Finite Group Databasebeta

← C31×#9201 #21465960

Subgroups — C31×#9201 #21465960

Order 7936.

612 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2626C211yesmaximal, characteristic
2C31×#2312C211yesmaximal, characteristic
3C31×#1520C212yesmaximal, complemented
4C2×C31×#323C211yesmaximal, characteristic
5C31×#1289C212yesmaximal, complemented
6C2×C31×C22⋊Q8C2211yescharacteristic
7C22×C22⋊C4×C31C2211yescharacteristic
8C2×C31×C22⋊C8C2212yes—
9C31×#399C412yescomplemented
10C31×#396—22no—
11C31×#393C412yescomplemented
12C31×#388—24no—
13C31×#385—21no—
14C31×#380C2211yescharacteristic
15C31×#380C2212yes—
16C31×#379—21no—
17C31×#379—21no—
18C4×C22⋊C4×C31—21no—
19C31×#328—24no—
20C31×#323—22no—
21C25×C31D411yescharacteristic
22C22×Q8×C31C2×C411yescharacteristic
23C23×C4×C31D411yescharacteristic
24C23×C4×C31C2311yescharacteristic
25C22×C8×C31—22no—
26C31×C22⋊Q8—22no—
27C31×C22⋊Q8—22no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31—22no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31C2×C411yescharacteristic
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—21no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31C2×C412yes—
43C2×C22⋊C4×C31D412yes—
44C2×C42×C31—22no—
45C31×C22⋊C8—22no—
46C31×C22⋊C8—22no—
47C31×#95—22no—
48C31×#95—22no—
49C31×#95C2×C412yes—
50C31×#95D414yes—
51C24×C31—21no—
52C24×C31—21no—
53C24×C31—22no—
54C24×C31—22no—
55C24×C31—81no—
56C24×C31—81no—
57C24×C31C22⋊C412yes—
58C24×C31C2×D411yescharacteristic
59C2×Q8×C31—22no—
60C2×Q8×C31—81no—
61C22×C4×C31—21no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—22no—
65C22×C4×C31—21no—
66C22×C4×C31—24no—
67C22×C4×C31—81no—
68C22×C4×C31—81no—
69C22×C4×C31C22⋊C412yes—
70C22×C4×C31C22×C411yescharacteristic
71C22×C4×C31C2×D412yes—
72C22×C4×C31C2×D411yescharacteristic
73C22×C4×C31#4112yes—
74C2×C8×C31—42no—
75C2×C8×C31—82no—
76C4⋊C4×C31—22no—
77C4⋊C4×C31—42no—
78C4⋊C4×C31—81no—
79C4⋊C4×C31—82no—
80C4⋊C4×C31—82no—
81C22⋊C4×C31—24no—
82C22⋊C4×C31—24no—
83C22⋊C4×C31—41no—
84C22⋊C4×C31—41no—
85C22⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—82no—
89C22⋊C4×C31—82no—
90C22⋊C4×C31—82no—
91C22⋊C4×C31—82no—
92C42×C31—82no—
93#9201C3111yesmaximal, characteristic, direct product factor, Sylow
94C23×C31—22no—
95C23×C31—22no—
96C23×C31—22no—
97C23×C31—41no—
98C23×C31—41no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31—42no—
104C23×C31—42no—
105C23×C31—81no—
106C23×C31—81no—
107C23×C31—84no—
108C23×C31—84no—
109C23×C31C2×C22⋊C411yescharacteristic
110C23×C31#12011yescharacteristic
111C23×C31#12311yescharacteristic
112C23×C31#13512yes—
113C23×C31#13712yes—
114Q8×C31—82no—
115C2×C4×C31—22no—
116C2×C4×C31—41no—
117C2×C4×C31—42no—
118C2×C4×C31—42no—
119C2×C4×C31—41no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—44no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—44no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—42no—
131C2×C4×C31—44no—
132C2×C4×C31—81no—
133C2×C4×C31C4×D412yes—
134C2×C4×C31#12112yes—
135C8×C31—82no—
136#2626C2×C3111yescharacteristic
137#2312C2×C3111yescharacteristic
138#1520C2×C3112yescomplemented
139C2×#323C2×C3111yescharacteristic
140#1289C2×C3112yescomplemented
141C22×C31—21no—
142C22×C31—21no—
143C22×C31—21no—
144C22×C31—21no—
145C22×C31—21no—
146C22×C31—21no—
147C22×C31—42no—
148C22×C31—42no—
149C22×C31—44no—
150C22×C31—42no—
151C22×C31—42no—
152C22×C31—42no—
153C22×C31—82no—
154C22×C31—84no—
155C22×C31—84no—
156C22×C31#38611yescharacteristic
157C22×C31#41611yescharacteristic
158C22×C31#41911yescharacteristic
159C22×C31#45212yes—
160C22×C31#48512yes—
161C4×C31—22no—
162C4×C31—22no—
163C4×C31—42no—
164C4×C31—82no—
165C4×C31—84no—
166C4×C31—81no—
167C4×C31—81no—
168C2×C22⋊Q8C22×C3111yescharacteristic
169C22×C22⋊C4C22×C3111yescharacteristic
170C2×C22⋊C8C22×C3112yes—
171#399C4×C3112yescomplemented
172#396—22no—
173#393C4×C3112yescomplemented
174#388—24no—
175#385—21no—
176#380C22×C3111yescharacteristic
177#380C22×C3112yes—
178#379—21no—
179#379—21no—
180C4×C22⋊C4—21no—
181#328—24no—
182#323—22no—
183C2×C31—21no—
184C2×C31—22no—
185C2×C31—21no—
186C2×C31—22no—
187C2×C31—82no—
188C2×C31#187311yescharacteristic
189C2×C31#191111yescharacteristic
190C2×C31#191311yescharacteristic
191C25D4×C3111yescharacteristic
192C22×Q8C2×C4×C3111yescharacteristic
193C23×C4D4×C3111yescharacteristic
194C23×C4C23×C3111yescharacteristic
195C22×C8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C2×C4×C3111yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4C2×C4×C3112yes—
213C2×C22⋊C4D4×C3112yes—
214C2×C42—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217#95—22no—
218#95—22no—
219#95C2×C4×C3112yes—
220#95D4×C3114yes—
221C31#920111yescharacteristic, direct product factor, Sylow
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24C22⋊C4×C3112yes—
229C24C2×D4×C3111yescharacteristic
230C2×Q8—22no—
231C2×Q8—81no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4C22⋊C4×C3112yes—
241C22×C4C22×C4×C3111yescharacteristic
242C22×C4C2×D4×C3112yes—
243C22×C4C2×D4×C3111yescharacteristic
244C22×C4#41×C3112yes—
245C2×C8—42no—
246C2×C8—82no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—82no—
250C4⋊C4—81no—
251C4⋊C4—82no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—82no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×C22⋊C4×C3111yescharacteristic
280C23C31×#12011yescharacteristic
281C23C31×#12311yescharacteristic
282C23C31×#13512yes—
283C23C31×#13712yes—
284Q8—82no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4C4×D4×C3112yes—
304C2×C4C31×#12112yes—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22C31×#38611yescharacteristic
322C22C31×#41611yescharacteristic
323C22C31×#41911yescharacteristic
324C22C31×#45212yes—
325C22C31×#48512yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—82no—
330C4—84no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2C31×#187311yescharacteristic
339C2C31×#191111yescharacteristic
340C2C31×#191311yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC25×C31D4
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
3. C31×#1520C2semidirect
5. C31×#1289C2semidirect
9. C31×#399C4semidirect
11. C31×#393C4semidirect
93. #9201C31direct (N × H)
138. #1520C2×C31semidirect
140. #1289C2×C31semidirect
171. #399C4×C31semidirect
173. #393C4×C31semidirect
221. C31#9201direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26262
C31×#23122
C31×#15202
C2×C31×#3232
C31×#12892
#920131