Finite Group Databasebeta

← C3×#21796 #10640403

Subgroups — C3×#21796 #10640403

Order 768.

1334 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 366 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×#537C211yesmaximal, characteristic
2C3×#2313C212yesmaximal, complemented
3C3×#1964C218yesmaximal, complemented
4C3×#1782C214yesmaximal, complemented
5#21796C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×#41C2212yes—
7C3×#537—28no—
8C2×C3×C22⋊Q8C414yescomplemented
9C2×C3×#121C414yescomplemented
10C2×C3×#121C2214yes—
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×C3×#99C22116yes—
13C3×#387—28no—
14C3×#381C2214yes—
15C3×#380C2214yes—
16C3×C4×C22⋊C4C2214yes—
17C2×#537C2×C311yescharacteristic
18#2313C2×C312yescomplemented
19#1964C2×C318yescomplemented
20#1782C2×C314yescomplemented
21C2×C3×#41—28no—
22C2×C3×#41—48no—
23C22×C3×Q8C2×C412yes—
24C22×C3×D4C2×C412yes—
25C22×C3×D4C2×C414yes—
26C22×C3×D4C2314yes—
27C23×C3×C4C2×C414yes—
28C23×C3×C4C2311yescharacteristic
29C23×C3×C4C2312yes—
30C3×C22⋊Q8—216no—
31C3×#121—28no—
32C3×#121—28no—
33C3×#121—216no—
34C2×C3×C4⋊C4—24no—
35C2×C3×C4⋊C4C2×C414yes—
36C2×C3×C4⋊C4C2×C414yes—
37C2×C3×C4⋊C4D418yes—
38C2×C3×C22⋊C4C2×C418yes—
39C2×C3×C22⋊C4C2318yes—
40C2×C3×C42—24no—
41C3×#99—416no—
42C3×#95—24no—
43C3×#95—24no—
44C3×#95—28no—
45C22×#41C22×C312yes—
46#537—28no—
47C2×C22⋊Q8C3×C414yescomplemented
48C2×#121C3×C414yescomplemented
49C2×#121C22×C314yes—
50C22×C4⋊C4C22×C311yescharacteristic
51C2×#99C22×C3116yes—
52#387—28no—
53#381C22×C314yes—
54#380C22×C314yes—
55C4×C22⋊C4C22×C314yes—
56C24×C3C22×C414yes—
57C24×C3C2411yescharacteristic
58C3×#41—432no—
59C2×C3×Q8—24no—
60C2×C3×Q8—44no—
61C2×C3×D4—28no—
62C2×C3×D4—216no—
63C2×C3×D4—24no—
64C2×C3×D4—28no—
65C2×C3×D4—44no—
66C2×C3×D4—84no—
67C22×C3×C4—24no—
68C22×C3×C4—28no—
69C22×C3×C4—28no—
70C22×C3×C4—48no—
71C22×C3×C4—82no—
72C22×C3×C4C22×C414yes—
73C22×C3×C4C22×C412yes—
74C22×C3×C4C22×C414yes—
75C22×C3×C4C2×D414yes—
76C22×C3×C4C2×D414yes—
77C22×C3×C4C2×D414yes—
78C3×C4⋊C4—28no—
79C3×C4⋊C4—84no—
80C3×C4⋊C4C22⋊C4116yes—
81C3×C22⋊C4—216no—
82C3×C22⋊C4—216no—
83C3×C42—84no—
84C2×#41—28no—
85C2×#41—48no—
86C22×Q8C2×C3×C412yes—
87C22×D4C2×C3×C412yes—
88C22×D4C2×C3×C414yes—
89C22×D4C23×C314yes—
90C23×C4C2×C3×C414yes—
91C23×C4C23×C311yescharacteristic
92C23×C4C23×C312yes—
93C22⋊Q8—216no—
94#121—28no—
95#121—28no—
96#121—216no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4C2×C3×C414yes—
99C2×C4⋊C4C2×C3×C414yes—
100C2×C4⋊C4C3×D418yes—
101C2×C22⋊C4C2×C3×C418yes—
102C2×C22⋊C4C23×C318yes—
103C2×C42—24no—
104#99—416no—
105#95—24no—
106#95—24no—
107#95—28no—
108C23×C3—24no—
109C23×C3—28no—
110C23×C3—41no—
111C23×C3—41no—
112C23×C3—44no—
113C23×C3—44no—
114C23×C3C23×C411yescharacteristic
115C23×C3C22×D412yes—
116C23×C3H(5,𝔽2)14yes—
117C3×Q8—48no—
118C3×D4—48no—
119C3×D4—416no—
120C3×D4—88no—
121C2×C3×C4—24no—
122C2×C3×C4—28no—
123C2×C3×C4—24no—
124C2×C3×C4—28no—
125C2×C3×C4—44no—
126C2×C3×C4—44no—
127C2×C3×C4—44no—
128C2×C3×C4—416no—
129C2×C3×C4—48no—
130C2×C3×C4—48no—
131C2×C3×C4—48no—
132C2×C3×C4C2×C22⋊C418yes—
133C2×C3×C4C2×C22⋊C414yes—
134C24C22×C3×C414yes—
135C24C24×C311yescharacteristic
136#41—432no—
137C2×Q8—24no—
138C2×Q8—44no—
139C2×D4—28no—
140C2×D4—216no—
141C2×D4—24no—
142C2×D4—28no—
143C2×D4—44no—
144C2×D4—84no—
145C22×C4—24no—
146C22×C4—28no—
147C22×C4—28no—
148C22×C4—48no—
149C22×C4—82no—
150C22×C4C22×C3×C414yes—
151C22×C4C22×C3×C412yes—
152C22×C4C22×C3×C414yes—
153C22×C4C2×C3×D414yes—
154C22×C4C2×C3×D414yes—
155C22×C4C2×C3×D414yes—
156C4⋊C4—28no—
157C4⋊C4—84no—
158C4⋊C4C3×C22⋊C4116yes—
159C22⋊C4—216no—
160C22⋊C4—216no—
161C42—84no—
162C22×C3—22no—
163C22×C3—21no—
164C22×C3—22no—
165C22×C3—21no—
166C22×C3—44no—
167C22×C3—44no—
168C22×C3—44no—
169C22×C3—44no—
170C22×C3—88no—
171C22×C3C22×C22⋊C411yescharacteristic
172C22×C3#51812yes—
173C22×C3#53514yes—
174C3×C4—24no—
175C3×C4—28no—
176C3×C4—44no—
177C3×C4—88no—
178C23—24no—
179C23—28no—
180C23—41no—
181C23—41no—
182C23—44no—
183C23—44no—
184C23C23×C3×C411yescharacteristic
185C23C22×C3×D412yes—
186C23C3×H(5,𝔽2)14yes—
187Q8—48no—
188D4—48no—
189D4—416no—
190D4—88no—
191C2×C4—24no—
192C2×C4—28no—
193C2×C4—24no—
194C2×C4—28no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—416no—
199C2×C4—48no—
200C2×C4—48no—
201C2×C4—48no—
202C2×C4C2×C3×C22⋊C418yes—
203C2×C4C2×C3×C22⋊C414yes—
204C2×C3—24no—
205C2×C3—22no—
206C2×C3—84no—
207C2×C3#230911yescharacteristic
208C2×C3#288311yescharacteristic
209C2×C3#288411yescharacteristic
210C22—22no—
211C22—21no—
212C22—22no—
213C22—21no—
214C22—44no—
215C22—44no—
216C22—44no—
217C22—44no—
218C22—88no—
219C22C22×C3×C22⋊C411yescharacteristic
220C22C3×#51812yes—
221C22C3×#53514yes—
222C4—28no—
223C4—24no—
224C4—44no—
225C4—88no—
226C3#2179611yescharacteristic, direct product factor, Sylow
227C2—24no—
228C2—22no—
229C2—84no—
230C2C3×#230911yescharacteristic
231C2C3×#288311yescharacteristic
232C2C3×#288411yescharacteristic
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC2×C3×#537C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2313C2semidirect
3. C3×#1964C2semidirect
4. C3×#1782C2semidirect
5. #21796C3direct (N × H)
8. C2×C3×C22⋊Q8C4semidirect
9. C2×C3×#121C4semidirect
18. #2313C2×C3semidirect
19. #1964C2×C3semidirect
20. #1782C2×C3semidirect
47. C2×C22⋊Q8C3×C4semidirect
48. C2×#121C3×C4semidirect
226. C3#21796direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5372
C3×#23132
C3×#19642
C3×#17822
#217963