⋊Q812yes—
201C23×C31C22×D411yescharacteristic
202C2×C4×C31—21no—
203C2×C4×C31—21no—
204C2×C4×C31—21no—
205C2×C4×C31—21no—
206C2×C4×C31—22no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—22no—
215C2×C4×C31—21no—
216C2×C4×C31—21no—
217C2×C4×C31—21no—
218C2×C4×C31—21no—
219C2×C4×C31—21no—
220C2×C4×C31—21no—
221C2×C4×C31—21no—
222C2×C4×C31—48no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—41no—
226C2×C4×C31—41no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—41no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—41no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—41no—
238C2×C4×C31—41no—
239C2×C4×C31—41no—
240C2×C4×C31—41no—
241C2×C4×C31—41no—
242C2×C4×C31—41no—
243C2×C4×C31—41no—
244C2×C4×C31C22⋊Q812yes—
245C2×C4×C31AGL(1,ℤ/8)12yes—
246C2×C4×C31C22×Q811yescharacteristic
247C2×C4×C31C2×#4111yescharacteristic
248C2×C4×C31C2×#4111yescharacteristic
249C2×C4×C31C2×#4111yescharacteristic
250C2×C4×C31H(5,𝔽2)11yescharacteristic
251C2×C4×C312−1+411yescharacteristic
252C8×C31—22no—
253C8×C31—41no—
254C8×C31—41no—
255C8×C31—41no—
256#2441C2×C3111yescharacteristic
257#2344C2×C3111yescharacteristic
258#2099C2×C3111yescharacteristic, complemented
259#2088C2×C3111yescharacteristic, complemented
260#2087C2×C3111yescharacteristic, complemented
261#2059C2×C3111yescharacteristic, complemented
262#1989C2×C3111yescharacteristic, complemented
263#1947C2×C3111yescharacteristic, complemented
264#1920C2×C3111yescharacteristic, complemented
265#1919C2×C3111yescharacteristic, complemented
266#1854C2×C3111yescharacteristic
267#1851C2×C3111yescharacteristic
268#1824C2×C3111yescharacteristic
269#1809C2×C3111yescharacteristic
270#1736C2×C3111yescharacteristic
271C22×C31—22no—
272C22×C31—22no—
273C22×C31—22no—
274C22×C31—41no—
275C22×C31—41no—
276C22×C31—41no—
277C22×C31—41no—
278C22×C31C2×C22⋊Q811yescharacteristic
279C22×C31#54611yescharacteristic
280C22×C31#54711yescharacteristic
281C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C31#57511yescharacteristic
283C22×C31#57711yescharacteristic
284C22×C31#57811yescharacteristic
285C4×C31—22no—
286C4×C31—22no—
287C4×C31—41no—
288C4×C31—42no—
289C4×C31—42no—
290C4×C31—41no—
291C4×C31—41no—
292C4×C31—42no—
293C4×C31—81no—
294C4×C31—81no—
295C4×C31—81no—
296C4×C31—81no—
297C4×C31—81no—
298C4×C31#53311yescharacteristic
299C4×C31#54411yescharacteristic
300C4×C31#55111yescharacteristic
301C4×C31#55211yescharacteristic
302C22×M16C22×C3111yescharacteristic
303C2×#125C22×C3111yescharacteristic
304C2×#117—21no—
305C2×#117C22×C3111yescharacteristic
306C22×C4⋊C4C22×C3111yescharacteristic
307C2×#107C22×C3111yescharacteristic
308C2×#107C22×C3111yescharacteristic
309C2×#106C22×C3111yescharacteristic
310C2×#106C22×C3111yescharacteristic
311C2×C4⋊C8C22×C3111yescharacteristic
312C2×C4⋊C8C22×C3111yescharacteristic
313C2×C22⋊C8C22×C3111yescharacteristic
314#399—21no—
315#396—21no—
316#389—21no—
317#389C22×C3111yescharacteristic
318#389C22×C3111yescharacteristic
319#389C22×C3111yescharacteristic
320#389C22×C3111yescharacteristic
321#387—21no—
322#387—21no—
323#387—21no—
324#384C22×C3111yescharacteristic
325#381—21no—
326#380C22×C3111yescharacteristic
327#380C22×C3111yescharacteristic
328C4×C4⋊C4C22×C3111yescharacteristic
329C4×C4⋊C4C22×C3111yescharacteristic
330C4×C22⋊C4C22×C3111yescharacteristic
331#340C22×C3111yescharacteristic
332#340C22×C3111yescharacteristic
333#340C22×C3111yescharacteristic
334#340C22×C3111yescharacteristic
335#340C22×C3111yescharacteristic
336#340C22×C3111yescharacteristic
337#340C22×C3111yescharacteristic
338#340C22×C3111yescharacteristic
339#340C22×C3111yescharacteristic
340#340C22×C3111yescharacteristic
341#340C22×C3111yescharacteristic
342#340C22×C3111yescharacteristic
343#336C22×C3111yescharacteristic
344#336C22×C3111yescharacteristic
345C2×C31—42no—
346C2×C31#243811yescharacteristic
347C2×C31#307711yescharacteristic
348C2×C31#308211yescharacteristic
349C2×C31#331511yescharacteristic
350C2×C31#333211yescharacteristic
351C2×C31#334311yescharacteristic
352C2×C31#335111yescharacteristic
353C23×C4—21no—
354C23×C4C23×C3111yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—41no—
358C22×C8C23×C3111yescharacteristic
359C22×C8C23×C3111yescharacteristic
360C22×C8C23×C3111yescharacteristic
361C22×C8C23×C3111yescharacteristic
362#125—22no—
363#125—41no—
364#117—22no—
365#117—44no—
366#117—41no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4Q8×C3114yes—
378C2×C4⋊C4C23×C3111yescharacteristic
379C2×C4⋊C4C23×C3111yescharacteristic
380C2×C4⋊C4C23×C3111yescharacteristic
381C2×C4⋊C4C23×C3111yescharacteristic
382C2×C4⋊C4C23×C3111yescharacteristic
383C2×C4⋊C4C23×C3111yescharacteristic
384C2×C4⋊C4C23×C3111yescharacteristic
385C2×C4⋊C4C23×C3111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4D4×C3111yescharacteristic
390C2×C22⋊C4D4×C3111yescharacteristic
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42C23×C3111yescharacteristic
395C2×C42C23×C3111yescharacteristic
396#107—41no—
397#107—41no—
398#106—22no—
399#106—22no—
400C4⋊C8—22no—
401C4⋊C8—41no—
402C22⋊C8—22no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95D4×C3111yescharacteristic
412#95D4×C3111yescharacteristic
413C31#2601211yescharacteristic, direct product factor, Sylow
414C24C2×D4×C3111yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4C2×D4×C3111yescharacteristic
431C22×C4C2×D4×C3111yescharacteristic
432C22×C4C2×D4×C3111yescharacteristic
433C22×C4C2×D4×C3111yescharacteristic
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×Q8×C3112yes—
436C22×C4#41×C3112yes—
437C22×C4C24×C3111yescharacteristic
438M16—44no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8#41×C3112yes—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4C2×Q8×C3112yes—
474C4⋊C4C2×Q8×C3112yes—
475C4⋊C4#41×C3112yes—
476C22⋊C4—22no—
477C22⋊C4—41no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—81no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—42no—
487C42—42no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23C31×C22⋊Q812yes—
495C23C22×D4×C3111yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—48no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4C31×C22⋊Q812yes—
539C2×C4C31×AGL(1,ℤ/8)12yes—
540C2×C4C22×Q8×C3111yescharacteristic
541C2×C4C2×#41×C3111yescharacteristic
542C2×C4C2×#41×C3111yescharacteristic
543C2×C4C2×#41×C3111yescharacteristic
544C2×C4C31×H(5,𝔽2)11yescharacteristic
545C2×C4C31×2−1+411yescharacteristic
546C8—22no—
547C8—41no—
548C8—41no—
549C8—41no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C31×C22⋊Q811yescharacteristic
558C22C31×#54611yescharacteristic
559C22C31×#54711yescharacteristic
560C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
561C22C31×#57511yescharacteristic
562C22C31×#57711yescharacteristic
563C22C31×#57811yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4C31×#53311yescharacteristic
578C4C31×#54411yescharacteristic
579C4C31×#55111yescharacteristic
580C4C31×#55211yescharacteristic
581C2—42no—
582C2C31×#243811yescharacteristic
583C2C31×#307711yescharacteristic
584C2C31×#308211yescharacteristic
585C2C31×#331511yescharacteristic
586C2C31×#333211yescharacteristic
587C2C31×#334311yescharacteristic
588C2C31×#335111yescharacteristic
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C4×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2099C2semidirect
4. C31×#2088C2semidirect
5. C31×#2087C2semidirect
6. C31×#2059C2semidirect
7. C31×#1989C2semidirect
8. C31×#1947C2semidirect
9. C31×#1920C2semidirect
10. C31×#1919C2semidirect
193. #26012C31direct (N × H)
258. #2099C2×C31semidirect
259. #2088C2×C31semidirect
260. #2087C2×C31semidirect
261. #2059C2×C31semidirect
262. #1989C2×C31semidirect
263. #1947C2×C31semidirect
264. #1920C2×C31semidirect
265. #1919C2×C31semidirect
413. C31#26012direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24412
C31×#23442
C31×#20992
C31×#20882
C31×#20872
C31×#20592
C31×#19892
C31×#19472
C31×#19202
C31×#19192
C31×#18542
C31×#18512
C31×#18242
C31×#18092
C31×#17362
#2601231