Finite Group Databasebeta

← C3×#12704 #10631311

Subgroups — C3×#12704 #10631311

Order 768.

962 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C3×#536C211yesmaximal, characteristic
2C2×C3×#356C214yesmaximal, complemented
3C3×#1785C212yesmaximal
4#12704C311yesmaximal, characteristic, direct product factor, Sylow
5C22×C3×#41C2211yescharacteristic
6C23×C3×Q8C411yescharacteristic
7C3×#536—28no—
8C2×C3×C4⋊Q8C2212yes—
9C2×C3×#124C412yes—
10C2×C3×#123—22no—
11C2×C3×C22⋊Q8—22no—
12C2×C3×#117C411yescharacteristic
13C2×C3×#101C2214yes—
14C2×C3×#100—22no—
15C3×#407—22no—
16C3×#356—28no—
17C2×#536C2×C311yescharacteristic
18C2×#356C2×C314yescomplemented
19#1785C2×C312yes—
20C2×C3×#41—24no—
21C2×C3×#41—42no—
22C2×C3×#41—42no—
23C22×C3×Q8—24no—
24C22×C3×Q8—21no—
25C22×C3×Q8—42no—
26C22×C3×Q8—42no—
27C22×C3×Q8C2×C412yes—
28C22×C3×Q8C2311yescharacteristic
29C22×C3×D4C2×C411yescharacteristic
30C22×C3×D4D412yes—
31C23×C3×C4—21no—
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4D412yes—
34C2×C3×M16—24no—
35C3×C4⋊Q8—24no—
36C3×C4⋊Q8D418yes—
37C3×#124—24no—
38C3×#124—24no—
39C3×#123—216no—
40C3×C22⋊Q8—216no—
41C3×#117—24no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—24no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—24no—
46C2×C3×C42C2×C412yes—
47C3×#101—28no—
48C3×#100—44no—
49C3×C22⋊C8—44no—
50C22×#41C22×C311yescharacteristic
51C23×Q8C3×C411yescharacteristic
52#536—28no—
53C2×C4⋊Q8C22×C312yes—
54C2×#124C3×C412yes—
55C2×#123—22no—
56C2×C22⋊Q8—22no—
57C2×#117C3×C411yescharacteristic
58C2×#101C22×C314yes—
59C2×#100—22no—
60#407—22no—
61#356—28no—
62C24×C3C22⋊C412yes—
63C24×C3C22⋊C411yescharacteristic
64C3×#41—416no—
65C2×C3×Q8—24no—
66C2×C3×Q8—22no—
67C2×C3×Q8—41no—
68C2×C3×Q8—41no—
69C2×C3×Q8—416no—
70C2×C3×Q8—42no—
71C2×C3×Q8—42no—
72C2×C3×Q8—42no—
73C2×C3×Q8—42no—
74C2×C3×Q8C22⋊C414yes—
75C2×C3×Q8C2×D414yes—
76C2×C3×D4—24no—
77C2×C3×D4—22no—
78C2×C3×D4—24no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—42no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—24no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D412yes—
96C3×M16—44no—
97C3×M16—44no—
98C2×C3×C8—44no—
99C3×C4⋊C4—24no—
100C3×C4⋊C4—216no—
101C3×C22⋊C4—24no—
102C3×C22⋊C4—216no—
103C3×C42—22no—
104C3×C42C22⋊C414yes—
105C2×#41—24no—
106C2×#41—42no—
107C2×#41—42no—
108C22×Q8—24no—
109C22×Q8—21no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8C2×C3×C412yes—
113C22×Q8C23×C311yescharacteristic
114C22×D4C2×C3×C411yescharacteristic
115C22×D4C3×D412yes—
116C23×C4—21no—
117C23×C4C2×C3×C411yescharacteristic
118C23×C4C3×D412yes—
119C2×M16—24no—
120C4⋊Q8—24no—
121C4⋊Q8C3×D418yes—
122#124—24no—
123#124—24no—
124#123—216no—
125C22⋊Q8—216no—
126#117—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—24no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—24no—
131C2×C42C2×C3×C412yes—
132#101—28no—
133#100—44no—
134C22⋊C8—44no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—42no—
141C23×C3—44no—
142C23×C3#9912yes—
143C23×C3C2×C22⋊C411yescharacteristic
144C3×Q8—44no—
145C3×Q8—44no—
146C3×Q8—48no—
147C3×Q8—44no—
148C3×D4—48no—
149C3×D4—44no—
150C2×C3×C4—28no—
151C2×C3×C4—21no—
152C2×C3×C4—22no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—22no—
156C2×C3×C4—21no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—48no—
160C2×C3×C4—44no—
161C2×C3×C4—48no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4#9912yes—
165C2×C3×C4C2×C22⋊C412yes—
166C2×C3×C4#12014yes—
167C3×C8—84no—
168C24C3×C22⋊C412yes—
169C24C3×C22⋊C411yescharacteristic
170#41—416no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—416no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8C3×C22⋊C414yes—
181C2×Q8C2×C3×D414yes—
182C2×D4—24no—
183C2×D4—22no—
184C2×D4—24no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—42no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C3×C22⋊C411yescharacteristic
200C22×C4C22×C3×C411yescharacteristic
201C22×C4C2×C3×D412yes—
202M16—44no—
203M16—44no—
204C2×C8—44no—
205C4⋊C4—24no—
206C4⋊C4—216no—
207C22⋊C4—24no—
208C22⋊C4—216no—
209C42—22no—
210C42C3×C22⋊C414yes—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—44no—
216C22×C3—42no—
217C22×C3—44no—
218C22×C3—42no—
219C22×C3—82no—
220C22×C3#37911yescharacteristic
221C22×C3C2×#9911yescharacteristic
222C22×C3C2×#9911yescharacteristic
223C3×C4—24no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—44no—
227C3×C4—44no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—42no—
234C23—44no—
235C23C3×#9912yes—
236C23C2×C3×C22⋊C411yescharacteristic
237Q8—44no—
238Q8—44no—
239Q8—48no—
240Q8—44no—
241D4—48no—
242D4—44no—
243C2×C4—28no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—48no—
253C2×C4—48no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4C3×#9912yes—
258C2×C4C2×C3×C22⋊C412yes—
259C2×C4C3×#12014yes—
260C8—84no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—42no—
264C2×C3—82no—
265C2×C3#178111yescharacteristic
266C2×C3#213512yes—
267C22—22no—
268C22—22no—
269C22—42no—
270C22—42no—
271C22—44no—
272C22—42no—
273C22—42no—
274C22—44no—
275C22—82no—
276C22C3×#37911yescharacteristic
277C22C2×C3×#9911yescharacteristic
278C22C2×C3×#9911yescharacteristic
279C4—24no—
280C4—22no—
281C4—22no—
282C4—44no—
283C4—44no—
284C3#1270411yescharacteristic, direct product factor, Sylow
285C2—21no—
286C2—21no—
287C2—42no—
288C2—82no—
289C2C3×#178111yescharacteristic
290C2C3×#213512yes—
subgroupG/H
center, socleC22×C3C2×#99
commutatorC22×C4C22×C3×C4
Frattini, cosocleC22×Q8C23×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#356C2semidirect
4. #12704C3direct (N × H)
18. C2×#356C2×C3semidirect
284. C3#12704direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#3562
C3×#17852
#127043