Finite Group Databasebeta

← C31×#61826 #21518585

Subgroups — C31×#61826 #21518585

Order 7936.

1726 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3502C211yesmaximal, characteristic, complemented
2C2×C31×#554C211yesmaximal, characteristic
3C31×#3385C218yesmaximal, complemented
4C31×#3372C218yesmaximal, complemented
5C31×#3236C214yesmaximal, complemented
6C31×#3230C212yesmaximal, complemented
7C31×#3110C214yesmaximal
8C31×#3101C212yesmaximal
9C31×#2983C211yesmaximal, characteristic, complemented
10Q82×C31C2214yescomplemented
11C31×#557—24no—
12C31×#557C2214yescomplemented
13C31×#554—24no—
14C31×#554C2218yes—
15C31×#544—21no—
16C31×#544—22no—
17C31×#533C2212yes—
18C31×#533C2214yes—
19C2×C31×C4⋊Q8C2211yescharacteristic
20C2×C31×#125C2212yes—
21C2×C31×C22⋊Q8—21no—
22C2×C31×C22⋊Q8—22no—
23C31×#519—21no—
24C2×C4×Q8×C31C2211yescharacteristic
25C2×C4×D4×C31C2211yescharacteristic
26C2×C4×D4×C31C2212yes—
27C22×C4⋊C4×C31—21no—
28C31×#501C2218yescomplemented
29C31×#500C2214yescomplemented
30C31×#491C2218yescomplemented
31C31×#487C2214yescomplemented
32C31×#484C2218yes—
33C31×#483C2214yes—
34C31×#479C22116yescomplemented
35C31×#477C2218yescomplemented
36C31×#476C2214yes—
37C31×#474C2218yes—
38C31×#461C2214yes—
39C31×#461C2218yes—
40C31×#450C2214yes—
41C31×#446C2218yescomplemented
42C31×#440C22116yes—
43C4×SD16×C31C2218yes—
44C31×#432C2214yes—
45C31×#431C2212yes—
46C22×Q8×C31—21no—
47C22×Q8×C31D411yescharacteristic
48C22×D4×C31C2311yescharacteristic
49C23×C4×C31—21no—
50C23×C4×C31—22no—
51C2×SD16×C31C2318yes—
52C31×C4⋊Q8—24no—
53C31×C4⋊Q8—28no—
54C31×C4⋊Q8—22no—
55C31×C4⋊Q8C2312yes—
56C31×C4⋊Q8C2314yes—
57C31×C4⋊Q8C2314yes—
58C31×C4⋊Q8C2312yes—
59C31×C4⋊Q8C2314yes—
60C31×#125—24no—
61C31×#125—28no—
62C31×#125—24no—
63C31×#125—24no—
64C31×#125—22no—
65C31×#125C2318yes—
66C31×C22⋊Q8—28no—
67C31×C22⋊Q8—22no—
68C31×C22⋊Q8—24no—
69C31×C22⋊Q8—216no—
70C31×C22⋊Q8C2314yes—
71C4×Q8×C31—24no—
72C4×Q8×C31—24no—
73C4×Q8×C31—24no—
74C4×Q8×C31—22no—
75C4×Q8×C31D414yes—
76C4×Q8×C31C2314yes—
77C4×Q8×C31C2318yes—
78C4×Q8×C31C2314yes—
79C4×D4×C31—24no—
80C4×D4×C31—22no—
81C4×D4×C31C2314yes—
82C4×D4×C31C2318yes—
83C31×#117—24no—
84C31×#117—22no—
85C2×C4⋊C4×C31—22no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—22no—
88C2×C4⋊C4×C31—28no—
89C2×C4⋊C4×C31D411yescharacteristic
90C2×C4⋊C4×C31D412yes—
91C2×C4⋊C4×C31C2311yescharacteristic
92C2×C4⋊C4×C31C2312yes—
93C2×C22⋊C4×C31—21no—
94C2×C22⋊C4×C31—22no—
95C2×C42×C31C2311yescharacteristic
96C2×C42×C31C2312yes—
97C31×#107C23116yes—
98C31×#106C2318yes—
99C31×C4⋊C8C2314yes—
100C31×C4⋊C8C2318yes—
101C31×#103C2318yes—
102C31×#103C23116yes—
103C31×#102C2318yes—
104C31×C22⋊C8C2314yes—
105C31×#97C2318yes—
106C4×C8×C31C2314yes—
107C24×C31—21no—
108C2×Q8×C31—24no—
109C2×Q8×C31—22no—
110C2×Q8×C31—24no—
111C2×Q8×C31—24no—
112C2×Q8×C31—82no—
113C2×Q8×C31C2×D414yes—
114C2×Q8×C31C2414yes—
115C2×D4×C31—24no—
116C2×D4×C31—21no—
117C2×D4×C31C2412yes—
118C22×C4×C31—21no—
119C22×C4×C31—22no—
120C22×C4×C31—24no—
121C22×C4×C31—28no—
122C22×C4×C31—44no—
123C22×C4×C31—84no—
124C22×C4×C31C2×D412yes—
125C22×C4×C31C2×D411yescharacteristic
126C22×C4×C31C2×D412yes—
127C22×C4×C31C2×D411yescharacteristic
128C22×C4×C31C2411yescharacteristic
129SD16×C31—216no—
130C2×C8×C31C2418yes—
131C4⋊C4×C31—28no—
132C4⋊C4×C31—24no—
133C4⋊C4×C31—28no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—24no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—24no—
139C4⋊C4×C31—22no—
140C4⋊C4×C31—24no—
141C4⋊C4×C31—22no—
142C4⋊C4×C31—24no—
143C4⋊C4×C31—24no—
144C4⋊C4×C31—28no—
145C4⋊C4×C31C2×D414yes—
146C4⋊C4×C31C2×D418yes—
147C4⋊C4×C31C2414yes—
148C4⋊C4×C31C2412yes—
149C4⋊C4×C31C2414yes—
150C22⋊C4×C31—24no—
151C22⋊C4×C31—24no—
152C22⋊C4×C31—28no—
153C42×C31—28no—
154C42×C31—24no—
155C42×C31C2×D414yes—
156C42×C31C2×D412yes—
157C42×C31C2412yes—
158C42×C31C2414yes—
159#61826C3111yesmaximal, characteristic, direct product factor, Sylow
160C23×C31—24no—
161C23×C31—22no—
162C23×C31—42no—
163C23×C31—42no—
164C23×C31C22×D411yescharacteristic
165Q8×C31—28no—
166Q8×C31—28no—
167Q8×C31—84no—
168Q8×C31—82no—
169Q8×C31—84no—
170D4×C31—24no—
171D4×C31—22no—
172D4×C312−1+414yes—
173C2×C4×C31—24no—
174C2×C4×C31—22no—
175C2×C4×C31—24no—
176C2×C4×C31—24no—
177C2×C4×C31—22no—
178C2×C4×C31—24no—
179C2×C4×C31—42no—
180C2×C4×C31—41no—
181C2×C4×C31—44no—
182C2×C4×C31—42no—
183C2×C4×C31—48no—
184C2×C4×C31—88no—
185C2×C4×C31#13714yes—
186C2×C4×C31C22×D414yes—
187C2×C4×C31C22×D411yescharacteristic
188C2×C4×C31C22×D412yes—
189C2×C4×C31C22×D414yes—
190C2×C4×C31C22×D412yes—
191C2×C4×C31C2511yescharacteristic
192C8×C31—28no—
193#3502C2×C3111yescharacteristic, complemented
194C2×#554C2×C3111yescharacteristic
195#3385C2×C3118yescomplemented
196#3372C2×C3118yescomplemented
197#3236C2×C3114yescomplemented
198#3230C2×C3112yescomplemented
199#3110C2×C3114yes—
200#3101C2×C3112yes—
201#2983C2×C3111yescharacteristic, complemented
202C22×C31—21no—
203C22×C31—21no—
204C22×C31—24no—
205C22×C31—48no—
206C22×C31—42no—
207C22×C31—42no—
208C22×C31C2×#13712yes—
209C22×C31C23×D411yescharacteristic
210C4×C31—24no—
211C4×C31—22no—
212C4×C31—21no—
213C4×C31—24no—
214C4×C31—44no—
215C4×C31—44no—
216C4×C31—44no—
217C4×C31—42no—
218C4×C31C2×#13714yes—
219C4×C31C2×2−1+412yes—
220Q82C22×C3114yescomplemented
221#557—24no—
222#557C22×C3114yescomplemented
223#554—24no—
224#554C22×C3118yes—
225#544—21no—
226#544—22no—
227#533C22×C3112yes—
228#533C22×C3114yes—
229C2×C4⋊Q8C22×C3111yescharacteristic
230C2×#125C22×C3112yes—
231C2×C22⋊Q8—21no—
232C2×C22⋊Q8—22no—
233#519—21no—
234C2×C4×Q8C22×C3111yescharacteristic
235C2×C4×D4C22×C3111yescharacteristic
236C2×C4×D4C22×C3112yes—
237C22×C4⋊C4—21no—
238#501C22×C3118yescomplemented
239#500C22×C3114yescomplemented
240#491C22×C3118yescomplemented
241#487C22×C3114yescomplemented
242#484C22×C3118yes—
243#483C22×C3114yes—
244#479C22×C31116yescomplemented
245#477C22×C3118yescomplemented
246#476C22×C3114yes—
247#474C22×C3118yes—
248#461C22×C3114yes—
249#461C22×C3118yes—
250#450C22×C3114yes—
251#446C22×C3118yescomplemented
252#440C22×C31116yes—
253C4×SD16C22×C3118yes—
254#432C22×C3114yes—
255#431C22×C3112yes—
256C2×C31—22no—
257C2×C31—44no—
258C2×C31#348611yescharacteristic
259C2×C31C22×#13711yescharacteristic
260C2×C31#358511yescharacteristic
261C22×Q8—21no—
262C22×Q8D4×C3111yescharacteristic
263C22×D4C23×C3111yescharacteristic
264C23×C4—21no—
265C23×C4—22no—
266C2×SD16C23×C3118yes—
267C4⋊Q8—24no—
268C4⋊Q8—28no—
269C4⋊Q8—22no—
270C4⋊Q8C23×C3112yes—
271C4⋊Q8C23×C3114yes—
272C4⋊Q8C23×C3114yes—
273C4⋊Q8C23×C3114yes—
274C4⋊Q8C23×C3112yes—
275#125—24no—
276#125—28no—
277#125—24no—
278#125—24no—
279#125—22no—
280#125C23×C3118yes—
281C22⋊Q8—28no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8—216no—
285C22⋊Q8C23×C3114yes—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8D4×C3114yes—
291C4×Q8C23×C3114yes—
292C4×Q8C23×C3118yes—
293C4×Q8C23×C3114yes—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4C23×C3114yes—
297C4×D4C23×C3118yes—
298#117—24no—
299#117—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4D4×C3111yescharacteristic
305C2×C4⋊C4D4×C3112yes—
306C2×C4⋊C4C23×C3111yescharacteristic
307C2×C4⋊C4C23×C3112yes—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C42C23×C3112yes—
311C2×C42C23×C3111yescharacteristic
312#107C23×C31116yes—
313#106C23×C3118yes—
314C4⋊C8C23×C3114yes—
315C4⋊C8C23×C3118yes—
316#103C23×C3118yes—
317#103C23×C31116yes—
318#102C23×C3118yes—
319C22⋊C8C23×C3114yes—
320#97C23×C3118yes—
321C4×C8C23×C3114yes—
322C31#6182611yescharacteristic, direct product factor, Sylow
323C24—21no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—82no—
329C2×Q8C2×D4×C3114yes—
330C2×Q8C24×C3114yes—
331C2×D4—24no—
332C2×D4—21no—
333C2×D4C24×C3112yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—44no—
339C22×C4—84no—
340C22×C4C2×D4×C3112yes—
341C22×C4C2×D4×C3111yescharacteristic
342C22×C4C2×D4×C3112yes—
343C22×C4C2×D4×C3111yescharacteristic
344C22×C4C24×C3111yescharacteristic
345SD16—216no—
346C2×C8C24×C3118yes—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4C2×D4×C3114yes—
362C4⋊C4C2×D4×C3118yes—
363C4⋊C4C24×C3114yes—
364C4⋊C4C24×C3112yes—
365C4⋊C4C24×C3114yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—28no—
369C42—28no—
370C42—24no—
371C42C2×D4×C3114yes—
372C42C2×D4×C3112yes—
373C42C24×C3112yes—
374C42C24×C3114yes—
375C23—24no—
376C23—22no—
377C23—42no—
378C23—42no—
379C23C22×D4×C3111yescharacteristic
380Q8—28no—
381Q8—28no—
382Q8—84no—
383Q8—82no—
384Q8—84no—
385D4—24no—
386D4—22no—
387D4C31×2−1+414yes—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—48no—
399C2×C4—88no—
400C2×C4C31×#13714yes—
401C2×C4C22×D4×C3114yes—
402C2×C4C22×D4×C3111yescharacteristic
403C2×C4C22×D4×C3112yes—
404C2×C4C22×D4×C3114yes—
405C2×C4C22×D4×C3112yes—
406C2×C4C25×C3111yescharacteristic
407C8—28no—
408C22—21no—
409C22—21no—
410C22—24no—
411C22—48no—
412C22—42no—
413C22—42no—
414C22C2×C31×#13712yes—
415C22C23×D4×C3111yescharacteristic
416C4—21no—
417C4—24no—
418C4—24no—
419C4—22no—
420C4—44no—
421C4—42no—
422C4—44no—
423C4—44no—
424C4C2×C31×#13714yes—
425C4C2×C31×2−1+412yes—
426C2—22no—
427C2—44no—
428C2C31×#348611yescharacteristic
429C2C22×C31×#13711yescharacteristic
430C2C31×#358511yescharacteristic
subgroupG/H
center, socleC22×C31C23×D4
commutator, Frattini, cosocleC2×C4C25×C31
prime radicalC22×D4×C31C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3502C2semidirect
3. C31×#3385C2semidirect
4. C31×#3372C2semidirect
5. C31×#3236C2semidirect
6. C31×#3230C2semidirect
9. C31×#2983C2semidirect
10. Q82×C31C22semidirect
12. C31×#557C22semidirect
28. C31×#501C22semidirect
29. C31×#500C22semidirect
30. C31×#491C22semidirect
31. C31×#487C22semidirect
34. C31×#479C22semidirect
35. C31×#477C22semidirect
41. C31×#446C22semidirect
159. #61826C31direct (N × H)
193. #3502C2×C31semidirect
195. #3385C2×C31semidirect
196. #3372C2×C31semidirect
197. #3236C2×C31semidirect
198. #3230C2×C31semidirect
201. #2983C2×C31semidirect
220. Q82C22×C31semidirect
222. #557C22×C31semidirect
238. #501C22×C31semidirect
239. #500C22×C31semidirect
240. #491C22×C31semidirect
241. #487C22×C31semidirect
244. #479C22×C31semidirect
245. #477C22×C31semidirect
251. #446C22×C31semidirect
322. C31#61826direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35022
C2×C31×#5542
C31×#33852
C31×#33722
C31×#32362
C31×#32302
C31×#31102
C31×#31012
C31×#29832
#6182631