Finite Group Databasebeta

← C31×#8550 #21465309

Subgroups — C31×#8550 #21465309

Order 7936.

1134 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C22×C31×C22⋊Q8C211yesmaximal, characteristic
2C31×#1784C212yesmaximal, complemented
3C31×#1463C212yesmaximal, complemented
4C31×#1462C212yesmaximal
5C23×Q8×C31C2211yescharacteristic
6C24×C4×C31C411yescharacteristic
7C2×C31×C22⋊Q8—24no—
8C2×C31×C22⋊Q8—24no—
9C2×C31×C22⋊Q8—24no—
10C22×C4⋊C4×C31C411yescharacteristic
11C22×C4⋊C4×C31C411yescharacteristic
12C22×C4⋊C4×C31C2211yescharacteristic
13C22×C22⋊C4×C31C411yescharacteristic
14C22×C22⋊C4×C31C2211yescharacteristic
15C2×C31×#101—24no—
16C31×#407C2214yes—
17C31×#324—24no—
18C31×#323—24no—
19C25×C31C2×C411yescharacteristic
20C22×Q8×C31—22no—
21C22×Q8×C31—22no—
22C22×Q8×C31—44no—
23C22×Q8×C31D414yes—
24C23×C4×C31—22no—
25C23×C4×C31—22no—
26C23×C4×C31—22no—
27C23×C4×C31—44no—
28C23×C4×C31C2×C411yescharacteristic
29C23×C4×C31C2×C411yescharacteristic
30C23×C4×C31C2×C411yescharacteristic
31C23×C4×C31C2×C411yescharacteristic
32C23×C4×C31C2×C411yescharacteristic
33C23×C4×C31C2311yescharacteristic
34C2×M16×C31—24no—
35C31×C22⋊Q8—232no—
36C2×C4⋊C4×C31—22no—
37C2×C4⋊C4×C31—22no—
38C2×C4⋊C4×C31—22no—
39C2×C4⋊C4×C31—22no—
40C2×C4⋊C4×C31—22no—
41C2×C4⋊C4×C31—22no—
42C2×C4⋊C4×C31—22no—
43C2×C4⋊C4×C31—22no—
44C2×C4⋊C4×C31D414yes—
45C2×C22⋊C4×C31—22no—
46C2×C22⋊C4×C31—22no—
47C2×C22⋊C4×C31—22no—
48C2×C22⋊C4×C31—22no—
49C2×C22⋊C4×C31—22no—
50C2×C22⋊C4×C31D414yes—
51C31×#101—48no—
52C31×C22⋊C8—24no—
53C31×C22⋊C8—24no—
54C24×C31—24no—
55C24×C31—21no—
56C24×C31—21no—
57C24×C31—42no—
58C24×C31—42no—
59C24×C31C22⋊C412yes—
60C24×C31C22×C411yescharacteristic
61C2×Q8×C31—28no—
62C2×Q8×C31—28no—
63C2×Q8×C31—44no—
64C2×Q8×C31—48no—
65C2×Q8×C31—48no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—24no—
69C22×C4×C31—28no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—42no—
79C22×C4×C31—42no—
80C22×C4×C31—42no—
81C22×C4×C31—48no—
82C22×C4×C31—48no—
83C22×C4×C31—44no—
84C22×C4×C31—44no—
85C22×C4×C31—42no—
86C22×C4×C31—42no—
87C22×C4×C31C22⋊C412yes—
88C22×C4×C31C22⋊C412yes—
89C22×C4×C31C22⋊C412yes—
90C22×C4×C31C22⋊C412yes—
91C22×C4×C31C22⋊C412yes—
92C22×C4×C31C2×D412yes—
93C22×C4×C31C2×D412yes—
94C22×C4×C31C2×D412yes—
95M16×C31—48no—
96C2×C8×C31—44no—
97C4⋊C4×C31—28no—
98C4⋊C4×C31—28no—
99C4⋊C4×C31—28no—
100C22⋊C4×C31—28no—
101C22⋊C4×C31—28no—
102#8550C3111yesmaximal, characteristic, direct product factor, Sylow
103C23×C31—22no—
104C23×C31—22no—
105C23×C31—22no—
106C23×C31—24no—
107C23×C31—24no—
108C23×C31—22no—
109C23×C31—22no—
110C23×C31—42no—
111C23×C31—42no—
112C23×C31—42no—
113C23×C31—416no—
114C23×C31—42no—
115C23×C31—42no—
116C23×C31—42no—
117C23×C31#10114yescomplemented
118C23×C31C2×C22⋊C411yescharacteristic
119C23×C31C2×C22⋊C411yescharacteristic
120C23×C31C2×C22⋊C411yescharacteristic
121Q8×C31—416no—
122C2×C4×C31—24no—
123C2×C4×C31—24no—
124C2×C4×C31—22no—
125C2×C4×C31—24no—
126C2×C4×C31—22no—
127C2×C4×C31—22no—
128C2×C4×C31—22no—
129C2×C4×C31—22no—
130C2×C4×C31—22no—
131C2×C4×C31—24no—
132C2×C4×C31—24no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31—416no—
136C2×C4×C31—42no—
137C2×C4×C31—42no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—42no—
141C2×C4×C31—42no—
142C2×C4×C31—42no—
143C2×C4×C31—42no—
144C2×C4×C31—42no—
145C2×C4×C31—42no—
146C2×C4×C31#9914yes—
147C2×C4×C31#12014yes—
148C8×C31—84no—
149C22×C22⋊Q8C2×C3111yescharacteristic
150#1784C2×C3112yescomplemented
151#1463C2×C3112yescomplemented
152#1462C2×C3112yes—
153C22×C31—28no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—21no—
160C22×C31—24no—
161C22×C31—44no—
162C22×C31—48no—
163C22×C31—44no—
164C22×C31—48no—
165C22×C31—42no—
166C22×C31—42no—
167C22×C31#37911yescharacteristic
168C22×C31C2×#9912yes—
169C22×C31C2×#10112yes—
170C22×C31#41312yes—
171C4×C31—24no—
172C4×C31—24no—
173C4×C31—42no—
174C4×C31—42no—
175C4×C31—42no—
176C4×C31—42no—
177C23×Q8C22×C3111yescharacteristic
178C24×C4C4×C3111yescharacteristic
179C2×C22⋊Q8—24no—
180C2×C22⋊Q8—24no—
181C2×C22⋊Q8—24no—
182C22×C4⋊C4C4×C3111yescharacteristic
183C22×C4⋊C4C4×C3111yescharacteristic
184C22×C4⋊C4C22×C3111yescharacteristic
185C22×C22⋊C4C4×C3111yescharacteristic
186C22×C22⋊C4C22×C3111yescharacteristic
187C2×#101—24no—
188#407C22×C3114yes—
189#324—24no—
190#323—24no—
191C2×C31—22no—
192C2×C31—22no—
193C2×C31—22no—
194C2×C31—44no—
195C2×C31#178111yescharacteristic
196C2×C31#178411yescharacteristic
197C2×C31#178511yescharacteristic
198C25C2×C4×C3111yescharacteristic
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—44no—
202C22×Q8D4×C3114yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4C2×C4×C3111yescharacteristic
208C23×C4C2×C4×C3111yescharacteristic
209C23×C4C2×C4×C3111yescharacteristic
210C23×C4C2×C4×C3111yescharacteristic
211C23×C4C2×C4×C3111yescharacteristic
212C23×C4C23×C3111yescharacteristic
213C2×M16—24no—
214C22⋊Q8—232no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4D4×C3114yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4D4×C3114yes—
230#101—48no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C31#855011yescharacteristic, direct product factor, Sylow
234C24—24no—
235C24—21no—
236C24—21no—
237C24—42no—
238C24—42no—
239C24C22⋊C4×C3112yes—
240C24C22×C4×C3111yescharacteristic
241C2×Q8—28no—
242C2×Q8—28no—
243C2×Q8—44no—
244C2×Q8—48no—
245C2×Q8—48no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C22⋊C4×C3112yes—
268C22×C4C22⋊C4×C3112yes—
269C22×C4C22⋊C4×C3112yes—
270C22×C4C22⋊C4×C3112yes—
271C22×C4C22⋊C4×C3112yes—
272C22×C4C2×D4×C3112yes—
273C22×C4C2×D4×C3112yes—
274C22×C4C2×D4×C3112yes—
275M16—48no—
276C2×C8—44no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—24no—
286C23—24no—
287C23—22no—
288C23—22no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—416no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23C31×#10114yescomplemented
297C23C2×C22⋊C4×C3111yescharacteristic
298C23C2×C22⋊C4×C3111yescharacteristic
299C23C2×C22⋊C4×C3111yescharacteristic
300Q8—416no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—416no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4C31×#9914yes—
326C2×C4C31×#12014yes—
327C8—84no—
328C22—28no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—24no—
336C22—44no—
337C22—48no—
338C22—44no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22C31×#37911yescharacteristic
343C22C2×C31×#9912yes—
344C22C2×C31×#10112yes—
345C22C31×#41312yes—
346C4—24no—
347C4—24no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C2—22no—
353C2—22no—
354C2—22no—
355C2—44no—
356C2C31×#178111yescharacteristic
357C2C31×#178411yescharacteristic
358C2C31×#178511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC24C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC25×C31C2×C4
LCSG ▷ C24 ▷ 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×#1784C2semidirect
3. C31×#1463C2semidirect
102. #8550C31direct (N × H)
117. C23×C31#101semidirect
150. #1784C2×C31semidirect
151. #1463C2×C31semidirect
233. C31#8550direct (N × H)
296. C23C31×#101semidirect

Maximal subgroups

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

subgroupindex
C22×C31×C22⋊Q82
C31×#17842
C31×#14632
C31×#14622
#855031