Finite Group Databasebeta

← C31×#8534 #21465293

Subgroups — C31×#8534 #21465293

Order 7936.

1408 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3484C211yesmaximal, characteristic
2C31×#1898C214yesmaximal, complemented
3C31×#1781C211yesmaximal, characteristic, complemented
4C31×#1438C211yesmaximal, characteristic, complemented
5C2×C31×H(5,𝔽2)C411yescharacteristic, complemented
6C23×D4×C31C2211yescharacteristic
7C31×#546—24no—
8D42×C31—24no—
9C31×#535—21no—
10C31×#534—21no—
11C31×#534C412yescomplemented
12C31×#525—21no—
13C2×C31×#120—21no—
14C2×C31×#120C2212yes—
15C31×#518C411yescharacteristic, complemented
16C22×C22⋊C4×C31—21no—
17C22×C22⋊C4×C31C2212yescomplemented
18C2×C31×#99—24no—
19C2×C31×#99—21no—
20C2×C31×#99—21no—
21C31×#379C2212yescomplemented
22C2×C31×#95—21no—
23C31×#342—24no—
24C25×C31D411yescharacteristic
25C25×C31C2311yescharacteristic
26C31×H(5,𝔽2)—42no—
27C31×H(5,𝔽2)—42no—
28C2×#41×C31—21no—
29C2×#41×C31—22no—
30C22×D4×C31—21no—
31C22×D4×C31—22no—
32C22×D4×C31—24no—
33C22×D4×C31—22no—
34C22×D4×C31—42no—
35C22×D4×C31—42no—
36C22×D4×C31C2×C411yescharacteristic
37C22×D4×C31C2×C411yescharacteristic
38C22×D4×C31C2×C412yes—
39C22×D4×C31D411yescharacteristic
40C22×D4×C31D411yescharacteristic
41C22×D4×C31D414yes—
42C23×C4×C31—22no—
43C23×C4×C31D411yescharacteristic
44C31×#127—22no—
45C31×#124—22no—
46C31×#123—22no—
47C31×#123—22no—
48C31×#123—22no—
49C31×C22⋊Q8—22no—
50C31×#121—22no—
51C31×#121—24no—
52C31×#121—24no—
53C31×#121—24no—
54C31×#120—24no—
55C31×#120—22no—
56C31×#120—22no—
57C31×#120—22no—
58C31×#120—22no—
59C31×#120—22no—
60C31×#120—24no—
61C4×D4×C31—24no—
62C4×D4×C31—24no—
63C31×#117—21no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—21no—
66C2×C22⋊C4×C31—24no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—24no—
69C2×C22⋊C4×C31—44no—
70C2×C22⋊C4×C31—44no—
71C2×C22⋊C4×C31—42no—
72C2×C22⋊C4×C31C2×C412yes—
73C2×C22⋊C4×C31D414yes—
74C31×#99—48no—
75C31×#99—42no—
76C31×#99—42no—
77C31×#95—42no—
78C24×C31—21no—
79C24×C31—22no—
80C24×C31—22no—
81C24×C31—42no—
82C24×C31—41no—
83C24×C31—41no—
84C24×C31—41no—
85C24×C31—42no—
86C24×C31—42no—
87C24×C31—42no—
88C24×C31C22⋊C411yescharacteristic
89C24×C31C22⋊C411yescharacteristic
90C24×C31C22⋊C412yescomplemented
91C24×C31C22×C411yescharacteristic
92C24×C31C2×D411yescharacteristic
93C24×C31C2×D414yes—
94C24×C31C2×D411yescharacteristic
95#41×C31—42no—
96#41×C31—48no—
97#41×C31—42no—
98C2×Q8×C31—21no—
99C2×D4×C31—22no—
100C2×D4×C31—22no—
101C2×D4×C31—22no—
102C2×D4×C31—21no—
103C2×D4×C31—22no—
104C2×D4×C31—22no—
105C2×D4×C31—22no—
106C2×D4×C31—22no—
107C2×D4×C31—24no—
108C2×D4×C31—22no—
109C2×D4×C31—22no—
110C2×D4×C31—24no—
111C2×D4×C31—24no—
112C2×D4×C31—22no—
113C2×D4×C31—22no—
114C2×D4×C31—41no—
115C2×D4×C31—41no—
116C2×D4×C31—41no—
117C2×D4×C31—41no—
118C2×D4×C31—416no—
119C2×D4×C31—48no—
120C2×D4×C31—42no—
121C2×D4×C31—42no—
122C2×D4×C31—42no—
123C2×D4×C31—42no—
124C2×D4×C31—44no—
125C2×D4×C31—44no—
126C2×D4×C31—82no—
127C2×D4×C31—82no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—22no—
132C22×C4×C31—22no—
133C22×C4×C31—24no—
134C22×C4×C31—41no—
135C22×C4×C31—42no—
136C22×C4×C31—42no—
137C22×C4×C31—42no—
138C22×C4×C31—42no—
139C22×C4×C31—42no—
140C22×C4×C31—42no—
141C22×C4×C31—42no—
142C22×C4×C31C22⋊C411yescharacteristic
143C22×C4×C31C22⋊C411yescharacteristic
144C22×C4×C31C22⋊C414yescomplemented
145C22×C4×C31C22⋊C412yescomplemented
146C4⋊C4×C31—22no—
147C4⋊C4×C31—24no—
148C22⋊C4×C31—22no—
149C22⋊C4×C31—24no—
150C22⋊C4×C31—24no—
151C22⋊C4×C31—22no—
152C22⋊C4×C31—22no—
153C22⋊C4×C31—24no—
154C22⋊C4×C31—24no—
155C22⋊C4×C31—28no—
156C22⋊C4×C31—44no—
157C22⋊C4×C31—48no—
158C22⋊C4×C31—42no—
159C22⋊C4×C31—42no—
160C22⋊C4×C31—44no—
161C22⋊C4×C31—44no—
162C42×C31—22no—
163#8534C3111yesmaximal, characteristic, direct product factor, Sylow
164C23×C31—21no—
165C23×C31—21no—
166C23×C31—22no—
167C23×C31—22no—
168C23×C31—21no—
169C23×C31—21no—
170C23×C31—21no—
171C23×C31—21no—
172C23×C31—21no—
173C23×C31—21no—
174C23×C31—24no—
175C23×C31—24no—
176C23×C31—21no—
177C23×C31—22no—
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—44no—
181C23×C31—42no—
182C23×C31—44no—
183C23×C31—42no—
184C23×C31—44no—
185C23×C31—42no—
186C23×C31—42no—
187C23×C31—42no—
188C23×C31—44no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—44no—
193C23×C31—42no—
194C23×C31—42no—
195C23×C31—82no—
196C23×C31—81no—
197C23×C31—82no—
198C23×C31—82no—
199C23×C31—81no—
200C23×C31—81no—
201C23×C31—82no—
202C23×C31—81no—
203C23×C31—81no—
204C23×C31—81no—
205C23×C31—82no—
206C23×C31#9912yescomplemented
207C23×C31C2×C22⋊C412yes—
208C23×C31C2×C22⋊C411yescharacteristic
209C23×C31#12012yes—
210C23×C31#12012yes—
211Q8×C31—42no—
212D4×C31—28no—
213D4×C31—44no—
214D4×C31—42no—
215D4×C31—44no—
216D4×C31—44no—
217D4×C31—44no—
218D4×C31—44no—
219D4×C31—42no—
220D4×C31—42no—
221D4×C31—44no—
222D4×C31—44no—
223D4×C31—84no—
224D4×C31—84no—
225C2×C4×C31—21no—
226C2×C4×C31—21no—
227C2×C4×C31—24no—
228C2×C4×C31—22no—
229C2×C4×C31—21no—
230C2×C4×C31—21no—
231C2×C4×C31—21no—
232C2×C4×C31—44no—
233C2×C4×C31—44no—
234C2×C4×C31—42no—
235C2×C4×C31—44no—
236C2×C4×C31—42no—
237C2×C4×C31—44no—
238C2×C4×C31—44no—
239C2×C4×C31—44no—
240C2×C4×C31—82no—
241C2×C4×C31—82no—
242C2×C4×C31—82no—
243C2×C4×C31—82no—
244C2×C4×C31—82no—
245C2×C4×C31—82no—
246C2×C4×C31—84no—
247C2×C4×C31#9912yescomplemented
248#3484C2×C3111yescharacteristic
249#1898C2×C3114yescomplemented
250#1781C2×C3111yescharacteristic, complemented
251#1438C2×C3111yescharacteristic, complemented
252C22×C31—21no—
253C22×C31—22no—
254C22×C31—21no—
255C22×C31—22no—
256C22×C31—21no—
257C22×C31—21no—
258C22×C31—22no—
259C22×C31—22no—
260C22×C31—22no—
261C22×C31—22no—
262C22×C31—22no—
263C22×C31—22no—
264C22×C31—24no—
265C22×C31—24no—
266C22×C31—42no—
267C22×C31—42no—
268C22×C31—42no—
269C22×C31—42no—
270C22×C31—41no—
271C22×C31—44no—
272C22×C31—42no—
273C22×C31—84no—
274C22×C31—81no—
275C22×C31—81no—
276C22×C31—81no—
277C22×C31—81no—
278C22×C31—81no—
279C22×C31—84no—
280C22×C31—84no—
281C22×C31—84no—
282C22×C31—81no—
283C22×C31—162no—
284C22×C31—162no—
285C22×C31#37911yescharacteristic
286C22×C31C2×#9911yescharacteristic
287C22×C31C2×#9911yescharacteristic
288C22×C31B(4,𝔽2)12yes—
289C22×C31#45812yes—
290C4×C31—22no—
291C4×C31—41no—
292C4×C31—42no—
293C4×C31—44no—
294C4×C31—84no—
295C4×C31—162no—
296C2×H(5,𝔽2)C4×C3111yescharacteristic, complemented
297C23×D4C22×C3111yescharacteristic
298#546—24no—
299D42—24no—
300#535—21no—
301#534—21no—
302#534C4×C3112yescomplemented
303#525—21no—
304C2×#120—21no—
305C2×#120C22×C3112yes—
306#518C4×C3111yescharacteristic, complemented
307C22×C22⋊C4—21no—
308C22×C22⋊C4C22×C3112yescomplemented
309C2×#99—24no—
310C2×#99—21no—
311C2×#99—21no—
312#379C22×C3112yescomplemented
313C2×#95—21no—
314#342—24no—
315C2×C31—22no—
316C2×C31—21no—
317C2×C31—21no—
318C2×C31—22no—
319C2×C31—42no—
320C2×C31—42no—
321C2×C31—42no—
322C2×C31—81no—
323C2×C31—82no—
324C2×C31#178111yescharacteristic
325C2×C31#179012yes—
326C25D4×C3111yescharacteristic
327C25C23×C3111yescharacteristic
328H(5,𝔽2)—42no—
329H(5,𝔽2)—42no—
330C2×#41—21no—
331C2×#41—22no—
332C22×D4—21no—
333C22×D4—22no—
334C22×D4—24no—
335C22×D4—22no—
336C22×D4—42no—
337C22×D4—42no—
338C22×D4C2×C4×C3111yescharacteristic
339C22×D4C2×C4×C3111yescharacteristic
340C22×D4C2×C4×C3112yes—
341C22×D4D4×C3111yescharacteristic
342C22×D4D4×C3111yescharacteristic
343C22×D4D4×C3114yes—
344C23×C4—22no—
345C23×C4D4×C3111yescharacteristic
346#127—22no—
347#124—22no—
348#123—22no—
349#123—22no—
350#123—22no—
351C22⋊Q8—22no—
352#121—22no—
353#121—24no—
354#121—24no—
355#121—24no—
356#120—24no—
357#120—22no—
358#120—22no—
359#120—22no—
360#120—22no—
361#120—22no—
362#120—24no—
363C4×D4—24no—
364C4×D4—24no—
365#117—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—42no—
374C2×C22⋊C4C2×C4×C3112yes—
375C2×C22⋊C4D4×C3114yes—
376#99—48no—
377#99—42no—
378#99—42no—
379#95—42no—
380C31#853411yescharacteristic, direct product factor, Sylow
381C24—21no—
382C24—22no—
383C24—22no—
384C24—42no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22⋊C4×C3111yescharacteristic
392C24C22⋊C4×C3111yescharacteristic
393C24C22⋊C4×C3112yescomplemented
394C24C22×C4×C3111yescharacteristic
395C24C2×D4×C3111yescharacteristic
396C24C2×D4×C3114yes—
397C24C2×D4×C3111yescharacteristic
398#41—42no—
399#41—48no—
400#41—42no—
401C2×Q8—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—416no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—82no—
430C2×D4—82no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4C22⋊C4×C3111yescharacteristic
446C22×C4C22⋊C4×C3111yescharacteristic
447C22×C4C22⋊C4×C3114yescomplemented
448C22×C4C22⋊C4×C3112yescomplemented
449C4⋊C4—22no—
450C4⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C42—22no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—24no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—21no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—82no—
498C23—81no—
499C23—82no—
500C23—82no—
501C23—81no—
502C23—81no—
503C23—82no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—82no—
508C23C31×#9912yescomplemented
509C23C2×C22⋊C4×C3112yes—
510C23C2×C22⋊C4×C3111yescharacteristic
511C23C31×#12012yes—
512C23C31×#12012yes—
513Q8—42no—
514D4—28no—
515D4—44no—
516D4—42no—
517D4—44no—
518D4—44no—
519D4—44no—
520D4—44no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—44no—
525D4—84no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4C31×#9912yescomplemented
550C22—21no—
551C22—22no—
552C22—21no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—41no—
569C22—44no—
570C22—42no—
571C22—84no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—81no—
581C22—162no—
582C22—162no—
583C22C31×#37911yescharacteristic
584C22C2×C31×#9911yescharacteristic
585C22C2×C31×#9911yescharacteristic
586C22C31×B(4,𝔽2)12yes—
587C22C31×#45812yes—
588C4—22no—
589C4—41no—
590C4—42no—
591C4—44no—
592C4—84no—
593C4—162no—
594C2—22no—
595C2—21no—
596C2—21no—
597C2—22no—
598C2—42no—
599C2—42no—
600C2—42no—
601C2—81no—
602C2—82no—
603C2C31×#178111yescharacteristic
604C2C31×#179012yes—
subgroupG/H
center, socleC22×C31#379
commutatorC24C22×C4×C31
Frattini, cosocleC25C23×C31
prime radicalC31×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C25×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1898C2semidirect
3. C31×#1781C2semidirect
4. C31×#1438C2semidirect
5. C2×C31×H(5,𝔽2)C4semidirect
11. C31×#534C4semidirect
15. C31×#518C4semidirect
17. C22×C22⋊C4×C31C22semidirect
21. C31×#379C22semidirect
90. C24×C31C22⋊C4semidirect
144. C22×C4×C31C22⋊C4semidirect
145. C22×C4×C31C22⋊C4semidirect
163. #8534C31direct (N × H)
206. C23×C31#99semidirect
247. C2×C4×C31#99semidirect
249. #1898C2×C31semidirect
250. #1781C2×C31semidirect
251. #1438C2×C31semidirect
296. C2×H(5,𝔽2)C4×C31semidirect
302. #534C4×C31semidirect
306. #518C4×C31semidirect
308. C22×C22⋊C4C22×C31semidirect
312. #379C22×C31semidirect
380. C31#8534direct (N × H)
393. C24C22⋊C4×C31semidirect
447. C22×C4C22⋊C4×C31semidirect
448. C22×C4C22⋊C4×C31semidirect
508. C23C31×#99semidirect
549. C2×C4C31×#99semidirect

Maximal subgroups

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

subgroupindex
C31×#34842
C31×#18982
C31×#17812
C31×#14382
#853431