Finite Group Databasebeta

← C31×#8463 #21465222

Subgroups — C31×#8463 #21465222

Order 7936.

718 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3493C211yesmaximal, characteristic
2C31×#1820C211yesmaximal, characteristic, complemented
3C31×#1818C211yesmaximal, characteristic, complemented
4C31×#1516C212yesmaximal, complemented
5C31×#1516C212yesmaximal, complemented
6C31×#556—22no—
7C31×#555—22no—
8C31×#553—22no—
9C31×#552—22no—
10C31×#544C411yescharacteristic, complemented
11C31×#538—22no—
12C31×#537C411yescharacteristic, complemented
13C31×#529—22no—
14C31×#525C2212yes—
15C31×#520C412yescomplemented
16C2×C31×#117C2211yescharacteristic
17C2×C31×C22⋊C8C2211yescharacteristic
18C2×C31×C22⋊C8C2211yescharacteristic
19C31×#381C2212yes—
20C31×#342—22no—
21C31×#336—22no—
22C31×#328—22no—
23C31×#328—22no—
24C31×#327—22no—
25C31×#327—22no—
26C2×#41×C31C2×C412yes—
27C23×C4×C31C2311yescharacteristic
28C22×C8×C31—21no—
29C22×C8×C31—21no—
30C31×C4⋊Q8—22no—
31C31×#126—24no—
32C31×#126—24no—
33C31×#125—22no—
34C31×#125—22no—
35C31×#124—22no—
36C31×#124—24no—
37C31×#123—24no—
38C31×#123—22no—
39C31×C22⋊Q8—22no—
40C31×C22⋊Q8—22no—
41C31×C22⋊Q8D412yes—
42C31×C22⋊Q8D412yes—
43C31×#121—24no—
44C31×#121—22no—
45C31×#121—22no—
46C31×#121D412yes—
47C31×#121D412yes—
48C31×#120—22no—
49C4×Q8×C31—22no—
50C4×Q8×C31—22no—
51C4×D4×C31—22no—
52C4×D4×C31—22no—
53C4×D4×C31—24no—
54C4×D4×C31—22no—
55C4×D4×C31—22no—
56C31×#117—24no—
57C31×#117—22no—
58C31×#117—22no—
59C31×#117—22no—
60C31×#117C2×C412yes—
61C2×C4⋊C4×C31—22no—
62C2×C4⋊C4×C31C2×C411yescharacteristic
63C2×C4⋊C4×C31C2×C411yescharacteristic
64C2×C22⋊C4×C31—22no—
65C2×C22⋊C4×C31D412yes—
66C2×C42×C31D412yes—
67C31×C22⋊C8—22no—
68C31×C22⋊C8—22no—
69C31×#95—22no—
70C31×#95—22no—
71C24×C31C2×D411yescharacteristic
72#41×C31—44no—
73C2×Q8×C31—22no—
74C2×Q8×C31C22⋊C412yes—
75C2×D4×C31—22no—
76C2×D4×C31—22no—
77C2×D4×C31—22no—
78C2×D4×C31—22no—
79C2×D4×C31C22⋊C412yes—
80C22×C4×C31—24no—
81C22×C4×C31—22no—
82C22×C4×C31—42no—
83C22×C4×C31—42no—
84C22×C4×C31C22⋊C412yes—
85C22×C4×C31C22⋊C412yes—
86C22×C4×C31C22×C411yescharacteristic
87C22×C4×C31C2×D411yescharacteristic
88C22×C4×C31C2×D412yes—
89C22×C4×C31C2×D412yes—
90C2×C8×C31—42no—
91C2×C8×C31—42no—
92C2×C8×C31—41no—
93C2×C8×C31—41no—
94C4⋊C4×C31—24no—
95C4⋊C4×C31—24no—
96C4⋊C4×C31—22no—
97C4⋊C4×C31—22no—
98C4⋊C4×C31—22no—
99C4⋊C4×C31—22no—
100C4⋊C4×C31—22no—
101C4⋊C4×C31—22no—
102C4⋊C4×C31C22⋊C412yes—
103C4⋊C4×C31C22⋊C412yes—
104C22⋊C4×C31—24no—
105C22⋊C4×C31—24no—
106C22⋊C4×C31—22no—
107C22⋊C4×C31—24no—
108C22⋊C4×C31—22no—
109C22⋊C4×C31—22no—
110C22⋊C4×C31—22no—
111C22⋊C4×C31—22no—
112C22⋊C4×C31—44no—
113C42×C31—24no—
114C42×C31—22no—
115C42×C31—22no—
116C42×C31—22no—
117#8463C3111yesmaximal, characteristic, direct product factor, Sylow
118C23×C31—22no—
119C23×C31—22no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31C2×C22⋊C411yescharacteristic
123C23×C31#12012yes—
124Q8×C31—42no—
125Q8×C31—82no—
126D4×C31—42no—
127D4×C31—44no—
128D4×C31—82no—
129D4×C31—82no—
130D4×C31—82no—
131C2×C4×C31—22no—
132C2×C4×C31—22no—
133C2×C4×C31—22no—
134C2×C4×C31—22no—
135C2×C4×C31—22no—
136C2×C4×C31—22no—
137C2×C4×C31—22no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—44no—
141C2×C4×C31—42no—
142C2×C4×C31—42no—
143C2×C4×C31—84no—
144C2×C4×C31—82no—
145C2×C4×C31—82no—
146C2×C4×C31—82no—
147C2×C4×C31C2×C22⋊C412yes—
148C2×C4×C31#12011yescharacteristic
149C2×C4×C31#12011yescharacteristic
150C8×C31—81no—
151C8×C31—81no—
152#3493C2×C3111yescharacteristic
153#1820C2×C3111yescharacteristic, complemented
154#1818C2×C3111yescharacteristic, complemented
155#1516C2×C3112yescomplemented
156#1516C2×C3112yescomplemented
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—22no—
160C22×C31—22no—
161C22×C31—22no—
162C22×C31—42no—
163C22×C31—42no—
164C22×C31—42no—
165C22×C31—82no—
166C22×C31#37911yescharacteristic
167C22×C31#41012yes—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—22no—
171C4×C31—42no—
172C4×C31—42no—
173C4×C31—42no—
174C4×C31—42no—
175C4×C31—42no—
176C4×C31—162no—
177#556—22no—
178#555—22no—
179#553—22no—
180#552—22no—
181#544C4×C3111yescharacteristic, complemented
182#538—22no—
183#537C4×C3111yescharacteristic, complemented
184#529—22no—
185#525C22×C3112yes—
186#520C4×C3112yescomplemented
187C2×#117C22×C3111yescharacteristic
188C2×C22⋊C8C22×C3111yescharacteristic
189C2×C22⋊C8C22×C3111yescharacteristic
190#381C22×C3112yes—
191#342—22no—
192#336—22no—
193#328—22no—
194#328—22no—
195#327—22no—
196#327—22no—
197C2×C31—22no—
198C2×C31—42no—
199C2×C31—82no—
200C2×C31#178211yescharacteristic
201C2×C31#179411yescharacteristic
202C2×C31#179411yescharacteristic
203C2×#41C2×C4×C3112yes—
204C23×C4C23×C3111yescharacteristic
205C22×C8—21no—
206C22×C8—21no—
207C4⋊Q8—22no—
208#126—24no—
209#126—24no—
210#125—22no—
211#125—22no—
212#124—22no—
213#124—24no—
214#123—24no—
215#123—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8D4×C3112yes—
219C22⋊Q8D4×C3112yes—
220#121—24no—
221#121—22no—
222#121—22no—
223#121D4×C3112yes—
224#121D4×C3112yes—
225#120—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—24no—
231C4×D4—22no—
232C4×D4—22no—
233#117—24no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117C2×C4×C3112yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C2×C4×C3111yescharacteristic
240C2×C4⋊C4C2×C4×C3111yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4D4×C3112yes—
243C2×C42D4×C3112yes—
244C22⋊C8—22no—
245C22⋊C8—22no—
246#95—22no—
247#95—22no—
248C31#846311yescharacteristic, direct product factor, Sylow
249C24C2×D4×C3111yescharacteristic
250#41—44no—
251C2×Q8—22no—
252C2×Q8C22⋊C4×C3112yes—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊C4×C3112yes—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C22⋊C4×C3112yes—
263C22×C4C22⋊C4×C3112yes—
264C22×C4C22×C4×C3111yescharacteristic
265C22×C4C2×D4×C3111yescharacteristic
266C22×C4C2×D4×C3112yes—
267C22×C4C2×D4×C3112yes—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4C22⋊C4×C3112yes—
281C4⋊C4C22⋊C4×C3112yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—22no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—44no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊C4×C3111yescharacteristic
300C23C31×#12012yes—
301Q8—42no—
302Q8—82no—
303D4—42no—
304D4—44no—
305D4—82no—
306D4—82no—
307D4—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—84no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4C2×C22⋊C4×C3112yes—
325C2×C4C31×#12011yescharacteristic
326C2×C4C31×#12011yescharacteristic
327C8—81no—
328C8—81no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22C31×#37911yescharacteristic
339C22C31×#41012yes—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—162no—
349C2—22no—
350C2—42no—
351C2—82no—
352C2C31×#178211yescharacteristic
353C2C31×#179411yescharacteristic
354C2C31×#179411yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3493C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1820C2semidirect
3. C31×#1818C2semidirect
4. C31×#1516C2semidirect
5. C31×#1516C2semidirect
10. C31×#544C4semidirect
12. C31×#537C4semidirect
15. C31×#520C4semidirect
117. #8463C31direct (N × H)
153. #1820C2×C31semidirect
154. #1818C2×C31semidirect
155. #1516C2×C31semidirect
156. #1516C2×C31semidirect
181. #544C4×C31semidirect
183. #537C4×C31semidirect
186. #520C4×C31semidirect
248. C31#8463direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34932
C31×#18202
C31×#18182
C31×#15162
#846331