Finite Group Databasebeta

← C31×#9938 #21466697

Subgroups — C31×#9938 #21466697

Order 7936.

1008 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1008 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×#3006C211yesmaximal, characteristic, complemented
2C31×#2539C211yesmaximal, characteristic, complemented
3C31×#2404C211yesmaximal, characteristic, complemented
4C31×#1789C211yesmaximal, characteristic, complemented
5C31×#1510C211yesmaximal, characteristic, complemented
6C31×#1462C211yesmaximal, characteristic, complemented
7C31×#1284C211yesmaximal, characteristic, complemented
8C22×#41×C31C2211yescharacteristic
9C2×C31×AGL(1,ℤ/8)—21no—
10C2×C31×#123—21no—
11C2×C31×#121C2211yescharacteristic
12C2×C31×#120—21no—
13C2×C31×#120—21no—
14C22×C22⋊C4×C31C2211yescharacteristic
15C31×#449—22no—
16C31×#448—21no—
17C31×#447—21no—
18C2×C31×#104—22no—
19C31×#417—21no—
20C31×#407C2211yescharacteristic
21C31×#407C2211yescharacteristic
22C31×#399—21no—
23C31×#393—21no—
24C31×#392—21no—
25C31×#392—21no—
26C31×#390—22no—
27C31×#386—21no—
28C31×#386—22no—
29C31×#381C2211yescharacteristic
30C4×C22⋊C4×C31C2211yescharacteristic
31C31×#327—22no—
32C31×#323—22no—
33C25×C31D411yescharacteristic
34C2×#41×C31—21no—
35C2×#41×C31—21no—
36C2×#41×C31—21no—
37C2×#41×C31—21no—
38C2×#41×C31—44no—
39C22×Q8×C31D411yescharacteristic, complemented
40C22×D4×C31—21no—
41C22×D4×C31—21no—
42C22×D4×C31—22no—
43C22×D4×C31—22no—
44C22×D4×C31D411yescharacteristic, complemented
45C22×D4×C31D411yescharacteristic, complemented
46C23×C4×C31—21no—
47C23×C4×C31D411yescharacteristic
48C23×C4×C31C2311yescharacteristic
49C31×AGL(1,ℤ/8)—44no—
50C2×SD16×C31—22no—
51C2×D8×C31—22no—
52C2×M16×C31—21no—
53C2×M16×C31—21no—
54C31×#123—44no—
55C31×#121—21no—
56C31×#121—21no—
57C31×#121—21no—
58C31×#121—21no—
59C31×#120—44no—
60C31×#120—44no—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31D411yescharacteristic, complemented
63C2×C22⋊C4×C31—21no—
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31—22no—
66C2×C22⋊C4×C31—22no—
67C2×C22⋊C4×C31—21no—
68C2×C22⋊C4×C31—21no—
69C2×C22⋊C4×C31—21no—
70C2×C22⋊C4×C31—21no—
71C2×C22⋊C4×C31—21no—
72C2×C22⋊C4×C31—21no—
73C2×C22⋊C4×C31—21no—
74C2×C22⋊C4×C31—21no—
75C2×C22⋊C4×C31—22no—
76C2×C42×C31D412yescomplemented
77C31×#104—44no—
78C31×#103—22no—
79C31×#102—22no—
80C31×C22⋊C8—21no—
81C31×C22⋊C8—21no—
82C31×C22⋊C8—21no—
83C31×C22⋊C8D412yescomplemented
84C31×#95—22no—
85C31×#95—21no—
86C31×#95—21no—
87C31×#95D412yescomplemented
88C24×C31—21no—
89C24×C31—21no—
90C24×C31—21no—
91C24×C31—21no—
92C24×C31—21no—
93C24×C31—42no—
94C24×C31—42no—
95C24×C31—42no—
96C24×C31C2×D411yescharacteristic
97#41×C31—24no—
98#41×C31—24no—
99#41×C31—42no—
100#41×C31—42no—
101#41×C31—42no—
102#41×C31—42no—
103#41×C31—42no—
104#41×C31—42no—
105C2×Q8×C31—21no—
106C2×Q8×C31—21no—
107C2×Q8×C31—42no—
108C2×D4×C31—24no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31—21no—
112C2×D4×C31—21no—
113C2×D4×C31—42no—
114C2×D4×C31—44no—
115C2×D4×C31—42no—
116C2×D4×C31—42no—
117C2×D4×C31—41no—
118C2×D4×C31—41no—
119C2×D4×C31—42no—
120C2×D4×C31—44no—
121C2×D4×C31—42no—
122C2×D4×C31—44no—
123C2×D4×C31—44no—
124C2×D4×C31—41no—
125C2×D4×C31—41no—
126C2×D4×C31—82no—
127C2×D4×C31—82no—
128C22×C4×C31—21no—
129C22×C4×C31—22no—
130C22×C4×C31—22no—
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—21no—
134C22×C4×C31—21no—
135C22×C4×C31—42no—
136C22×C4×C31—42no—
137C22×C4×C31—42no—
138C22×C4×C31—41no—
139C22×C4×C31—41no—
140C22×C4×C31—42no—
141C22×C4×C31C2×D411yescharacteristic
142C22×C4×C31C2×D411yescharacteristic
143C22×C4×C31C2×D411yescharacteristic
144C22×C4×C31C2×D411yescharacteristic
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31#4111yescharacteristic
147SD16×C31—44no—
148D8×C31—44no—
149M16×C31—42no—
150M16×C31—42no—
151C2×C8×C31—22no—
152C2×C8×C31—41no—
153C4⋊C4×C31—21no—
154C4⋊C4×C31—21no—
155C4⋊C4×C31—44no—
156C22⋊C4×C31—24no—
157C22⋊C4×C31—44no—
158C22⋊C4×C31—44no—
159C22⋊C4×C31—44no—
160C22⋊C4×C31—42no—
161C22⋊C4×C31—41no—
162C22⋊C4×C31—41no—
163C22⋊C4×C31—41no—
164C22⋊C4×C31—41no—
165C22⋊C4×C31—41no—
166C22⋊C4×C31—41no—
167C22⋊C4×C31—82no—
168C42×C31—42no—
169#9938C3111yesmaximal, characteristic, direct product factor, Sylow
170C23×C31—22no—
171C23×C31—22no—
172C23×C31—22no—
173C23×C31—22no—
174C23×C31—22no—
175C23×C31—22no—
176C23×C31—41no—
177C23×C31—42no—
178C23×C31—42no—
179C23×C31—41no—
180C23×C31—42no—
181C23×C31—41no—
182C23×C31—41no—
183C23×C31—41no—
184C23×C31—41no—
185C23×C31—41no—
186C23×C31—42no—
187C23×C31—41no—
188C23×C31—41no—
189C23×C31—42no—
190C23×C31—42no—
191C23×C31—42no—
192C23×C31—41no—
193C23×C31—42no—
194C23×C31—81no—
195C23×C31—81no—
196C23×C31—88no—
197C23×C31#12011yescharacteristic
198C23×C31#12011yescharacteristic
199C23×C31#12111yescharacteristic
200Q8×C31—22no—
201Q8×C31—22no—
202Q8×C31—41no—
203Q8×C31—41no—
204D4×C31—22no—
205D4×C31—22no—
206D4×C31—41no—
207D4×C31—41no—
208D4×C31—42no—
209D4×C31—44no—
210D4×C31—44no—
211D4×C31—81no—
212D4×C31—88no—
213D4×C31—82no—
214D4×C31—88no—
215D4×C31—82no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—24no—
219C2×C4×C31—22no—
220C2×C4×C31—41no—
221C2×C4×C31—41no—
222C2×C4×C31—41no—
223C2×C4×C31—41no—
224C2×C4×C31—44no—
225C2×C4×C31—44no—
226C2×C4×C31—41no—
227C2×C4×C31—42no—
228C2×C4×C31—41no—
229C2×C4×C31—42no—
230C2×C4×C31—42no—
231C2×C4×C31—41no—
232C2×C4×C31—42no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—41no—
236C2×C4×C31—41no—
237C2×C4×C31—42no—
238C2×C4×C31—42no—
239C2×C4×C31—81no—
240C2×C4×C31#12011yescharacteristic
241C2×C4×C31#12111yescharacteristic
242C2×C4×C31#12111yescharacteristic
243C2×C4×C31#12711yescharacteristic
244C8×C31—42no—
245C8×C31—81no—
246#3006C2×C3111yescharacteristic, complemented
247#2539C2×C3111yescharacteristic, complemented
248#2404C2×C3111yescharacteristic, complemented
249#1789C2×C3111yescharacteristic, complemented
250#1510C2×C3111yescharacteristic, complemented
251#1462C2×C3111yescharacteristic, complemented
252#1284C2×C3111yescharacteristic, complemented
253C22×C31—21no—
254C22×C31—21no—
255C22×C31—21no—
256C22×C31—21no—
257C22×C31—21no—
258C22×C31—21no—
259C22×C31—21no—
260C22×C31—44no—
261C22×C31—44no—
262C22×C31—44no—
263C22×C31—44no—
264C22×C31—42no—
265C22×C31—42no—
266C22×C31—82no—
267C22×C31—82no—
268C22×C31—82no—
269C22×C31—82no—
270C22×C31—82no—
271C22×C31—82no—
272C22×C31—82no—
273C22×C31—81no—
274C22×C31—81no—
275C22×C31—81no—
276C22×C31—81no—
277C22×C31—81no—
278C22×C31—81no—
279C22×C31—81no—
280C22×C31#39211yescharacteristic
281C22×C31#45312yes—
282C22×C31B(4,𝔽2)12yes—
283C4×C31—21no—
284C4×C31—21no—
285C4×C31—21no—
286C4×C31—21no—
287C4×C31—44no—
288C4×C31—42no—
289C4×C31—82no—
290C4×C31—81no—
291C4×C31—81no—
292C22×#41C22×C3111yescharacteristic
293C2×AGL(1,ℤ/8)—21no—
294C2×#123—21no—
295C2×#121C22×C3111yescharacteristic
296C2×#120—21no—
297C2×#120—21no—
298C22×C22⋊C4C22×C3111yescharacteristic
299#449—22no—
300#448—21no—
301#447—21no—
302C2×#104—22no—
303#417—21no—
304#407C22×C3111yescharacteristic
305#407C22×C3111yescharacteristic
306#399—21no—
307#393—21no—
308#392—21no—
309#392—21no—
310#390—22no—
311#386—21no—
312#386—22no—
313#381C22×C3111yescharacteristic
314C4×C22⋊C4C22×C3111yescharacteristic
315#327—22no—
316#323—22no—
317C2×C31—22no—
318C2×C31—22no—
319C2×C31—41no—
320C2×C31—82no—
321C2×C31—82no—
322C2×C31—161no—
323C2×C31#200211yescharacteristic
324C2×C31#201011yescharacteristic
325C2×C31#202111yescharacteristic
326C25D4×C3111yescharacteristic
327C2×#41—21no—
328C2×#41—21no—
329C2×#41—21no—
330C2×#41—21no—
331C2×#41—44no—
332C22×Q8D4×C3111yescharacteristic, complemented
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—22no—
336C22×D4—22no—
337C22×D4D4×C3111yescharacteristic, complemented
338C22×D4D4×C3111yescharacteristic, complemented
339C23×C4—21no—
340C23×C4D4×C3111yescharacteristic
341C23×C4C23×C3111yescharacteristic
342AGL(1,ℤ/8)—44no—
343C2×SD16—22no—
344C2×D8—22no—
345C2×M16—21no—
346C2×M16—21no—
347#123—44no—
348#121—21no—
349#121—21no—
350#121—21no—
351#121—21no—
352#120—44no—
353#120—44no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4D4×C3111yescharacteristic, complemented
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C42D4×C3112yescomplemented
370#104—44no—
371#103—22no—
372#102—22no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8D4×C3112yescomplemented
377#95—22no—
378#95—21no—
379#95—21no—
380#95D4×C3112yescomplemented
381C31#993811yescharacteristic, direct product factor, Sylow
382C24—21no—
383C24—21no—
384C24—21no—
385C24—21no—
386C24—21no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24C2×D4×C3111yescharacteristic
391#41—24no—
392#41—24no—
393#41—42no—
394#41—42no—
395#41—42no—
396#41—42no—
397#41—42no—
398#41—42no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—82no—
421C2×D4—82no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4C2×D4×C3111yescharacteristic
436C22×C4C2×D4×C3111yescharacteristic
437C22×C4C2×D4×C3111yescharacteristic
438C22×C4C2×D4×C3111yescharacteristic
439C22×C4C2×D4×C3111yescharacteristic
440C22×C4#41×C3111yescharacteristic
441SD16—44no—
442D8—44no—
443M16—42no—
444M16—42no—
445C2×C8—22no—
446C2×C8—41no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—44no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—82no—
462C42—42no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—81no—
488C23—81no—
489C23—88no—
490C23C31×#12011yescharacteristic
491C23C31×#12011yescharacteristic
492C23C31×#12111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—22no—
498D4—22no—
499D4—41no—
500D4—41no—
501D4—42no—
502D4—44no—
503D4—44no—
504D4—81no—
505D4—88no—
506D4—82no—
507D4—88no—
508D4—82no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—81no—
533C2×C4C31×#12011yescharacteristic
534C2×C4C31×#12111yescharacteristic
535C2×C4C31×#12111yescharacteristic
536C2×C4C31×#12711yescharacteristic
537C8—42no—
538C8—81no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22C31×#39211yescharacteristic
567C22C31×#45312yes—
568C22C31×B(4,𝔽2)12yes—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—44no—
574C4—42no—
575C4—82no—
576C4—81no—
577C4—81no—
578C2—22no—
579C2—22no—
580C2—41no—
581C2—82no—
582C2—82no—
583C2—161no—
584C2C31×#200211yescharacteristic
585C2C31×#201011yescharacteristic
586C2C31×#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×#3006C2semidirect
2. C31×#2539C2semidirect
3. C31×#2404C2semidirect
4. C31×#1789C2semidirect
5. C31×#1510C2semidirect
6. C31×#1462C2semidirect
7. C31×#1284C2semidirect
39. C22×Q8×C31D4semidirect
44. C22×D4×C31D4semidirect
45. C22×D4×C31D4semidirect
62. C2×C4⋊C4×C31D4semidirect
76. C2×C42×C31D4semidirect
83. C31×C22⋊C8D4semidirect
87. C31×#95D4semidirect
169. #9938C31direct (N × H)
246. #3006C2×C31semidirect
247. #2539C2×C31semidirect
248. #2404C2×C31semidirect
249. #1789C2×C31semidirect
250. #1510C2×C31semidirect
251. #1462C2×C31semidirect
252. #1284C2×C31semidirect
332. C22×Q8D4×C31semidirect
337. C22×D4D4×C31semidirect
338. C22×D4D4×C31semidirect
355. C2×C4⋊C4D4×C31semidirect
369. C2×C42D4×C31semidirect
376. C22⋊C8D4×C31semidirect
380. #95D4×C31semidirect
381. C31#9938direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30062
C31×#25392
C31×#24042
C31×#17892
C31×#15102
C31×#14622
C31×#12842
#993831