Finite Group Databasebeta

← C3×#12700 #10631307

Subgroups — C3×#12700 #10631307

Order 768.

1018 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 158 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×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#356C214yesmaximal, complemented
3C3×#1784C212yesmaximal
4#12700C311yesmaximal, characteristic, direct product factor, Sylow
5C23×C3×Q8C411yescharacteristic
6C23×C3×Q8C2211yescharacteristic
7C2×C3×C4⋊Q8—22no—
8C2×C3×C4⋊Q8—28no—
9C2×C3×C4⋊Q8C412yes—
10C2×C3×C4⋊Q8C2212yes—
11C22×C3×C4⋊C4—22no—
12C22×C3×C42C411yescharacteristic
13C2×C3×#101—22no—
14C2×C3×#101C2214yes—
15C3×#407—22no—
16C3×#356—28no—
17C22×C4⋊Q8C2×C311yescharacteristic
18C2×#356C2×C314yescomplemented
19#1784C2×C312yes—
20C22×C3×Q8—24no—
21C22×C3×Q8—24no—
22C22×C3×Q8—21no—
23C22×C3×Q8—42no—
24C22×C3×Q8—42no—
25C22×C3×Q8—42no—
26C22×C3×Q8—42no—
27C22×C3×Q8C2×C412yes—
28C22×C3×Q8C2×C411yescharacteristic
29C22×C3×Q8D412yes—
30C22×C3×Q8C2311yescharacteristic
31C23×C3×C4—21no—
32C23×C3×C4—21no—
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4D412yes—
35C2×C3×M16—24no—
36C3×C4⋊Q8—216no—
37C3×C4⋊Q8—24no—
38C3×C4⋊Q8—24no—
39C3×C4⋊Q8—24no—
40C3×C4⋊Q8D418yes—
41C2×C3×C4⋊C4—216no—
42C2×C3×C4⋊C4—24no—
43C2×C3×C4⋊C4—24no—
44C2×C3×C42—21no—
45C2×C3×C42—24no—
46C2×C3×C42C2×C412yes—
47C3×#101—28no—
48C3×#101—44no—
49C3×C22⋊C8—44no—
50C23×Q8C3×C411yescharacteristic
51C23×Q8C22×C311yescharacteristic
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—28no—
54C2×C4⋊Q8C3×C412yes—
55C2×C4⋊Q8C22×C312yes—
56C22×C4⋊C4—22no—
57C22×C42C3×C411yescharacteristic
58C2×#101—22no—
59C2×#101C22×C314yes—
60#407—22no—
61#356—28no—
62C24×C3C22⋊C411yescharacteristic
63C2×C3×Q8—24no—
64C2×C3×Q8—24no—
65C2×C3×Q8—22no—
66C2×C3×Q8—22no—
67C2×C3×Q8—24no—
68C2×C3×Q8—41no—
69C2×C3×Q8—41no—
70C2×C3×Q8—42no—
71C2×C3×Q8—41no—
72C2×C3×Q8—41no—
73C2×C3×Q8—416no—
74C2×C3×Q8—416no—
75C2×C3×Q8—42no—
76C2×C3×Q8—42no—
77C2×C3×Q8—42no—
78C2×C3×Q8—42no—
79C2×C3×Q8C22⋊C414yes—
80C2×C3×Q8C2×D414yes—
81C22×C3×C4—24no—
82C22×C3×C4—28no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—24no—
88C22×C3×C4—24no—
89C22×C3×C4—22no—
90C22×C3×C4—24no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4C22⋊C412yes—
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D412yes—
98C3×M16—44no—
99C3×M16—44no—
100C2×C3×C8—44no—
101C3×C4⋊C4—216no—
102C3×C4⋊C4—216no—
103C3×C42—24no—
104C3×C42—22no—
105C3×C42C22⋊C414yes—
106C22×Q8—24no—
107C22×Q8—24no—
108C22×Q8—21no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8C2×C3×C412yes—
114C22×Q8C2×C3×C411yescharacteristic
115C22×Q8C3×D412yes—
116C22×Q8C23×C311yescharacteristic
117C23×C4—21no—
118C23×C4—21no—
119C23×C4C2×C3×C411yescharacteristic
120C23×C4C3×D412yes—
121C2×M16—24no—
122C4⋊Q8—216no—
123C4⋊Q8—24no—
124C4⋊Q8—24no—
125C4⋊Q8—24no—
126C4⋊Q8C3×D418yes—
127C2×C4⋊C4—216no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C42—21no—
131C2×C42—24no—
132C2×C42C2×C3×C412yes—
133#101—28no—
134#101—44no—
135C22⋊C8—44no—
136C23×C3—24no—
137C23×C3—21no—
138C23×C3—21no—
139C23×C3#9912yes—
140C23×C3C2×C22⋊C411yescharacteristic
141C3×Q8—44no—
142C3×Q8—48no—
143C3×Q8—44no—
144C3×Q8—48no—
145C3×Q8—44no—
146C3×Q8—44no—
147C2×C3×C4—216no—
148C2×C3×C4—28no—
149C2×C3×C4—28no—
150C2×C3×C4—21no—
151C2×C3×C4—24no—
152C2×C3×C4—22no—
153C2×C3×C4—21no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—48no—
162C2×C3×C4—48no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4#9912yes—
166C2×C3×C4C2×C22⋊C412yes—
167C2×C3×C4#12014yes—
168C3×C8—84no—
169C24C3×C22⋊C411yescharacteristic
170C2×Q8—24no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—416no—
181C2×Q8—416no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8C3×C22⋊C414yes—
187C2×Q8C2×C3×D414yes—
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4C3×C22⋊C412yes—
202C22×C4C3×C22⋊C411yescharacteristic
203C22×C4C22×C3×C411yescharacteristic
204C22×C4C2×C3×D412yes—
205M16—44no—
206M16—44no—
207C2×C8—44no—
208C4⋊C4—216no—
209C4⋊C4—216no—
210C42—24no—
211C42—22no—
212C42C3×C22⋊C414yes—
213C22×C3—24no—
214C22×C3—24no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3#35614yescomplemented
219C22×C3#37911yescharacteristic
220C22×C3C2×#9911yescharacteristic
221C22×C3C2×#9911yescharacteristic
222C3×C4—28no—
223C3×C4—24no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—42no—
227C3×C4—44no—
228C23—21no—
229C23—21no—
230C23—24no—
231C23C3×#9912yes—
232C23C2×C3×C22⋊C411yescharacteristic
233Q8—44no—
234Q8—48no—
235Q8—44no—
236Q8—48no—
237Q8—44no—
238Q8—44no—
239C2×C4—28no—
240C2×C4—216no—
241C2×C4—28no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—48no—
254C2×C4—48no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4C3×#9912yes—
258C2×C4C2×C3×C22⋊C412yes—
259C2×C4C3×#12014yes—
260C8—84no—
261C2×C3—21no—
262C2×C3—24no—
263C2×C3—21no—
264C2×C3#178111yescharacteristic
265C2×C3C2×#35612yes—
266C22—24no—
267C22—24no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22C3×#35614yescomplemented
272C22C3×#37911yescharacteristic
273C22C2×C3×#9911yescharacteristic
274C22C2×C3×#9911yescharacteristic
275C4—28no—
276C4—24no—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—44no—
281C3#1270011yescharacteristic, direct product factor, Sylow
282C2—21no—
283C2—21no—
284C2—24no—
285C2C3×#178111yescharacteristic
286C2C2×C3×#35612yes—
subgroupG/H
center, socleC22×C3C2×#99
commutatorC22×C4C22×C3×C4
Frattini, cosocleC22×Q8C23×C3
prime radicalC24×C3C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#356C2semidirect
4. #12700C3direct (N × H)
18. C2×#356C2×C3semidirect
218. C22×C3#356semidirect
271. C22C3×#356semidirect
281. C3#12700direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#3562
C3×#17842
#127003