Finite Group Databasebeta

← C31×#61378 #21518137

Subgroups — C31×#61378 #21518137

Order 7936.

1810 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3501C212yesmaximal, complemented
2C31×#3389C218yesmaximal, complemented
3C31×#3384C218yesmaximal, complemented
4C31×#3163C212yesmaximal, complemented
5C31×#3162C211yesmaximal, characteristic, complemented
6C31×#3081C214yesmaximal, complemented
7C31×#3077C212yesmaximal, complemented
8C31×#2961C211yesmaximal, characteristic, complemented
9C31×#2948C212yesmaximal
10C2×C31×#442C211yesmaximal, characteristic
11C22×D8×C31C2211yescharacteristic
12C31×#554—22no—
13C31×#554—24no—
14C31×#554C2218yescomplemented
15C31×#554C2214yescomplemented
16D4×Q8×C31—22no—
17D4×Q8×C31C2214yescomplemented
18C31×#544C2214yes—
19C31×#543—24no—
20C31×#533C2212yes—
21C31×#533C2211yescharacteristic
22C2×C31×C22⋊Q8—24no—
23C31×#518C2214yes—
24C2×C4×D4×C31C2212yes—
25C31×#501C2214yescomplemented
26C31×#501C2218yescomplemented
27C31×#501C2214yescomplemented
28C31×#499C2214yescomplemented
29C31×#498C2214yescomplemented
30C31×#478C2218yescomplemented
31C31×#478C2218yescomplemented
32C31×#478C2218yescomplemented
33C31×#476C2214yescomplemented
34C31×#476C2218yescomplemented
35C31×#474C2214yescomplemented
36C31×#474C2218yescomplemented
37C31×#446C2218yescomplemented
38C31×#442C2214yes—
39C31×#442C2218yes—
40C31×#442C2214yes—
41C4×D8×C31C2218yes—
42C31×#428C2214yes—
43C31×#427C2212yes—
44C31×#427C2212yes—
45C2×C31×#106C2211yescharacteristic
46C31×#424C2212yes—
47C31×#423C2211yescharacteristic
48C31×#418C2214yes—
49C2×C31×#102C2212yes—
50C31×#405C2212yes—
51C2×C31×#97C2211yescharacteristic
52C22×Q8×C31—24no—
53C22×D4×C31C2312yes—
54C23×C4×C31—22no—
55C2×D8×C31Q818yescomplemented
56C2×D8×C31C2314yes—
57C2×M16×C31C2312yes—
58C22×C8×C31C2311yescharacteristic
59C22×C8×C31C2311yescharacteristic
60C31×C4⋊Q8—24no—
61C31×C4⋊Q8D412yes—
62C31×C4⋊Q8C2312yes—
63C31×C4⋊Q8C2318yes—
64C31×C4⋊Q8C2312yes—
65C31×#125—24no—
66C31×#125D412yes—
67C31×#125C2318yes—
68C31×#125C2314yes—
69C31×C22⋊Q8—24no—
70C31×C22⋊Q8—28no—
71C31×C22⋊Q8—22no—
72C31×C22⋊Q8—22no—
73C31×C22⋊Q8—24no—
74C31×C22⋊Q8—28no—
75C31×C22⋊Q8—28no—
76C31×C22⋊Q8—28no—
77C31×C22⋊Q8—28no—
78C31×C22⋊Q8—28no—
79C31×C22⋊Q8D414yes—
80C4×Q8×C31—22no—
81C4×Q8×C31—22no—
82C4×Q8×C31C2314yes—
83C4×D4×C31—24no—
84C4×D4×C31—24no—
85C4×D4×C31—22no—
86C4×D4×C31—22no—
87C4×D4×C31C2318yes—
88C4×D4×C31C2318yes—
89C4×D4×C31C2314yes—
90C4×D4×C31C2314yes—
91C31×#117C2314yes—
92C31×#117C2312yes—
93C2×C4⋊C4×C31—28no—
94C2×C4⋊C4×C31—24no—
95C2×C4⋊C4×C31C2312yes—
96C2×C22⋊C4×C31—22no—
97C2×C22⋊C4×C31—28no—
98C2×C42×C31C2311yescharacteristic
99C31×#107C2314yes—
100C31×#107C2314yes—
101C31×#107C2318yes—
102C31×#106C2314yes—
103C31×#106C2312yes—
104C31×#106C2312yes—
105C31×#106C2312yes—
106C31×#106C2312yes—
107C31×#106C2318yes—
108C31×C4⋊C8C2314yes—
109C31×C4⋊C8C2318yes—
110C31×#102C2318yes—
111C31×#102C2318yes—
112C31×#102C2318yes—
113C31×#97C2314yes—
114C31×#97C2314yes—
115C4×C8×C31C2314yes—
116C24×C31—22no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—28no—
120C2×Q8×C31—88no—
121C2×Q8×C31C2×D414yes—
122C2×D4×C31—22no—
123C2×D4×C31—24no—
124C2×D4×C31C2×Q818yescomplemented
125C2×D4×C31C2414yes—
126C22×C4×C31—22no—
127C22×C4×C31—24no—
128C22×C4×C31—28no—
129C22×C4×C31—28no—
130C22×C4×C31—24no—
131C22×C4×C31—84no—
132C22×C4×C31C2×D412yes—
133C22×C4×C31C2411yescharacteristic
134D8×C31C2×Q8116yescomplemented
135M16×C31—24no—
136C2×C8×C31—22no—
137C2×C8×C31C2×Q814yescomplemented
138C2×C8×C31C2414yes—
139C2×C8×C31C2412yes—
140C2×C8×C31C2412yes—
141C4⋊C4×C31—24no—
142C4⋊C4×C31—22no—
143C4⋊C4×C31—28no—
144C4⋊C4×C31—28no—
145C4⋊C4×C31—28no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—24no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31C2×D418yes—
150C4⋊C4×C31C2×D414yes—
151C4⋊C4×C31C2×D414yes—
152C4⋊C4×C31C2414yes—
153C4⋊C4×C31C2414yes—
154C4⋊C4×C31C2414yes—
155C22⋊C4×C31—24no—
156C22⋊C4×C31—28no—
157C22⋊C4×C31—28no—
158C22⋊C4×C31—24no—
159C22⋊C4×C31—24no—
160C22⋊C4×C31C2×D414yes—
161C42×C31—24no—
162C42×C31C2×D411yescharacteristic
163C42×C31C2×D411yescharacteristic
164C42×C31C2414yes—
165C42×C31C2412yes—
166#61378C3111yesmaximal, characteristic, direct product factor, Sylow
167C23×C31—24no—
168C23×C31—28no—
169C23×C31—82no—
170C23×C31—82no—
171C23×C31C22×D411yescharacteristic
172Q8×C31—44no—
173Q8×C31—44no—
174Q8×C31—82no—
175Q8×C31—88no—
176D4×C31—28no—
177D4×C31—24no—
178D4×C31C22×Q818yes—
179C2×C4×C31—24no—
180C2×C4×C31—24no—
181C2×C4×C31—24no—
182C2×C4×C31—42no—
183C2×C4×C31—48no—
184C2×C4×C31—88no—
185C2×C4×C31—84no—
186C2×C4×C31—88no—
187C2×C4×C31—82no—
188C2×C4×C31C22×D412yes—
189C2×C4×C31C22×D412yes—
190C2×C4×C31C22×D418yes—
191C2×C4×C31C22×D411yescharacteristic
192C2×C4×C31C22×Q812yes—
193C2×C4×C312−1+412yes—
194C2×C4×C31C2511yescharacteristic
195C8×C31—22no—
196C8×C31—24no—
197C8×C31C22×Q814yes—
198#3501C2×C3112yescomplemented
199#3389C2×C3118yescomplemented
200#3384C2×C3118yescomplemented
201#3163C2×C3112yescomplemented
202#3162C2×C3111yescharacteristic, complemented
203#3081C2×C3114yescomplemented
204#3077C2×C3112yescomplemented
205#2961C2×C3111yescharacteristic, complemented
206#2948C2×C3112yes—
207C2×#442C2×C3111yescharacteristic
208C22×C31—22no—
209C22×C31—28no—
210C22×C31—44no—
211C22×C31—44no—
212C22×C31—88no—
213C22×C31D4×Q812yes—
214C22×C31C23×D411yescharacteristic
215C4×C31—22no—
216C4×C31—22no—
217C4×C31—28no—
218C4×C31—44no—
219C4×C31—44no—
220C4×C31—44no—
221C4×C31D4×Q812yes—
222C4×C31C23×Q811yescharacteristic
223C4×C31C2×2−1+411yescharacteristic
224C22×D8C22×C3111yescharacteristic
225#554—22no—
226#554—24no—
227#554C22×C3118yescomplemented
228#554C22×C3114yescomplemented
229D4×Q8—22no—
230D4×Q8C22×C3114yescomplemented
231#544C22×C3114yes—
232#543—24no—
233#533C22×C3112yes—
234#533C22×C3111yescharacteristic
235C2×C22⋊Q8—24no—
236#518C22×C3114yes—
237C2×C4×D4C22×C3112yes—
238#501C22×C3114yescomplemented
239#501C22×C3118yescomplemented
240#501C22×C3114yescomplemented
241#499C22×C3114yescomplemented
242#498C22×C3114yescomplemented
243#478C22×C3118yescomplemented
244#478C22×C3118yescomplemented
245#478C22×C3118yescomplemented
246#476C22×C3114yescomplemented
247#476C22×C3118yescomplemented
248#474C22×C3114yescomplemented
249#474C22×C3118yescomplemented
250#446C22×C3118yescomplemented
251#442C22×C3114yes—
252#442C22×C3118yes—
253#442C22×C3114yes—
254C4×D8C22×C3118yes—
255#428C22×C3114yes—
256#427C22×C3112yes—
257#427C22×C3112yes—
258C2×#106C22×C3111yescharacteristic
259#424C22×C3112yes—
260#423C22×C3111yescharacteristic
261#418C22×C3114yes—
262C2×#102C22×C3112yes—
263#405C22×C3112yes—
264C2×#97C22×C3111yescharacteristic
265C2×C31—22no—
266C2×C31—48no—
267C2×C31C2×D4×Q811yescharacteristic
268C2×C31#358512yes—
269C22×Q8—24no—
270C22×D4C23×C3112yes—
271C23×C4—22no—
272C2×D8Q8×C3118yescomplemented
273C2×D8C23×C3114yes—
274C2×M16C23×C3112yes—
275C22×C8C23×C3111yescharacteristic
276C22×C8C23×C3111yescharacteristic
277C4⋊Q8—24no—
278C4⋊Q8D4×C3112yes—
279C4⋊Q8C23×C3112yes—
280C4⋊Q8C23×C3112yes—
281C4⋊Q8C23×C3118yes—
282#125—24no—
283#125D4×C3112yes—
284#125C23×C3114yes—
285#125C23×C3118yes—
286C22⋊Q8—24no—
287C22⋊Q8—28no—
288C22⋊Q8—22no—
289C22⋊Q8—22no—
290C22⋊Q8—24no—
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8D4×C3114yes—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8C23×C3114yes—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4C23×C3114yes—
305C4×D4C23×C3114yes—
306C4×D4C23×C3118yes—
307C4×D4C23×C3118yes—
308#117C23×C3114yes—
309#117C23×C3112yes—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4C23×C3112yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—28no—
315C2×C42C23×C3111yescharacteristic
316#107C23×C3114yes—
317#107C23×C3114yes—
318#107C23×C3118yes—
319#106C23×C3114yes—
320#106C23×C3112yes—
321#106C23×C3112yes—
322#106C23×C3112yes—
323#106C23×C3112yes—
324#106C23×C3118yes—
325C4⋊C8C23×C3114yes—
326C4⋊C8C23×C3118yes—
327#102C23×C3118yes—
328#102C23×C3118yes—
329#102C23×C3118yes—
330#97C23×C3114yes—
331#97C23×C3114yes—
332C4×C8C23×C3114yes—
333C31#6137811yescharacteristic, direct product factor, Sylow
334C24—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—28no—
338C2×Q8—88no—
339C2×Q8C2×D4×C3114yes—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4C2×Q8×C3118yescomplemented
343C2×D4C24×C3114yes—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—84no—
350C22×C4C2×D4×C3112yes—
351C22×C4C24×C3111yescharacteristic
352D8C2×Q8×C31116yescomplemented
353M16—24no—
354C2×C8—22no—
355C2×C8C2×Q8×C3114yescomplemented
356C2×C8C24×C3114yes—
357C2×C8C24×C3112yes—
358C2×C8C24×C3112yes—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—28no—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4C2×D4×C3118yes—
368C4⋊C4C2×D4×C3114yes—
369C4⋊C4C2×D4×C3114yes—
370C4⋊C4C24×C3114yes—
371C4⋊C4C24×C3114yes—
372C4⋊C4C24×C3114yes—
373C22⋊C4—24no—
374C22⋊C4—28no—
375C22⋊C4—28no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4C2×D4×C3114yes—
379C42—24no—
380C42C2×D4×C3111yescharacteristic
381C42C2×D4×C3111yescharacteristic
382C42C24×C3114yes—
383C42C24×C3112yes—
384C23—24no—
385C23—28no—
386C23—82no—
387C23—82no—
388C23C22×D4×C3111yescharacteristic
389Q8—44no—
390Q8—44no—
391Q8—82no—
392Q8—88no—
393D4—28no—
394D4—24no—
395D4C22×Q8×C3118yes—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—88no—
402C2×C4—84no—
403C2×C4—88no—
404C2×C4—82no—
405C2×C4C22×D4×C3112yes—
406C2×C4C22×D4×C3112yes—
407C2×C4C22×D4×C3118yes—
408C2×C4C22×D4×C3111yescharacteristic
409C2×C4C22×Q8×C3112yes—
410C2×C4C31×2−1+412yes—
411C2×C4C25×C3111yescharacteristic
412C8—22no—
413C8—24no—
414C8C22×Q8×C3114yes—
415C22—22no—
416C22—28no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22D4×Q8×C3112yes—
421C22C23×D4×C3111yescharacteristic
422C4—22no—
423C4—22no—
424C4—28no—
425C4—44no—
426C4—44no—
427C4—44no—
428C4D4×Q8×C3112yes—
429C4C23×Q8×C3111yescharacteristic
430C4C2×C31×2−1+411yescharacteristic
431C2—22no—
432C2—48no—
433C2C2×D4×Q8×C3111yescharacteristic
434C2C31×#358512yes—
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC22×D8×C31C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C31×#533 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3501C2semidirect
2. C31×#3389C2semidirect
3. C31×#3384C2semidirect
4. C31×#3163C2semidirect
5. C31×#3162C2semidirect
6. C31×#3081C2semidirect
7. C31×#3077C2semidirect
8. C31×#2961C2semidirect
14. C31×#554C22semidirect
15. C31×#554C22semidirect
17. D4×Q8×C31C22semidirect
25. C31×#501C22semidirect
26. C31×#501C22semidirect
27. C31×#501C22semidirect
28. C31×#499C22semidirect
29. C31×#498C22semidirect
30. C31×#478C22semidirect
31. C31×#478C22semidirect
32. C31×#478C22semidirect
33. C31×#476C22semidirect
34. C31×#476C22semidirect
35. C31×#474C22semidirect
36. C31×#474C22semidirect
37. C31×#446C22semidirect
55. C2×D8×C31Q8semidirect
124. C2×D4×C31C2×Q8semidirect
134. D8×C31C2×Q8semidirect
137. C2×C8×C31C2×Q8semidirect
166. #61378C31direct (N × H)
198. #3501C2×C31semidirect
199. #3389C2×C31semidirect
200. #3384C2×C31semidirect
201. #3163C2×C31semidirect
202. #3162C2×C31semidirect
203. #3081C2×C31semidirect
204. #3077C2×C31semidirect
205. #2961C2×C31semidirect
227. #554C22×C31semidirect
228. #554C22×C31semidirect
230. D4×Q8C22×C31semidirect
238. #501C22×C31semidirect
239. #501C22×C31semidirect
240. #501C22×C31semidirect
241. #499C22×C31semidirect
242. #498C22×C31semidirect
243. #478C22×C31semidirect
244. #478C22×C31semidirect
245. #478C22×C31semidirect
246. #476C22×C31semidirect
247. #476C22×C31semidirect
248. #474C22×C31semidirect
249. #474C22×C31semidirect
250. #446C22×C31semidirect
272. C2×D8Q8×C31semidirect
333. C31#61378direct (N × H)
342. C2×D4C2×Q8×C31semidirect
352. D8C2×Q8×C31semidirect
355. C2×C8C2×Q8×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#35012
C31×#33892
C31×#33842
C31×#31632
C31×#31622
C31×#30812
C31×#30772
C31×#29612
C31×#29482
C2×C31×#4422
#6137831