Finite Group Databasebeta

← C31×#24077 #21480836

Subgroups — C31×#24077 #21480836

Order 7936.

1086 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3492C211yesmaximal, characteristic
2C31×#3092C212yesmaximal, complemented
3C2×C31×#382C211yesmaximal, characteristic, complemented
4C31×#2061C212yesmaximal, complemented
5C31×#2040C212yesmaximal, complemented
6C31×#1896C212yesmaximal, complemented
7C31×#1860C214yesmaximal, complemented
8C31×#1821C211yesmaximal, characteristic, complemented
9C31×#554—22no—
10C31×#553—22no—
11C31×#552—22no—
12C31×#547—22no—
13C31×#544C2211yescharacteristic
14C31×#539—22no—
15C31×#537C2211yescharacteristic
16C31×#525—22no—
17C31×#525C2212yes—
18C31×#520C2212yes—
19C22×C4⋊C4×C31C2211yescharacteristic
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C42×C31C2211yescharacteristic
22C31×#484—22no—
23C31×#483—22no—
24C31×#481—22no—
25C31×#480—22no—
26C31×#428C2214yes—
27C2×C31×#104C2214yes—
28C31×#417C2214yes—
29C31×#407C2212yes—
30C31×#382—22no—
31C31×#382C2212yes—
32C31×#382C2212yes—
33C2×C31×#95—22no—
34C31×#343C2212yes—
35C31×#343C2212yes—
36C31×#339C2214yes—
37C2×#41×C31—22no—
38C2×#41×C31C2312yes—
39C23×C4×C31—21no—
40C23×C4×C31—21no—
41C23×C4×C31D412yes—
42C23×C4×C31C2311yescharacteristic
43C2×M16×C31C2314yes—
44C31×C4⋊Q8—22no—
45C31×#126—24no—
46C31×#125—24no—
47C31×#125—22no—
48C31×#123—24no—
49C31×#123—24no—
50C31×#123—22no—
51C31×C22⋊Q8—22no—
52C31×C22⋊Q8—24no—
53C31×C22⋊Q8—22no—
54C31×C22⋊Q8—22no—
55C31×C22⋊Q8D412yes—
56C31×C22⋊Q8D412yes—
57C31×#121—22no—
58C31×#121—22no—
59C31×#121—22no—
60C31×#121D412yes—
61C31×#121D412yes—
62C31×#120—22no—
63C4×Q8×C31—22no—
64C4×D4×C31—24no—
65C4×D4×C31—22no—
66C4×D4×C31—24no—
67C4×D4×C31—22no—
68C4×D4×C31—22no—
69C31×#117—22no—
70C31×#117—22no—
71C31×#117C2312yes—
72C2×C4⋊C4×C31—22no—
73C2×C4⋊C4×C31—22no—
74C2×C4⋊C4×C31—22no—
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—22no—
77C2×C4⋊C4×C31—21no—
78C2×C4⋊C4×C31D412yes—
79C2×C4⋊C4×C31C2312yes—
80C2×C4⋊C4×C31C2311yescharacteristic
81C2×C4⋊C4×C31C2311yescharacteristic
82C2×C42×C31—21no—
83C2×C42×C31—24no—
84C2×C42×C31C2312yes—
85C31×#107—24no—
86C31×#106—24no—
87C31×#104—28no—
88C31×#103—24no—
89C31×#102—24no—
90C31×C22⋊C8—22no—
91C31×C22⋊C8—22no—
92C31×#95—24no—
93C31×#95—24no—
94C24×C31C2×D411yescharacteristic
95#41×C31—24no—
96#41×C31—24no—
97#41×C31—84no—
98C2×Q8×C31—22no—
99C2×Q8×C31C2×D412yes—
100C2×D4×C31—22no—
101C2×D4×C31—22no—
102C2×D4×C31—22no—
103C2×D4×C31—22no—
104C2×D4×C31C2×D412yes—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—24no—
108C22×C4×C31—22no—
109C22×C4×C31—22no—
110C22×C4×C31—24no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—22no—
114C22×C4×C31—42no—
115C22×C4×C31—42no—
116C22×C4×C31—44no—
117C22×C4×C31—44no—
118C22×C4×C31—41no—
119C22×C4×C31—41no—
120C22×C4×C31C2×D411yescharacteristic
121C22×C4×C31C2×D412yes—
122C22×C4×C31C2×D412yes—
123C22×C4×C31C2×D412yes—
124C22×C4×C31C2×D412yes—
125C22×C4×C31C2411yescharacteristic
126M16×C31—24no—
127M16×C31—24no—
128C2×C8×C31—24no—
129C4⋊C4×C31—24no—
130C4⋊C4×C31—22no—
131C4⋊C4×C31—22no—
132C4⋊C4×C31—24no—
133C4⋊C4×C31—24no—
134C4⋊C4×C31—22no—
135C4⋊C4×C31—22no—
136C4⋊C4×C31—22no—
137C4⋊C4×C31—22no—
138C4⋊C4×C31—24no—
139C4⋊C4×C31—42no—
140C4⋊C4×C31C2×D412yes—
141C4⋊C4×C31C2×D412yes—
142C22⋊C4×C31—24no—
143C22⋊C4×C31—24no—
144C22⋊C4×C31—22no—
145C22⋊C4×C31—22no—
146C22⋊C4×C31—22no—
147C22⋊C4×C31—22no—
148C22⋊C4×C31—22no—
149C42×C31—24no—
150C42×C31—24no—
151C42×C31—22no—
152C42×C31—22no—
153C42×C31#4114yescomplemented
154#24077C3111yesmaximal, characteristic, direct product factor, Sylow
155C23×C31—21no—
156C23×C31—21no—
157C23×C31—22no—
158C23×C31—22no—
159C23×C31—22no—
160C23×C31#12011yescharacteristic
161C23×C31#12011yescharacteristic
162C23×C31C22×D411yescharacteristic
163Q8×C31—24no—
164Q8×C31—82no—
165D4×C31—24no—
166D4×C31—42no—
167D4×C31—42no—
168D4×C31—44no—
169D4×C31—44no—
170D4×C31—82no—
171C2×C4×C31—21no—
172C2×C4×C31—21no—
173C2×C4×C31—24no—
174C2×C4×C31—24no—
175C2×C4×C31—22no—
176C2×C4×C31—22no—
177C2×C4×C31—22no—
178C2×C4×C31—22no—
179C2×C4×C31—22no—
180C2×C4×C31—24no—
181C2×C4×C31—48no—
182C2×C4×C31—42no—
183C2×C4×C31—42no—
184C2×C4×C31—44no—
185C2×C4×C31—44no—
186C2×C4×C31—42no—
187C2×C4×C31—42no—
188C2×C4×C31—44no—
189C2×C4×C31—42no—
190C2×C4×C31—82no—
191C2×C4×C31—84no—
192C2×C4×C31—82no—
193C2×C4×C31—82no—
194C2×C4×C31#12012yes—
195C2×C4×C31C22×D412yes—
196C2×C4×C31C2×#4112yes—
197C2×C4×C312−1+412yes—
198C8×C31—44no—
199#3492C2×C3111yescharacteristic
200#3092C2×C3112yescomplemented
201C2×#382C2×C3111yescharacteristic, complemented
202#2061C2×C3112yescomplemented
203#2040C2×C3112yescomplemented
204#1896C2×C3112yescomplemented
205#1860C2×C3114yescomplemented
206#1821C2×C3111yescharacteristic, complemented
207C22×C31—24no—
208C22×C31—24no—
209C22×C31—21no—
210C22×C31—21no—
211C22×C31—21no—
212C22×C31—21no—
213C22×C31—21no—
214C22×C31—21no—
215C22×C31—42no—
216C22×C31—42no—
217C22×C31—42no—
218C22×C31#45412yes—
219C22×C31#45512yes—
220C22×C31C2×#12011yescharacteristic
221C22×C31#52511yescharacteristic
222C22×C31#53611yescharacteristic
223C4×C31—22no—
224C4×C31—44no—
225C4×C31—42no—
226C4×C31—42no—
227C4×C31—42no—
228C4×C31—42no—
229C4×C31—42no—
230C4×C31—82no—
231C4×C31#54714yes—
232#554—22no—
233#553—22no—
234#552—22no—
235#547—22no—
236#544C22×C3111yescharacteristic
237#539—22no—
238#537C22×C3111yescharacteristic
239#525—22no—
240#525C22×C3112yes—
241#520C22×C3112yes—
242C22×C4⋊C4C22×C3111yescharacteristic
243C22×C4⋊C4C22×C3111yescharacteristic
244C22×C42C22×C3111yescharacteristic
245#484—22no—
246#483—22no—
247#481—22no—
248#480—22no—
249#428C22×C3114yes—
250C2×#104C22×C3114yes—
251#417C22×C3114yes—
252#407C22×C3112yes—
253#382—22no—
254#382C22×C3112yes—
255#382C22×C3112yes—
256C2×#95—22no—
257#343C22×C3112yes—
258#343C22×C3112yes—
259#339C22×C3114yes—
260C2×C31—21no—
261C2×C31—21no—
262C2×C31—22no—
263C2×C31—22no—
264C2×C31—82no—
265C2×C31#240611yescharacteristic
266C2×C31C2×#45511yescharacteristic
267C2×C31C2×#45411yescharacteristic
268C2×#41—22no—
269C2×#41C23×C3112yes—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4D4×C3112yes—
273C23×C4C23×C3111yescharacteristic
274C2×M16C23×C3114yes—
275C4⋊Q8—22no—
276#126—24no—
277#125—24no—
278#125—22no—
279#123—24no—
280#123—24no—
281#123—22no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8D4×C3112yes—
287C22⋊Q8D4×C3112yes—
288#121—22no—
289#121—22no—
290#121—22no—
291#121D4×C3112yes—
292#121D4×C3112yes—
293#120—22no—
294C4×Q8—22no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300#117—22no—
301#117—22no—
302#117C23×C3112yes—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4D4×C3112yes—
310C2×C4⋊C4C23×C3112yes—
311C2×C4⋊C4C23×C3111yescharacteristic
312C2×C4⋊C4C23×C3111yescharacteristic
313C2×C42—21no—
314C2×C42—24no—
315C2×C42C23×C3112yes—
316#107—24no—
317#106—24no—
318#104—28no—
319#103—24no—
320#102—24no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323#95—24no—
324#95—24no—
325C31#2407711yescharacteristic, direct product factor, Sylow
326C24C2×D4×C3111yescharacteristic
327#41—24no—
328#41—24no—
329#41—84no—
330C2×Q8—22no—
331C2×Q8C2×D4×C3112yes—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4C2×D4×C3112yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C2×D4×C3111yescharacteristic
353C22×C4C2×D4×C3112yes—
354C22×C4C2×D4×C3112yes—
355C22×C4C2×D4×C3112yes—
356C22×C4C2×D4×C3112yes—
357C22×C4C24×C3111yescharacteristic
358M16—24no—
359M16—24no—
360C2×C8—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—42no—
372C4⋊C4C2×D4×C3112yes—
373C4⋊C4C2×D4×C3112yes—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C42—24no—
382C42—24no—
383C42—22no—
384C42—22no—
385C42#41×C3114yescomplemented
386C23—22no—
387C23—22no—
388C23—21no—
389C23—21no—
390C23—22no—
391C23C31×#12011yescharacteristic
392C23C31×#12011yescharacteristic
393C23C22×D4×C3111yescharacteristic
394Q8—24no—
395Q8—82no—
396D4—24no—
397D4—44no—
398D4—44no—
399D4—42no—
400D4—42no—
401D4—82no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4C31×#12012yes—
426C2×C4C22×D4×C3112yes—
427C2×C4C2×#41×C3112yes—
428C2×C4C31×2−1+412yes—
429C8—44no—
430C22—24no—
431C22—24no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22C31×#45412yes—
442C22C31×#45512yes—
443C22C2×C31×#12011yescharacteristic
444C22C31×#52511yescharacteristic
445C22C31×#53611yescharacteristic
446C4—22no—
447C4—44no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—82no—
454C4C31×#54714yes—
455C2—21no—
456C2—22no—
457C2—22no—
458C2—21no—
459C2—82no—
460C2C31×#240611yescharacteristic
461C2C2×C31×#45511yescharacteristic
462C2C2×C31×#45411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC31×#3492C2
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×#3092C2semidirect
3. C2×C31×#382C2semidirect
4. C31×#2061C2semidirect
5. C31×#2040C2semidirect
6. C31×#1896C2semidirect
7. C31×#1860C2semidirect
8. C31×#1821C2semidirect
153. C42×C31#41semidirect
154. #24077C31direct (N × H)
200. #3092C2×C31semidirect
201. C2×#382C2×C31semidirect
202. #2061C2×C31semidirect
203. #2040C2×C31semidirect
204. #1896C2×C31semidirect
205. #1860C2×C31semidirect
206. #1821C2×C31semidirect
325. C31#24077direct (N × H)
385. C42#41×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#34922
C31×#30922
C2×C31×#3822
C31×#20612
C31×#20402
C31×#18962
C31×#18602
C31×#18212
#2407731