Finite Group Databasebeta

← C31×#25988 #21482747

Subgroups — C31×#25988 #21482747

Order 7936.

802 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2440C211yesmaximal, characteristic
2C31×#2344C211yesmaximal, characteristic
3C31×#2099C211yesmaximal, characteristic, complemented
4C31×#2086C211yesmaximal, characteristic, complemented
5C31×#2086C211yesmaximal, characteristic, complemented
6C31×#2057C211yesmaximal, characteristic, complemented
7C31×#1988C211yesmaximal, characteristic, complemented
8C31×#1947C211yesmaximal, characteristic, complemented
9C31×#1918C211yesmaximal, characteristic, complemented
10C31×#1918C211yesmaximal, characteristic, complemented
11C31×#1852C211yesmaximal, characteristic
12C31×#1850C211yesmaximal, characteristic
13C31×#1823C211yesmaximal, characteristic
14C31×#1808C211yesmaximal, characteristic
15C2×C31×#340C211yesmaximal, characteristic
16C23×C8×C31C2211yescharacteristic
17C2×C31×C4⋊Q8C2211yescharacteristic
18C2×C31×C22⋊Q8—21no—
19C2×C31×#117—21no—
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C4⋊C4×C31C2211yescharacteristic
22C2×C31×#107C2211yescharacteristic
23C2×C31×#107C2211yescharacteristic
24C2×C31×#107C2211yescharacteristic
25C2×C31×#107C2211yescharacteristic
26C2×C31×C4⋊C8C2211yescharacteristic
27C2×C31×C4⋊C8C2211yescharacteristic
28C2×C31×C22⋊C8C2211yescharacteristic
29C31×#398—21no—
30C31×#395—21no—
31C31×#389C2211yescharacteristic
32C31×#389C2211yescharacteristic
33C31×#389C2211yescharacteristic
34C31×#389C2211yescharacteristic
35C31×#387—21no—
36C31×#387—21no—
37C31×#385—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
59C22×Q8×C31—21no—
60C23×C4×C31—21no—
61C23×C4×C31—21no—
62C23×C4×C31C2311yescharacteristic
63C22×C8×C31—22no—
64C22×C8×C31—22no—
65C22×C8×C31—41no—
66C22×C8×C31C2311yescharacteristic
67C22×C8×C31C2311yescharacteristic
68C22×C8×C31C2311yescharacteristic
69C22×C8×C31C2311yescharacteristic
70C31×C4⋊Q8—22no—
71C31×C4⋊Q8—41no—
72C31×C22⋊Q8—44no—
73C31×#117—44no—
74C2×C4⋊C4×C31—21no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—22no—
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×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—41no—
86C2×C4⋊C4×C31Q814yescomplemented
87C2×C4⋊C4×C31C2311yescharacteristic
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C4⋊C4×C31C2311yescharacteristic
94C2×C4⋊C4×C31C2311yescharacteristic
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C22⋊C4×C31D411yescharacteristic
99C2×C42×C31—21no—
100C2×C42×C31—21no—
101C2×C42×C31C2311yescharacteristic
102C2×C42×C31C2311yescharacteristic
103C31×#107—22no—
104C31×#107—22no—
105C31×#107—41no—
106C31×#107—41no—
107C31×C4⋊C8—22no—
108C31×C4⋊C8—41no—
109C31×C22⋊C8—22no—
110C31×#95—21no—
111C31×#95—21no—
112C31×#95—21no—
113C31×#95—21no—
114C31×#95—21no—
115C31×#95—21no—
116C31×#95—21no—
117C31×#95—21no—
118C31×#95D411yescharacteristic
119C31×#95D411yescharacteristic
120C24×C31C2×D411yescharacteristic
121C2×Q8×C31—42no—
122C2×Q8×C31—42no—
123C2×Q8×C31—81no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—24no—
129C22×C4×C31—21no—
130C22×C4×C31—21no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—42no—
138C22×C4×C31—42no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31—81no—
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D411yescharacteristic
147C22×C4×C31C2×Q812yes—
148C22×C4×C31#4112yes—
149C22×C4×C31C2411yescharacteristic
150C2×C8×C31—21no—
151C2×C8×C31—21no—
152C2×C8×C31—21no—
153C2×C8×C31—21no—
154C2×C8×C31—21no—
155C2×C8×C31—21no—
156C2×C8×C31—21no—
157C2×C8×C31—21no—
158C2×C8×C31—21no—
159C2×C8×C31—21no—
160C2×C8×C31—21no—
161C2×C8×C31—44no—
162C2×C8×C31#4112yes—
163C4⋊C4×C31—24no—
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—21no—
171C4⋊C4×C31—21no—
172C4⋊C4×C31—21no—
173C4⋊C4×C31—21no—
174C4⋊C4×C31—21no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—42no—
178C4⋊C4×C31—42no—
179C4⋊C4×C31—42no—
180C4⋊C4×C31—42no—
181C4⋊C4×C31—42no—
182C4⋊C4×C31—42no—
183C4⋊C4×C31—42no—
184C4⋊C4×C31—81no—
185C4⋊C4×C31C2×Q812yes—
186C4⋊C4×C31C2×Q812yes—
187C4⋊C4×C31#4112yes—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—42no—
190C22⋊C4×C31—42no—
191C22⋊C4×C31D814yes—
192C42×C31—21no—
193C42×C31—21no—
194C42×C31—21no—
195C42×C31—21no—
196C42×C31—42no—
197C42×C31—42no—
198#25988C3111yesmaximal, characteristic, direct product factor, Sylow
199C23×C31—21no—
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—21no—
203C23×C31C22⋊Q812yes—
204C23×C31C2×D812yes—
205C23×C31#13712yes—
206C23×C31C22×D411yescharacteristic
207Q8×C31—84no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—21no—
212C2×C4×C31—21no—
213C2×C4×C31—21no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—21no—
223C2×C4×C31—21no—
224C2×C4×C31—21no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
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—48no—
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×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—41no—
247C2×C4×C31—41no—
248C2×C4×C31—41no—
249C2×C4×C31—84no—
250C2×C4×C31C22⋊Q812yes—
251C2×C4×C31C2×D812yes—
252C2×C4×C31C2×D812yes—
253C2×C4×C31C22×Q811yescharacteristic
254C2×C4×C31C2×#4111yescharacteristic
255C2×C4×C31C2×#4111yescharacteristic
256C2×C4×C31C2×#4111yescharacteristic
257C2×C4×C31H(5,𝔽2)11yescharacteristic
258C2×C4×C312−1+411yescharacteristic
259C8×C31—22no—
260C8×C31—41no—
261C8×C31—41no—
262C8×C31—41no—
263#2440C2×C3111yescharacteristic
264#2344C2×C3111yescharacteristic
265#2099C2×C3111yescharacteristic, complemented
266#2086C2×C3111yescharacteristic, complemented
267#2086C2×C3111yescharacteristic, complemented
268#2057C2×C3111yescharacteristic, complemented
269#1988C2×C3111yescharacteristic, complemented
270#1947C2×C3111yescharacteristic, complemented
271#1918C2×C3111yescharacteristic, complemented
272#1918C2×C3111yescharacteristic, complemented
273#1852C2×C3111yescharacteristic
274#1850C2×C3111yescharacteristic
275#1823C2×C3111yescharacteristic
276#1808C2×C3111yescharacteristic
277C2×#340C2×C3111yescharacteristic
278C22×C31—22no—
279C22×C31—22no—
280C22×C31—22no—
281C22×C31—22no—
282C22×C31—22no—
283C22×C31—22no—
284C22×C31#47414yes—
285C22×C31C2×C22⋊Q811yescharacteristic
286C22×C31#54611yescharacteristic
287C22×C31#54711yescharacteristic
288C22×C31C22×D811yescharacteristic
289C22×C31C2×#13711yescharacteristic
290C22×C31#57711yescharacteristic
291C22×C31#57811yescharacteristic
292C4×C31—22no—
293C4×C31—22no—
294C4×C31—22no—
295C4×C31—41no—
296C4×C31—42no—
297C4×C31—41no—
298C4×C31—42no—
299C4×C31—42no—
300C4×C31—81no—
301C4×C31—81no—
302C4×C31—81no—
303C4×C31—81no—
304C4×C31—81no—
305C4×C31#53311yescharacteristic
306C4×C31#54411yescharacteristic
307C4×C31#55111yescharacteristic
308C4×C31#55211yescharacteristic
309C23×C8C22×C3111yescharacteristic
310C2×C4⋊Q8C22×C3111yescharacteristic
311C2×C22⋊Q8—21no—
312C2×#117—21no—
313C22×C4⋊C4C22×C3111yescharacteristic
314C22×C4⋊C4C22×C3111yescharacteristic
315C2×#107C22×C3111yescharacteristic
316C2×#107C22×C3111yescharacteristic
317C2×#107C22×C3111yescharacteristic
318C2×#107C22×C3111yescharacteristic
319C2×C4⋊C8C22×C3111yescharacteristic
320C2×C4⋊C8C22×C3111yescharacteristic
321C2×C22⋊C8C22×C3111yescharacteristic
322#398—21no—
323#395—21no—
324#389C22×C3111yescharacteristic
325#389C22×C3111yescharacteristic
326#389C22×C3111yescharacteristic
327#389C22×C3111yescharacteristic
328#387—21no—
329#387—21no—
330#385—21no—
331#384C22×C3111yescharacteristic
332#381—21no—
333#380C22×C3111yescharacteristic
334#380C22×C3111yescharacteristic
335C4×C4⋊C4C22×C3111yescharacteristic
336C4×C4⋊C4C22×C3111yescharacteristic
337C4×C22⋊C4C22×C3111yescharacteristic
338#340C22×C3111yescharacteristic
339#340C22×C3111yescharacteristic
340#340C22×C3111yescharacteristic
341#340C22×C3111yescharacteristic
342#340C22×C3111yescharacteristic
343#340C22×C3111yescharacteristic
344#340C22×C3111yescharacteristic
345#340C22×C3111yescharacteristic
346#340C22×C3111yescharacteristic
347#340C22×C3111yescharacteristic
348#340C22×C3111yescharacteristic
349#340C22×C3111yescharacteristic
350#336C22×C3111yescharacteristic
351#336C22×C3111yescharacteristic
352C2×C31—24no—
353C2×C31#243811yescharacteristic
354C2×C31C2×#47411yescharacteristic
355C2×C31#308211yescharacteristic
356C2×C31#329411yescharacteristic
357C2×C31#331211yescharacteristic
358C2×C31#333411yescharacteristic
359C2×C31#334411yescharacteristic
360C22×Q8—21no—
361C23×C4—21no—
362C23×C4—21no—
363C23×C4C23×C3111yescharacteristic
364C22×C8—22no—
365C22×C8—22no—
366C22×C8—41no—
367C22×C8C23×C3111yescharacteristic
368C22×C8C23×C3111yescharacteristic
369C22×C8C23×C3111yescharacteristic
370C22×C8C23×C3111yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—41no—
373C22⋊Q8—44no—
374#117—44no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—41no—
387C2×C4⋊C4Q8×C3114yescomplemented
388C2×C4⋊C4C23×C3111yescharacteristic
389C2×C4⋊C4C23×C3111yescharacteristic
390C2×C4⋊C4C23×C3111yescharacteristic
391C2×C4⋊C4C23×C3111yescharacteristic
392C2×C4⋊C4C23×C3111yescharacteristic
393C2×C4⋊C4C23×C3111yescharacteristic
394C2×C4⋊C4C23×C3111yescharacteristic
395C2×C4⋊C4C23×C3111yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4D4×C3111yescharacteristic
399C2×C22⋊C4D4×C3111yescharacteristic
400C2×C42—21no—
401C2×C42—21no—
402C2×C42C23×C3111yescharacteristic
403C2×C42C23×C3111yescharacteristic
404#107—22no—
405#107—22no—
406#107—41no—
407#107—41no—
408C4⋊C8—22no—
409C4⋊C8—41no—
410C22⋊C8—22no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95D4×C3111yescharacteristic
420#95D4×C3111yescharacteristic
421C31#2598811yescharacteristic, direct product factor, Sylow
422C24C2×D4×C3111yescharacteristic
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—81no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4C2×D4×C3111yescharacteristic
445C22×C4C2×D4×C3111yescharacteristic
446C22×C4C2×D4×C3111yescharacteristic
447C22×C4C2×D4×C3111yescharacteristic
448C22×C4C2×D4×C3111yescharacteristic
449C22×C4C2×Q8×C3112yes—
450C22×C4#41×C3112yes—
451C22×C4C24×C3111yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—44no—
464C2×C8#41×C3112yes—
465C4⋊C4—24no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—81no—
487C4⋊C4C2×Q8×C3112yes—
488C4⋊C4C2×Q8×C3112yes—
489C4⋊C4#41×C3112yes—
490C22⋊C4—22no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4D8×C3114yes—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—42no—
499C42—42no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C31×C22⋊Q812yes—
505C23C2×D8×C3112yes—
506C23C31×#13712yes—
507C23C22×D4×C3111yescharacteristic
508Q8—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—48no—
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×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4C31×C22⋊Q812yes—
552C2×C4C2×D8×C3112yes—
553C2×C4C2×D8×C3112yes—
554C2×C4C22×Q8×C3111yescharacteristic
555C2×C4C2×#41×C3111yescharacteristic
556C2×C4C2×#41×C3111yescharacteristic
557C2×C4C2×#41×C3111yescharacteristic
558C2×C4C31×H(5,𝔽2)11yescharacteristic
559C2×C4C31×2−1+411yescharacteristic
560C8—22no—
561C8—41no—
562C8—41no—
563C8—41no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22C31×#47414yes—
571C22C2×C31×C22⋊Q811yescharacteristic
572C22C31×#54611yescharacteristic
573C22C31×#54711yescharacteristic
574C22C22×D8×C3111yescharacteristic
575C22C2×C31×#13711yescharacteristic
576C22C31×#57711yescharacteristic
577C22C31×#57811yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4C31×#53311yescharacteristic
592C4C31×#54411yescharacteristic
593C4C31×#55111yescharacteristic
594C4C31×#55211yescharacteristic
595C2—24no—
596C2C31×#243811yescharacteristic
597C2C2×C31×#47411yescharacteristic
598C2C31×#308211yescharacteristic
599C2C31×#329411yescharacteristic
600C2C31×#331211yescharacteristic
601C2C31×#333411yescharacteristic
602C2C31×#334411yescharacteristic
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×#2086C2semidirect
5. C31×#2086C2semidirect
6. C31×#2057C2semidirect
7. C31×#1988C2semidirect
8. C31×#1947C2semidirect
9. C31×#1918C2semidirect
10. C31×#1918C2semidirect
86. C2×C4⋊C4×C31Q8semidirect
198. #25988C31direct (N × H)
265. #2099C2×C31semidirect
266. #2086C2×C31semidirect
267. #2086C2×C31semidirect
268. #2057C2×C31semidirect
269. #1988C2×C31semidirect
270. #1947C2×C31semidirect
271. #1918C2×C31semidirect
272. #1918C2×C31semidirect
387. C2×C4⋊C4Q8×C31semidirect
421. C31#25988direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#24402
C31×#23442
C31×#20992
C31×#20862
C31×#20572
C31×#19882
C31×#19472
C31×#19182
C31×#18522
C31×#18502
C31×#18232
C31×#18082
C2×C31×#3402
#2598831