Finite Group Databasebeta

← C31×#14717 #21471476

Subgroups — C31×#14717 #21471476

Order 7936.

3654 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×C22⋊Q8C211yesmaximal, characteristic, complemented
2C23×C22⋊C4×C31C211yesmaximal, characteristic
3C2×C31×#393C212yesmaximal, complemented
4C2×C31×#379C212yesmaximal
5C31×#1439C214yesmaximal, complemented
6C31×#1439C212yesmaximal, complemented
7C31×C22⋊#95C211yesmaximal, characteristic, complemented
8C31×C22⋊#95C212yesmaximal, complemented
9C26×C31C2211yescharacteristic
10C23×Q8×C31C2211yescharacteristic, complemented
11C24×C4×C31C2211yescharacteristic
12C24×C4×C31C2211yescharacteristic
13C2×C31×C22⋊Q8—24no—
14C2×C31×C22⋊Q8—28no—
15C22×C4⋊C4×C31C2211yescharacteristic, complemented
16C22×C4⋊C4×C31C2212yescomplemented
17C22×C22⋊C4×C31—24no—
18C22×C22⋊C4×C31—28no—
19C22×C22⋊C4×C31C2212yes—
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2214yes—
22C22×C22⋊C4×C31C2214yes—
23C22×C22⋊C4×C31C2212yes—
24C22×C22⋊C4×C31C2212yes—
25C31×#393—28no—
26C31×#379—28no—
27C2×C31×#95C2212yescomplemented
28C2×C31×#95C2214yescomplemented
29C2×C31×#95C2212yescomplemented
30C2×C31×#95C2214yescomplemented
31C25×C31—24no—
32C25×C31—22no—
33C25×C31—24no—
34C25×C31—22no—
35C25×C31—24no—
36C25×C31—24no—
37C25×C31—42no—
38C25×C31—42no—
39C25×C31Q814yescomplemented
40C25×C31C2312yes—
41C25×C31C2311yescharacteristic
42C22×Q8×C31—24no—
43C22×Q8×C31—22no—
44C22×Q8×C31—24no—
45C22×Q8×C31—42no—
46C23×C4×C31—22no—
47C23×C4×C31—24no—
48C23×C4×C31—24no—
49C23×C4×C31—24no—
50C23×C4×C31—24no—
51C23×C4×C31—42no—
52C23×C4×C31—42no—
53C23×C4×C31D414yes—
54C23×C4×C31C2314yes—
55C23×C4×C31C2312yes—
56C23×C4×C31C2312yes—
57C23×C4×C31C2311yescharacteristic
58C23×C4×C31C2312yes—
59C23×C4×C31C2311yescharacteristic
60C31×C22⋊Q8—232no—
61C2×C4⋊C4×C31—24no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—24no—
64C2×C4⋊C4×C31—24no—
65C2×C4⋊C4×C31—24no—
66C2×C22⋊C4×C31—232no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—24no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—24no—
74C2×C22⋊C4×C31—24no—
75C2×C22⋊C4×C31—28no—
76C2×C22⋊C4×C31—28no—
77C2×C22⋊C4×C31—24no—
78C2×C22⋊C4×C31—24no—
79C2×C22⋊C4×C31—28no—
80C2×C22⋊C4×C31—28no—
81C2×C22⋊C4×C31—44no—
82C2×C22⋊C4×C31—44no—
83C2×C22⋊C4×C31D418yes—
84C2×C22⋊C4×C31D418yes—
85C31×#95—28no—
86C31×#95—24no—
87C31×#95—44no—
88C31×#95—42no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—28no—
94C24×C31—28no—
95C24×C31—216no—
96C24×C31—28no—
97C24×C31—216no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—24no—
101C24×C31—22no—
102C24×C31—22no—
103C24×C31—24no—
104C24×C31—24no—
105C24×C31—24no—
106C24×C31—216no—
107C24×C31—48no—
108C24×C31—416no—
109C24×C31—416no—
110C24×C31—416no—
111C24×C31—416no—
112C24×C31—44no—
113C24×C31—44no—
114C24×C31—48no—
115C24×C31—44no—
116C24×C31—48no—
117C24×C31—44no—
118C24×C31C2×D414yes—
119C24×C31C2×D414yes—
120C24×C31C2×D412yes—
121C24×C31C2×Q814yes—
122C24×C31C2×Q812yes—
123C24×C31#4114yes—
124C24×C31#4112yes—
125C24×C31#4114yes—
126C24×C31C2411yescharacteristic
127C2×Q8×C31—28no—
128C2×Q8×C31—416no—
129C2×Q8×C31—44no—
130C2×Q8×C31—44no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—28no—
134C22×C4×C31—28no—
135C22×C4×C31—24no—
136C22×C4×C31—22no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—22no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—22no—
144C22×C4×C31—24no—
145C22×C4×C31—24no—
146C22×C4×C31—24no—
147C22×C4×C31—22no—
148C22×C4×C31—24no—
149C22×C4×C31—24no—
150C22×C4×C31—28no—
151C22×C4×C31—24no—
152C22×C4×C31—24no—
153C22×C4×C31—24no—
154C22×C4×C31—24no—
155C22×C4×C31—24no—
156C22×C4×C31—24no—
157C22×C4×C31—24no—
158C22×C4×C31—44no—
159C22×C4×C31—416no—
160C22×C4×C31—416no—
161C22×C4×C31—44no—
162C22×C4×C31—44no—
163C22×C4×C31C2×D414yes—
164C22×C4×C31C2×D414yes—
165C22×C4×C31C2×D414yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D414yes—
168C22×C4×C31C2×D412yes—
169C4⋊C4×C31—28no—
170C4⋊C4×C31—216no—
171C22⋊C4×C31—216no—
172C22⋊C4×C31—216no—
173C22⋊C4×C31—216no—
174C22⋊C4×C31—416no—
175C22⋊C4×C31—416no—
176C22⋊C4×C31—48no—
177#14717C3111yesmaximal, characteristic, direct product factor, Sylow
178C23×C31—22no—
179C23×C31—24no—
180C23×C31—22no—
181C23×C31—24no—
182C23×C31—22no—
183C23×C31—22no—
184C23×C31—24no—
185C23×C31—24no—
186C23×C31—28no—
187C23×C31—28no—
188C23×C31—28no—
189C23×C31—28no—
190C23×C31—232no—
191C23×C31—24no—
192C23×C31—24no—
193C23×C31—24no—
194C23×C31—28no—
195C23×C31—24no—
196C23×C31—24no—
197C23×C31—24no—
198C23×C31—28no—
199C23×C31—28no—
200C23×C31—28no—
201C23×C31—28no—
202C23×C31—28no—
203C23×C31—464no—
204C23×C31—416no—
205C23×C31—416no—
206C23×C31—416no—
207C23×C31—416no—
208C23×C31—416no—
209C23×C31—464no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—44no—
213C23×C31—42no—
214C23×C31—42no—
215C23×C31—44no—
216C23×C31—44no—
217C23×C31—44no—
218C23×C31—44no—
219C23×C31—44no—
220C23×C31—416no—
221C23×C31#12014yes—
222C23×C31C22⋊Q814yes—
223C23×C31C22⋊Q818yes—
224C23×C31C22⋊Q818yescomplemented
225C23×C31#12418yes—
226C23×C31#12414yes—
227C23×C31C22×D411yescharacteristic
228C23×C31C22×D412yes—
229C23×C31C22×D412yes—
230C23×C31C22×Q811yescharacteristic
231C23×C31C2×#4111yescharacteristic
232C23×C31C2×#4112yes—
233C23×C31C2×#4112yes—
234C23×C31H(5,𝔽2)12yes—
235C23×C31H(5,𝔽2)12yes—
236Q8×C31—416no—
237C2×C4×C31—22no—
238C2×C4×C31—22no—
239C2×C4×C31—28no—
240C2×C4×C31—22no—
241C2×C4×C31—24no—
242C2×C4×C31—24no—
243C2×C4×C31—24no—
244C2×C4×C31—22no—
245C2×C4×C31—22no—
246C2×C4×C31—28no—
247C2×C4×C31—28no—
248C2×C4×C31—28no—
249C2×C4×C31—416no—
250C2×C4×C31—416no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—42no—
257C2×C4×C31—44no—
258C2×C4×C31—44no—
259C2×C4×C31—44no—
260C2×C4×C31—44no—
261C2×C4×C31—44no—
262C2×C4×C31—44no—
263C2×C4×C31—44no—
264C2×C4×C31—44no—
265C2×C4×C31—44no—
266C2×C4×C31—44no—
267C2×C4×C31—44no—
268C2×C4×C31#12014yes—
269C22×C22⋊Q8C2×C3111yescharacteristic, complemented
270C23×C22⋊C4C2×C3111yescharacteristic
271C2×#393C2×C3112yescomplemented
272C2×#379C2×C3112yes—
273#1439C2×C3114yescomplemented
274#1439C2×C3112yescomplemented
275C22⋊#95C2×C3111yescharacteristic, complemented
276C22⋊#95C2×C3112yescomplemented
277C22×C31—24no—
278C22×C31—24no—
279C22×C31—24no—
280C22×C31—24no—
281C22×C31—24no—
282C22×C31—24no—
283C22×C31—216no—
284C22×C31—216no—
285C22×C31—28no—
286C22×C31—216no—
287C22×C31—464no—
288C22×C31—44no—
289C22×C31—44no—
290C22×C31—44no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—48no—
295C22×C31—48no—
296C22×C31—48no—
297C22×C31#39318yescomplemented
298C22×C31C2×#12011yescharacteristic
299C22×C31C2×#12011yescharacteristic
300C22×C31C2×C22⋊Q811yescharacteristic
301C22×C31C2×C22⋊Q812yes—
302C22×C31C2×C22⋊Q812yes—
303C22×C31#52511yescharacteristic
304C22×C31#52512yes—
305C22×C31C2×#12411yescharacteristic
306C22×C31C2×#12412yes—
307C22×C31#53411yescharacteristic
308C22×C31#53511yescharacteristic
309C22×C31#53812yes—
310C22×C31#54312yes—
311C22×C31D4214yes—
312C22×C31#54614yes—
313C22×C31#54614yes—
314C22×C31#55114yes—
315C4×C31—24no—
316C4×C31—42no—
317C4×C31—44no—
318C4×C31—44no—
319C4×C31—44no—
320C4×C31—84no—
321C26C22×C3111yescharacteristic
322C23×Q8C22×C3111yescharacteristic, complemented
323C24×C4C22×C3111yescharacteristic
324C24×C4C22×C3111yescharacteristic
325C2×C22⋊Q8—24no—
326C2×C22⋊Q8—28no—
327C22×C4⋊C4C22×C3111yescharacteristic, complemented
328C22×C4⋊C4C22×C3112yescomplemented
329C22×C22⋊C4—24no—
330C22×C22⋊C4—28no—
331C22×C22⋊C4C22×C3112yes—
332C22×C22⋊C4C22×C3112yes—
333C22×C22⋊C4C22×C3114yes—
334C22×C22⋊C4C22×C3114yes—
335C22×C22⋊C4C22×C3112yes—
336C22×C22⋊C4C22×C3112yes—
337#393—28no—
338#379—28no—
339C2×#95C22×C3112yescomplemented
340C2×#95C22×C3114yescomplemented
341C2×#95C22×C3112yescomplemented
342C2×#95C22×C3114yescomplemented
343C2×C31—28no—
344C2×C31—48no—
345C2×C31C2×#39311yescharacteristic
346C2×C31#240311yescharacteristic
347C2×C31#240411yescharacteristic
348C2×C31#243511yescharacteristic
349C2×C31#243714yes—
350C2×C31#247214yes—
351C2×C31#256111yescharacteristic
352C2×C31#262611yescharacteristic
353C2×C31#284811yescharacteristic
354C25—24no—
355C25—22no—
356C25—24no—
357C25—22no—
358C25—24no—
359C25—24no—
360C25—42no—
361C25—42no—
362C25Q8×C3114yescomplemented
363C25C23×C3112yes—
364C25C23×C3111yescharacteristic
365C22×Q8—24no—
366C22×Q8—22no—
367C22×Q8—24no—
368C22×Q8—42no—
369C23×C4—22no—
370C23×C4—24no—
371C23×C4—24no—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—42no—
375C23×C4—42no—
376C23×C4D4×C3114yes—
377C23×C4C23×C3114yes—
378C23×C4C23×C3112yes—
379C23×C4C23×C3112yes—
380C23×C4C23×C3111yescharacteristic
381C23×C4C23×C3112yes—
382C23×C4C23×C3111yescharacteristic
383C22⋊Q8—232no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C22⋊C4—232no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—28no—
399C2×C22⋊C4—28no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—28no—
403C2×C22⋊C4—28no—
404C2×C22⋊C4—44no—
405C2×C22⋊C4—44no—
406C2×C22⋊C4D4×C3118yes—
407C2×C22⋊C4D4×C3118yes—
408#95—28no—
409#95—24no—
410#95—44no—
411#95—42no—
412C31#1471711yescharacteristic, direct product factor, Sylow
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—28no—
418C24—28no—
419C24—216no—
420C24—28no—
421C24—216no—
422C24—22no—
423C24—22no—
424C24—24no—
425C24—22no—
426C24—22no—
427C24—24no—
428C24—24no—
429C24—24no—
430C24—216no—
431C24—48no—
432C24—416no—
433C24—416no—
434C24—416no—
435C24—416no—
436C24—44no—
437C24—44no—
438C24—48no—
439C24—44no—
440C24—48no—
441C24—44no—
442C24C2×D4×C3114yes—
443C24C2×D4×C3114yes—
444C24C2×D4×C3112yes—
445C24C2×Q8×C3114yes—
446C24C2×Q8×C3112yes—
447C24#41×C3114yes—
448C24#41×C3112yes—
449C24#41×C3114yes—
450C24C24×C3111yescharacteristic
451C2×Q8—28no—
452C2×Q8—416no—
453C2×Q8—44no—
454C2×Q8—44no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—28no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—44no—
483C22×C4—416no—
484C22×C4—416no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4C2×D4×C3114yes—
488C22×C4C2×D4×C3114yes—
489C22×C4C2×D4×C3114yes—
490C22×C4C2×D4×C3112yes—
491C22×C4C2×D4×C3114yes—
492C22×C4C2×D4×C3112yes—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—232no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—464no—
527C23—416no—
528C23—416no—
529C23—416no—
530C23—416no—
531C23—464no—
532C23—416no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—416no—
544C23C31×#12014yes—
545C23C31×C22⋊Q814yes—
546C23C31×C22⋊Q818yes—
547C23C31×C22⋊Q818yescomplemented
548C23C31×#12418yes—
549C23C31×#12414yes—
550C23C22×D4×C3111yescharacteristic
551C23C22×D4×C3112yes—
552C23C22×D4×C3112yes—
553C23C22×Q8×C3111yescharacteristic
554C23C2×#41×C3111yescharacteristic
555C23C2×#41×C3112yes—
556C23C2×#41×C3112yes—
557C23C31×H(5,𝔽2)12yes—
558C23C31×H(5,𝔽2)12yes—
559Q8—416no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—416no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4C31×#12014yes—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—216no—
598C22—216no—
599C22—24no—
600C22—28no—
601C22—216no—
602C22—464no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—48no—
610C22—48no—
611C22—44no—
612C22C31×#39318yescomplemented
613C22C2×C31×#12011yescharacteristic
614C22C2×C31×#12011yescharacteristic
615C22C2×C31×C22⋊Q811yescharacteristic
616C22C2×C31×C22⋊Q812yes—
617C22C2×C31×C22⋊Q812yes—
618C22C31×#52511yescharacteristic
619C22C31×#52512yes—
620C22C2×C31×#12411yescharacteristic
621C22C2×C31×#12412yes—
622C22C31×#53411yescharacteristic
623C22C31×#53511yescharacteristic
624C22C31×#53812yes—
625C22C31×#54312yes—
626C22D42×C3114yes—
627C22C31×#54614yes—
628C22C31×#54614yes—
629C22C31×#55114yes—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—84no—
636C2—28no—
637C2—48no—
638C2C2×C31×#39311yescharacteristic
639C2C31×#240311yescharacteristic
640C2C31×#240411yescharacteristic
641C2C31×#243511yescharacteristic
642C2C31×#243714yes—
643C2C31×#247214yes—
644C2C31×#256111yescharacteristic
645C2C31×#262611yescharacteristic
646C2C31×#284811yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC26×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×C22⋊Q8C2semidirect
3. C2×C31×#393C2semidirect
5. C31×#1439C2semidirect
6. C31×#1439C2semidirect
7. C31×C22⋊#95C2semidirect
8. C31×C22⋊#95C2semidirect
10. C23×Q8×C31C22semidirect
15. C22×C4⋊C4×C31C22semidirect
16. C22×C4⋊C4×C31C22semidirect
27. C2×C31×#95C22semidirect
28. C2×C31×#95C22semidirect
29. C2×C31×#95C22semidirect
30. C2×C31×#95C22semidirect
39. C25×C31Q8semidirect
177. #14717C31direct (N × H)
224. C23×C31C22⋊Q8semidirect
269. C22×C22⋊Q8C2×C31semidirect
271. C2×#393C2×C31semidirect
273. #1439C2×C31semidirect
274. #1439C2×C31semidirect
275. C22⋊#95C2×C31semidirect
276. C22⋊#95C2×C31semidirect
297. C22×C31#393semidirect
322. C23×Q8C22×C31semidirect
327. C22×C4⋊C4C22×C31semidirect
328. C22×C4⋊C4C22×C31semidirect
339. C2×#95C22×C31semidirect
340. C2×#95C22×C31semidirect
341. C2×#95C22×C31semidirect
342. C2×#95C22×C31semidirect
362. C25Q8×C31semidirect
412. C31#14717direct (N × H)
547. C23C31×C22⋊Q8semidirect
612. C22C31×#393semidirect

Maximal subgroups

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

subgroupindex
C22×C31×C22⋊Q82
C23×C22⋊C4×C312
C2×C31×#3932
C2×C31×#3792
C31×#14392
C31×C22⋊#952
#1471731