Finite Group Databasebeta

← C31×#8346 #21465105

Subgroups — C31×#8346 #21465105

Order 7936.

1052 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3490C211yesmaximal, characteristic
2C31×#1896C212yesmaximal, complemented
3C31×#1504C212yesmaximal, complemented
4C31×#1462C211yesmaximal, characteristic, complemented
5C31×#1437C211yesmaximal, characteristic, complemented
6C31×#551—22no—
7C31×#551—22no—
8C31×#546—22no—
9C31×#546—22no—
10C31×#543C411yescharacteristic, complemented
11C31×#538—21no—
12C31×#538C412yescomplemented
13C31×#534C411yescharacteristic, complemented
14C31×#525—21no—
15C31×#525C2212yes—
16C2×C31×#120—21no—
17C31×#518—21no—
18C22×C22⋊C4×C31—21no—
19C22×C22⋊C4×C31C2211yescharacteristic
20C22×C42×C31C2211yescharacteristic, complemented
21C2×C31×#104—22no—
22C31×#407C2212yescomplemented
23C2×C31×#95—21no—
24C2×C31×#95C2211yescharacteristic, complemented
25C31×#339—22no—
26C31×#328—22no—
27C31×#327—22no—
28C31×#323—21no—
29C31×#323—21no—
30C25×C31D411yescharacteristic
31C2×#41×C31—21no—
32C2×#41×C31D412yes—
33C22×D4×C31—21no—
34C22×D4×C31—21no—
35C22×D4×C31—21no—
36C23×C4×C31—21no—
37C23×C4×C31—21no—
38C23×C4×C31D411yescharacteristic
39C23×C4×C31C2311yescharacteristic
40C2×M16×C31—22no—
41C31×#126—22no—
42C31×#126—22no—
43C31×#124—22no—
44C31×#124—22no—
45C31×#123—22no—
46C31×#123—22no—
47C31×#123—22no—
48C31×#123—22no—
49C31×#123—22no—
50C31×C22⋊Q8—21no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8—21no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8C2×C412yes—
55C31×C22⋊Q8D412yes—
56C31×#121—21no—
57C31×#121—22no—
58C31×#121—22no—
59C31×#121—21no—
60C31×#121C2×C412yes—
61C31×#121D412yes—
62C31×#120—22no—
63C31×#120—22no—
64C31×#120—22no—
65C31×#120—22no—
66C31×#120—22no—
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C4×D4×C31—22no—
70C4×D4×C31—22no—
71C4×D4×C31—22no—
72C4×D4×C31—22no—
73C31×#117—22no—
74C31×#117—21no—
75C31×#117D412yes—
76C2×C22⋊C4×C31—22no—
77C2×C22⋊C4×C31—21no—
78C2×C22⋊C4×C31—21no—
79C2×C22⋊C4×C31—21no—
80C2×C22⋊C4×C31—22no—
81C2×C22⋊C4×C31—22no—
82C2×C22⋊C4×C31—24no—
83C2×C22⋊C4×C31—42no—
84C2×C22⋊C4×C31—42no—
85C2×C22⋊C4×C31C2×C411yescharacteristic
86C2×C22⋊C4×C31C2×C411yescharacteristic
87C2×C22⋊C4×C31D411yescharacteristic
88C2×C22⋊C4×C31D411yescharacteristic
89C2×C42×C31—22no—
90C2×C42×C31—42no—
91C31×#104—44no—
92C31×C22⋊C8—22no—
93C31×C22⋊C8—22no—
94C31×#95—22no—
95C31×#95—42no—
96C24×C31—21no—
97C24×C31—22no—
98C24×C31—21no—
99C24×C31—21no—
100C24×C31—41no—
101C24×C31—41no—
102C24×C31—41no—
103C24×C31—41no—
104C24×C31—41no—
105C24×C31C22⋊C411yescharacteristic
106C24×C31C22⋊C411yescharacteristic
107C24×C31C2×D411yescharacteristic
108#41×C31—42no—
109#41×C31—42no—
110#41×C31—81no—
111#41×C31—81no—
112C2×Q8×C31—21no—
113C2×Q8×C31C22⋊C412yescomplemented
114C2×D4×C31—22no—
115C2×D4×C31—22no—
116C2×D4×C31—22no—
117C2×D4×C31—22no—
118C2×D4×C31—21no—
119C2×D4×C31—22no—
120C2×D4×C31—22no—
121C2×D4×C31—42no—
122C2×D4×C31—42no—
123C2×D4×C31—81no—
124C2×D4×C31—81no—
125C2×D4×C31—81no—
126C2×D4×C31—81no—
127C2×D4×C31C22⋊C412yescomplemented
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—21no—
131C22×C4×C31—22no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—41no—
140C22×C4×C31—41no—
141C22×C4×C31—41no—
142C22×C4×C31—41no—
143C22×C4×C31—41no—
144C22×C4×C31—41no—
145C22×C4×C31—41no—
146C22×C4×C31—41no—
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×C31C2×D412yes—
157C22×C4×C31C2×D412yes—
158C22×C4×C31C2×D411yescharacteristic
159M16×C31—42no—
160M16×C31—42no—
161C2×C8×C31—42no—
162C4⋊C4×C31—22no—
163C4⋊C4×C31—21no—
164C4⋊C4×C31—21no—
165C4⋊C4×C31—22no—
166C4⋊C4×C31—22no—
167C4⋊C4×C31—22no—
168C4⋊C4×C31C22⋊C412yescomplemented
169C4⋊C4×C31C22⋊C412yescomplemented
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C22⋊C4×C31—22no—
181C22⋊C4×C31—22no—
182C22⋊C4×C31—22no—
183C22⋊C4×C31—22no—
184C22⋊C4×C31—48no—
185C42×C31—22no—
186C42×C31—22no—
187C42×C31—24no—
188C42×C31—42no—
189#8346C3111yesmaximal, characteristic, direct product factor, Sylow
190C23×C31—21no—
191C23×C31—21no—
192C23×C31—21no—
193C23×C31—21no—
194C23×C31—21no—
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—21no—
198C23×C31—21no—
199C23×C31—22no—
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—41no—
205C23×C31—41no—
206C23×C31—41no—
207C23×C31—41no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31—41no—
214C23×C31—41no—
215C23×C31—41no—
216C23×C31—41no—
217C23×C31—81no—
218C23×C31—81no—
219C23×C31—81no—
220C23×C31—81no—
221C23×C31—81no—
222C23×C31—81no—
223C23×C31—81no—
224C23×C31—81no—
225C23×C31#10412yescomplemented
226C23×C31C2×C22⋊C411yescharacteristic
227C23×C31#12011yescharacteristic
228C23×C31#12011yescharacteristic
229Q8×C31—42no—
230Q8×C31—81no—
231D4×C31—42no—
232D4×C31—44no—
233D4×C31—42no—
234D4×C31—42no—
235D4×C31—42no—
236D4×C31—42no—
237D4×C31—42no—
238D4×C31—42no—
239D4×C31—82no—
240D4×C31—82no—
241D4×C31—81no—
242C2×C4×C31—22no—
243C2×C4×C31—22no—
244C2×C4×C31—24no—
245C2×C4×C31—22no—
246C2×C4×C31—21no—
247C2×C4×C31—22no—
248C2×C4×C31—22no—
249C2×C4×C31—21no—
250C2×C4×C31—41no—
251C2×C4×C31—41no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—42no—
258C2×C4×C31—42no—
259C2×C4×C31—41no—
260C2×C4×C31—41no—
261C2×C4×C31—42no—
262C2×C4×C31—41no—
263C2×C4×C31—41no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—82no—
268C2×C4×C31—81no—
269C2×C4×C31—81no—
270C2×C4×C31—81no—
271C2×C4×C31—82no—
272C2×C4×C31—81no—
273C2×C4×C31—81no—
274C2×C4×C31—81no—
275C2×C4×C31—81no—
276C2×C4×C31C2×C22⋊C412yes—
277C2×C4×C31#12012yes—
278C8×C31—82no—
279#3490C2×C3111yescharacteristic
280#1896C2×C3112yescomplemented
281#1504C2×C3112yescomplemented
282#1462C2×C3111yescharacteristic, complemented
283#1437C2×C3111yescharacteristic, complemented
284C22×C31—21no—
285C22×C31—21no—
286C22×C31—21no—
287C22×C31—21no—
288C22×C31—21no—
289C22×C31—21no—
290C22×C31—21no—
291C22×C31—21no—
292C22×C31—21no—
293C22×C31—21no—
294C22×C31—21no—
295C22×C31—21no—
296C22×C31—21no—
297C22×C31—21no—
298C22×C31—22no—
299C22×C31—42no—
300C22×C31—42no—
301C22×C31—41no—
302C22×C31—41no—
303C22×C31—42no—
304C22×C31—41no—
305C22×C31—42no—
306C22×C31—41no—
307C22×C31—42no—
308C22×C31—41no—
309C22×C31—81no—
310C22×C31—82no—
311C22×C31—82no—
312C22×C31—82no—
313C22×C31—82no—
314C22×C31—81no—
315C22×C31—162no—
316C22×C31#37911yescharacteristic
317C22×C31C2×#10411yescharacteristic
318C22×C31#42111yescharacteristic
319C22×C31#45311yescharacteristic
320C22×C31#45511yescharacteristic
321C22×C31B(4,𝔽2)11yescharacteristic
322C22×C31#45811yescharacteristic
323C4×C31—22no—
324C4×C31—22no—
325C4×C31—44no—
326C4×C31—41no—
327C4×C31—41no—
328C4×C31—42no—
329C4×C31—42no—
330C4×C31—42no—
331C4×C31—82no—
332#551—22no—
333#551—22no—
334#546—22no—
335#546—22no—
336#543C4×C3111yescharacteristic, complemented
337#538—21no—
338#538C4×C3112yescomplemented
339#534C4×C3111yescharacteristic, complemented
340#525—21no—
341#525C22×C3112yes—
342C2×#120—21no—
343#518—21no—
344C22×C22⋊C4—21no—
345C22×C22⋊C4C22×C3111yescharacteristic
346C22×C42C22×C3111yescharacteristic, complemented
347C2×#104—22no—
348#407C22×C3112yescomplemented
349C2×#95—21no—
350C2×#95C22×C3111yescharacteristic, complemented
351#339—22no—
352#328—22no—
353#327—22no—
354#323—21no—
355#323—21no—
356C2×C31—21no—
357C2×C31—21no—
358C2×C31—21no—
359C2×C31—21no—
360C2×C31—21no—
361C2×C31—21no—
362C2×C31—42no—
363C2×C31—81no—
364C2×C31—82no—
365C2×C31#178911yescharacteristic
366C2×C31#179011yescharacteristic
367C2×C31#179211yescharacteristic
368C25D4×C3111yescharacteristic
369C2×#41—21no—
370C2×#41D4×C3112yes—
371C22×D4—21no—
372C22×D4—21no—
373C22×D4—21no—
374C23×C4—21no—
375C23×C4—21no—
376C23×C4D4×C3111yescharacteristic
377C23×C4C23×C3111yescharacteristic
378C2×M16—22no—
379#126—22no—
380#126—22no—
381#124—22no—
382#124—22no—
383#123—22no—
384#123—22no—
385#123—22no—
386#123—22no—
387#123—22no—
388C22⋊Q8—21no—
389C22⋊Q8—22no—
390C22⋊Q8—21no—
391C22⋊Q8—22no—
392C22⋊Q8C2×C4×C3112yes—
393C22⋊Q8D4×C3112yes—
394#121—21no—
395#121—22no—
396#121—22no—
397#121—21no—
398#121C2×C4×C3112yes—
399#121D4×C3112yes—
400#120—22no—
401#120—22no—
402#120—22no—
403#120—22no—
404#120—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411#117—22no—
412#117—21no—
413#117D4×C3112yes—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—42no—
422C2×C22⋊C4—42no—
423C2×C22⋊C4C2×C4×C3111yescharacteristic
424C2×C22⋊C4C2×C4×C3111yescharacteristic
425C2×C22⋊C4D4×C3111yescharacteristic
426C2×C22⋊C4D4×C3111yescharacteristic
427C2×C42—22no—
428C2×C42—42no—
429#104—44no—
430C22⋊C8—22no—
431C22⋊C8—22no—
432#95—22no—
433#95—42no—
434C31#834611yescharacteristic, direct product factor, Sylow
435C24—21no—
436C24—22no—
437C24—21no—
438C24—21no—
439C24—41no—
440C24—41no—
441C24—41no—
442C24—41no—
443C24—41no—
444C24C22⋊C4×C3111yescharacteristic
445C24C22⋊C4×C3111yescharacteristic
446C24C2×D4×C3111yescharacteristic
447#41—42no—
448#41—42no—
449#41—81no—
450#41—81no—
451C2×Q8—21no—
452C2×Q8C22⋊C4×C3112yescomplemented
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—81no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4—81no—
466C2×D4C22⋊C4×C3112yescomplemented
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4C22⋊C4×C3111yescharacteristic
493C22×C4C22⋊C4×C3111yescharacteristic
494C22×C4C22×C4×C3111yescharacteristic
495C22×C4C2×D4×C3112yes—
496C22×C4C2×D4×C3112yes—
497C22×C4C2×D4×C3111yescharacteristic
498M16—42no—
499M16—42no—
500C2×C8—42no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4C22⋊C4×C3112yescomplemented
508C4⋊C4C22⋊C4×C3112yescomplemented
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—48no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—42no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—21no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—81no—
556C23—81no—
557C23—81no—
558C23—81no—
559C23—81no—
560C23—81no—
561C23—81no—
562C23—81no—
563C23C31×#10412yescomplemented
564C23C2×C22⋊C4×C3111yescharacteristic
565C23C31×#12011yescharacteristic
566C23C31×#12011yescharacteristic
567Q8—42no—
568Q8—81no—
569D4—42no—
570D4—44no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—82no—
578D4—82no—
579D4—81no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—82no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4C2×C22⋊C4×C3112yes—
615C2×C4C31×#12012yes—
616C8—82no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—41no—
642C22—81no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22—162no—
649C22C31×#37911yescharacteristic
650C22C2×C31×#10411yescharacteristic
651C22C31×#42111yescharacteristic
652C22C31×#45311yescharacteristic
653C22C31×#45511yescharacteristic
654C22C31×B(4,𝔽2)11yescharacteristic
655C22C31×#45811yescharacteristic
656C4—22no—
657C4—22no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—41no—
663C4—42no—
664C4—82no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—21no—
671C2—42no—
672C2—81no—
673C2—82no—
674C2C31×#178911yescharacteristic
675C2C31×#179011yescharacteristic
676C2C31×#179211yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3490C2
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×#1896C2semidirect
3. C31×#1504C2semidirect
4. C31×#1462C2semidirect
5. C31×#1437C2semidirect
10. C31×#543C4semidirect
12. C31×#538C4semidirect
13. C31×#534C4semidirect
20. C22×C42×C31C22semidirect
22. C31×#407C22semidirect
24. C2×C31×#95C22semidirect
113. C2×Q8×C31C22⋊C4semidirect
127. C2×D4×C31C22⋊C4semidirect
168. C4⋊C4×C31C22⋊C4semidirect
169. C4⋊C4×C31C22⋊C4semidirect
189. #8346C31direct (N × H)
225. C23×C31#104semidirect
280. #1896C2×C31semidirect
281. #1504C2×C31semidirect
282. #1462C2×C31semidirect
283. #1437C2×C31semidirect
336. #543C4×C31semidirect
338. #538C4×C31semidirect
339. #534C4×C31semidirect
346. C22×C42C22×C31semidirect
348. #407C22×C31semidirect
350. C2×#95C22×C31semidirect
434. C31#8346direct (N × H)
452. C2×Q8C22⋊C4×C31semidirect
466. C2×D4C22⋊C4×C31semidirect
507. C4⋊C4C22⋊C4×C31semidirect
508. C4⋊C4C22⋊C4×C31semidirect
563. C23C31×#104semidirect

Maximal subgroups

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

subgroupindex
C31×#34902
C31×#18962
C31×#15042
C31×#14622
C31×#14372
#834631