Finite Group Databasebeta

← C31×#15261 #21472020

Subgroups — C31×#15261 #21472020

Order 7936.

2942 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#127C211yesmaximal, characteristic
2C22×C31×#123C211yesmaximal, characteristic, complemented
3C2×C31×#394C212yesmaximal, complemented
4C2×C31×#386C212yesmaximal, complemented
5C31×#1438C214yesmaximal, complemented
6C31×#1438C212yesmaximal, complemented
7C31×#1438C211yesmaximal, characteristic, complemented
8C31×#1436C212yesmaximal, complemented
9C23×D4×C31C2211yescharacteristic
10C23×D4×C31C2212yes—
11C23×D4×C31C2211yescharacteristic
12C23×D4×C31C2212yes—
13C24×C4×C31C2211yescharacteristic
14C2×C31×#127—24no—
15C2×C31×#127—28no—
16C2×C31×#123—24no—
17C2×C31×#123—28no—
18C22×C4⋊C4×C31C2212yes—
19C22×C22⋊C4×C31C2211yescharacteristic
20C22×C22⋊C4×C31C2212yes—
21C22×C22⋊C4×C31C2214yes—
22C22×C22⋊C4×C31C2214yes—
23C22×C22⋊C4×C31C2212yes—
24C22×C22⋊C4×C31C2212yes—
25C22×C42×C31C2211yescharacteristic
26C31×#394—28no—
27C31×#386—28no—
28C2×C31×#95C2212yes—
29C2×C31×#95C2214yes—
30C2×C31×#95C2214yes—
31C25×C31C2311yescharacteristic
32C25×C31C2312yes—
33C25×C31C2311yescharacteristic
34C22×D4×C31—24no—
35C22×D4×C31—24no—
36C22×D4×C31—24no—
37C22×D4×C31—24no—
38C22×D4×C31—24no—
39C22×D4×C31—24no—
40C22×D4×C31—24no—
41C22×D4×C31—22no—
42C22×D4×C31—24no—
43C22×D4×C31—24no—
44C22×D4×C31—28no—
45C22×D4×C31—28no—
46C22×D4×C31—44no—
47C22×D4×C31—42no—
48C22×D4×C31—42no—
49C22×D4×C31—44no—
50C22×D4×C31D414yes—
51C22×D4×C31D418yes—
52C23×C4×C31—22no—
53C23×C4×C31—24no—
54C23×C4×C31—24no—
55C23×C4×C31—42no—
56C23×C4×C31C2314yes—
57C23×C4×C31C2312yes—
58C23×C4×C31C2312yes—
59C23×C4×C31C2311yescharacteristic
60C23×C4×C31C2312yes—
61C31×#127—232no—
62C31×#123—232no—
63C2×C4⋊C4×C31—24no—
64C2×C4⋊C4×C31—24no—
65C2×C4⋊C4×C31D418yescomplemented
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—24no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—24no—
70C2×C22⋊C4×C31—24no—
71C2×C22⋊C4×C31—28no—
72C2×C22⋊C4×C31—24no—
73C2×C22⋊C4×C31—28no—
74C2×C22⋊C4×C31—28no—
75C2×C22⋊C4×C31—44no—
76C2×C22⋊C4×C31—44no—
77C2×C22⋊C4×C31—44no—
78C2×C22⋊C4×C31—44no—
79C2×C22⋊C4×C31—44no—
80C2×C22⋊C4×C31—42no—
81C2×C42×C31—22no—
82C2×C42×C31—24no—
83C31×#95—28no—
84C31×#95—24no—
85C31×#95—28no—
86C24×C31—22no—
87C24×C31—24no—
88C24×C31—24no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—24no—
92C24×C31—22no—
93C24×C31—22no—
94C24×C31—22no—
95C24×C31—24no—
96C24×C31—24no—
97C24×C31—24no—
98C24×C31—24no—
99C24×C31—22no—
100C24×C31—24no—
101C24×C31—24no—
102C24×C31—24no—
103C24×C31C2×D412yes—
104C24×C31C2×D414yes—
105C24×C31C2×D412yes—
106C24×C31C2×D414yes—
107C24×C31C2411yescharacteristic
108C2×D4×C31—28no—
109C2×D4×C31—216no—
110C2×D4×C31—28no—
111C2×D4×C31—216no—
112C2×D4×C31—28no—
113C2×D4×C31—216no—
114C2×D4×C31—48no—
115C2×D4×C31—44no—
116C2×D4×C31—416no—
117C2×D4×C31—416no—
118C2×D4×C31—416no—
119C2×D4×C31—416no—
120C2×D4×C31—416no—
121C2×D4×C31—416no—
122C2×D4×C31—44no—
123C2×D4×C31—48no—
124C2×D4×C31—44no—
125C2×D4×C31—48no—
126C22×C4×C31—22no—
127C22×C4×C31—22no—
128C22×C4×C31—22no—
129C22×C4×C31—22no—
130C22×C4×C31—24no—
131C22×C4×C31—28no—
132C22×C4×C31—24no—
133C22×C4×C31—22no—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—22no—
137C22×C4×C31—24no—
138C22×C4×C31—24no—
139C22×C4×C31—24no—
140C22×C4×C31—24no—
141C22×C4×C31—24no—
142C22×C4×C31—24no—
143C22×C4×C31—24no—
144C22×C4×C31—24no—
145C22×C4×C31—24no—
146C22×C4×C31—24no—
147C22×C4×C31—416no—
148C22×C4×C31—44no—
149C22×C4×C31—44no—
150C22×C4×C31C2×D414yes—
151C22×C4×C31C2×D414yes—
152C22×C4×C31C2×D414yes—
153C22×C4×C31C2×D412yes—
154C22×C4×C31C2×D414yes—
155C22×C4×C31#4118yes—
156C4⋊C4×C31—216no—
157C22⋊C4×C31—28no—
158C22⋊C4×C31—216no—
159C22⋊C4×C31—416no—
160C22⋊C4×C31—416no—
161C22⋊C4×C31—48no—
162C22⋊C4×C31—48no—
163C42×C31—28no—
164#15261C3111yesmaximal, characteristic, direct product factor, Sylow
165C23×C31—22no—
166C23×C31—24no—
167C23×C31—24no—
168C23×C31—22no—
169C23×C31—28no—
170C23×C31—28no—
171C23×C31—28no—
172C23×C31—22no—
173C23×C31—22no—
174C23×C31—24no—
175C23×C31—24no—
176C23×C31—22no—
177C23×C31—24no—
178C23×C31—28no—
179C23×C31—28no—
180C23×C31—28no—
181C23×C31—42no—
182C23×C31—42no—
183C23×C31—44no—
184C23×C31—44no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—42no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—44no—
192C23×C31—42no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—44no—
196C23×C31—42no—
197C23×C31—44no—
198C23×C31—44no—
199C23×C31—44no—
200C23×C31—44no—
201C23×C31—44no—
202C23×C31—44no—
203C23×C31—44no—
204C23×C31—44no—
205C23×C31—44no—
206C23×C31—44no—
207C23×C31—44no—
208C23×C31—44no—
209C23×C31—44no—
210C23×C31—42no—
211C23×C31—42no—
212C23×C31—44no—
213C23×C31#12014yes—
214C23×C31C22×D411yescharacteristic
215C23×C31C22×D412yes—
216C23×C31C22×D412yes—
217C23×C31C2×#4114yes—
218C23×C31H(5,𝔽2)14yes—
219C23×C31H(5,𝔽2)12yes—
220D4×C31—416no—
221D4×C31—432no—
222D4×C31—416no—
223D4×C31—432no—
224C2×C4×C31—22no—
225C2×C4×C31—24no—
226C2×C4×C31—22no—
227C2×C4×C31—24no—
228C2×C4×C31—28no—
229C2×C4×C31—28no—
230C2×C4×C31—28no—
231C2×C4×C31—24no—
232C2×C4×C31—24no—
233C2×C4×C31—24no—
234C2×C4×C31—24no—
235C2×C4×C31—28no—
236C2×C4×C31—416no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—44no—
240C2×C4×C31—44no—
241C2×C4×C31—44no—
242C2×C4×C31—44no—
243C2×C4×C31—44no—
244C2×C4×C31—44no—
245C2×C4×C31—44no—
246C2×C4×C31—44no—
247C2×C4×C31—44no—
248C2×C4×C31—44no—
249C2×C4×C31#12014yes—
250C2×C4×C31#12318yescomplemented
251C22×#127C2×C3111yescharacteristic
252C22×#123C2×C3111yescharacteristic, complemented
253C2×#394C2×C3112yescomplemented
254C2×#386C2×C3112yescomplemented
255#1438C2×C3114yescomplemented
256#1438C2×C3112yescomplemented
257#1438C2×C3111yescharacteristic, complemented
258#1436C2×C3112yescomplemented
259C22×C31—24no—
260C22×C31—28no—
261C22×C31—48no—
262C22×C31—44no—
263C22×C31—48no—
264C22×C31—44no—
265C22×C31—42no—
266C22×C31—44no—
267C22×C31—44no—
268C22×C31—44no—
269C22×C31—44no—
270C22×C31—48no—
271C22×C31—44no—
272C22×C31—48no—
273C22×C31—44no—
274C22×C31—84no—
275C22×C31—84no—
276C22×C31—84no—
277C22×C31—84no—
278C22×C31—84no—
279C22×C31—84no—
280C22×C31C2×#12011yescharacteristic
281C22×C31C2×#12011yescharacteristic
282C22×C31C2×#12312yes—
283C22×C31#53212yes—
284C22×C31#53411yescharacteristic
285C22×C31#53412yes—
286C22×C31#53512yes—
287C22×C31#54014yes—
288C22×C31D4214yes—
289C22×C31#54614yes—
290C22×C31#54614yes—
291C22×C31#55014yes—
292C22×C31#55912yes—
293C22×C31#56012yes—
294C4×C31—24no—
295C4×C31—28no—
296C4×C31—44no—
297C4×C31—44no—
298C4×C31—84no—
299C23×D4C22×C3111yescharacteristic
300C23×D4C22×C3112yes—
301C23×D4C22×C3111yescharacteristic
302C23×D4C22×C3112yes—
303C24×C4C22×C3111yescharacteristic
304C2×#127—24no—
305C2×#127—28no—
306C2×#123—24no—
307C2×#123—28no—
308C22×C4⋊C4C22×C3112yes—
309C22×C22⋊C4C22×C3111yescharacteristic
310C22×C22⋊C4C22×C3112yes—
311C22×C22⋊C4C22×C3114yes—
312C22×C22⋊C4C22×C3114yes—
313C22×C22⋊C4C22×C3112yes—
314C22×C22⋊C4C22×C3112yes—
315C22×C42C22×C3111yescharacteristic
316#394—28no—
317#386—28no—
318C2×#95C22×C3112yes—
319C2×#95C22×C3114yes—
320C2×#95C22×C3114yes—
321C2×C31—44no—
322C2×C31—84no—
323C2×C31—82no—
324C2×C31#240311yescharacteristic
325C2×C31#240512yes—
326C2×C31#243112yes—
327C2×C31#244514yes—
328C2×C31#267314yes—
329C2×C31#270912yes—
330C25C23×C3111yescharacteristic
331C25C23×C3112yes—
332C25C23×C3111yescharacteristic
333C22×D4—24no—
334C22×D4—24no—
335C22×D4—24no—
336C22×D4—24no—
337C22×D4—24no—
338C22×D4—24no—
339C22×D4—24no—
340C22×D4—22no—
341C22×D4—24no—
342C22×D4—24no—
343C22×D4—28no—
344C22×D4—28no—
345C22×D4—44no—
346C22×D4—42no—
347C22×D4—42no—
348C22×D4—44no—
349C22×D4D4×C3114yes—
350C22×D4D4×C3118yes—
351C23×C4—22no—
352C23×C4—24no—
353C23×C4—24no—
354C23×C4—42no—
355C23×C4C23×C3114yes—
356C23×C4C23×C3112yes—
357C23×C4C23×C3112yes—
358C23×C4C23×C3111yescharacteristic
359C23×C4C23×C3112yes—
360#127—232no—
361#123—232no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4D4×C3118yescomplemented
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—28no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—28no—
373C2×C22⋊C4—28no—
374C2×C22⋊C4—44no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4—44no—
377C2×C22⋊C4—44no—
378C2×C22⋊C4—44no—
379C2×C22⋊C4—42no—
380C2×C42—22no—
381C2×C42—24no—
382#95—28no—
383#95—24no—
384#95—28no—
385C31#1526111yescharacteristic, direct product factor, Sylow
386C24—22no—
387C24—24no—
388C24—24no—
389C24—22no—
390C24—22no—
391C24—24no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—24no—
396C24—24no—
397C24—24no—
398C24—24no—
399C24—22no—
400C24—24no—
401C24—24no—
402C24—24no—
403C24C2×D4×C3112yes—
404C24C2×D4×C3114yes—
405C24C2×D4×C3112yes—
406C24C2×D4×C3114yes—
407C24C24×C3111yescharacteristic
408C2×D4—28no—
409C2×D4—216no—
410C2×D4—28no—
411C2×D4—216no—
412C2×D4—28no—
413C2×D4—216no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—416no—
417C2×D4—416no—
418C2×D4—416no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—44no—
423C2×D4—48no—
424C2×D4—44no—
425C2×D4—48no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—416no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4C2×D4×C3114yes—
451C22×C4C2×D4×C3114yes—
452C22×C4C2×D4×C3114yes—
453C22×C4C2×D4×C3112yes—
454C22×C4C2×D4×C3114yes—
455C22×C4#41×C3118yes—
456C4⋊C4—216no—
457C22⋊C4—28no—
458C22⋊C4—216no—
459C22⋊C4—416no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C42—28no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23C31×#12014yes—
513C23C22×D4×C3111yescharacteristic
514C23C22×D4×C3112yes—
515C23C22×D4×C3112yes—
516C23C2×#41×C3114yes—
517C23C31×H(5,𝔽2)14yes—
518C23C31×H(5,𝔽2)12yes—
519D4—416no—
520D4—432no—
521D4—416no—
522D4—432no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—416no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4C31×#12014yes—
549C2×C4C31×#12318yescomplemented
550C22—24no—
551C22—28no—
552C22—48no—
553C22—44no—
554C22—48no—
555C22—44no—
556C22—42no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—48no—
562C22—44no—
563C22—48no—
564C22—44no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22C2×C31×#12011yescharacteristic
572C22C2×C31×#12011yescharacteristic
573C22C2×C31×#12312yes—
574C22C31×#53212yes—
575C22C31×#53411yescharacteristic
576C22C31×#53412yes—
577C22C31×#53512yes—
578C22C31×#54014yes—
579C22D42×C3114yes—
580C22C31×#54614yes—
581C22C31×#54614yes—
582C22C31×#55014yes—
583C22C31×#55912yes—
584C22C31×#56012yes—
585C4—24no—
586C4—28no—
587C4—44no—
588C4—44no—
589C4—84no—
590C2—44no—
591C2—84no—
592C2—82no—
593C2C31×#240311yescharacteristic
594C2C31×#240512yes—
595C2C31×#243112yes—
596C2C31×#244514yes—
597C2C31×#267314yes—
598C2C31×#270912yes—
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC22×C31×#127C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C31×#123C2semidirect
3. C2×C31×#394C2semidirect
4. C2×C31×#386C2semidirect
5. C31×#1438C2semidirect
6. C31×#1438C2semidirect
7. C31×#1438C2semidirect
8. C31×#1436C2semidirect
65. C2×C4⋊C4×C31D4semidirect
164. #15261C31direct (N × H)
250. C2×C4×C31#123semidirect
252. C22×#123C2×C31semidirect
253. C2×#394C2×C31semidirect
254. C2×#386C2×C31semidirect
255. #1438C2×C31semidirect
256. #1438C2×C31semidirect
257. #1438C2×C31semidirect
258. #1436C2×C31semidirect
364. C2×C4⋊C4D4×C31semidirect
385. C31#15261direct (N × H)
549. C2×C4C31×#123semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1272
C22×C31×#1232
C2×C31×#3942
C2×C31×#3862
C31×#14382
C31×#14362
#1526131