Finite Group Databasebeta

← C31×#24078 #21480837

Subgroups — C31×#24078 #21480837

Order 7936.

1038 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic
2C31×#3093C212yesmaximal, complemented
3C31×#2285C211yesmaximal, characteristic, complemented
4C31×#2061C211yesmaximal, characteristic, complemented
5C31×#2060C211yesmaximal, characteristic, complemented
6C31×#2040C211yesmaximal, characteristic, complemented
7C31×#2039C211yesmaximal, characteristic, complemented
8C31×#1896C212yesmaximal, complemented
9C31×#1860C212yesmaximal, complemented
10C31×#1859C212yesmaximal, complemented
11C31×#1822C211yesmaximal, characteristic, complemented
12C31×#556—22no—
13C31×#555—22no—
14C31×#553—22no—
15C31×#552—22no—
16C31×#544C2211yescharacteristic
17C31×#538—22no—
18C31×#537C2211yescharacteristic
19C31×#529—22no—
20C31×#525C2212yes—
21C31×#520C2212yes—
22C2×C31×#117C2211yescharacteristic
23C2×C31×#117C2211yescharacteristic
24C22×C42×C31C2211yescharacteristic
25C31×#485—22no—
26C31×#485—22no—
27C31×#482—22no—
28C31×#482—22no—
29C31×#428C2212yes—
30C31×#428C2212yes—
31C2×C31×#104C2212yes—
32C2×C31×#104C2212yes—
33C31×#417C2212yes—
34C31×#417C2212yes—
35C31×#407C2212yes—
36C31×#384C2211yescharacteristic
37C31×#384C2211yescharacteristic
38C31×#382C2211yescharacteristic
39C31×#382C2211yescharacteristic
40C31×#380—22no—
41C4×C4⋊C4×C31—22no—
42C31×#343C2211yescharacteristic
43C31×#343C2211yescharacteristic
44C31×#343C2212yes—
45C31×#339C2212yes—
46C31×#339C2212yes—
47C2×#41×C31C2312yes—
48C23×C4×C31—21no—
49C23×C4×C31C2311yescharacteristic
50C2×M16×C31C2312yes—
51C2×M16×C31C2312yes—
52C31×C4⋊Q8—22no—
53C31×#126—24no—
54C31×#126—24no—
55C31×#125—22no—
56C31×#125—22no—
57C31×#124—22no—
58C31×#124—24no—
59C31×#123—24no—
60C31×#123—22no—
61C31×C22⋊Q8—22no—
62C31×C22⋊Q8—22no—
63C31×C22⋊Q8D412yes—
64C31×C22⋊Q8D412yes—
65C31×#121—24no—
66C31×#121—22no—
67C31×#121—22no—
68C31×#121D412yes—
69C31×#121D412yes—
70C31×#120—22no—
71C4×Q8×C31—22no—
72C4×Q8×C31—22no—
73C4×D4×C31—22no—
74C4×D4×C31—22no—
75C4×D4×C31—24no—
76C4×D4×C31—22no—
77C4×D4×C31—22no—
78C31×#117—24no—
79C31×#117—22no—
80C31×#117—22no—
81C31×#117—22no—
82C31×#117—22no—
83C31×#117—22no—
84C31×#117C2312yes—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31—22no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31C2311yescharacteristic
89C2×C4⋊C4×C31C2311yescharacteristic
90C2×C4⋊C4×C31C2311yescharacteristic
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C22⋊C4×C31—21no—
93C2×C22⋊C4×C31D412yes—
94C2×C42×C31—21no—
95C2×C42×C31—22no—
96C2×C42×C31—22no—
97C2×C42×C31—21no—
98C2×C42×C31D412yes—
99C2×C42×C31C2311yescharacteristic
100C2×C42×C31C2311yescharacteristic
101C31×#107—22no—
102C31×#107—22no—
103C31×#106—22no—
104C31×#106—22no—
105C31×#104—24no—
106C31×#104—24no—
107C31×#103—22no—
108C31×#103—22no—
109C31×#102—22no—
110C31×#102—22no—
111C31×C22⋊C8—22no—
112C31×C22⋊C8—22no—
113C31×#95—22no—
114C31×#95—22no—
115C24×C31C2×D411yescharacteristic
116#41×C31—22no—
117#41×C31—22no—
118#41×C31—22no—
119#41×C31—22no—
120C2×Q8×C31—22no—
121C2×Q8×C31C2×D412yes—
122C2×D4×C31—22no—
123C2×D4×C31—22no—
124C2×D4×C31—22no—
125C2×D4×C31—22no—
126C2×D4×C31C2×D412yes—
127C22×C4×C31—21no—
128C22×C4×C31—21no—
129C22×C4×C31—21no—
130C22×C4×C31—22no—
131C22×C4×C31—24no—
132C22×C4×C31—21no—
133C22×C4×C31—24no—
134C22×C4×C31—22no—
135C22×C4×C31—21no—
136C22×C4×C31—21no—
137C22×C4×C31—42no—
138C22×C4×C31—42no—
139C22×C4×C31C2×D411yescharacteristic
140C22×C4×C31C2×D412yes—
141C22×C4×C31C2×D412yes—
142C22×C4×C31C2×D412yes—
143C22×C4×C31C2×D412yes—
144C22×C4×C31C2411yescharacteristic
145M16×C31—22no—
146M16×C31—22no—
147M16×C31—22no—
148M16×C31—22no—
149C2×C8×C31—22no—
150C2×C8×C31—22no—
151C4⋊C4×C31—24no—
152C4⋊C4×C31—24no—
153C4⋊C4×C31—22no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—44no—
162C4⋊C4×C31—44no—
163C4⋊C4×C31C2×D412yes—
164C4⋊C4×C31C2×D412yes—
165C22⋊C4×C31—24no—
166C22⋊C4×C31—24no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—42no—
174C42×C31—24no—
175C42×C31—24no—
176C42×C31—22no—
177C42×C31—22no—
178C42×C31—22no—
179C42×C31—21no—
180C42×C31—21no—
181C42×C31—41no—
182C42×C31—41no—
183C42×C31#4112yes—
184C42×C31#4112yes—
185#24078C3111yesmaximal, characteristic, direct product factor, Sylow
186C23×C31—21no—
187C23×C31—21no—
188C23×C31—22no—
189C23×C31—41no—
190C23×C31—41no—
191C23×C31#12012yes—
192C23×C31C22×D411yescharacteristic
193Q8×C31—22no—
194Q8×C31—22no—
195Q8×C31—82no—
196D4×C31—22no—
197D4×C31—22no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—82no—
201D4×C31—82no—
202D4×C31—82no—
203C2×C4×C31—22no—
204C2×C4×C31—22no—
205C2×C4×C31—21no—
206C2×C4×C31—22no—
207C2×C4×C31—21no—
208C2×C4×C31—22no—
209C2×C4×C31—22no—
210C2×C4×C31—22no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—48no—
219C2×C4×C31—41no—
220C2×C4×C31—41no—
221C2×C4×C31—42no—
222C2×C4×C31—42no—
223C2×C4×C31—42no—
224C2×C4×C31—42no—
225C2×C4×C31—42no—
226C2×C4×C31—42no—
227C2×C4×C31—41no—
228C2×C4×C31—41no—
229C2×C4×C31—84no—
230C2×C4×C31#12011yescharacteristic
231C2×C4×C31#12011yescharacteristic
232C2×C4×C31C22×D412yes—
233C2×C4×C31C2×#4111yescharacteristic
234C2×C4×C31C2×#4111yescharacteristic
235C2×C4×C312−1+412yes—
236C8×C31—42no—
237C8×C31—42no—
238#3493C2×C3111yescharacteristic
239#3093C2×C3112yescomplemented
240#2285C2×C3111yescharacteristic, complemented
241#2061C2×C3111yescharacteristic, complemented
242#2060C2×C3111yescharacteristic, complemented
243#2040C2×C3111yescharacteristic, complemented
244#2039C2×C3111yescharacteristic, complemented
245#1896C2×C3112yescomplemented
246#1860C2×C3112yescomplemented
247#1859C2×C3112yescomplemented
248#1822C2×C3111yescharacteristic, complemented
249C22×C31—21no—
250C22×C31—21no—
251C22×C31—22no—
252C22×C31—22no—
253C22×C31—22no—
254C22×C31—42no—
255C22×C31—42no—
256C22×C31—42no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31C2×#12011yescharacteristic
260C22×C31#52511yescharacteristic
261C22×C31#53611yescharacteristic
262C4×C31—24no—
263C4×C31—21no—
264C4×C31—21no—
265C4×C31—42no—
266C4×C31—42no—
267C4×C31—42no—
268C4×C31—42no—
269C4×C31—42no—
270C4×C31—42no—
271C4×C31—81no—
272C4×C31—81no—
273C4×C31#54712yes—
274C4×C31#54712yes—
275#556—22no—
276#555—22no—
277#553—22no—
278#552—22no—
279#544C22×C3111yescharacteristic
280#538—22no—
281#537C22×C3111yescharacteristic
282#529—22no—
283#525C22×C3112yes—
284#520C22×C3112yes—
285C2×#117C22×C3111yescharacteristic
286C2×#117C22×C3111yescharacteristic
287C22×C42C22×C3111yescharacteristic
288#485—22no—
289#485—22no—
290#482—22no—
291#482—22no—
292#428C22×C3112yes—
293#428C22×C3112yes—
294C2×#104C22×C3112yes—
295C2×#104C22×C3112yes—
296#417C22×C3112yes—
297#417C22×C3112yes—
298#407C22×C3112yes—
299#384C22×C3111yescharacteristic
300#384C22×C3111yescharacteristic
301#382C22×C3111yescharacteristic
302#382C22×C3111yescharacteristic
303#380—22no—
304C4×C4⋊C4—22no—
305#343C22×C3111yescharacteristic
306#343C22×C3111yescharacteristic
307#343C22×C3112yes—
308#339C22×C3112yes—
309#339C22×C3112yes—
310C2×C31—21no—
311C2×C31—21no—
312C2×C31—42no—
313C2×C31—82no—
314C2×C31#240611yescharacteristic
315C2×C31#301811yescharacteristic
316C2×C31#301811yescharacteristic
317C2×#41C23×C3112yes—
318C23×C4—21no—
319C23×C4C23×C3111yescharacteristic
320C2×M16C23×C3112yes—
321C2×M16C23×C3112yes—
322C4⋊Q8—22no—
323#126—24no—
324#126—24no—
325#125—22no—
326#125—22no—
327#124—22no—
328#124—24no—
329#123—24no—
330#123—22no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8D4×C3112yes—
334C22⋊Q8D4×C3112yes—
335#121—24no—
336#121—22no—
337#121—22no—
338#121D4×C3112yes—
339#121D4×C3112yes—
340#120—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—24no—
346C4×D4—22no—
347C4×D4—22no—
348#117—24no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117—22no—
354#117C23×C3112yes—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4C23×C3111yescharacteristic
359C2×C4⋊C4C23×C3111yescharacteristic
360C2×C4⋊C4C23×C3111yescharacteristic
361C2×C4⋊C4C23×C3111yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4D4×C3112yes—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42D4×C3112yes—
369C2×C42C23×C3111yescharacteristic
370C2×C42C23×C3111yescharacteristic
371#107—22no—
372#107—22no—
373#106—22no—
374#106—22no—
375#104—24no—
376#104—24no—
377#103—22no—
378#103—22no—
379#102—22no—
380#102—22no—
381C22⋊C8—22no—
382C22⋊C8—22no—
383#95—22no—
384#95—22no—
385C31#2407811yescharacteristic, direct product factor, Sylow
386C24C2×D4×C3111yescharacteristic
387#41—22no—
388#41—22no—
389#41—22no—
390#41—22no—
391C2×Q8—22no—
392C2×Q8C2×D4×C3112yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4C2×D4×C3112yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4C2×D4×C3111yescharacteristic
411C22×C4C2×D4×C3112yes—
412C22×C4C2×D4×C3112yes—
413C22×C4C2×D4×C3112yes—
414C22×C4C2×D4×C3112yes—
415C22×C4C24×C3111yescharacteristic
416M16—22no—
417M16—22no—
418M16—22no—
419M16—22no—
420C2×C8—22no—
421C2×C8—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4C2×D4×C3112yes—
435C4⋊C4C2×D4×C3112yes—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—42no—
445C42—24no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—41no—
453C42—41no—
454C42#41×C3112yes—
455C42#41×C3112yes—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—41no—
460C23—41no—
461C23C31×#12012yes—
462C23C22×D4×C3111yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—82no—
466D4—22no—
467D4—22no—
468D4—42no—
469D4—42no—
470D4—82no—
471D4—82no—
472D4—82no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—48no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—84no—
500C2×C4C31×#12011yescharacteristic
501C2×C4C31×#12011yescharacteristic
502C2×C4C22×D4×C3112yes—
503C2×C4C2×#41×C3111yescharacteristic
504C2×C4C2×#41×C3111yescharacteristic
505C2×C4C31×2−1+412yes—
506C8—42no—
507C8—42no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22C2×C31×#12011yescharacteristic
519C22C31×#52511yescharacteristic
520C22C31×#53611yescharacteristic
521C4—24no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—81no—
531C4—81no—
532C4C31×#54712yes—
533C4C31×#54712yes—
534C2—21no—
535C2—21no—
536C2—42no—
537C2—82no—
538C2C31×#240611yescharacteristic
539C2C31×#301811yescharacteristic
540C2C31×#301811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3493C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3093C2semidirect
3. C31×#2285C2semidirect
4. C31×#2061C2semidirect
5. C31×#2060C2semidirect
6. C31×#2040C2semidirect
7. C31×#2039C2semidirect
8. C31×#1896C2semidirect
9. C31×#1860C2semidirect
10. C31×#1859C2semidirect
11. C31×#1822C2semidirect
185. #24078C31direct (N × H)
239. #3093C2×C31semidirect
240. #2285C2×C31semidirect
241. #2061C2×C31semidirect
242. #2060C2×C31semidirect
243. #2040C2×C31semidirect
244. #2039C2×C31semidirect
245. #1896C2×C31semidirect
246. #1860C2×C31semidirect
247. #1859C2×C31semidirect
248. #1822C2×C31semidirect
385. C31#24078direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34932
C31×#30932
C31×#22852
C31×#20612
C31×#20602
C31×#20402
C31×#20392
C31×#18962
C31×#18602
C31×#18592
C31×#18222
#2407831