Finite Group Databasebeta

← C31×#32965 #21489724

Subgroups — C31×#32965 #21489724

Order 7936.

1282 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#520C211yesmaximal, characteristic
2C2×C31×#475C211yesmaximal, characteristic, complemented
3C2×C31×#476C211yesmaximal, characteristic, complemented
4C31×#2628C211yesmaximal, characteristic, complemented
5C31×#1865C212yesmaximal, complemented
6C31×#1864C212yesmaximal, complemented
7C31×#1853C212yesmaximal, complemented
8C31×#1805C211yesmaximal, characteristic, complemented
9C31×#1804C211yesmaximal, characteristic, complemented
10C31×#1798C211yesmaximal, characteristic, complemented
11C31×#1788C212yesmaximal
12C22×#41×C31C2211yescharacteristic
13C2×C31×C4⋊Q8C2211yescharacteristic
14C31×#520—24no—
15C31×#520—24no—
16C2×C4×Q8×C31C2211yescharacteristic
17C2×C4×Q8×C31C2211yescharacteristic
18C2×C4×D4×C31—21no—
19C2×C4×D4×C31—21no—
20C2×C4×D4×C31C2211yescharacteristic
21C2×C4×D4×C31C2211yescharacteristic
22C2×C31×#117—21no—
23C2×C31×#117C2211yescharacteristic
24C22×C4⋊C4×C31—21no—
25C22×C4⋊C4×C31C2211yescharacteristic
26C31×#476—24no—
27C31×#475—24no—
28C2×C31×#106C2212yes—
29C2×C31×#106C2212yes—
30C2×C31×C4⋊C8C2211yescharacteristic
31C2×C31×C4⋊C8C2211yescharacteristic
32C2×C31×#103C2212yes—
33C2×C31×#103C2212yes—
34C2×C31×#102C2212yes—
35C2×C31×#102C2212yes—
36C2×C31×C22⋊C8C2212yes—
37C31×#400—21no—
38C31×#399—21no—
39C31×#395—21no—
40C31×#393—21no—
41C31×#389C2212yes—
42C31×#382C2211yescharacteristic
43C31×#380C2212yes—
44C31×#340C2212yes—
45C31×#340C2212yes—
46C31×#336C2212yes—
47C2×#41×C31—22no—
48C2×#41×C31—22no—
49C2×#41×C31—22no—
50C2×#41×C31—22no—
51C2×#41×C31—22no—
52C2×#41×C31Q814yescomplemented
53C22×Q8×C31—21no—
54C22×Q8×C31C2311yescharacteristic
55C22×D4×C31—21no—
56C22×D4×C31C2311yescharacteristic
57C23×C4×C31—21no—
58C23×C4×C31—21no—
59C23×C4×C31—21no—
60C23×C4×C31—21no—
61C23×C4×C31C2311yescharacteristic
62C22×C8×C31C2312yes—
63C22×C8×C31C2312yes—
64C31×C4⋊Q8—22no—
65C31×C4⋊Q8—22no—
66C4×Q8×C31—22no—
67C4×Q8×C31—22no—
68C4×Q8×C31—22no—
69C4×Q8×C31D414yes—
70C4×D4×C31—28no—
71C4×D4×C31—28no—
72C4×D4×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31D414yes—
76C31×#117—28no—
77C31×#117—22no—
78C31×#117—22no—
79C2×C4⋊C4×C31—28no—
80C2×C4⋊C4×C31—21no—
81C2×C4⋊C4×C31—21no—
82C2×C4⋊C4×C31—21no—
83C2×C4⋊C4×C31—22no—
84C2×C4⋊C4×C31—22no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—22no—
87C2×C4⋊C4×C31D411yescharacteristic
88C2×C4⋊C4×C31D411yescharacteristic
89C2×C4⋊C4×C31C2312yes—
90C2×C4⋊C4×C31C2312yes—
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C4⋊C4×C31C2311yescharacteristic
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—21no—
95C2×C22⋊C4×C31—22no—
96C2×C22⋊C4×C31D411yescharacteristic
97C2×C22⋊C4×C31D411yescharacteristic
98C2×C42×C31—21no—
99C2×C42×C31—21no—
100C2×C42×C31C2311yescharacteristic
101C2×C42×C31C2311yescharacteristic
102C31×#106—24no—
103C31×#106—24no—
104C31×C4⋊C8—22no—
105C31×C4⋊C8—22no—
106C31×#103—24no—
107C31×#103—24no—
108C31×#102—24no—
109C31×#102—24no—
110C31×C22⋊C8—24no—
111C31×#95—22no—
112C31×#95—21no—
113C31×#95—21no—
114C31×#95—22no—
115C24×C31—21no—
116C24×C31C2×D411yescharacteristic
117#41×C31—28no—
118#41×C31—28no—
119#41×C31—28no—
120#41×C31—28no—
121C2×Q8×C31—21no—
122C2×Q8×C31—21no—
123C2×Q8×C31—21no—
124C2×Q8×C31—42no—
125C2×Q8×C31—82no—
126C2×Q8×C31C2×D412yes—
127C2×Q8×C31C2×Q812yes—
128C2×Q8×C31#4112yes—
129C2×D4×C31—24no—
130C2×D4×C31—24no—
131C2×D4×C31—24no—
132C2×D4×C31—21no—
133C2×D4×C31—21no—
134C2×D4×C31—21no—
135C2×D4×C31C2×D412yes—
136C2×D4×C31C2×Q812yes—
137C2×D4×C31#4112yes—
138C22×C4×C31—24no—
139C22×C4×C31—24no—
140C22×C4×C31—21no—
141C22×C4×C31—24no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—21no—
145C22×C4×C31—21no—
146C22×C4×C31—21no—
147C22×C4×C31—21no—
148C22×C4×C31—24no—
149C22×C4×C31—24no—
150C22×C4×C31—24no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—82no—
154C22×C4×C31—82no—
155C22×C4×C31—82no—
156C22×C4×C31C2×D411yescharacteristic
157C22×C4×C31C2×D411yescharacteristic
158C22×C4×C31C2×D411yescharacteristic
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×Q812yes—
162C22×C4×C31#4112yes—
163C22×C4×C31C2411yescharacteristic
164C2×C8×C31—22no—
165C2×C8×C31—22no—
166C2×C8×C31—22no—
167C2×C8×C31—22no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C4⋊C4×C31—24no—
171C4⋊C4×C31—24no—
172C4⋊C4×C31—22no—
173C4⋊C4×C31—24no—
174C4⋊C4×C31—22no—
175C4⋊C4×C31—21no—
176C4⋊C4×C31—21no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—22no—
179C4⋊C4×C31—22no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—42no—
182C4⋊C4×C31—82no—
183C4⋊C4×C31C2×D412yes—
184C4⋊C4×C31C2×D412yes—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—22no—
188C22⋊C4×C31—22no—
189C22⋊C4×C31—82no—
190C42×C31—24no—
191C42×C31—24no—
192C42×C31—21no—
193C42×C31—21no—
194C42×C31C2×D412yes—
195C42×C31C2×D412yes—
196#32965C3111yesmaximal, characteristic, direct product factor, Sylow
197C23×C31—21no—
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—21no—
201C23×C31—22no—
202C23×C31—22no—
203C23×C31—22no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31C22⋊Q812yes—
207C23×C31C22×D411yescharacteristic
208Q8×C31—22no—
209Q8×C31—22no—
210Q8×C31—22no—
211Q8×C31—84no—
212Q8×C31C22⋊Q814yes—
213D4×C31—216no—
214D4×C31—22no—
215D4×C31—22no—
216D4×C31—22no—
217D4×C31C22⋊Q814yes—
218C2×C4×C31—216no—
219C2×C4×C31—22no—
220C2×C4×C31—21no—
221C2×C4×C31—22no—
222C2×C4×C31—21no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—22no—
235C2×C4×C31—22no—
236C2×C4×C31—42no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—84no—
245C2×C4×C31—84no—
246C2×C4×C31—84no—
247C2×C4×C31#12012yes—
248C2×C4×C31#12012yes—
249C2×C4×C31C22⋊Q812yes—
250C2×C4×C31C22×D411yescharacteristic
251C2×C4×C31C22×D411yescharacteristic
252C2×C4×C31C22×Q811yescharacteristic
253C2×C4×C31C2×#4111yescharacteristic
254C2×C4×C31C2×#4111yescharacteristic
255C2×C4×C31C2×#4111yescharacteristic
256C8×C31—42no—
257C8×C31—42no—
258C2×#520C2×C3111yescharacteristic
259C2×#475C2×C3111yescharacteristic, complemented
260C2×#476C2×C3111yescharacteristic, complemented
261#2628C2×C3111yescharacteristic, complemented
262#1865C2×C3112yescomplemented
263#1864C2×C3112yescomplemented
264#1853C2×C3112yescomplemented
265#1805C2×C3111yescharacteristic, complemented
266#1804C2×C3111yescharacteristic, complemented
267#1798C2×C3111yescharacteristic, complemented
268#1788C2×C3112yes—
269C22×C31—24no—
270C22×C31—22no—
271C22×C31—24no—
272C22×C31—44no—
273C22×C31—42no—
274C22×C31—44no—
275C22×C31—41no—
276C22×C31—42no—
277C22×C31—41no—
278C22×C31—42no—
279C22×C31C2×#12011yescharacteristic
280C22×C31C2×C22⋊Q811yescharacteristic
281C22×C31#52511yescharacteristic
282C22×C31#57712yes—
283C22×C31#57712yes—
284C4×C31—22no—
285C4×C31—24no—
286C4×C31—22no—
287C4×C31—22no—
288C4×C31—41no—
289C4×C31—42no—
290C4×C31—42no—
291C4×C31—41no—
292C4×C31—82no—
293C4×C31—82no—
294C4×C31C2×C22⋊Q811yescharacteristic
295C4×C31C2×C22⋊Q811yescharacteristic
296C4×C31#52511yescharacteristic
297C4×C31#52511yescharacteristic
298C22×#41C22×C3111yescharacteristic
299C2×C4⋊Q8C22×C3111yescharacteristic
300#520—24no—
301#520—24no—
302C2×C4×Q8C22×C3111yescharacteristic
303C2×C4×Q8C22×C3111yescharacteristic
304C2×C4×D4—21no—
305C2×C4×D4—21no—
306C2×C4×D4C22×C3111yescharacteristic
307C2×C4×D4C22×C3111yescharacteristic
308C2×#117—21no—
309C2×#117C22×C3111yescharacteristic
310C22×C4⋊C4—21no—
311C22×C4⋊C4C22×C3111yescharacteristic
312#476—24no—
313#475—24no—
314C2×#106C22×C3112yes—
315C2×#106C22×C3112yes—
316C2×C4⋊C8C22×C3111yescharacteristic
317C2×C4⋊C8C22×C3111yescharacteristic
318C2×#103C22×C3112yes—
319C2×#103C22×C3112yes—
320C2×#102C22×C3112yes—
321C2×#102C22×C3112yes—
322C2×C22⋊C8C22×C3112yes—
323#400—21no—
324#399—21no—
325#395—21no—
326#393—21no—
327#389C22×C3112yes—
328#382C22×C3111yescharacteristic
329#380C22×C3112yes—
330#340C22×C3112yes—
331#340C22×C3112yes—
332#336C22×C3112yes—
333C2×C31—42no—
334C2×C31—44no—
335C2×C31#284811yescharacteristic
336C2×C31#300911yescharacteristic
337C2×C31#301011yescharacteristic
338C2×C31#308412yes—
339C2×C31#312512yes—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41Q8×C3114yescomplemented
346C22×Q8—21no—
347C22×Q8C23×C3111yescharacteristic
348C22×D4—21no—
349C22×D4C23×C3111yescharacteristic
350C23×C4—21no—
351C23×C4—21no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4C23×C3111yescharacteristic
355C22×C8C23×C3112yes—
356C22×C8C23×C3112yes—
357C4⋊Q8—22no—
358C4⋊Q8—22no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8D4×C3114yes—
363C4×D4—28no—
364C4×D4—28no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4D4×C3114yes—
369#117—28no—
370#117—22no—
371#117—22no—
372C2×C4⋊C4—28no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4D4×C3111yescharacteristic
381C2×C4⋊C4D4×C3111yescharacteristic
382C2×C4⋊C4C23×C3112yes—
383C2×C4⋊C4C23×C3112yes—
384C2×C4⋊C4C23×C3111yescharacteristic
385C2×C4⋊C4C23×C3111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4D4×C3111yescharacteristic
390C2×C22⋊C4D4×C3111yescharacteristic
391C2×C42—21no—
392C2×C42—21no—
393C2×C42C23×C3111yescharacteristic
394C2×C42C23×C3111yescharacteristic
395#106—24no—
396#106—24no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399#103—24no—
400#103—24no—
401#102—24no—
402#102—24no—
403C22⋊C8—24no—
404#95—22no—
405#95—21no—
406#95—21no—
407#95—22no—
408C31#3296511yescharacteristic, direct product factor, Sylow
409C24—21no—
410C24C2×D4×C3111yescharacteristic
411#41—28no—
412#41—28no—
413#41—28no—
414#41—28no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—42no—
419C2×Q8—82no—
420C2×Q8C2×D4×C3112yes—
421C2×Q8C2×Q8×C3112yes—
422C2×Q8#41×C3112yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×D4×C3112yes—
430C2×D4C2×Q8×C3112yes—
431C2×D4#41×C3112yes—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4C2×D4×C3111yescharacteristic
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3111yescharacteristic
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C2×Q8×C3112yes—
456C22×C4#41×C3112yes—
457C22×C4C24×C3111yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—82no—
477C4⋊C4C2×D4×C3112yes—
478C4⋊C4C2×D4×C3112yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—82no—
484C42—24no—
485C42—24no—
486C42—21no—
487C42—21no—
488C42C2×D4×C3112yes—
489C42C2×D4×C3112yes—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—21no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23C31×C22⋊Q812yes—
500C23C22×D4×C3111yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—22no—
504Q8—84no—
505Q8C31×C22⋊Q814yes—
506D4—216no—
507D4—22no—
508D4—22no—
509D4—22no—
510D4C31×C22⋊Q814yes—
511C2×C4—216no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4C31×#12012yes—
541C2×C4C31×#12012yes—
542C2×C4C31×C22⋊Q812yes—
543C2×C4C22×D4×C3111yescharacteristic
544C2×C4C22×D4×C3111yescharacteristic
545C2×C4C22×Q8×C3111yescharacteristic
546C2×C4C2×#41×C3111yescharacteristic
547C2×C4C2×#41×C3111yescharacteristic
548C2×C4C2×#41×C3111yescharacteristic
549C8—42no—
550C8—42no—
551C22—24no—
552C22—22no—
553C22—24no—
554C22—44no—
555C22—42no—
556C22—44no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22C2×C31×#12011yescharacteristic
562C22C2×C31×C22⋊Q811yescharacteristic
563C22C31×#52511yescharacteristic
564C22C31×#57712yes—
565C22C31×#57712yes—
566C4—22no—
567C4—24no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—42no—
572C4—42no—
573C4—41no—
574C4—82no—
575C4—82no—
576C4C2×C31×C22⋊Q811yescharacteristic
577C4C2×C31×C22⋊Q811yescharacteristic
578C4C31×#52511yescharacteristic
579C4C31×#52511yescharacteristic
580C2—42no—
581C2—44no—
582C2C31×#284811yescharacteristic
583C2C31×#300911yescharacteristic
584C2C31×#301011yescharacteristic
585C2C31×#308412yes—
586C2C31×#312512yes—
subgroupG/H
center, socleC23×C31C22×D4
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#475C2semidirect
3. C2×C31×#476C2semidirect
4. C31×#2628C2semidirect
5. C31×#1865C2semidirect
6. C31×#1864C2semidirect
7. C31×#1853C2semidirect
8. C31×#1805C2semidirect
9. C31×#1804C2semidirect
10. C31×#1798C2semidirect
52. C2×#41×C31Q8semidirect
196. #32965C31direct (N × H)
259. C2×#475C2×C31semidirect
260. C2×#476C2×C31semidirect
261. #2628C2×C31semidirect
262. #1865C2×C31semidirect
263. #1864C2×C31semidirect
264. #1853C2×C31semidirect
265. #1805C2×C31semidirect
266. #1804C2×C31semidirect
267. #1798C2×C31semidirect
345. C2×#41Q8×C31semidirect
408. C31#32965direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5202
C2×C31×#4752
C2×C31×#4762
C31×#26282
C31×#18652
C31×#18642
C31×#18532
C31×#18052
C31×#18042
C31×#17982
C31×#17882
#3296531