Finite Group Databasebeta

← C31×#15506 #21472265

Subgroups — C31×#15506 #21472265

Order 7936.

1954 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#393C212yesmaximal, complemented
2C2×C31×#386C212yesmaximal
3C31×#1446C211yesmaximal, characteristic, complemented
4C31×#1445C211yesmaximal, characteristic, complemented
5C31×#1443C212yesmaximal, complemented
6C31×#1439C212yesmaximal, complemented
7C31×#1438C211yesmaximal, characteristic
8C31×C22⋊#95C212yesmaximal, complemented
9C31×C22⋊#95C212yesmaximal, complemented
10C23×Q8×C31C2211yescharacteristic
11C23×D4×C31C2211yescharacteristic
12C24×C4×C31C2212yes—
13C22×C4⋊C4×C31C2211yescharacteristic
14C22×C22⋊C4×C31C2212yes—
15C22×C22⋊C4×C31C2212yes—
16C22×C22⋊C4×C31C2212yes—
17C22×C22⋊C4×C31C2212yes—
18C22×C22⋊C4×C31C2212yes—
19C31×#393—28no—
20C31×#386—28no—
21C2×C31×#95C2212yes—
22C2×C31×#95C2212yes—
23C2×C31×#95C2212yes—
24C2×C31×#95C2212yes—
25C2×C31×#95C2212yes—
26C2×C31×#95C2212yes—
27C2×C31×#95C2212yes—
28C2×C31×#95C2211yescharacteristic
29C2×C31×#95C2211yescharacteristic
30C2×C31×#95C2212yes—
31C2×C31×#95C2211yescharacteristic
32C2×C31×#95C2211yescharacteristic
33C25×C31C2312yes—
34C22×Q8×C31—24no—
35C22×Q8×C31—22no—
36C22×Q8×C31—41no—
37C22×Q8×C31—42no—
38C22×Q8×C31—41no—
39C22×D4×C31—24no—
40C22×D4×C31—24no—
41C22×D4×C31—41no—
42C22×D4×C31—41no—
43C22×D4×C31Q814yescomplemented
44C23×C4×C31—24no—
45C23×C4×C31—24no—
46C23×C4×C31—24no—
47C23×C4×C31—42no—
48C23×C4×C31—42no—
49C23×C4×C31—42no—
50C23×C4×C31—42no—
51C23×C4×C31C2312yes—
52C23×C4×C31C2312yes—
53C23×C4×C31C2312yes—
54C23×C4×C31C2312yes—
55C23×C4×C31C2311yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C23×C4×C31C2311yescharacteristic
58C23×C4×C31C2312yes—
59C2×C4⋊C4×C31—22no—
60C2×C4⋊C4×C31—41no—
61C2×C4⋊C4×C31—41no—
62C2×C22⋊C4×C31—24no—
63C2×C22⋊C4×C31—24no—
64C2×C22⋊C4×C31—24no—
65C2×C22⋊C4×C31—24no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—42no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31—42no—
73C2×C22⋊C4×C31—42no—
74C2×C22⋊C4×C31—42no—
75C2×C22⋊C4×C31—42no—
76C2×C22⋊C4×C31D418yes—
77C31×#95—24no—
78C31×#95—24no—
79C31×#95—24no—
80C31×#95—24no—
81C31×#95—24no—
82C31×#95—24no—
83C31×#95—42no—
84C31×#95—41no—
85C31×#95—41no—
86C31×#95—41no—
87C31×#95—42no—
88C31×#95—41no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—22no—
96C24×C31—22no—
97C24×C31—22no—
98C24×C31—22no—
99C24×C31—22no—
100C24×C31—22no—
101C24×C31C2×D414yes—
102C24×C31C2×Q814yes—
103C24×C31#4114yes—
104C24×C31C2411yescharacteristic
105C2×Q8×C31—44no—
106C2×Q8×C31—48no—
107C2×Q8×C31—48no—
108C2×Q8×C31—44no—
109C2×Q8×C31—44no—
110C2×D4×C31—216no—
111C2×D4×C31—44no—
112C2×D4×C31—44no—
113C2×D4×C31—48no—
114C2×D4×C31—44no—
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—22no—
124C22×C4×C31—22no—
125C22×C4×C31—22no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
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—22no—
150C22×C4×C31—22no—
151C22×C4×C31—22no—
152C22×C4×C31—22no—
153C22×C4×C31—22no—
154C22×C4×C31—22no—
155C22×C4×C31—22no—
156C22×C4×C31—22no—
157C22×C4×C31—22no—
158C22×C4×C31—22no—
159C22×C4×C31—48no—
160C22×C4×C31—48no—
161C22×C4×C31—48no—
162C22×C4×C31—48no—
163C22×C4×C31—48no—
164C22×C4×C31—48no—
165C22×C4×C31—48no—
166C22×C4×C31C2×D414yes—
167C22×C4×C31C2×D414yes—
168C22×C4×C31C2×Q812yes—
169C22×C4×C31#4114yes—
170C22×C4×C31#4112yes—
171C22×C4×C31#4112yes—
172C4⋊C4×C31—44no—
173C22⋊C4×C31—48no—
174C22⋊C4×C31—48no—
175C22⋊C4×C31—48no—
176C22⋊C4×C31—48no—
177C22⋊C4×C31—48no—
178#15506C3111yesmaximal, characteristic, direct product factor, Sylow
179C23×C31—24no—
180C23×C31—24no—
181C23×C31—24no—
182C23×C31—24no—
183C23×C31—24no—
184C23×C31—24no—
185C23×C31—24no—
186C23×C31—24no—
187C23×C31—24no—
188C23×C31—24no—
189C23×C31—24no—
190C23×C31—24no—
191C23×C31—24no—
192C23×C31—24no—
193C23×C31—24no—
194C23×C31—24no—
195C23×C31—24no—
196C23×C31—24no—
197C23×C31—42no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31—42no—
202C23×C31—42no—
203C23×C31—42no—
204C23×C31—42no—
205C23×C31—42no—
206C23×C31—42no—
207C23×C31—42no—
208C23×C31—42no—
209C23×C31—42no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—42no—
213C23×C31C22⋊Q818yes—
214C23×C31C22×D412yes—
215C23×C31C22×Q811yescharacteristic
216C23×C31C2×#4112yes—
217C23×C31C2×#4111yescharacteristic
218C23×C31C2×#4112yes—
219C23×C31C2×#4111yescharacteristic
220C23×C31H(5,𝔽2)11yescharacteristic
221C23×C31H(5,𝔽2)12yes—
222C23×C31H(5,𝔽2)11yescharacteristic
223C23×C31H(5,𝔽2)11yescharacteristic
224C23×C31H(5,𝔽2)11yescharacteristic
225Q8×C31—416no—
226D4×C31—416no—
227C2×C4×C31—24no—
228C2×C4×C31—24no—
229C2×C4×C31—24no—
230C2×C4×C31—22no—
231C2×C4×C31—24no—
232C2×C4×C31—22no—
233C2×C4×C31—22no—
234C2×C4×C31—24no—
235C2×C4×C31—24no—
236C2×C4×C31—24no—
237C2×C4×C31—24no—
238C2×C4×C31—24no—
239C2×C4×C31—24no—
240C2×C4×C31—24no—
241C2×C4×C31—24no—
242C2×C4×C31—432no—
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—41no—
252C2×C4×C31—41no—
253C2×C4×C31—42no—
254C2×C4×C31—42no—
255C2×C4×C31—42no—
256C2×C4×C31—41no—
257C2×C4×C31—42no—
258C2×C4×C31—41no—
259C2×C4×C31—41no—
260C2×C4×C31—42no—
261C2×C4×C31—41no—
262C2×C4×C31—42no—
263C2×C4×C31—42no—
264C2×C4×C31—42no—
265C2×C4×C31—42no—
266C2×C4×C31—42no—
267C2×C4×C31—42no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—42no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—42no—
276C2×C4×C31—42no—
277C2×C4×C31—42no—
278C2×#393C2×C3112yescomplemented
279C2×#386C2×C3112yes—
280#1446C2×C3111yescharacteristic, complemented
281#1445C2×C3111yescharacteristic, complemented
282#1443C2×C3112yescomplemented
283#1439C2×C3112yescomplemented
284#1438C2×C3111yescharacteristic
285C22⋊#95C2×C3112yescomplemented
286C22⋊#95C2×C3112yescomplemented
287C22×C31—28no—
288C22×C31—28no—
289C22×C31—28no—
290C22×C31—44no—
291C22×C31—44no—
292C22×C31—44no—
293C22×C31—44no—
294C22×C31—44no—
295C22×C31—44no—
296C22×C31—44no—
297C22×C31—44no—
298C22×C31—44no—
299C22×C31—44no—
300C22×C31—44no—
301C22×C31—44no—
302C22×C31C2×C22⋊Q812yes—
303C22×C31#52912yes—
304C22×C31#53212yes—
305C22×C31#53412yes—
306C22×C31#53811yescharacteristic
307C22×C31#53812yes—
308C22×C31#53812yes—
309C22×C31#53811yescharacteristic
310C22×C31#54311yescharacteristic
311C22×C31D4211yescharacteristic
312C22×C31#54612yes—
313C22×C31#54612yes—
314C22×C31#54612yes—
315C22×C31#55112yes—
316C22×C31#55112yes—
317C22×C31#55111yescharacteristic
318C22×C31#55411yescharacteristic
319C22×C31#55412yes—
320C22×C31#55411yescharacteristic
321C22×C31#55511yescharacteristic
322C22×C31#55911yescharacteristic
323C22×C31#56011yescharacteristic
324C22×C31H(3,𝔽4)11yescharacteristic
325C4×C31—44no—
326C4×C31—42no—
327C4×C31—44no—
328C4×C31—44no—
329C4×C31—42no—
330C4×C31—42no—
331C4×C31—82no—
332C4×C31—82no—
333C23×Q8C22×C3111yescharacteristic
334C23×D4C22×C3111yescharacteristic
335C24×C4C22×C3112yes—
336C22×C4⋊C4C22×C3111yescharacteristic
337C22×C22⋊C4C22×C3112yes—
338C22×C22⋊C4C22×C3112yes—
339C22×C22⋊C4C22×C3112yes—
340C22×C22⋊C4C22×C3112yes—
341C22×C22⋊C4C22×C3112yes—
342#393—28no—
343#386—28no—
344C2×#95C22×C3112yes—
345C2×#95C22×C3112yes—
346C2×#95C22×C3112yes—
347C2×#95C22×C3112yes—
348C2×#95C22×C3112yes—
349C2×#95C22×C3112yes—
350C2×#95C22×C3112yes—
351C2×#95C22×C3111yescharacteristic
352C2×#95C22×C3111yescharacteristic
353C2×#95C22×C3112yes—
354C2×#95C22×C3111yescharacteristic
355C2×#95C22×C3111yescharacteristic
356C2×C31—48no—
357C2×C31#243711yescharacteristic
358C2×C31#245012yes—
359C2×C31#255511yescharacteristic
360C2×C31#255712yes—
361C2×C31#256412yes—
362C2×C31#257211yescharacteristic
363C2×C31#266811yescharacteristic
364C2×C31#268412yes—
365C2×C31#276011yescharacteristic
366C2×C31#278411yescharacteristic
367C2×C31#280411yescharacteristic
368C25C23×C3112yes—
369C22×Q8—24no—
370C22×Q8—22no—
371C22×Q8—41no—
372C22×Q8—42no—
373C22×Q8—41no—
374C22×D4—24no—
375C22×D4—24no—
376C22×D4—41no—
377C22×D4—41no—
378C22×D4Q8×C3114yescomplemented
379C23×C4—24no—
380C23×C4—24no—
381C23×C4—24no—
382C23×C4—42no—
383C23×C4—42no—
384C23×C4—42no—
385C23×C4—42no—
386C23×C4C23×C3112yes—
387C23×C4C23×C3112yes—
388C23×C4C23×C3112yes—
389C23×C4C23×C3112yes—
390C23×C4C23×C3111yescharacteristic
391C23×C4C23×C3111yescharacteristic
392C23×C4C23×C3111yescharacteristic
393C23×C4C23×C3112yes—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—41no—
396C2×C4⋊C4—41no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—42no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4—42no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4D4×C3118yes—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—42no—
419#95—41no—
420#95—41no—
421#95—41no—
422#95—42no—
423#95—41no—
424C31#1550611yescharacteristic, direct product factor, Sylow
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24C2×D4×C3114yes—
438C24C2×Q8×C3114yes—
439C24#41×C3114yes—
440C24C24×C3111yescharacteristic
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—48no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×D4—216no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—48no—
450C2×D4—44no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4C2×D4×C3114yes—
503C22×C4C2×D4×C3114yes—
504C22×C4C2×Q8×C3112yes—
505C22×C4#41×C3114yes—
506C22×C4#41×C3112yes—
507C22×C4#41×C3112yes—
508C4⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23C31×C22⋊Q818yes—
549C23C22×D4×C3112yes—
550C23C22×Q8×C3111yescharacteristic
551C23C2×#41×C3112yes—
552C23C2×#41×C3111yescharacteristic
553C23C2×#41×C3112yes—
554C23C2×#41×C3111yescharacteristic
555C23C31×H(5,𝔽2)11yescharacteristic
556C23C31×H(5,𝔽2)12yes—
557C23C31×H(5,𝔽2)11yescharacteristic
558C23C31×H(5,𝔽2)11yescharacteristic
559C23C31×H(5,𝔽2)11yescharacteristic
560Q8—416no—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—432no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C22—28no—
614C22—28no—
615C22—28no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22C2×C31×C22⋊Q812yes—
629C22C31×#52912yes—
630C22C31×#53212yes—
631C22C31×#53412yes—
632C22C31×#53811yescharacteristic
633C22C31×#53812yes—
634C22C31×#53812yes—
635C22C31×#53811yescharacteristic
636C22C31×#54311yescharacteristic
637C22D42×C3111yescharacteristic
638C22C31×#54612yes—
639C22C31×#54612yes—
640C22C31×#54612yes—
641C22C31×#55112yes—
642C22C31×#55112yes—
643C22C31×#55111yescharacteristic
644C22C31×#55411yescharacteristic
645C22C31×#55412yes—
646C22C31×#55411yescharacteristic
647C22C31×#55511yescharacteristic
648C22C31×#55911yescharacteristic
649C22C31×#56011yescharacteristic
650C22C31×H(3,𝔽4)11yescharacteristic
651C4—44no—
652C4—42no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C2—48no—
660C2C31×#243711yescharacteristic
661C2C31×#245012yes—
662C2C31×#255511yescharacteristic
663C2C31×#255712yes—
664C2C31×#256412yes—
665C2C31×#257211yescharacteristic
666C2C31×#266811yescharacteristic
667C2C31×#268412yes—
668C2C31×#276011yescharacteristic
669C2C31×#278411yescharacteristic
670C2C31×#280411yescharacteristic
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC23×D4×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#393C2semidirect
3. C31×#1446C2semidirect
4. C31×#1445C2semidirect
5. C31×#1443C2semidirect
6. C31×#1439C2semidirect
8. C31×C22⋊#95C2semidirect
9. C31×C22⋊#95C2semidirect
43. C22×D4×C31Q8semidirect
178. #15506C31direct (N × H)
278. C2×#393C2×C31semidirect
280. #1446C2×C31semidirect
281. #1445C2×C31semidirect
282. #1443C2×C31semidirect
283. #1439C2×C31semidirect
285. C22⋊#95C2×C31semidirect
286. C22⋊#95C2×C31semidirect
378. C22×D4Q8×C31semidirect
424. C31#15506direct (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×#3862
C31×#14462
C31×#14452
C31×#14432
C31×#14392
C31×#14382
C31×C22⋊#952
#1550631