Finite Group Databasebeta

← C31×#8905 #21465664

Subgroups — C31×#8905 #21465664

Order 7936.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2925C211yesmaximal, characteristic
2C31×#1806C211yesmaximal, characteristic
3C31×#1571C211yesmaximal, characteristic, complemented
4C31×#1566C211yesmaximal, characteristic, complemented
5C31×#1476C211yesmaximal, characteristic
6C31×#1303C211yesmaximal, characteristic, complemented
7C31×#1279C211yesmaximal, characteristic, complemented
8C4×#41×C31C2211yescharacteristic
9C31×#433—22no—
10C31×#432—21no—
11C31×#431—21no—
12C31×#431C2211yescharacteristic
13C31×#423C2211yescharacteristic
14C31×#423C2211yescharacteristic
15C2×C31×C4⋊C8C2211yescharacteristic
16C2×C31×#104—21no—
17C31×#408—21no—
18C31×#405—21no—
19C2×C4×C8×C31C2211yescharacteristic
20C4×C4⋊C4×C31C2211yescharacteristic
21C31×#339—21no—
22C31×#335—21no—
23C31×#335—21no—
24C31×#334—22no—
25C31×Q8⋊C8C411yescharacteristic, complemented
26C31×Q8⋊C8C411yescharacteristic, complemented
27C31×D4⋊C8C411yescharacteristic, complemented
28C31×D4⋊C8C411yescharacteristic, complemented
29C2×#41×C31D411yescharacteristic
30C2×M16×C31—21no—
31C2×M16×C31—21no—
32C22×C8×C31—21no—
33C22×C8×C31—21no—
34C4×Q8×C31C2×C411yescharacteristic
35C4×Q8×C31Q811yescharacteristic
36C4×D4×C31—21no—
37C4×D4×C31—21no—
38C4×D4×C31C2×C411yescharacteristic
39C4×D4×C31Q811yescharacteristic
40C31×#117—21no—
41C31×#117D411yescharacteristic
42C2×C4⋊C4×C31—21no—
43C2×C42×C31—21no—
44C2×C42×C31—21no—
45C2×C42×C31C2311yescharacteristic
46C31×C4⋊C8—21no—
47C31×C4⋊C8—21no—
48C31×C4⋊C8—21no—
49C31×C4⋊C8—21no—
50C31×C4⋊C8C2×C411yescharacteristic
51C31×C4⋊C8C2×C411yescharacteristic
52C31×C4⋊C8D411yescharacteristic
53C31×C4⋊C8D411yescharacteristic
54C31×C4⋊C8D412yes—
55C31×#104—42no—
56C31×C22⋊C8—22no—
57C31×C22⋊C8—22no—
58C31×C22⋊C8—21no—
59C31×#97—22no—
60C4×C8×C31—22no—
61C4×C8×C31—21no—
62C4×C8×C31—21no—
63C4×C8×C31C2×C411yescharacteristic
64C4×C8×C31C2×C411yescharacteristic
65C31×#95—21no—
66#41×C31—21no—
67#41×C31—21no—
68#41×C31—21no—
69#41×C31—21no—
70C2×Q8×C31C4⋊C411yescharacteristic
71C2×D4×C31—21no—
72C2×D4×C31C4⋊C411yescharacteristic
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—41no—
77C22×C4×C31C2×D411yescharacteristic
78C22×C4×C31#4111yescharacteristic
79C22×C4×C31#4111yescharacteristic
80M16×C31—41no—
81M16×C31—41no—
82M16×C31—41no—
83M16×C31—41no—
84C2×C8×C31—21no—
85C2×C8×C31—21no—
86C2×C8×C31—21no—
87C2×C8×C31—21no—
88C2×C8×C31—21no—
89C2×C8×C31—22no—
90C2×C8×C31—21no—
91C2×C8×C31—42no—
92C2×C8×C31—41no—
93C2×C8×C31—41no—
94C2×C8×C31—41no—
95C4⋊C4×C31—21no—
96C4⋊C4×C31—21no—
97C4⋊C4×C31—22no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31C4⋊C411yescharacteristic
100C4⋊C4×C31C4⋊C411yescharacteristic
101C22⋊C4×C31—21no—
102C22⋊C4×C31—21no—
103C22⋊C4×C31—21no—
104C42×C31—21no—
105C42×C31—21no—
106C42×C31—21no—
107C42×C31—41no—
108C42×C31—41no—
109C42×C31C22×C411yescharacteristic
110C42×C31C2×D411yescharacteristic
111C42×C31C2×D411yescharacteristic
112C42×C31C2×Q811yescharacteristic
113#8905C3111yesmaximal, characteristic, direct product factor, Sylow
114C23×C31—21no—
115C23×C31#12311yescharacteristic
116Q8×C31—21no—
117Q8×C31—21no—
118D4×C31—21no—
119D4×C31—21no—
120D4×C31—41no—
121D4×C31—41no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—42no—
133C2×C4×C31—81no—
134C2×C4×C31—81no—
135C2×C4×C31—81no—
136C2×C4×C31C2×C4⋊C411yescharacteristic
137C2×C4×C31C4×D411yescharacteristic
138C2×C4×C31C4×D411yescharacteristic
139C2×C4×C31#12011yescharacteristic
140C2×C4×C31C22⋊Q811yescharacteristic
141C2×C4×C31C22⋊Q811yescharacteristic
142C2×C4×C31#13511yescharacteristic
143C2×C4×C31#13511yescharacteristic
144C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C31#13711yescharacteristic
146C8×C31—22no—
147C8×C31—41no—
148C8×C31—41no—
149C8×C31—41no—
150C8×C31—41no—
151C8×C31—41no—
152C8×C31—42no—
153C8×C31—81no—
154#2925C2×C3111yescharacteristic
155#1806C2×C3111yescharacteristic
156#1571C2×C3111yescharacteristic, complemented
157#1566C2×C3111yescharacteristic, complemented
158#1476C2×C3111yescharacteristic
159#1303C2×C3111yescharacteristic, complemented
160#1279C2×C3111yescharacteristic, complemented
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31#38511yescharacteristic
167C22×C31#48211yescharacteristic
168C22×C31#48511yescharacteristic
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—21no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31—82no—
176C4×C31#42711yescharacteristic
177C4×C31#42811yescharacteristic
178C4×C31#44311yescharacteristic
179C4×C31#44411yescharacteristic
180C4×C31#44911yescharacteristic
181C4×C31#45211yescharacteristic
182C4×#41C22×C3111yescharacteristic
183#433—22no—
184#432—21no—
185#431—21no—
186#431C22×C3111yescharacteristic
187#423C22×C3111yescharacteristic
188#423C22×C3111yescharacteristic
189C2×C4⋊C8C22×C3111yescharacteristic
190C2×#104—21no—
191#408—21no—
192#405—21no—
193C2×C4×C8C22×C3111yescharacteristic
194C4×C4⋊C4C22×C3111yescharacteristic
195#339—21no—
196#335—21no—
197#335—21no—
198#334—22no—
199Q8⋊C8C4×C3111yescharacteristic, complemented
200Q8⋊C8C4×C3111yescharacteristic, complemented
201D4⋊C8C4×C3111yescharacteristic, complemented
202D4⋊C8C4×C3111yescharacteristic, complemented
203C2×C31—21no—
204C2×C31—21no—
205C2×C31—81no—
206C2×C31#185411yescharacteristic
207C2×C31#186111yescharacteristic
208C2×C31#186611yescharacteristic
209C2×#41D4×C3111yescharacteristic
210C2×M16—21no—
211C2×M16—21no—
212C22×C8—21no—
213C22×C8—21no—
214C4×Q8C2×C4×C3111yescharacteristic
215C4×Q8Q8×C3111yescharacteristic
216C4×D4—21no—
217C4×D4—21no—
218C4×D4C2×C4×C3111yescharacteristic
219C4×D4Q8×C3111yescharacteristic
220#117—21no—
221#117D4×C3111yescharacteristic
222C2×C4⋊C4—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42C23×C3111yescharacteristic
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8C2×C4×C3111yescharacteristic
231C4⋊C8C2×C4×C3111yescharacteristic
232C4⋊C8D4×C3112yes—
233C4⋊C8D4×C3111yescharacteristic
234C4⋊C8D4×C3111yescharacteristic
235#104—42no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—21no—
239#97—22no—
240C4×C8—22no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8C2×C4×C3111yescharacteristic
244C4×C8C2×C4×C3111yescharacteristic
245#95—21no—
246C31#890511yescharacteristic, direct product factor, Sylow
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8C4⋊C4×C3111yescharacteristic
252C2×D4—21no—
253C2×D4C4⋊C4×C3111yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4C2×D4×C3111yescharacteristic
259C22×C4#41×C3111yescharacteristic
260C22×C4#41×C3111yescharacteristic
261M16—41no—
262M16—41no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—22no—
271C2×C8—21no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4C4⋊C4×C3111yescharacteristic
281C4⋊C4C4⋊C4×C3111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42C22×C4×C3111yescharacteristic
291C42C2×D4×C3111yescharacteristic
292C42C2×D4×C3111yescharacteristic
293C42C2×Q8×C3111yescharacteristic
294C23—21no—
295C23C31×#12311yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C2×C4⋊C4×C3111yescharacteristic
317C2×C4C4×D4×C3111yescharacteristic
318C2×C4C4×D4×C3111yescharacteristic
319C2×C4C31×#12011yescharacteristic
320C2×C4C31×C22⋊Q811yescharacteristic
321C2×C4C31×C22⋊Q811yescharacteristic
322C2×C4C31×#13511yescharacteristic
323C2×C4C31×#13511yescharacteristic
324C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
325C2×C4C31×#13711yescharacteristic
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—41no—
331C8—41no—
332C8—42no—
333C8—81no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22C31×#38511yescharacteristic
340C22C31×#48211yescharacteristic
341C22C31×#48511yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—82no—
349C4C31×#42711yescharacteristic
350C4C31×#42811yescharacteristic
351C4C31×#44311yescharacteristic
352C4C31×#44411yescharacteristic
353C4C31×#44911yescharacteristic
354C4C31×#45211yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2C31×#185411yescharacteristic
359C2C31×#186111yescharacteristic
360C2C31×#186611yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#385
prime radicalC2×#41×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1571C2semidirect
4. C31×#1566C2semidirect
6. C31×#1303C2semidirect
7. C31×#1279C2semidirect
25. C31×Q8⋊C8C4semidirect
26. C31×Q8⋊C8C4semidirect
27. C31×D4⋊C8C4semidirect
28. C31×D4⋊C8C4semidirect
113. #8905C31direct (N × H)
156. #1571C2×C31semidirect
157. #1566C2×C31semidirect
159. #1303C2×C31semidirect
160. #1279C2×C31semidirect
199. Q8⋊C8C4×C31semidirect
200. Q8⋊C8C4×C31semidirect
201. D4⋊C8C4×C31semidirect
202. D4⋊C8C4×C31semidirect
246. C31#8905direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#29252
C31×#18062
C31×#15712
C31×#15662
C31×#14762
C31×#13032
C31×#12792
#890531