Finite Group Databasebeta

← C31×#16666 #21473425

Subgroups — C31×#16666 #21473425

Order 7936.

1462 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#393C211yesmaximal, characteristic
2C2×C31×#381C211yesmaximal, characteristic
3C31×#1446C212yesmaximal, complemented
4C31×#1445C212yesmaximal, complemented
5C31×#1443C212yesmaximal, complemented
6C31×#1441C212yesmaximal, complemented
7C31×#1439C212yesmaximal
8C31×#1439C211yesmaximal, characteristic
9C31×C22⋊#95C212yesmaximal
10C23×Q8×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C4⋊C4×C31C2212yes—
14C22×C22⋊C4×C31C2212yes—
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2211yescharacteristic
17C22×C22⋊C4×C31C2211yescharacteristic
18C31×#393—24no—
19C31×#381—24no—
20C2×C31×#95C2212yes—
21C2×C31×#95C2211yescharacteristic
22C2×C31×#95C2212yes—
23C2×C31×#95C2212yes—
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2212yes—
29C2×C31×#95C2212yes—
30C2×C31×#95C2211yescharacteristic
31C2×C31×#95C2212yes—
32C2×C31×#95C2212yes—
33C25×C31C2311yescharacteristic
34C22×Q8×C31—22no—
35C22×Q8×C31—24no—
36C22×Q8×C31—42no—
37C22×Q8×C31—41no—
38C22×Q8×C31—41no—
39C23×C4×C31—22no—
40C23×C4×C31—22no—
41C23×C4×C31—41no—
42C23×C4×C31—41no—
43C23×C4×C31—41no—
44C23×C4×C31—41no—
45C23×C4×C31Q814yescomplemented
46C23×C4×C31C2312yes—
47C23×C4×C31C2312yes—
48C23×C4×C31C2312yes—
49C23×C4×C31C2311yescharacteristic
50C23×C4×C31C2311yescharacteristic
51C23×C4×C31C2312yes—
52C23×C4×C31C2312yes—
53C23×C4×C31C2312yes—
54C2×C4⋊C4×C31—24no—
55C2×C4⋊C4×C31—24no—
56C2×C4⋊C4×C31—41no—
57C2×C4⋊C4×C31—42no—
58C2×C4⋊C4×C31—42no—
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—24no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—41no—
65C2×C22⋊C4×C31—41no—
66C2×C22⋊C4×C31—42no—
67C2×C22⋊C4×C31—42no—
68C2×C22⋊C4×C31—41no—
69C2×C22⋊C4×C31—41no—
70C2×C22⋊C4×C31D414yes—
71C31×#95—24no—
72C31×#95—24no—
73C31×#95—24no—
74C31×#95—22no—
75C31×#95—24no—
76C31×#95—42no—
77C31×#95—41no—
78C31×#95—42no—
79C31×#95—42no—
80C31×#95—42no—
81C31×#95—42no—
82C31×#95—42no—
83C31×#95—42no—
84C24×C31—22no—
85C24×C31—22no—
86C24×C31—22no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—21no—
90C24×C31—21no—
91C24×C31—22no—
92C24×C31C2×D412yes—
93C24×C31C2×Q812yes—
94C24×C31#4112yes—
95C24×C31C2411yescharacteristic
96C2×Q8×C31—48no—
97C2×Q8×C31—44no—
98C2×Q8×C31—44no—
99C2×Q8×C31—48no—
100C2×Q8×C31—44no—
101C22×C4×C31—28no—
102C22×C4×C31—22no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—22no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—21no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—22no—
115C22×C4×C31—22no—
116C22×C4×C31—22no—
117C22×C4×C31—22no—
118C22×C4×C31—22no—
119C22×C4×C31—22no—
120C22×C4×C31—22no—
121C22×C4×C31—22no—
122C22×C4×C31—22no—
123C22×C4×C31—21no—
124C22×C4×C31—21no—
125C22×C4×C31—21no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
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—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—22no—
149C22×C4×C31—44no—
150C22×C4×C31—44no—
151C22×C4×C31—44no—
152C22×C4×C31—44no—
153C22×C4×C31—44no—
154C22×C4×C31—44no—
155C22×C4×C31C2×D412yes—
156C22×C4×C31C2×D412yes—
157C22×C4×C31C2×Q814yes—
158C22×C4×C31#4114yes—
159C22×C4×C31#4114yes—
160C4⋊C4×C31—44no—
161C4⋊C4×C31—48no—
162C22⋊C4×C31—44no—
163C22⋊C4×C31—48no—
164C22⋊C4×C31—48no—
165C22⋊C4×C31—44no—
166#16666C3111yesmaximal, characteristic, direct product factor, Sylow
167C23×C31—24no—
168C23×C31—22no—
169C23×C31—22no—
170C23×C31—22no—
171C23×C31—22no—
172C23×C31—22no—
173C23×C31—22no—
174C23×C31—24no—
175C23×C31—24no—
176C23×C31—24no—
177C23×C31—24no—
178C23×C31—24no—
179C23×C31—42no—
180C23×C31—42no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—42no—
186C23×C31—41no—
187C23×C31—42no—
188C23×C31—41no—
189C23×C31C22⋊Q814yes—
190C23×C31C22×D411yescharacteristic
191C23×C31C22×Q811yescharacteristic
192C23×C31C2×#4111yescharacteristic
193C23×C31C2×#4112yes—
194C23×C31C2×#4112yes—
195C23×C31H(5,𝔽2)11yescharacteristic
196C23×C31H(5,𝔽2)11yescharacteristic
197C23×C31H(5,𝔽2)11yescharacteristic
198C23×C31H(5,𝔽2)11yescharacteristic
199C23×C31H(5,𝔽2)12yes—
200C23×C312−1+412yes—
201Q8×C31—416no—
202C2×C4×C31—24no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—24no—
206C2×C4×C31—24no—
207C2×C4×C31—24no—
208C2×C4×C31—24no—
209C2×C4×C31—24no—
210C2×C4×C31—24no—
211C2×C4×C31—24no—
212C2×C4×C31—24no—
213C2×C4×C31—24no—
214C2×C4×C31—24no—
215C2×C4×C31—416no—
216C2×C4×C31—42no—
217C2×C4×C31—42no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—42no—
229C2×C4×C31—41no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—42no—
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—42no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—42no—
249C2×C4×C31—42no—
250C2×C4×C31—42no—
251C2×C4×C31—42no—
252C2×C4×C31—42no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×#393C2×C3111yescharacteristic
256C2×#381C2×C3111yescharacteristic
257#1446C2×C3112yescomplemented
258#1445C2×C3112yescomplemented
259#1443C2×C3112yescomplemented
260#1441C2×C3112yescomplemented
261#1439C2×C3112yes—
262#1439C2×C3111yescharacteristic
263C22⋊#95C2×C3112yes—
264C22×C31—24no—
265C22×C31—28no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—44no—
269C22×C31—42no—
270C22×C31—44no—
271C22×C31—42no—
272C22×C31—44no—
273C22×C31—42no—
274C22×C31—42no—
275C22×C31C2×C22⋊Q811yescharacteristic
276C22×C31#53312yes—
277C22×C31#53411yescharacteristic
278C22×C31#53511yescharacteristic
279C22×C31#53811yescharacteristic
280C22×C31#53811yescharacteristic
281C22×C31#53812yes—
282C22×C31#53812yes—
283C22×C31#54212yes—
284C22×C31#54212yes—
285C22×C31#54311yescharacteristic
286C22×C31#54311yescharacteristic
287C22×C31#54412yes—
288C22×C31#54612yes—
289C22×C31#54712yes—
290C22×C31#55111yescharacteristic
291C22×C31#55111yescharacteristic
292C22×C31#55112yes—
293C22×C31#55212yes—
294C22×C31#55912yes—
295C22×C31#56011yescharacteristic
296C22×C31#56212yes—
297C22×C31#56311yescharacteristic
298C4×C31—44no—
299C4×C31—42no—
300C4×C31—44no—
301C4×C31—42no—
302C4×C31—44no—
303C4×C31—82no—
304C4×C31—82no—
305C4×C31—82no—
306C23×Q8C22×C3111yescharacteristic
307C24×C4C22×C3111yescharacteristic
308C22×C4⋊C4C22×C3111yescharacteristic
309C22×C4⋊C4C22×C3112yes—
310C22×C22⋊C4C22×C3112yes—
311C22×C22⋊C4C22×C3112yes—
312C22×C22⋊C4C22×C3111yescharacteristic
313C22×C22⋊C4C22×C3111yescharacteristic
314#393—24no—
315#381—24no—
316C2×#95C22×C3112yes—
317C2×#95C22×C3111yescharacteristic
318C2×#95C22×C3112yes—
319C2×#95C22×C3112yes—
320C2×#95C22×C3112yes—
321C2×#95C22×C3112yes—
322C2×#95C22×C3112yes—
323C2×#95C22×C3112yes—
324C2×#95C22×C3112yes—
325C2×#95C22×C3112yes—
326C2×#95C22×C3111yescharacteristic
327C2×#95C22×C3112yes—
328C2×#95C22×C3112yes—
329C2×C31—44no—
330C2×C31#243812yes—
331C2×C31#260711yescharacteristic
332C2×C31#262611yescharacteristic
333C2×C31#266512yes—
334C2×C31#269711yescharacteristic
335C2×C31#271012yes—
336C2×C31#271711yescharacteristic
337C2×C31#275112yes—
338C2×C31#276011yescharacteristic
339C2×C31#276411yescharacteristic
340C2×C31#284111yescharacteristic
341C25C23×C3111yescharacteristic
342C22×Q8—22no—
343C22×Q8—24no—
344C22×Q8—42no—
345C22×Q8—41no—
346C22×Q8—41no—
347C23×C4—22no—
348C23×C4—22no—
349C23×C4—41no—
350C23×C4—41no—
351C23×C4—41no—
352C23×C4—41no—
353C23×C4Q8×C3114yescomplemented
354C23×C4C23×C3112yes—
355C23×C4C23×C3112yes—
356C23×C4C23×C3112yes—
357C23×C4C23×C3111yescharacteristic
358C23×C4C23×C3111yescharacteristic
359C23×C4C23×C3112yes—
360C23×C4C23×C3112yes—
361C23×C4C23×C3112yes—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—42no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—41no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4—42no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4D4×C3114yes—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95—42no—
385#95—41no—
386#95—42no—
387#95—42no—
388#95—42no—
389#95—42no—
390#95—42no—
391#95—42no—
392C31#1666611yescharacteristic, direct product factor, Sylow
393C24—22no—
394C24—22no—
395C24—22no—
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—22no—
401C24C2×D4×C3112yes—
402C24C2×Q8×C3112yes—
403C24#41×C3112yes—
404C24C24×C3111yescharacteristic
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—48no—
409C2×Q8—44no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4C2×D4×C3112yes—
465C22×C4C2×D4×C3112yes—
466C22×C4C2×Q8×C3114yes—
467C22×C4#41×C3114yes—
468C22×C4#41×C3114yes—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—44no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23C31×C22⋊Q814yes—
498C23C22×D4×C3111yescharacteristic
499C23C22×Q8×C3111yescharacteristic
500C23C2×#41×C3111yescharacteristic
501C23C2×#41×C3112yes—
502C23C2×#41×C3112yes—
503C23C31×H(5,𝔽2)11yescharacteristic
504C23C31×H(5,𝔽2)11yescharacteristic
505C23C31×H(5,𝔽2)11yescharacteristic
506C23C31×H(5,𝔽2)11yescharacteristic
507C23C31×H(5,𝔽2)12yes—
508C23C31×2−1+412yes—
509Q8—416no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—416no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C22—24no—
564C22—28no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—42no—
569C22—44no—
570C22—42no—
571C22—44no—
572C22—42no—
573C22—42no—
574C22C2×C31×C22⋊Q811yescharacteristic
575C22C31×#53312yes—
576C22C31×#53411yescharacteristic
577C22C31×#53511yescharacteristic
578C22C31×#53811yescharacteristic
579C22C31×#53811yescharacteristic
580C22C31×#53812yes—
581C22C31×#53812yes—
582C22C31×#54212yes—
583C22C31×#54212yes—
584C22C31×#54311yescharacteristic
585C22C31×#54311yescharacteristic
586C22C31×#54412yes—
587C22C31×#54612yes—
588C22C31×#54712yes—
589C22C31×#55111yescharacteristic
590C22C31×#55111yescharacteristic
591C22C31×#55112yes—
592C22C31×#55212yes—
593C22C31×#55912yes—
594C22C31×#56011yescharacteristic
595C22C31×#56212yes—
596C22C31×#56311yescharacteristic
597C4—44no—
598C4—42no—
599C4—44no—
600C4—42no—
601C4—44no—
602C4—82no—
603C4—82no—
604C4—82no—
605C2—44no—
606C2C31×#243812yes—
607C2C31×#260711yescharacteristic
608C2C31×#262611yescharacteristic
609C2C31×#266512yes—
610C2C31×#269711yescharacteristic
611C2C31×#271012yes—
612C2C31×#271711yescharacteristic
613C2C31×#275112yes—
614C2C31×#276011yescharacteristic
615C2C31×#276411yescharacteristic
616C2C31×#284111yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC25×C31C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1446C2semidirect
4. C31×#1445C2semidirect
5. C31×#1443C2semidirect
6. C31×#1441C2semidirect
45. C23×C4×C31Q8semidirect
166. #16666C31direct (N × H)
257. #1446C2×C31semidirect
258. #1445C2×C31semidirect
259. #1443C2×C31semidirect
260. #1441C2×C31semidirect
353. C23×C4Q8×C31semidirect
392. C31#16666direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3932
C2×C31×#3812
C31×#14462
C31×#14452
C31×#14432
C31×#14412
C31×#14392
C31×C22⋊#952
#1666631