Finite Group Databasebeta

← C31×#20908 #21477667

Subgroups — C31×#20908 #21477667

Order 7936.

1986 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#525C211yesmaximal, characteristic
2C22×C31×#117C211yesmaximal, characteristic, complemented
3C2×C31×#416C212yesmaximal, complemented
4C22×C31×C22⋊C8C211yesmaximal, characteristic, complemented
5C31×#1875C212yesmaximal, complemented
6C31×#1874C212yesmaximal, complemented
7C31×#1873C212yesmaximal, complemented
8C31×#1872C212yesmaximal, complemented
9C31×#1735C212yesmaximal, complemented
10C22×#41×C31C2211yescharacteristic
11C24×C4×C31C2211yescharacteristic
12C23×C8×C31C2212yes—
13C31×#525—24no—
14C31×#525C418yescomplemented
15C2×C31×#123—21no—
16C2×C31×C22⋊Q8C2211yescharacteristic
17C2×C31×C22⋊Q8C2211yescharacteristic
18C2×C31×#121C2211yescharacteristic
19C2×C31×#121C2211yescharacteristic
20C2×C31×#120—21no—
21C2×C4×D4×C31—21no—
22C2×C4×D4×C31—21no—
23C2×C31×#117—21no—
24C2×C31×#117—21no—
25C2×C31×#117—28no—
26C2×C31×#117C2212yes—
27C2×C31×#117C2211yescharacteristic
28C2×C31×#117C2212yes—
29C22×C4⋊C4×C31C2212yes—
30C22×C22⋊C4×C31—21no—
31C22×C42×C31—21no—
32C31×#416—28no—
33C2×C31×#103C2214yescomplemented
34C2×C31×#102C2214yescomplemented
35C2×C31×C22⋊C8—24no—
36C2×C31×C22⋊C8C2212yes—
37C2×C31×C22⋊C8C2212yes—
38C31×#340C2214yescomplemented
39C31×#340C2214yescomplemented
40C25×C31D411yescharacteristic
41C2×#41×C31—22no—
42C2×#41×C31—24no—
43C2×#41×C31—24no—
44C2×#41×C31C2×C414yescomplemented
45C22×Q8×C31C2311yescharacteristic
46C22×D4×C31—21no—
47C22×D4×C31—21no—
48C22×D4×C31—21no—
49C22×D4×C31C2311yescharacteristic
50C23×C4×C31—24no—
51C23×C4×C31—24no—
52C23×C4×C31—22no—
53C23×C4×C31—21no—
54C23×C4×C31C2×C414yes—
55C23×C4×C31D411yescharacteristic
56C23×C4×C31D411yescharacteristic
57C23×C4×C31D411yescharacteristic
58C23×C4×C31D414yes—
59C23×C4×C31C2311yescharacteristic
60C23×C4×C31C2311yescharacteristic
61C23×C4×C31C2311yescharacteristic
62C22×C8×C31—24no—
63C22×C8×C31—24no—
64C22×C8×C31—24no—
65C22×C8×C31C2314yes—
66C31×#123—28no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—22no—
69C31×C22⋊Q8C2×C414yes—
70C31×C22⋊Q8C2×C414yescomplemented
71C31×#121—22no—
72C31×#121—22no—
73C31×#121C2×C414yes—
74C31×#121C2×C414yescomplemented
75C31×#120—28no—
76C4×D4×C31—28no—
77C4×D4×C31—28no—
78C31×#117—28no—
79C31×#117—28no—
80C31×#117—24no—
81C31×#117—22no—
82C31×#117—28no—
83C31×#117C2×C414yescomplemented
84C31×#117D418yescomplemented
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—28no—
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C4⋊C4×C31C2314yes—
92C2×C22⋊C4×C31—28no—
93C2×C22⋊C4×C31—22no—
94C2×C22⋊C4×C31—22no—
95C2×C22⋊C4×C31—21no—
96C2×C22⋊C4×C31—21no—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C42×C31—28no—
101C2×C42×C31—22no—
102C2×C42×C31—22no—
103C2×C42×C31—21no—
104C31×#103—28no—
105C31×#102—28no—
106C31×C22⋊C8—24no—
107C31×C22⋊C8—24no—
108C24×C31—22no—
109C24×C31—22no—
110C24×C31—24no—
111C24×C31—21no—
112C24×C31—42no—
113C24×C31C22⋊C414yes—
114C24×C31C2×D411yescharacteristic
115C24×C31C2×D411yescharacteristic
116C24×C31C2×D411yescharacteristic
117#41×C31—216no—
118#41×C31—48no—
119C2×Q8×C31—21no—
120C2×Q8×C31—22no—
121C2×Q8×C31—22no—
122C2×Q8×C31C22×C412yes—
123C2×D4×C31—24no—
124C2×D4×C31—24no—
125C2×D4×C31—24no—
126C2×D4×C31—22no—
127C2×D4×C31—21no—
128C2×D4×C31—22no—
129C2×D4×C31—44no—
130C2×D4×C31—42no—
131C2×D4×C31—42no—
132C2×D4×C31—44no—
133C2×D4×C31C22×C412yes—
134C22×C4×C31—22no—
135C22×C4×C31—22no—
136C22×C4×C31—216no—
137C22×C4×C31—22no—
138C22×C4×C31—22no—
139C22×C4×C31—216no—
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—28no—
147C22×C4×C31—24no—
148C22×C4×C31—21no—
149C22×C4×C31—21no—
150C22×C4×C31—21no—
151C22×C4×C31—24no—
152C22×C4×C31—44no—
153C22×C4×C31—44no—
154C22×C4×C31—42no—
155C22×C4×C31—42no—
156C22×C4×C31C22⋊C414yes—
157C22×C4×C31C22⋊C414yes—
158C22×C4×C31C22⋊C414yes—
159C22×C4×C31C22×C412yes—
160C22×C4×C31C22×C412yes—
161C22×C4×C31C22×C412yes—
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D412yes—
165C22×C4×C31C2×D412yes—
166C22×C4×C31C2×D412yes—
167C22×C4×C31C2×D414yes—
168C22×C4×C31C2×D411yescharacteristic
169C22×C4×C31#4114yes—
170C22×C4×C31C2411yescharacteristic
171C2×C8×C31—24no—
172C2×C8×C31—24no—
173C2×C8×C31—24no—
174C2×C8×C31—48no—
175C4⋊C4×C31—28no—
176C4⋊C4×C31—24no—
177C4⋊C4×C31—24no—
178C4⋊C4×C31—24no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—21no—
181C4⋊C4×C31C22×C412yes—
182C4⋊C4×C31C22×C412yes—
183C4⋊C4×C31C2×D418yes—
184C22⋊C4×C31—28no—
185C22⋊C4×C31—24no—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—28no—
189C22⋊C4×C31—24no—
190C22⋊C4×C31—24no—
191C42×C31—28no—
192C42×C31—24no—
193C42×C31—28no—
194#20908C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—21no—
196C23×C31—21no—
197C23×C31—22no—
198C23×C31—22no—
199C23×C31—22no—
200C23×C31—21no—
201C23×C31—21no—
202C23×C31—22no—
203C23×C31—22no—
204C23×C31—28no—
205C23×C31—28no—
206C23×C31—22no—
207C23×C31—44no—
208C23×C31—48no—
209C23×C31—48no—
210C23×C31—41no—
211C23×C31—41no—
212C23×C31—42no—
213C23×C31—42no—
214C23×C31C2×C22⋊C412yes—
215C23×C31C2×C22⋊C412yes—
216C23×C31C2×C22⋊C412yes—
217C23×C31#12012yes—
218C23×C31#12312yes—
219C23×C31#13514yes—
220C23×C31C22×D411yescharacteristic
221Q8×C31—24no—
222Q8×C31—42no—
223D4×C31—24no—
224D4×C31—48no—
225D4×C31—48no—
226D4×C31—48no—
227D4×C31—42no—
228C2×C4×C31—21no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—28no—
233C2×C4×C31—24no—
234C2×C4×C31—24no—
235C2×C4×C31—24no—
236C2×C4×C31—24no—
237C2×C4×C31—28no—
238C2×C4×C31—28no—
239C2×C4×C31—22no—
240C2×C4×C31—22no—
241C2×C4×C31—22no—
242C2×C4×C31—28no—
243C2×C4×C31—416no—
244C2×C4×C31—41no—
245C2×C4×C31—41no—
246C2×C4×C31—44no—
247C2×C4×C31—42no—
248C2×C4×C31—44no—
249C2×C4×C31—42no—
250C2×C4×C31—48no—
251C2×C4×C31C2×C22⋊C412yes—
252C2×C4×C31C2×C22⋊C412yes—
253C2×C4×C31C2×C22⋊C412yes—
254C2×C4×C31C4×D418yes—
255C2×C4×C31#12012yes—
256C2×C4×C31#12118yes—
257C2×C4×C31#12312yes—
258C2×C4×C31C23×C411yescharacteristic
259C2×C4×C31C22×D412yes—
260C2×C4×C31C22×D411yescharacteristic
261C2×C4×C31C2×#4112yes—
262C8×C31—44no—
263C2×#525C2×C3111yescharacteristic
264C22×#117C2×C3111yescharacteristic, complemented
265C2×#416C2×C3112yescomplemented
266C22×C22⋊C8C2×C3111yescharacteristic, complemented
267#1875C2×C3112yescomplemented
268#1874C2×C3112yescomplemented
269#1873C2×C3112yescomplemented
270#1872C2×C3112yescomplemented
271#1735C2×C3112yescomplemented
272C22×C31—24no—
273C22×C31—24no—
274C22×C31—22no—
275C22×C31—22no—
276C22×C31—22no—
277C22×C31—24no—
278C22×C31—24no—
279C22×C31—22no—
280C22×C31—22no—
281C22×C31—22no—
282C22×C31—22no—
283C22×C31—416no—
284C22×C31—42no—
285C22×C31—42no—
286C22×C31—42no—
287C22×C31—42no—
288C22×C31—42no—
289C22×C31—82no—
290C22×C31—82no—
291C22×C31#38614yes—
292C22×C31#41614yescomplemented
293C22×C31C22×C22⋊C411yescharacteristic
294C22×C31C2×#12011yescharacteristic
295C22×C31C2×#12311yescharacteristic
296C22×C31C2×#13512yes—
297C22×C31#57512yes—
298C4×C31—22no—
299C4×C31—22no—
300C4×C31—42no—
301C4×C31—42no—
302C4×C31—48no—
303C4×C31—42no—
304C4×C31#38614yes—
305C4×C31C2×C4×D412yes—
306C4×C31C2×#12112yes—
307C22×#41C22×C3111yescharacteristic
308C24×C4C22×C3111yescharacteristic
309C23×C8C22×C3112yes—
310#525—24no—
311#525C4×C3118yescomplemented
312C2×#123—21no—
313C2×C22⋊Q8C22×C3111yescharacteristic
314C2×C22⋊Q8C22×C3111yescharacteristic
315C2×#121C22×C3111yescharacteristic
316C2×#121C22×C3111yescharacteristic
317C2×#120—21no—
318C2×C4×D4—21no—
319C2×C4×D4—21no—
320C2×#117—21no—
321C2×#117—21no—
322C2×#117—28no—
323C2×#117C22×C3112yes—
324C2×#117C22×C3111yescharacteristic
325C2×#117C22×C3112yes—
326C22×C4⋊C4C22×C3112yes—
327C22×C22⋊C4—21no—
328C22×C42—21no—
329#416—28no—
330C2×#103C22×C3114yescomplemented
331C2×#102C22×C3114yescomplemented
332C2×C22⋊C8—24no—
333C2×C22⋊C8C22×C3112yes—
334C2×C22⋊C8C22×C3112yes—
335#340C22×C3114yescomplemented
336#340C22×C3114yescomplemented
337C2×C31—24no—
338C2×C31—24no—
339C2×C31—42no—
340C2×C31—82no—
341C2×C31C2×#38611yescharacteristic
342C2×C31C2×#41611yescharacteristic
343C2×C31#289611yescharacteristic
344C2×C31#300212yes—
345C2×C31#309112yes—
346C25D4×C3111yescharacteristic
347C2×#41—22no—
348C2×#41—24no—
349C2×#41—24no—
350C2×#41C2×C4×C3114yescomplemented
351C22×Q8C23×C3111yescharacteristic
352C22×D4—21no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4C23×C3111yescharacteristic
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—22no—
359C23×C4—21no—
360C23×C4C2×C4×C3114yes—
361C23×C4D4×C3111yescharacteristic
362C23×C4D4×C3111yescharacteristic
363C23×C4D4×C3111yescharacteristic
364C23×C4D4×C3114yes—
365C23×C4C23×C3111yescharacteristic
366C23×C4C23×C3111yescharacteristic
367C23×C4C23×C3111yescharacteristic
368C22×C8—24no—
369C22×C8—24no—
370C22×C8—24no—
371C22×C8C23×C3114yes—
372#123—28no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8C2×C4×C3114yes—
376C22⋊Q8C2×C4×C3114yescomplemented
377#121—22no—
378#121—22no—
379#121C2×C4×C3114yes—
380#121C2×C4×C3114yescomplemented
381#120—28no—
382C4×D4—28no—
383C4×D4—28no—
384#117—28no—
385#117—28no—
386#117—24no—
387#117—22no—
388#117—28no—
389#117C2×C4×C3114yescomplemented
390#117D4×C3118yescomplemented
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—28no—
395C2×C4⋊C4C23×C3111yescharacteristic
396C2×C4⋊C4C23×C3111yescharacteristic
397C2×C4⋊C4C23×C3114yes—
398C2×C22⋊C4—28no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—22no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C42—28no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42—21no—
410#103—28no—
411#102—28no—
412C22⋊C8—24no—
413C22⋊C8—24no—
414C31#2090811yescharacteristic, direct product factor, Sylow
415C24—22no—
416C24—22no—
417C24—24no—
418C24—21no—
419C24—42no—
420C24C22⋊C4×C3114yes—
421C24C2×D4×C3111yescharacteristic
422C24C2×D4×C3111yescharacteristic
423C24C2×D4×C3111yescharacteristic
424#41—216no—
425#41—48no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8C22×C4×C3112yes—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—44no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4C22×C4×C3112yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—216no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—216no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4C22⋊C4×C3114yes—
464C22×C4C22⋊C4×C3114yes—
465C22×C4C22⋊C4×C3114yes—
466C22×C4C22×C4×C3112yes—
467C22×C4C22×C4×C3112yes—
468C22×C4C22×C4×C3112yes—
469C22×C4C2×D4×C3111yescharacteristic
470C22×C4C2×D4×C3111yescharacteristic
471C22×C4C2×D4×C3112yes—
472C22×C4C2×D4×C3112yes—
473C22×C4C2×D4×C3111yescharacteristic
474C22×C4C2×D4×C3112yes—
475C22×C4C2×D4×C3114yes—
476C22×C4#41×C3114yes—
477C22×C4C24×C3111yescharacteristic
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—48no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4C22×C4×C3112yes—
489C4⋊C4C22×C4×C3112yes—
490C4⋊C4C2×D4×C3118yes—
491C22⋊C4—28no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—28no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C42—28no—
499C42—24no—
500C42—28no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—28no—
512C23—22no—
513C23—44no—
514C23—48no—
515C23—48no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23C2×C22⋊C4×C3112yes—
521C23C2×C22⋊C4×C3112yes—
522C23C2×C22⋊C4×C3112yes—
523C23C31×#12012yes—
524C23C31×#12312yes—
525C23C31×#13514yes—
526C23C22×D4×C3111yescharacteristic
527Q8—24no—
528Q8—42no—
529D4—24no—
530D4—48no—
531D4—48no—
532D4—48no—
533D4—42no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—416no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—48no—
557C2×C4C2×C22⋊C4×C3112yes—
558C2×C4C2×C22⋊C4×C3112yes—
559C2×C4C2×C22⋊C4×C3112yes—
560C2×C4C4×D4×C3118yes—
561C2×C4C31×#12012yes—
562C2×C4C31×#12118yes—
563C2×C4C31×#12312yes—
564C2×C4C23×C4×C3111yescharacteristic
565C2×C4C22×D4×C3112yes—
566C2×C4C22×D4×C3111yescharacteristic
567C2×C4C2×#41×C3112yes—
568C8—44no—
569C22—24no—
570C22—24no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—416no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22C31×#38614yes—
589C22C31×#41614yescomplemented
590C22C22×C22⋊C4×C3111yescharacteristic
591C22C2×C31×#12011yescharacteristic
592C22C2×C31×#12311yescharacteristic
593C22C2×C31×#13512yes—
594C22C31×#57512yes—
595C4—22no—
596C4—22no—
597C4—42no—
598C4—42no—
599C4—48no—
600C4—42no—
601C4C31×#38614yes—
602C4C2×C4×D4×C3112yes—
603C4C2×C31×#12112yes—
604C2—24no—
605C2—24no—
606C2—42no—
607C2—82no—
608C2C2×C31×#38611yescharacteristic
609C2C2×C31×#41611yescharacteristic
610C2C31×#289611yescharacteristic
611C2C31×#300212yes—
612C2C31×#309112yes—
subgroupG/H
centerC22×C4×C31C2×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
socleC23×C31C22×D4
prime radicalC2×C31×#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C31×#117C2semidirect
3. C2×C31×#416C2semidirect
4. C22×C31×C22⋊C8C2semidirect
5. C31×#1875C2semidirect
6. C31×#1874C2semidirect
7. C31×#1873C2semidirect
8. C31×#1872C2semidirect
9. C31×#1735C2semidirect
14. C31×#525C4semidirect
33. C2×C31×#103C22semidirect
34. C2×C31×#102C22semidirect
38. C31×#340C22semidirect
39. C31×#340C22semidirect
44. C2×#41×C31C2×C4semidirect
70. C31×C22⋊Q8C2×C4semidirect
74. C31×#121C2×C4semidirect
83. C31×#117C2×C4semidirect
84. C31×#117D4semidirect
194. #20908C31direct (N × H)
264. C22×#117C2×C31semidirect
265. C2×#416C2×C31semidirect
266. C22×C22⋊C8C2×C31semidirect
267. #1875C2×C31semidirect
268. #1874C2×C31semidirect
269. #1873C2×C31semidirect
270. #1872C2×C31semidirect
271. #1735C2×C31semidirect
292. C22×C31#416semidirect
311. #525C4×C31semidirect
330. C2×#103C22×C31semidirect
331. C2×#102C22×C31semidirect
335. #340C22×C31semidirect
336. #340C22×C31semidirect
350. C2×#41C2×C4×C31semidirect
376. C22⋊Q8C2×C4×C31semidirect
380. #121C2×C4×C31semidirect
389. #117C2×C4×C31semidirect
390. #117D4×C31semidirect
414. C31#20908direct (N × H)
589. C22C31×#416semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#5252
C22×C31×#1172
C2×C31×#4162
C22×C31×C22⋊C82
C31×#18752
C31×#18742
C31×#18732
C31×#18722
C31×#17352
#2090831