Finite Group Databasebeta

← C31×#18244 #21475003

Subgroups — C31×#18244 #21475003

Order 7936.

956 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3559C211yesmaximal, characteristic, complemented
2C31×#3114C211yesmaximal, characteristic, complemented
3C31×#3100C211yesmaximal, characteristic, complemented
4C31×#2633C211yesmaximal, characteristic, complemented
5C31×#1610C212yesmaximal, complemented
6C31×#1609C212yesmaximal, complemented
7C31×#1598C212yesmaximal, complemented
8C31×#1597C212yesmaximal, complemented
9C31×#1521C211yesmaximal, characteristic, complemented
10C31×#1514C211yesmaximal, characteristic, complemented
11C31×#1470C211yesmaximal, characteristic, complemented
12C31×#563—21no—
13C31×#562—21no—
14C31×#554—21no—
15C31×#553—21no—
16C31×#553—21no—
17C31×#547—22no—
18C31×#547—21no—
19C31×#544—21no—
20C31×#540—21no—
21C31×#539—22no—
22C31×#539C2212yes—
23C31×#537C2212yes—
24C31×#532C2211yescharacteristic
25C31×#529C2211yescharacteristic
26C2×C31×#125C2211yescharacteristic
27C2×C4×D4×C31C2211yescharacteristic
28C31×#483C2211yescharacteristic
29C31×#480C2212yes—
30C31×#480C2211yescharacteristic
31C31×#478—21no—
32C31×#474—21no—
33C31×#463—21no—
34C31×#459—21no—
35C31×#450C2211yescharacteristic
36C31×#447C2212yes—
37C31×#447C2211yescharacteristic
38C31×#442—21no—
39C31×#440—21no—
40C4×D8×C31—22no—
41C31×#432C2211yescharacteristic
42C31×#431C2211yescharacteristic
43C31×#399—21no—
44C31×#398C2212yes—
45C31×#397—21no—
46C31×#394C2212yes—
47C31×#385—22no—
48C31×#383C2211yescharacteristic
49C31×#328C2212yes—
50C31×#327C2212yes—
51C31×#327C2212yes—
52C31×#327C2212yes—
53C31×#324C2212yes—
54C31×#324C2212yes—
55C2×#41×C31—21no—
56C2×#41×C31—22no—
57C22×D4×C31C2311yescharacteristic
58C23×C4×C31—21no—
59C2×SD16×C31—21no—
60C2×D8×C31—22no—
61C2×D8×C31—21no—
62C31×C4⋊Q8—21no—
63C31×C4⋊Q8—22no—
64C31×C4⋊Q8D411yescharacteristic
65C31×#127D411yescharacteristic
66C31×#126—22no—
67C31×#126—22no—
68C31×#126—21no—
69C31×#125—21no—
70C31×#125—22no—
71C31×#125—21no—
72C31×#125—21no—
73C31×#125—21no—
74C31×#125—21no—
75C31×#125D411yescharacteristic
76C31×#124—21no—
77C31×#124D411yescharacteristic
78C31×#123—22no—
79C31×#123—22no—
80C31×#123—22no—
81C31×#123—22no—
82C31×#123—22no—
83C31×#123—21no—
84C31×#123—22no—
85C31×C22⋊Q8—22no—
86C31×C22⋊Q8—22no—
87C31×C22⋊Q8—22no—
88C31×C22⋊Q8—22no—
89C31×C22⋊Q8—22no—
90C31×C22⋊Q8—22no—
91C31×C22⋊Q8C2311yescharacteristic
92C31×#121—21no—
93C31×#121—22no—
94C31×#121—21no—
95C31×#121—22no—
96C31×#121—22no—
97C31×#121—21no—
98C31×#121—22no—
99C31×#121C2311yescharacteristic
100C31×#121C2312yes—
101C4×Q8×C31D411yescharacteristic
102C4×D4×C31—22no—
103C4×D4×C31—21no—
104C4×D4×C31—21no—
105C4×D4×C31—21no—
106C4×D4×C31—22no—
107C4×D4×C31—21no—
108C4×D4×C31—21no—
109C4×D4×C31—21no—
110C4×D4×C31—21no—
111C4×D4×C31—21no—
112C4×D4×C31D412yes—
113C4×D4×C31D411yescharacteristic
114C31×#117—21no—
115C31×#117—21no—
116C2×C4⋊C4×C31—22no—
117C2×C4⋊C4×C31—21no—
118C2×C4⋊C4×C31—22no—
119C2×C4⋊C4×C31D411yescharacteristic
120C2×C4⋊C4×C31D411yescharacteristic
121C2×C4⋊C4×C31D412yes—
122C2×C4⋊C4×C31C2311yescharacteristic
123C2×C4⋊C4×C31C2312yes—
124C2×C22⋊C4×C31—21no—
125C2×C22⋊C4×C31—22no—
126C2×C22⋊C4×C31—22no—
127C2×C42×C31C2311yescharacteristic
128C31×#107—22no—
129C31×#107—21no—
130C31×#106—21no—
131C31×C4⋊C8—21no—
132C31×C4⋊C8—21no—
133C31×#103—21no—
134C31×#102—22no—
135C31×#102—21no—
136C31×#102—22no—
137C31×#102—21no—
138C31×#102—21no—
139C31×C22⋊C8C2312yes—
140C31×C22⋊C8C2311yescharacteristic
141C31×C22⋊C8C2311yescharacteristic
142C31×#97—21no—
143C4×C8×C31—21no—
144C31×#95—22no—
145C31×#95—21no—
146C31×#95—21no—
147C31×#95C2312yes—
148C24×C31—21no—
149#41×C31—82no—
150#41×C31—82no—
151#41×C31—82no—
152C2×Q8×C31—22no—
153C2×Q8×C31—21no—
154C2×Q8×C31C2×D411yescharacteristic
155C2×D4×C31—21no—
156C2×D4×C31—22no—
157C2×D4×C31—22no—
158C2×D4×C31—22no—
159C2×D4×C31—21no—
160C2×D4×C31—21no—
161C2×D4×C31—21no—
162C2×D4×C31—21no—
163C2×D4×C31—42no—
164C2×D4×C31C2×D411yescharacteristic
165C2×D4×C31C2×D412yes—
166C22×C4×C31—22no—
167C22×C4×C31—22no—
168C22×C4×C31—22no—
169C22×C4×C31—21no—
170C22×C4×C31—22no—
171C22×C4×C31—22no—
172C22×C4×C31—21no—
173C22×C4×C31—41no—
174C22×C4×C31—41no—
175C22×C4×C31—82no—
176C22×C4×C31C2×D412yes—
177C22×C4×C31C2×D411yescharacteristic
178C22×C4×C31C2×D412yes—
179C22×C4×C31C2×D411yescharacteristic
180C22×C4×C31C2411yescharacteristic
181SD16×C31—42no—
182D8×C31—44no—
183D8×C31—42no—
184C2×C8×C31—22no—
185C2×C8×C31—21no—
186C2×C8×C31—21no—
187C4⋊C4×C31—22no—
188C4⋊C4×C31—22no—
189C4⋊C4×C31—22no—
190C4⋊C4×C31—22no—
191C4⋊C4×C31—21no—
192C4⋊C4×C31—21no—
193C4⋊C4×C31—22no—
194C4⋊C4×C31—21no—
195C4⋊C4×C31—21no—
196C4⋊C4×C31—22no—
197C4⋊C4×C31—21no—
198C4⋊C4×C31—21no—
199C4⋊C4×C31—21no—
200C4⋊C4×C31—21no—
201C4⋊C4×C31—22no—
202C4⋊C4×C31—22no—
203C4⋊C4×C31—21no—
204C4⋊C4×C31—21no—
205C4⋊C4×C31—21no—
206C4⋊C4×C31—82no—
207C4⋊C4×C31C2×D411yescharacteristic
208C4⋊C4×C31C2×D412yes—
209C4⋊C4×C31C2×D411yescharacteristic
210C22⋊C4×C31—22no—
211C22⋊C4×C31—22no—
212C22⋊C4×C31—22no—
213C22⋊C4×C31—22no—
214C22⋊C4×C31—22no—
215C22⋊C4×C31—21no—
216C22⋊C4×C31—21no—
217C22⋊C4×C31—21no—
218C22⋊C4×C31—21no—
219C22⋊C4×C31—22no—
220C22⋊C4×C31—21no—
221C22⋊C4×C31—22no—
222C22⋊C4×C31—44no—
223C22⋊C4×C31—41no—
224C22⋊C4×C31—41no—
225C22⋊C4×C31—82no—
226C42×C31—22no—
227C42×C31—21no—
228C42×C31C2×D411yescharacteristic
229C42×C31C2×D411yescharacteristic
230C42×C31C2×D411yescharacteristic
231C42×C31C2×D411yescharacteristic
232#18244C3111yesmaximal, characteristic, direct product factor, Sylow
233C23×C31—22no—
234C23×C31—21no—
235C23×C31—22no—
236C23×C31—41no—
237C23×C31—42no—
238C23×C31—41no—
239C23×C31—42no—
240C23×C31C22×D411yescharacteristic
241Q8×C31—41no—
242Q8×C31—81no—
243Q8×C31—82no—
244D4×C31—22no—
245D4×C31—22no—
246D4×C31—42no—
247D4×C31—41no—
248D4×C31—42no—
249D4×C31—41no—
250D4×C31—41no—
251D4×C31—42no—
252D4×C31—42no—
253D4×C31—42no—
254D4×C31—81no—
255D4×C31—82no—
256D4×C31—82no—
257C2×C4×C31—22no—
258C2×C4×C31—21no—
259C2×C4×C31—21no—
260C2×C4×C31—22no—
261C2×C4×C31—22no—
262C2×C4×C31—21no—
263C2×C4×C31—22no—
264C2×C4×C31—22no—
265C2×C4×C31—22no—
266C2×C4×C31—41no—
267C2×C4×C31—41no—
268C2×C4×C31—42no—
269C2×C4×C31—42no—
270C2×C4×C31—42no—
271C2×C4×C31—42no—
272C2×C4×C31—41no—
273C2×C4×C31—42no—
274C2×C4×C31—42no—
275C2×C4×C31—82no—
276C2×C4×C31—82no—
277C2×C4×C31—82no—
278C2×C4×C31—82no—
279C2×C4×C31—82no—
280C2×C4×C31—82no—
281C2×C4×C31#12011yescharacteristic
282C2×C4×C31#12011yescharacteristic
283C2×C4×C31#12011yescharacteristic
284C2×C4×C31#12011yescharacteristic
285C2×C4×C31C22×D411yescharacteristic
286C2×C4×C31C22×D411yescharacteristic
287C8×C31—42no—
288C8×C31—41no—
289C8×C31—41no—
290#3559C2×C3111yescharacteristic, complemented
291#3114C2×C3111yescharacteristic, complemented
292#3100C2×C3111yescharacteristic, complemented
293#2633C2×C3111yescharacteristic, complemented
294#1610C2×C3112yescomplemented
295#1609C2×C3112yescomplemented
296#1598C2×C3112yescomplemented
297#1597C2×C3112yescomplemented
298#1521C2×C3111yescharacteristic, complemented
299#1514C2×C3111yescharacteristic, complemented
300#1470C2×C3111yescharacteristic, complemented
301C22×C31—21no—
302C22×C31—21no—
303C22×C31—44no—
304C22×C31—42no—
305C22×C31—41no—
306C22×C31—41no—
307C22×C31—42no—
308C22×C31—42no—
309C22×C31—42no—
310C22×C31—41no—
311C22×C31—82no—
312C22×C31—81no—
313C22×C31—81no—
314C22×C31C2×#12011yescharacteristic
315C22×C31#57612yes—
316C4×C31—21no—
317C4×C31—21no—
318C4×C31—21no—
319C4×C31—21no—
320C4×C31—21no—
321C4×C31—21no—
322C4×C31—42no—
323C4×C31—42no—
324C4×C31—41no—
325C4×C31—41no—
326C4×C31—42no—
327C4×C31—41no—
328C4×C31—82no—
329C4×C31—81no—
330#563—21no—
331#562—21no—
332#554—21no—
333#553—21no—
334#553—21no—
335#547—22no—
336#547—21no—
337#544—21no—
338#540—21no—
339#539—22no—
340#539C22×C3112yes—
341#537C22×C3112yes—
342#532C22×C3111yescharacteristic
343#529C22×C3111yescharacteristic
344C2×#125C22×C3111yescharacteristic
345C2×C4×D4C22×C3111yescharacteristic
346#483C22×C3111yescharacteristic
347#480C22×C3112yes—
348#480C22×C3111yescharacteristic
349#478—21no—
350#474—21no—
351#463—21no—
352#459—21no—
353#450C22×C3111yescharacteristic
354#447C22×C3112yes—
355#447C22×C3111yescharacteristic
356#442—21no—
357#440—21no—
358C4×D8—22no—
359#432C22×C3111yescharacteristic
360#431C22×C3111yescharacteristic
361#399—21no—
362#398C22×C3112yes—
363#397—21no—
364#394C22×C3112yes—
365#385—22no—
366#383C22×C3111yescharacteristic
367#328C22×C3112yes—
368#327C22×C3112yes—
369#327C22×C3112yes—
370#327C22×C3112yes—
371#324C22×C3112yes—
372#324C22×C3112yes—
373C2×C31—22no—
374C2×C31—81no—
375C2×C31—82no—
376C2×C31—82no—
377C2×C31#300811yescharacteristic
378C2×C31#302011yescharacteristic
379C2×C31#302511yescharacteristic
380C2×#41—21no—
381C2×#41—22no—
382C22×D4C23×C3111yescharacteristic
383C23×C4—21no—
384C2×SD16—21no—
385C2×D8—22no—
386C2×D8—21no—
387C4⋊Q8—21no—
388C4⋊Q8—22no—
389C4⋊Q8D4×C3111yescharacteristic
390#127D4×C3111yescharacteristic
391#126—22no—
392#126—22no—
393#126—21no—
394#125—21no—
395#125—22no—
396#125—21no—
397#125—21no—
398#125—21no—
399#125—21no—
400#125D4×C3111yescharacteristic
401#124—21no—
402#124D4×C3111yescharacteristic
403#123—22no—
404#123—22no—
405#123—22no—
406#123—22no—
407#123—22no—
408#123—21no—
409#123—22no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8C23×C3111yescharacteristic
417#121—21no—
418#121—22no—
419#121—21no—
420#121—22no—
421#121—22no—
422#121—21no—
423#121—22no—
424#121C23×C3111yescharacteristic
425#121C23×C3112yes—
426C4×Q8D4×C3111yescharacteristic
427C4×D4—22no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—22no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4—21no—
437C4×D4D4×C3111yescharacteristic
438C4×D4D4×C3112yes—
439#117—21no—
440#117—21no—
441C2×C4⋊C4—22no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4D4×C3112yes—
445C2×C4⋊C4D4×C3111yescharacteristic
446C2×C4⋊C4D4×C3111yescharacteristic
447C2×C4⋊C4C23×C3111yescharacteristic
448C2×C4⋊C4C23×C3112yes—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—22no—
451C2×C22⋊C4—22no—
452C2×C42C23×C3111yescharacteristic
453#107—22no—
454#107—21no—
455#106—21no—
456C4⋊C8—21no—
457C4⋊C8—21no—
458#103—21no—
459#102—22no—
460#102—21no—
461#102—22no—
462#102—21no—
463#102—21no—
464C22⋊C8C23×C3111yescharacteristic
465C22⋊C8C23×C3111yescharacteristic
466C22⋊C8C23×C3112yes—
467#97—21no—
468C4×C8—21no—
469#95—21no—
470#95—21no—
471#95—22no—
472#95C23×C3112yes—
473C31#1824411yescharacteristic, direct product factor, Sylow
474C24—21no—
475#41—82no—
476#41—82no—
477#41—82no—
478C2×Q8—22no—
479C2×Q8—21no—
480C2×Q8C2×D4×C3111yescharacteristic
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—42no—
490C2×D4C2×D4×C3111yescharacteristic
491C2×D4C2×D4×C3112yes—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—82no—
502C22×C4C2×D4×C3112yes—
503C22×C4C2×D4×C3111yescharacteristic
504C22×C4C2×D4×C3112yes—
505C22×C4C2×D4×C3111yescharacteristic
506C22×C4C24×C3111yescharacteristic
507SD16—42no—
508D8—44no—
509D8—42no—
510C2×C8—22no—
511C2×C8—21no—
512C2×C8—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—82no—
533C4⋊C4C2×D4×C3111yescharacteristic
534C4⋊C4C2×D4×C3112yes—
535C4⋊C4C2×D4×C3111yescharacteristic
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—44no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—82no—
552C42—22no—
553C42—21no—
554C42C2×D4×C3111yescharacteristic
555C42C2×D4×C3111yescharacteristic
556C42C2×D4×C3111yescharacteristic
557C42C2×D4×C3111yescharacteristic
558C23—22no—
559C23—21no—
560C23—22no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—42no—
565C23C22×D4×C3111yescharacteristic
566Q8—41no—
567Q8—81no—
568Q8—82no—
569D4—22no—
570D4—22no—
571D4—42no—
572D4—41no—
573D4—42no—
574D4—41no—
575D4—41no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—81no—
580D4—82no—
581D4—82no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4C31×#12011yescharacteristic
607C2×C4C31×#12011yescharacteristic
608C2×C4C31×#12011yescharacteristic
609C2×C4C31×#12011yescharacteristic
610C2×C4C22×D4×C3111yescharacteristic
611C2×C4C22×D4×C3111yescharacteristic
612C8—42no—
613C8—41no—
614C8—41no—
615C22—21no—
616C22—21no—
617C22—44no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—82no—
626C22—81no—
627C22—81no—
628C22C2×C31×#12011yescharacteristic
629C22C31×#57612yes—
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—82no—
643C4—81no—
644C2—22no—
645C2—82no—
646C2—81no—
647C2—82no—
648C2C31×#300811yescharacteristic
649C2C31×#302011yescharacteristic
650C2C31×#302511yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3559C2semidirect
2. C31×#3114C2semidirect
3. C31×#3100C2semidirect
4. C31×#2633C2semidirect
5. C31×#1610C2semidirect
6. C31×#1609C2semidirect
7. C31×#1598C2semidirect
8. C31×#1597C2semidirect
9. C31×#1521C2semidirect
10. C31×#1514C2semidirect
11. C31×#1470C2semidirect
232. #18244C31direct (N × H)
290. #3559C2×C31semidirect
291. #3114C2×C31semidirect
292. #3100C2×C31semidirect
293. #2633C2×C31semidirect
294. #1610C2×C31semidirect
295. #1609C2×C31semidirect
296. #1598C2×C31semidirect
297. #1597C2×C31semidirect
298. #1521C2×C31semidirect
299. #1514C2×C31semidirect
300. #1470C2×C31semidirect
473. C31#18244direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35592
C31×#31142
C31×#31002
C31×#26332
C31×#16102
C31×#16092
C31×#15982
C31×#15972
C31×#15212
C31×#15142
C31×#14702
#1824431