Finite Group Databasebeta

← C31×#34512 #21491271

Subgroups — C31×#34512 #21491271

Order 7936.

2576 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×H(3,𝔽4)C211yesmaximal, characteristic
2C31×#2375C213yesmaximal, complemented
3C31×#2372C213yesmaximal, complemented
4C31×#2367C216yesmaximal, complemented
5C31×#2358C213yesmaximal, complemented
6C31×#2337C216yesmaximal, complemented
7C31×#2326C213yesmaximal, complemented
8C31×#2321C213yesmaximal, complemented
9C4×C31×#120C213yesmaximal, complemented
10C24×C4×C31C2211yescharacteristic, complemented
11C31×H(3,𝔽4)C4116yescomplemented
12C2×C31×#124C2213yes—
13C2×C31×#124C2216yes—
14C2×C31×#120C2213yes—
15C2×C31×#120C2213yes—
16C2×C4×Q8×C31C2213yescomplemented
17C2×C4×D4×C31C2213yes—
18C2×C4×D4×C31C2216yes—
19C22×C22⋊C4×C31C2213yescomplemented
20C31×#391C2213yescomplemented
21C31×#391C2216yescomplemented
22C31×#390C2216yes—
23C31×#390C2213yes—
24C31×#388C2216yescomplemented
25C31×#388C2216yescomplemented
26C31×#388C2216yescomplemented
27C31×#388C2216yescomplemented
28C31×#388C2216yescomplemented
29C31×#388C2216yescomplemented
30C31×#386C2216yes—
31C31×#386C2213yes—
32C31×#386C2216yes—
33C31×#386C2213yes—
34C31×#385C2213yescomplemented
35C31×#385C2216yescomplemented
36C31×#385C2216yescomplemented
37C31×#385C2213yescomplemented
38C31×#384C2211yescharacteristic, complemented
39C31×#383C2212yescomplemented
40C31×#379C2211yescharacteristic, complemented
41C31×#379C2213yescomplemented
42C4×C4⋊C4×C31C2216yescomplemented
43C4×C22⋊C4×C31C2213yescomplemented
44C4×C22⋊C4×C31C2213yescomplemented
45C4×C22⋊C4×C31C2216yescomplemented
46C4×C22⋊C4×C31C2216yescomplemented
47C31×#376C2213yescomplemented
48C25×C31C2311yescharacteristic
49C25×C31C2311yescharacteristic
50C22×Q8×C31C2313yes—
51C22×D4×C31C2313yes—
52C22×D4×C31C2316yes—
53C23×C4×C31—46no—
54C23×C4×C31C2313yes—
55C23×C4×C31C2313yes—
56C23×C4×C31C2313yes—
57C23×C4×C31C2313yes—
58C31×#124C2×C4124yescomplemented
59C31×#124C2×C4148yescomplemented
60C31×#120C2×C4124yescomplemented
61C31×#120C2×C4124yescomplemented
62C4×Q8×C31—46no—
63C4×D4×C31—46no—
64C4×D4×C31—412no—
65C2×C4⋊C4×C31C2316yes—
66C2×C4⋊C4×C31C2313yes—
67C2×C4⋊C4×C31C2313yes—
68C2×C4⋊C4×C31C2316yes—
69C2×C22⋊C4×C31—46no—
70C2×C22⋊C4×C31C2316yes—
71C2×C22⋊C4×C31C2316yes—
72C2×C22⋊C4×C31C2313yes—
73C2×C22⋊C4×C31C2316yes—
74C2×C22⋊C4×C31C2313yes—
75C2×C22⋊C4×C31C2313yes—
76C2×C22⋊C4×C31C2316yes—
77C2×C22⋊C4×C31C2313yes—
78C2×C22⋊C4×C31C2316yes—
79C2×C22⋊C4×C31C2313yes—
80C2×C22⋊C4×C31C2316yes—
81C2×C22⋊C4×C31C2313yes—
82C2×C42×C31C2313yes—
83C2×C42×C31C2316yes—
84C2×C42×C31C2316yes—
85C2×C42×C31C2313yes—
86C2×C42×C31C2311yescharacteristic
87C2×C42×C31C2312yes—
88C31×#95C2316yes—
89C31×#95C2316yes—
90C31×#95C2316yes—
91C31×#95C2316yes—
92C31×#95C2316yes—
93C31×#95C2316yes—
94C24×C31—43no—
95C24×C31—43no—
96C24×C31—43no—
97C24×C31—43no—
98C24×C31C22×C414yes—
99C24×C31C22×C414yescomplemented
100C24×C31C2413yes—
101C24×C31C2413yes—
102C2×Q8×C31—43no—
103C2×Q8×C31—43no—
104C2×Q8×C31C22×C4112yescomplemented
105C2×D4×C31—43no—
106C2×D4×C31—46no—
107C2×D4×C31—43no—
108C2×D4×C31—46no—
109C2×D4×C31C22×C4112yes—
110C2×D4×C31C22×C4124yes—
111C22×C4×C31—212no—
112C22×C4×C31—26no—
113C22×C4×C31—26no—
114C22×C4×C31—43no—
115C22×C4×C31—46no—
116C22×C4×C31—43no—
117C22×C4×C31—46no—
118C22×C4×C31—46no—
119C22×C4×C31—416no—
120C22×C4×C31C2416yes—
121C22×C4×C31C2411yescharacteristic
122C22×C4×C31C2413yes—
123C22×C4×C31C2413yes—
124C22×C4×C31C2413yes—
125C22×C4×C31C2416yes—
126C22×C4×C31C2413yes—
127C4⋊C4×C31—212no—
128C4⋊C4×C31—46no—
129C4⋊C4×C31—43no—
130C4⋊C4×C31—43no—
131C22⋊C4×C31—212no—
132C22⋊C4×C31—46no—
133C22⋊C4×C31—46no—
134C22⋊C4×C31—43no—
135C22⋊C4×C31—43no—
136C22⋊C4×C31—43no—
137C22⋊C4×C31—43no—
138C22⋊C4×C31—46no—
139C22⋊C4×C31C22×C4124yes—
140C22⋊C4×C31C22×C4112yes—
141C22⋊C4×C31C22×C4124yescomplemented
142C22⋊C4×C31C22×C4112yescomplemented
143C42×C31—26no—
144C42×C31—43no—
145C42×C31—46no—
146C42×C31—46no—
147C42×C31C22×C414yescomplemented
148C42×C31C22×C418yescomplemented
149#34512C3111yesmaximal, characteristic, direct product factor, Sylow
150C23×C31—26no—
151C23×C31—23no—
152C23×C31—26no—
153C23×C31—23no—
154C23×C31—26no—
155C23×C31—23no—
156C23×C31—23no—
157C23×C31—26no—
158C23×C31—412no—
159C23×C31—412no—
160C23×C31—44no—
161C23×C31—412no—
162C23×C31—82no—
163C23×C31—86no—
164C23×C31C23×C416yes—
165C23×C31C23×C416yes—
166C23×C31C2511yescharacteristic
167Q8×C31—412no—
168D4×C31—412no—
169D4×C31—424no—
170C2×C4×C31—23no—
171C2×C4×C31—26no—
172C2×C4×C31—26no—
173C2×C4×C31—23no—
174C2×C4×C31—26no—
175C2×C4×C31—23no—
176C2×C4×C31—26no—
177C2×C4×C31—26no—
178C2×C4×C31—23no—
179C2×C4×C31—23no—
180C2×C4×C31—26no—
181C2×C4×C31—26no—
182C2×C4×C31—26no—
183C2×C4×C31—412no—
184C2×C4×C31—412no—
185C2×C4×C31—86no—
186C2×C4×C31—86no—
187C2×C4×C31C23×C416yes—
188C2×C4×C31C23×C4112yes—
189C2×C4×C31H(5,𝔽2)16yes—
190C2×H(3,𝔽4)C2×C3111yescharacteristic
191#2375C2×C3113yescomplemented
192#2372C2×C3113yescomplemented
193#2367C2×C3116yescomplemented
194#2358C2×C3113yescomplemented
195#2337C2×C3116yescomplemented
196#2326C2×C3113yescomplemented
197#2321C2×C3113yescomplemented
198C4×#120C2×C3113yescomplemented
199C22×C31—26no—
200C22×C31—212no—
201C22×C31—212no—
202C22×C31—26no—
203C22×C31—416no—
204C22×C31—43no—
205C22×C31—43no—
206C22×C31—43no—
207C22×C31—43no—
208C22×C31—46no—
209C22×C31—46no—
210C22×C31—88no—
211C22×C31C24×C411yescharacteristic
212C22×C31C2×H(5,𝔽2)13yes—
213C22×C31#58513yes—
214C4×C31—26no—
215C4×C31—212no—
216C4×C31—26no—
217C4×C31—41no—
218C4×C31—43no—
219C4×C31—43no—
220C4×C31—46no—
221C24×C4C22×C3111yescharacteristic, complemented
222H(3,𝔽4)C4×C31116yescomplemented
223C2×#124C22×C3113yes—
224C2×#124C22×C3116yes—
225C2×#120C22×C3113yes—
226C2×#120C22×C3113yes—
227C2×C4×Q8C22×C3113yescomplemented
228C2×C4×D4C22×C3113yes—
229C2×C4×D4C22×C3116yes—
230C22×C22⋊C4C22×C3113yescomplemented
231#391C22×C3113yescomplemented
232#391C22×C3116yescomplemented
233#390C22×C3116yes—
234#390C22×C3113yes—
235#388C22×C3116yescomplemented
236#388C22×C3116yescomplemented
237#388C22×C3116yescomplemented
238#388C22×C3116yescomplemented
239#388C22×C3116yescomplemented
240#388C22×C3116yescomplemented
241#386C22×C3116yes—
242#386C22×C3113yes—
243#386C22×C3116yes—
244#386C22×C3113yes—
245#385C22×C3113yescomplemented
246#385C22×C3116yescomplemented
247#385C22×C3116yescomplemented
248#385C22×C3113yescomplemented
249#384C22×C3111yescharacteristic, complemented
250#383C22×C3112yescomplemented
251#379C22×C3111yescharacteristic, complemented
252#379C22×C3113yescomplemented
253C4×C4⋊C4C22×C3116yescomplemented
254C4×C22⋊C4C22×C3113yescomplemented
255C4×C22⋊C4C22×C3113yescomplemented
256C4×C22⋊C4C22×C3116yescomplemented
257C4×C22⋊C4C22×C3116yescomplemented
258#376C22×C3113yescomplemented
259C2×C31—46no—
260C2×C31—46no—
261C2×C31C4×H(5,𝔽2)13yes—
262C2×C31#353211yescharacteristic
263C2×C31#354013yes—
264C25C23×C3111yescharacteristic
265C25C23×C3111yescharacteristic
266C22×Q8C23×C3113yes—
267C22×D4C23×C3113yes—
268C22×D4C23×C3116yes—
269C23×C4—46no—
270C23×C4C23×C3113yes—
271C23×C4C23×C3113yes—
272C23×C4C23×C3113yes—
273C23×C4C23×C3113yes—
274#124C2×C4×C31124yescomplemented
275#124C2×C4×C31148yescomplemented
276#120C2×C4×C31124yescomplemented
277#120C2×C4×C31124yescomplemented
278C4×Q8—46no—
279C4×D4—46no—
280C4×D4—412no—
281C2×C4⋊C4C23×C3116yes—
282C2×C4⋊C4C23×C3113yes—
283C2×C4⋊C4C23×C3113yes—
284C2×C4⋊C4C23×C3116yes—
285C2×C22⋊C4—46no—
286C2×C22⋊C4C23×C3116yes—
287C2×C22⋊C4C23×C3116yes—
288C2×C22⋊C4C23×C3113yes—
289C2×C22⋊C4C23×C3116yes—
290C2×C22⋊C4C23×C3113yes—
291C2×C22⋊C4C23×C3113yes—
292C2×C22⋊C4C23×C3116yes—
293C2×C22⋊C4C23×C3113yes—
294C2×C22⋊C4C23×C3116yes—
295C2×C22⋊C4C23×C3113yes—
296C2×C22⋊C4C23×C3116yes—
297C2×C22⋊C4C23×C3113yes—
298C2×C42C23×C3113yes—
299C2×C42C23×C3116yes—
300C2×C42C23×C3116yes—
301C2×C42C23×C3113yes—
302C2×C42C23×C3111yescharacteristic
303C2×C42C23×C3112yes—
304#95C23×C3116yes—
305#95C23×C3116yes—
306#95C23×C3116yes—
307#95C23×C3116yes—
308#95C23×C3116yes—
309#95C23×C3116yes—
310C31#3451211yescharacteristic, direct product factor, Sylow
311C24—43no—
312C24—43no—
313C24—43no—
314C24—43no—
315C24C22×C4×C3114yes—
316C24C22×C4×C3114yescomplemented
317C24C24×C3113yes—
318C24C24×C3113yes—
319C2×Q8—43no—
320C2×Q8—43no—
321C2×Q8C22×C4×C31112yescomplemented
322C2×D4—43no—
323C2×D4—46no—
324C2×D4—43no—
325C2×D4—46no—
326C2×D4C22×C4×C31112yes—
327C2×D4C22×C4×C31124yes—
328C22×C4—212no—
329C22×C4—26no—
330C22×C4—26no—
331C22×C4—43no—
332C22×C4—46no—
333C22×C4—43no—
334C22×C4—46no—
335C22×C4—46no—
336C22×C4—416no—
337C22×C4C24×C3116yes—
338C22×C4C24×C3111yescharacteristic
339C22×C4C24×C3113yes—
340C22×C4C24×C3113yes—
341C22×C4C24×C3113yes—
342C22×C4C24×C3116yes—
343C22×C4C24×C3113yes—
344C4⋊C4—212no—
345C4⋊C4—46no—
346C4⋊C4—43no—
347C4⋊C4—43no—
348C22⋊C4—212no—
349C22⋊C4—46no—
350C22⋊C4—46no—
351C22⋊C4—43no—
352C22⋊C4—43no—
353C22⋊C4—43no—
354C22⋊C4—43no—
355C22⋊C4—46no—
356C22⋊C4C22×C4×C31124yes—
357C22⋊C4C22×C4×C31112yes—
358C22⋊C4C22×C4×C31124yescomplemented
359C22⋊C4C22×C4×C31112yescomplemented
360C42—26no—
361C42—43no—
362C42—46no—
363C42—46no—
364C42C22×C4×C3114yescomplemented
365C42C22×C4×C3118yescomplemented
366C23—26no—
367C23—23no—
368C23—26no—
369C23—23no—
370C23—26no—
371C23—23no—
372C23—23no—
373C23—26no—
374C23—412no—
375C23—412no—
376C23—44no—
377C23—412no—
378C23—82no—
379C23—86no—
380C23C23×C4×C3116yes—
381C23C23×C4×C3116yes—
382C23C25×C3111yescharacteristic
383Q8—412no—
384D4—412no—
385D4—424no—
386C2×C4—23no—
387C2×C4—26no—
388C2×C4—26no—
389C2×C4—23no—
390C2×C4—26no—
391C2×C4—23no—
392C2×C4—26no—
393C2×C4—26no—
394C2×C4—23no—
395C2×C4—23no—
396C2×C4—26no—
397C2×C4—26no—
398C2×C4—26no—
399C2×C4—412no—
400C2×C4—412no—
401C2×C4—86no—
402C2×C4—86no—
403C2×C4C23×C4×C3116yes—
404C2×C4C23×C4×C31112yes—
405C2×C4C31×H(5,𝔽2)16yes—
406C22—26no—
407C22—212no—
408C22—212no—
409C22—26no—
410C22—416no—
411C22—43no—
412C22—43no—
413C22—43no—
414C22—43no—
415C22—46no—
416C22—46no—
417C22—88no—
418C22C24×C4×C3111yescharacteristic
419C22C2×C31×H(5,𝔽2)13yes—
420C22C31×#58513yes—
421C4—26no—
422C4—212no—
423C4—26no—
424C4—41no—
425C4—43no—
426C4—46no—
427C4—43no—
428C2—46no—
429C2—46no—
430C2C4×C31×H(5,𝔽2)13yes—
431C2C31×#353211yescharacteristic
432C2C31×#354013yes—
subgroupG/H
center, socleC23×C31C25
commutatorC22C24×C4×C31
Frattini, cosocleC23C25×C31
prime radicalC2×C31×H(3,𝔽4)C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2375C2semidirect
3. C31×#2372C2semidirect
4. C31×#2367C2semidirect
5. C31×#2358C2semidirect
6. C31×#2337C2semidirect
7. C31×#2326C2semidirect
8. C31×#2321C2semidirect
9. C4×C31×#120C2semidirect
10. C24×C4×C31C22semidirect
11. C31×H(3,𝔽4)C4semidirect
16. C2×C4×Q8×C31C22semidirect
19. C22×C22⋊C4×C31C22semidirect
20. C31×#391C22semidirect
21. C31×#391C22semidirect
24. C31×#388C22semidirect
25. C31×#388C22semidirect
26. C31×#388C22semidirect
27. C31×#388C22semidirect
28. C31×#388C22semidirect
29. C31×#388C22semidirect
34. C31×#385C22semidirect
35. C31×#385C22semidirect
36. C31×#385C22semidirect
37. C31×#385C22semidirect
38. C31×#384C22semidirect
39. C31×#383C22semidirect
40. C31×#379C22semidirect
41. C31×#379C22semidirect
42. C4×C4⋊C4×C31C22semidirect
43. C4×C22⋊C4×C31C22semidirect
44. C4×C22⋊C4×C31C22semidirect
45. C4×C22⋊C4×C31C22semidirect
46. C4×C22⋊C4×C31C22semidirect
47. C31×#376C22semidirect
58. C31×#124C2×C4semidirect
59. C31×#124C2×C4semidirect
60. C31×#120C2×C4semidirect
61. C31×#120C2×C4semidirect
99. C24×C31C22×C4semidirect
104. C2×Q8×C31C22×C4semidirect
141. C22⋊C4×C31C22×C4semidirect
142. C22⋊C4×C31C22×C4semidirect
147. C42×C31C22×C4semidirect
148. C42×C31C22×C4semidirect
149. #34512C31direct (N × H)
191. #2375C2×C31semidirect
192. #2372C2×C31semidirect
193. #2367C2×C31semidirect
194. #2358C2×C31semidirect
195. #2337C2×C31semidirect
196. #2326C2×C31semidirect
197. #2321C2×C31semidirect
198. C4×#120C2×C31semidirect
221. C24×C4C22×C31semidirect
222. H(3,𝔽4)C4×C31semidirect
227. C2×C4×Q8C22×C31semidirect
230. C22×C22⋊C4C22×C31semidirect
231. #391C22×C31semidirect
232. #391C22×C31semidirect
235. #388C22×C31semidirect
236. #388C22×C31semidirect
237. #388C22×C31semidirect
238. #388C22×C31semidirect
239. #388C22×C31semidirect
240. #388C22×C31semidirect
245. #385C22×C31semidirect
246. #385C22×C31semidirect
247. #385C22×C31semidirect
248. #385C22×C31semidirect
249. #384C22×C31semidirect
250. #383C22×C31semidirect
251. #379C22×C31semidirect
252. #379C22×C31semidirect
253. C4×C4⋊C4C22×C31semidirect
254. C4×C22⋊C4C22×C31semidirect
255. C4×C22⋊C4C22×C31semidirect
256. C4×C22⋊C4C22×C31semidirect
257. C4×C22⋊C4C22×C31semidirect
258. #376C22×C31semidirect
274. #124C2×C4×C31semidirect
275. #124C2×C4×C31semidirect
276. #120C2×C4×C31semidirect
277. #120C2×C4×C31semidirect
310. C31#34512direct (N × H)
316. C24C22×C4×C31semidirect
321. C2×Q8C22×C4×C31semidirect
358. C22⋊C4C22×C4×C31semidirect
359. C22⋊C4C22×C4×C31semidirect
364. C42C22×C4×C31semidirect
365. C42C22×C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×H(3,𝔽4)2
C31×#23752
C31×#23722
C31×#23672
C31×#23582
C31×#23372
C31×#23262
C31×#23212
C4×C31×#1202
#3451231