Finite Group Databasebeta

← C31×#9930 #21466689

Subgroups — C31×#9930 #21466689

Order 7936.

850 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3192C211yesmaximal, characteristic, complemented
2C31×#3006C211yesmaximal, characteristic, complemented
3C31×#2615C211yesmaximal, characteristic, complemented
4C31×#1964C211yesmaximal, characteristic, complemented
5C31×#1789C211yesmaximal, characteristic, complemented
6C31×#1507C211yesmaximal, characteristic, complemented
7C31×#1305C211yesmaximal, characteristic, complemented
8C22×#41×C31C2211yescharacteristic
9C2×C31×AGL(1,ℤ/8)—21no—
10C2×C31×#121—21no—
11C2×C31×#121C2211yescharacteristic
12C2×C31×#121C2211yescharacteristic
13C31×#481—21no—
14C31×#481—21no—
15C31×#469—22no—
16C31×#449—22no—
17C31×#448—21no—
18C31×#447—21no—
19C31×#447—21no—
20C31×#447—21no—
21C2×C31×#104—22no—
22C31×#417—21no—
23C2×C31×#99—22no—
24C31×#407C2211yescharacteristic
25C31×#407C2211yescharacteristic
26C31×#393—21no—
27C31×#392—21no—
28C31×#392—22no—
29C31×#380—21no—
30C31×#380C2211yescharacteristic
31C4×C22⋊C4×C31C2211yescharacteristic
32C31×#327—22no—
33C2×#41×C31—21no—
34C2×#41×C31—21no—
35C2×#41×C31—21no—
36C2×#41×C31—21no—
37C2×#41×C31—44no—
38C22×Q8×C31D411yescharacteristic
39C22×D4×C31—21no—
40C22×D4×C31—21no—
41C22×D4×C31—22no—
42C22×D4×C31—21no—
43C22×D4×C31D411yescharacteristic
44C22×D4×C31D411yescharacteristic
45C22×D4×C31D411yescharacteristic, complemented
46C23×C4×C31—21no—
47C23×C4×C31C2311yescharacteristic
48C31×AGL(1,ℤ/8)—44no—
49C2×SD16×C31—22no—
50C2×D8×C31—22no—
51C2×D8×C31—22no—
52C2×M16×C31—21no—
53C2×M16×C31D412yescomplemented
54C31×#121—21no—
55C31×#121—21no—
56C31×#121—21no—
57C31×#121—21no—
58C31×#121—21no—
59C31×#121—21no—
60C31×#121—21no—
61C31×#121—21no—
62C31×#121—44no—
63C2×C4⋊C4×C31—21no—
64C2×C4⋊C4×C31D411yescharacteristic
65C2×C4⋊C4×C31D411yescharacteristic, complemented
66C2×C22⋊C4×C31—21no—
67C2×C22⋊C4×C31—22no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—22no—
71C2×C22⋊C4×C31D412yes—
72C2×C42×C31—21no—
73C31×#106—22no—
74C31×#104—44no—
75C31×#103—22no—
76C31×#102—22no—
77C31×#102—22no—
78C31×#102—22no—
79C31×#99—44no—
80C31×C22⋊C8—21no—
81C31×C22⋊C8—21no—
82C31×C22⋊C8—21no—
83C31×C22⋊C8D412yescomplemented
84C31×#95—22no—
85C31×#95—21no—
86C31×#95—21no—
87C24×C31—21no—
88C24×C31—21no—
89C24×C31—21no—
90C24×C31C2×D411yescharacteristic
91#41×C31—24no—
92#41×C31—24no—
93#41×C31—42no—
94#41×C31—42no—
95#41×C31—42no—
96#41×C31—42no—
97#41×C31—42no—
98#41×C31—42no—
99C2×Q8×C31—21no—
100C2×Q8×C31—21no—
101C2×Q8×C31—42no—
102C2×D4×C31—21no—
103C2×D4×C31—21no—
104C2×D4×C31—21no—
105C2×D4×C31—21no—
106C2×D4×C31—21no—
107C2×D4×C31—21no—
108C2×D4×C31—42no—
109C2×D4×C31—42no—
110C2×D4×C31—42no—
111C2×D4×C31—42no—
112C2×D4×C31—42no—
113C2×D4×C31—41no—
114C2×D4×C31—41no—
115C2×D4×C31—42no—
116C2×D4×C31—44no—
117C2×D4×C31—42no—
118C2×D4×C31—41no—
119C2×D4×C31—41no—
120C2×D4×C31—41no—
121C2×D4×C31—41no—
122C2×D4×C31—82no—
123C2×D4×C31—82no—
124C2×D4×C31—82no—
125C22×C4×C31—24no—
126C22×C4×C31—21no—
127C22×C4×C31—21no—
128C22×C4×C31—22no—
129C22×C4×C31—21no—
130C22×C4×C31—42no—
131C22×C4×C31—42no—
132C22×C4×C31—41no—
133C22×C4×C31—41no—
134C22×C4×C31—41no—
135C22×C4×C31C2×D411yescharacteristic
136C22×C4×C31C2×D411yescharacteristic
137C22×C4×C31C2×D411yescharacteristic
138C22×C4×C31C2×D411yescharacteristic
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31#4111yescharacteristic
141SD16×C31—44no—
142D8×C31—44no—
143D8×C31—82no—
144M16×C31—42no—
145M16×C31—42no—
146C2×C8×C31—22no—
147C2×C8×C31—22no—
148C4⋊C4×C31—21no—
149C4⋊C4×C31—21no—
150C4⋊C4×C31—21no—
151C4⋊C4×C31—21no—
152C4⋊C4×C31—42no—
153C22⋊C4×C31—22no—
154C22⋊C4×C31—44no—
155C22⋊C4×C31—41no—
156C22⋊C4×C31—41no—
157C22⋊C4×C31—41no—
158C22⋊C4×C31—41no—
159C22⋊C4×C31—41no—
160C22⋊C4×C31—42no—
161C22⋊C4×C31—82no—
162C42×C31—42no—
163#9930C3111yesmaximal, characteristic, direct product factor, Sylow
164C23×C31—21no—
165C23×C31—21no—
166C23×C31—22no—
167C23×C31—41no—
168C23×C31—41no—
169C23×C31—41no—
170C23×C31—41no—
171C23×C31—41no—
172C23×C31—42no—
173C23×C31—42no—
174C23×C31—42no—
175C23×C31—41no—
176C23×C31—41no—
177C23×C31—41no—
178C23×C31—41no—
179C23×C31—41no—
180C23×C31—41no—
181C23×C31—41no—
182C23×C31—81no—
183C23×C31—81no—
184C23×C31#12011yescharacteristic
185C23×C31#12111yescharacteristic
186C23×C31#12711yescharacteristic
187Q8×C31—22no—
188Q8×C31—22no—
189Q8×C31—41no—
190Q8×C31—41no—
191D4×C31—22no—
192D4×C31—22no—
193D4×C31—41no—
194D4×C31—41no—
195D4×C31—42no—
196D4×C31—44no—
197D4×C31—44no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—81no—
201D4×C31—82no—
202D4×C31—82no—
203D4×C31—82no—
204D4×C31—88no—
205D4×C31—81no—
206D4×C31—81no—
207C2×C4×C31—22no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—41no—
212C2×C4×C31—41no—
213C2×C4×C31—41no—
214C2×C4×C31—44no—
215C2×C4×C31—41no—
216C2×C4×C31—41no—
217C2×C4×C31—44no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—41no—
224C2×C4×C31—41no—
225C2×C4×C31—42no—
226C2×C4×C31—81no—
227C2×C4×C31—81no—
228C2×C4×C31#12011yescharacteristic
229C2×C4×C31#12011yescharacteristic
230C2×C4×C31#12111yescharacteristic
231C2×C4×C31#12111yescharacteristic
232C8×C31—42no—
233C8×C31—42no—
234#3192C2×C3111yescharacteristic, complemented
235#3006C2×C3111yescharacteristic, complemented
236#2615C2×C3111yescharacteristic, complemented
237#1964C2×C3111yescharacteristic, complemented
238#1789C2×C3111yescharacteristic, complemented
239#1507C2×C3111yescharacteristic, complemented
240#1305C2×C3111yescharacteristic, complemented
241C22×C31—21no—
242C22×C31—21no—
243C22×C31—21no—
244C22×C31—21no—
245C22×C31—21no—
246C22×C31—21no—
247C22×C31—21no—
248C22×C31—21no—
249C22×C31—42no—
250C22×C31—42no—
251C22×C31—42no—
252C22×C31—81no—
253C22×C31—81no—
254C22×C31—81no—
255C22×C31—81no—
256C22×C31—82no—
257C22×C31—82no—
258C22×C31—81no—
259C22×C31—81no—
260C22×C31—81no—
261C22×C31—81no—
262C22×C31—81no—
263C22×C31—81no—
264C22×C31—81no—
265C22×C31—81no—
266C22×C31—162no—
267C22×C31#39211yescharacteristic
268C22×C31#45311yescharacteristic
269C22×C31#45411yescharacteristic
270C4×C31—21no—
271C4×C31—21no—
272C4×C31—21no—
273C4×C31—42no—
274C4×C31—82no—
275C4×C31—82no—
276C4×C31—81no—
277C4×C31—81no—
278C4×C31B(4,𝔽2)12yes—
279C22×#41C22×C3111yescharacteristic
280C2×AGL(1,ℤ/8)—21no—
281C2×#121—21no—
282C2×#121C22×C3111yescharacteristic
283C2×#121C22×C3111yescharacteristic
284#481—21no—
285#481—21no—
286#469—22no—
287#449—22no—
288#448—21no—
289#447—21no—
290#447—21no—
291#447—21no—
292C2×#104—22no—
293#417—21no—
294C2×#99—22no—
295#407C22×C3111yescharacteristic
296#407C22×C3111yescharacteristic
297#393—21no—
298#392—21no—
299#392—22no—
300#380—21no—
301#380C22×C3111yescharacteristic
302C4×C22⋊C4C22×C3111yescharacteristic
303#327—22no—
304C2×C31—21no—
305C2×C31—21no—
306C2×C31—41no—
307C2×C31—41no—
308C2×C31—82no—
309C2×C31—161no—
310C2×C31—161no—
311C2×C31#200611yescharacteristic
312C2×C31#200811yescharacteristic
313C2×C31#202111yescharacteristic
314C2×#41—21no—
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—21no—
318C2×#41—44no—
319C22×Q8D4×C3111yescharacteristic
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—22no—
323C22×D4—21no—
324C22×D4D4×C3111yescharacteristic
325C22×D4D4×C3111yescharacteristic
326C22×D4D4×C3111yescharacteristic, complemented
327C23×C4—21no—
328C23×C4C23×C3111yescharacteristic
329AGL(1,ℤ/8)—44no—
330C2×SD16—22no—
331C2×D8—22no—
332C2×D8—22no—
333C2×M16—21no—
334C2×M16D4×C3112yescomplemented
335#121—21no—
336#121—21no—
337#121—21no—
338#121—21no—
339#121—21no—
340#121—21no—
341#121—21no—
342#121—21no—
343#121—44no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4D4×C3111yescharacteristic
346C2×C4⋊C4D4×C3111yescharacteristic, complemented
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4D4×C3112yes—
353C2×C42—21no—
354#106—22no—
355#104—44no—
356#103—22no—
357#102—22no—
358#102—22no—
359#102—22no—
360#99—44no—
361C22⋊C8—21no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364C22⋊C8D4×C3112yescomplemented
365#95—22no—
366#95—21no—
367#95—21no—
368C31#993011yescharacteristic, direct product factor, Sylow
369C24—21no—
370C24—21no—
371C24—21no—
372C24C2×D4×C3111yescharacteristic
373#41—24no—
374#41—24no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—42no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—82no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4C2×D4×C3111yescharacteristic
418C22×C4C2×D4×C3111yescharacteristic
419C22×C4C2×D4×C3111yescharacteristic
420C22×C4C2×D4×C3111yescharacteristic
421C22×C4C2×D4×C3111yescharacteristic
422C22×C4#41×C3111yescharacteristic
423SD16—44no—
424D8—44no—
425D8—82no—
426M16—42no—
427M16—42no—
428C2×C8—22no—
429C2×C8—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—42no—
435C22⋊C4—22no—
436C22⋊C4—44no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—42no—
443C22⋊C4—82no—
444C42—42no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—81no—
464C23—81no—
465C23C31×#12011yescharacteristic
466C23C31×#12111yescharacteristic
467C23C31×#12711yescharacteristic
468Q8—22no—
469Q8—22no—
470Q8—41no—
471Q8—41no—
472D4—22no—
473D4—22no—
474D4—41no—
475D4—41no—
476D4—42no—
477D4—44no—
478D4—44no—
479D4—42no—
480D4—42no—
481D4—81no—
482D4—82no—
483D4—82no—
484D4—82no—
485D4—88no—
486D4—81no—
487D4—81no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4C31×#12011yescharacteristic
510C2×C4C31×#12011yescharacteristic
511C2×C4C31×#12111yescharacteristic
512C2×C4C31×#12111yescharacteristic
513C8—42no—
514C8—42no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—82no—
531C22—82no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—162no—
541C22C31×#39211yescharacteristic
542C22C31×#45311yescharacteristic
543C22C31×#45411yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—42no—
548C4—82no—
549C4—82no—
550C4—81no—
551C4—81no—
552C4C31×B(4,𝔽2)12yes—
553C2—21no—
554C2—21no—
555C2—41no—
556C2—41no—
557C2—82no—
558C2—161no—
559C2—161no—
560C2C31×#200611yescharacteristic
561C2C31×#200811yescharacteristic
562C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3192C2semidirect
2. C31×#3006C2semidirect
3. C31×#2615C2semidirect
4. C31×#1964C2semidirect
5. C31×#1789C2semidirect
6. C31×#1507C2semidirect
7. C31×#1305C2semidirect
45. C22×D4×C31D4semidirect
53. C2×M16×C31D4semidirect
65. C2×C4⋊C4×C31D4semidirect
83. C31×C22⋊C8D4semidirect
163. #9930C31direct (N × H)
234. #3192C2×C31semidirect
235. #3006C2×C31semidirect
236. #2615C2×C31semidirect
237. #1964C2×C31semidirect
238. #1789C2×C31semidirect
239. #1507C2×C31semidirect
240. #1305C2×C31semidirect
326. C22×D4D4×C31semidirect
334. C2×M16D4×C31semidirect
346. C2×C4⋊C4D4×C31semidirect
364. C22⋊C8D4×C31semidirect
368. C31#9930direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31922
C31×#30062
C31×#26152
C31×#19642
C31×#17892
C31×#15072
C31×#13052
#993031