Finite Group Databasebeta

← C31×#8352 #21465111

Subgroups — C31×#8352 #21465111

Order 7936.

860 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3505C211yesmaximal, characteristic
2C31×#1893C211yesmaximal, characteristic, complemented
3C31×#1892C211yesmaximal, characteristic, complemented
4C31×#1869C211yesmaximal, characteristic, complemented
5C31×#1799C211yesmaximal, characteristic, complemented
6C31×#1520C211yesmaximal, characteristic, complemented
7C31×#1519C211yesmaximal, characteristic, complemented
8C23×C8×C31C2211yescharacteristic
9C31×#553—22no—
10C31×#553—21no—
11C31×#552—21no—
12C31×#551—22no—
13C31×#548—21no—
14C31×#546—21no—
15C31×#539C412yescomplemented
16C31×#538C412yescomplemented
17C31×#534—21no—
18C31×#529—21no—
19C31×#525—21no—
20C2×C31×C22⋊Q8C2211yescharacteristic
21C2×C31×#121C2211yescharacteristic
22C31×#518—21no—
23C2×C4×D4×C31C2211yescharacteristic
24C2×C31×#103—21no—
25C2×C31×#102—21no—
26C2×C31×#99—21no—
27C2×C31×C22⋊C8—21no—
28C2×C31×C22⋊C8C2211yescharacteristic
29C31×#380C2211yescharacteristic
30C31×#380C2211yescharacteristic
31C31×#342—21no—
32C31×#340—21no—
33C31×#340—21no—
34C31×#336—21no—
35C31×#328—22no—
36C31×#327—22no—
37C2×#41×C31—21no—
38C22×Q8×C31D411yescharacteristic
39C22×D4×C31—21no—
40C22×D4×C31D411yescharacteristic
41C22×D4×C31D411yescharacteristic
42C23×C4×C31D411yescharacteristic
43C23×C4×C31C2311yescharacteristic
44C22×C8×C31—21no—
45C22×C8×C31—21no—
46C22×C8×C31—21no—
47C22×C8×C31—21no—
48C22×C8×C31—42no—
49C31×#126—22no—
50C31×#126—22no—
51C31×#126—22no—
52C31×#125—21no—
53C31×#125—22no—
54C31×#124—21no—
55C31×#124—22no—
56C31×#123—22no—
57C31×#123—22no—
58C31×#123—22no—
59C31×#123—22no—
60C31×#123—22no—
61C31×C22⋊Q8—21no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8—21no—
65C31×C22⋊Q8C2×C412yes—
66C31×C22⋊Q8D412yes—
67C31×#121—21no—
68C31×#121—22no—
69C31×#121—22no—
70C31×#121—22no—
71C31×#121—21no—
72C31×#121—21no—
73C31×#121C2×C412yes—
74C31×#121D412yes—
75C31×#120—22no—
76C31×#120—22no—
77C4×Q8×C31—21no—
78C4×D4×C31—22no—
79C4×D4×C31—22no—
80C4×D4×C31—22no—
81C4×D4×C31—21no—
82C4×D4×C31—22no—
83C4×D4×C31—22no—
84C4×D4×C31—21no—
85C4×D4×C31—21no—
86C4×D4×C31—21no—
87C4×D4×C31—21no—
88C31×#117—22no—
89C31×#117—21no—
90C31×#117—22no—
91C2×C4⋊C4×C31—21no—
92C2×C4⋊C4×C31—21no—
93C2×C4⋊C4×C31—21no—
94C2×C4⋊C4×C31C2×C411yescharacteristic
95C2×C4⋊C4×C31D411yescharacteristic
96C2×C4⋊C4×C31D411yescharacteristic
97C2×C22⋊C4×C31—22no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C22⋊C4×C31C2×C411yescharacteristic
103C2×C22⋊C4×C31D411yescharacteristic
104C2×C42×C31D411yescharacteristic
105C31×#103—42no—
106C31×#102—42no—
107C31×#99—42no—
108C31×C22⋊C8—21no—
109C31×C22⋊C8—21no—
110C31×C22⋊C8—42no—
111C31×#95—21no—
112C31×#95—21no—
113C24×C31—21no—
114C24×C31C22⋊C411yescharacteristic
115C24×C31C2×D411yescharacteristic
116#41×C31—81no—
117#41×C31—81no—
118C2×Q8×C31—21no—
119C2×Q8×C31—41no—
120C2×Q8×C31—41no—
121C2×Q8×C31C22⋊C412yes—
122C2×D4×C31—22no—
123C2×D4×C31—21no—
124C2×D4×C31—21no—
125C2×D4×C31—22no—
126C2×D4×C31—21no—
127C2×D4×C31—21no—
128C2×D4×C31—21no—
129C2×D4×C31—21no—
130C2×D4×C31—22no—
131C2×D4×C31—22no—
132C2×D4×C31—21no—
133C2×D4×C31—41no—
134C2×D4×C31—41no—
135C2×D4×C31—41no—
136C2×D4×C31—41no—
137C2×D4×C31—41no—
138C2×D4×C31—41no—
139C2×D4×C31C22⋊C412yes—
140C22×C4×C31—21no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—41no—
148C22×C4×C31—41no—
149C22×C4×C31—41no—
150C22×C4×C31—41no—
151C22×C4×C31—41no—
152C22×C4×C31—41no—
153C22×C4×C31C22⋊C411yescharacteristic
154C22×C4×C31C22⋊C411yescharacteristic
155C22×C4×C31C22⋊C411yescharacteristic
156C22×C4×C31C22×C411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D411yescharacteristic
162C2×C8×C31—22no—
163C2×C8×C31—22no—
164C2×C8×C31—22no—
165C2×C8×C31—41no—
166C2×C8×C31—41no—
167C2×C8×C31—41no—
168C2×C8×C31—41no—
169C2×C8×C31—41no—
170C2×C8×C31—41no—
171C2×C8×C31—41no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—22no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—21no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31—21no—
182C4⋊C4×C31—41no—
183C4⋊C4×C31—41no—
184C4⋊C4×C31—41no—
185C4⋊C4×C31—41no—
186C4⋊C4×C31C22⋊C412yes—
187C4⋊C4×C31C22⋊C412yes—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C22⋊C4×C31—22no—
194C22⋊C4×C31—22no—
195C22⋊C4×C31—21no—
196C22⋊C4×C31—22no—
197C22⋊C4×C31—21no—
198C22⋊C4×C31—22no—
199C22⋊C4×C31—21no—
200C22⋊C4×C31—21no—
201C22⋊C4×C31—22no—
202C22⋊C4×C31—41no—
203C22⋊C4×C31—41no—
204C22⋊C4×C31—41no—
205C22⋊C4×C31—41no—
206C42×C31—22no—
207C42×C31—22no—
208C42×C31—21no—
209C42×C31—21no—
210#8352C3111yesmaximal, characteristic, direct product factor, Sylow
211C23×C31—21no—
212C23×C31—22no—
213C23×C31—21no—
214C23×C31—21no—
215C23×C31—41no—
216C23×C31—41no—
217C23×C31—41no—
218C23×C31—41no—
219C23×C31—41no—
220C23×C31—41no—
221C23×C31—81no—
222C23×C31—81no—
223C23×C31C2×C22⋊C411yescharacteristic
224C23×C31#12011yescharacteristic
225C23×C31#12011yescharacteristic
226C23×C31#13511yescharacteristic
227C23×C31#13511yescharacteristic
228C23×C31AGL(1,ℤ/8)11yescharacteristic
229C23×C31#13711yescharacteristic
230Q8×C31—42no—
231Q8×C31—41no—
232Q8×C31—41no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—42no—
236D4×C31—41no—
237D4×C31—41no—
238D4×C31—42no—
239D4×C31—41no—
240D4×C31—41no—
241D4×C31—42no—
242D4×C31—41no—
243D4×C31—41no—
244D4×C31—82no—
245D4×C31—82no—
246C2×C4×C31—21no—
247C2×C4×C31—21no—
248C2×C4×C31—21no—
249C2×C4×C31—21no—
250C2×C4×C31—21no—
251C2×C4×C31—21no—
252C2×C4×C31—21no—
253C2×C4×C31—21no—
254C2×C4×C31—22no—
255C2×C4×C31—21no—
256C2×C4×C31—22no—
257C2×C4×C31—22no—
258C2×C4×C31—21no—
259C2×C4×C31—21no—
260C2×C4×C31—41no—
261C2×C4×C31—41no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—41no—
265C2×C4×C31—41no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—41no—
269C2×C4×C31—81no—
270C2×C4×C31—81no—
271C2×C4×C31—81no—
272C2×C4×C31—82no—
273C2×C4×C31—82no—
274C2×C4×C31—81no—
275C2×C4×C31—81no—
276C2×C4×C31—81no—
277C2×C4×C31—81no—
278C2×C4×C31—81no—
279C2×C4×C31—81no—
280C2×C4×C31C2×C22⋊C411yescharacteristic
281C2×C4×C31C2×C22⋊C411yescharacteristic
282C2×C4×C31#12011yescharacteristic
283C2×C4×C31#12011yescharacteristic
284C8×C31—42no—
285C8×C31—81no—
286#3505C2×C3111yescharacteristic
287#1893C2×C3111yescharacteristic, complemented
288#1892C2×C3111yescharacteristic, complemented
289#1869C2×C3111yescharacteristic, complemented
290#1799C2×C3111yescharacteristic, complemented
291#1520C2×C3111yescharacteristic, complemented
292#1519C2×C3111yescharacteristic, complemented
293C22×C31—21no—
294C22×C31—21no—
295C22×C31—21no—
296C22×C31—21no—
297C22×C31—21no—
298C22×C31—21no—
299C22×C31—41no—
300C22×C31—41no—
301C22×C31—41no—
302C22×C31—41no—
303C22×C31—42no—
304C22×C31—42no—
305C22×C31—41no—
306C22×C31—41no—
307C22×C31—42no—
308C22×C31—41no—
309C22×C31—42no—
310C22×C31—81no—
311C22×C31—161no—
312C22×C31—161no—
313C22×C31#37911yescharacteristic
314C22×C31#41611yescharacteristic
315C22×C31#41711yescharacteristic
316C22×C31#44911yescharacteristic
317C22×C31#44911yescharacteristic
318C22×C31#45211yescharacteristic
319C22×C31#45211yescharacteristic
320C4×C31—21no—
321C4×C31—21no—
322C4×C31—21no—
323C4×C31—41no—
324C4×C31—42no—
325C4×C31—41no—
326C4×C31—42no—
327C4×C31—41no—
328C4×C31—42no—
329C4×C31—161no—
330C4×C31—161no—
331C4×C31B(4,𝔽2)11yescharacteristic
332C4×C31#45811yescharacteristic
333C23×C8C22×C3111yescharacteristic
334#553—22no—
335#553—21no—
336#552—21no—
337#551—22no—
338#548—21no—
339#546—21no—
340#539C4×C3112yescomplemented
341#538C4×C3112yescomplemented
342#534—21no—
343#529—21no—
344#525—21no—
345C2×C22⋊Q8C22×C3111yescharacteristic
346C2×#121C22×C3111yescharacteristic
347#518—21no—
348C2×C4×D4C22×C3111yescharacteristic
349C2×#103—21no—
350C2×#102—21no—
351C2×#99—21no—
352C2×C22⋊C8—21no—
353C2×C22⋊C8C22×C3111yescharacteristic
354#380C22×C3111yescharacteristic
355#380C22×C3111yescharacteristic
356#342—21no—
357#340—21no—
358#340—21no—
359#336—21no—
360#328—22no—
361#327—22no—
362C2×C31—21no—
363C2×C31—21no—
364C2×C31—21no—
365C2×C31—21no—
366C2×C31—21no—
367C2×C31—21no—
368C2×C31—81no—
369C2×C31—82no—
370C2×C31#178811yescharacteristic
371C2×C31#179011yescharacteristic
372C2×C31#179411yescharacteristic
373C2×#41—21no—
374C22×Q8D4×C3111yescharacteristic
375C22×D4—21no—
376C22×D4D4×C3111yescharacteristic
377C22×D4D4×C3111yescharacteristic
378C23×C4D4×C3111yescharacteristic
379C23×C4C23×C3111yescharacteristic
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—21no—
383C22×C8—21no—
384C22×C8—42no—
385#126—22no—
386#126—22no—
387#126—22no—
388#125—21no—
389#125—22no—
390#124—21no—
391#124—22no—
392#123—22no—
393#123—22no—
394#123—22no—
395#123—22no—
396#123—22no—
397C22⋊Q8—21no—
398C22⋊Q8—22no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8C2×C4×C3112yes—
402C22⋊Q8D4×C3112yes—
403#121—21no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121—21no—
408#121—21no—
409#121C2×C4×C3112yes—
410#121D4×C3112yes—
411#120—22no—
412#120—22no—
413C4×Q8—21no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—21no—
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—21no—
424#117—22no—
425#117—21no—
426#117—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4C2×C4×C3111yescharacteristic
431C2×C4⋊C4D4×C3111yescharacteristic
432C2×C4⋊C4D4×C3111yescharacteristic
433C2×C22⋊C4—22no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4C2×C4×C3111yescharacteristic
439C2×C22⋊C4D4×C3111yescharacteristic
440C2×C42D4×C3111yescharacteristic
441#103—42no—
442#102—42no—
443#99—42no—
444C22⋊C8—21no—
445C22⋊C8—21no—
446C22⋊C8—42no—
447#95—21no—
448#95—21no—
449C31#835211yescharacteristic, direct product factor, Sylow
450C24—21no—
451C24C22⋊C4×C3111yescharacteristic
452C24C2×D4×C3111yescharacteristic
453#41—81no—
454#41—81no—
455C2×Q8—21no—
456C2×Q8—41no—
457C2×Q8—41no—
458C2×Q8C22⋊C4×C3112yes—
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4C22⋊C4×C3112yes—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4C22⋊C4×C3111yescharacteristic
491C22×C4C22⋊C4×C3111yescharacteristic
492C22×C4C22⋊C4×C3111yescharacteristic
493C22×C4C22×C4×C3111yescharacteristic
494C22×C4C2×D4×C3111yescharacteristic
495C22×C4C2×D4×C3111yescharacteristic
496C22×C4C2×D4×C3111yescharacteristic
497C22×C4C2×D4×C3111yescharacteristic
498C22×C4C2×D4×C3111yescharacteristic
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4C22⋊C4×C3112yes—
524C4⋊C4C22⋊C4×C3112yes—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—21no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C42—22no—
544C42—22no—
545C42—21no—
546C42—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—81no—
558C23—81no—
559C23C2×C22⋊C4×C3111yescharacteristic
560C23C31×#12011yescharacteristic
561C23C31×#12011yescharacteristic
562C23C31×#13511yescharacteristic
563C23C31×#13511yescharacteristic
564C23C31×AGL(1,ℤ/8)11yescharacteristic
565C23C31×#13711yescharacteristic
566Q8—42no—
567Q8—41no—
568Q8—41no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—41no—
573D4—41no—
574D4—42no—
575D4—41no—
576D4—41no—
577D4—42no—
578D4—41no—
579D4—41no—
580D4—82no—
581D4—82no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4C2×C22⋊C4×C3111yescharacteristic
617C2×C4C2×C22⋊C4×C3111yescharacteristic
618C2×C4C31×#12011yescharacteristic
619C2×C4C31×#12011yescharacteristic
620C8—42no—
621C8—81no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—81no—
640C22—161no—
641C22—161no—
642C22C31×#37911yescharacteristic
643C22C31×#41611yescharacteristic
644C22C31×#41711yescharacteristic
645C22C31×#44911yescharacteristic
646C22C31×#44911yescharacteristic
647C22C31×#45211yescharacteristic
648C22C31×#45211yescharacteristic
649C4—21no—
650C4—21no—
651C4—21no—
652C4—41no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—161no—
659C4—161no—
660C4C31×B(4,𝔽2)11yescharacteristic
661C4C31×#45811yescharacteristic
662C2—21no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—81no—
669C2—82no—
670C2C31×#178811yescharacteristic
671C2C31×#179011yescharacteristic
672C2C31×#179411yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3505C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1893C2semidirect
3. C31×#1892C2semidirect
4. C31×#1869C2semidirect
5. C31×#1799C2semidirect
6. C31×#1520C2semidirect
7. C31×#1519C2semidirect
15. C31×#539C4semidirect
16. C31×#538C4semidirect
210. #8352C31direct (N × H)
287. #1893C2×C31semidirect
288. #1892C2×C31semidirect
289. #1869C2×C31semidirect
290. #1799C2×C31semidirect
291. #1520C2×C31semidirect
292. #1519C2×C31semidirect
340. #539C4×C31semidirect
341. #538C4×C31semidirect
449. C31#8352direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35052
C31×#18932
C31×#18922
C31×#18692
C31×#17992
C31×#15202
C31×#15192
#835231