Finite Group Databasebeta

← #10804455

Subgroups — #10804455

Order 768.

1334 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 315 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
2#82155C212yesmaximal, complemented
3#72139C214yesmaximal, complemented
4#71686C218yesmaximal, complemented
5#21796—31nomaximal, Sylow
6C22×C3×#41C2212yes—
7C3×#537—28no—
8C2×C3×C22⋊Q8C414yescomplemented
9C2×C3×#121C414yescomplemented
10C2×C3×#121C2214yes—
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#834C22116yes—
13#5212C2214yes—
14#5211C2214yes—
15C4×#217C2214yes—
16#4980—28no—
17C2×#537D311yescharacteristic
18#2313—32no—
19#1964—38no—
20#1782—34no—
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⋊C4C2×C414yes—
35C2×C3×C4⋊C4C2×C414yes—
36C2×C3×C4⋊C4D418yes—
37C2×C3×C22⋊C4C2×C418yes—
38C2×#217C2318yes—
39C2×C4⋊Dic3—24no—
40C2×C4×Dic3—24no—
41#834—416no—
42C3⋊#95—24no—
43C3⋊#95—28no—
44C3⋊#95—24no—
45C22×#41C2×D312yes—
46#537—28no—
47C2×C22⋊Q8Dic314yescomplemented
48C2×#121Dic314yescomplemented
49C2×#121C2×D314yes—
50C22×C4⋊C4C2×D311yescharacteristic
51C2×#99—316no—
52#387—68no—
53#381—34no—
54#380—34no—
55C4×C22⋊C4—34no—
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—48no—
70C22×C3×C4—82no—
71C22×C3×C4C22×C414yes—
72C22×C3×C4C22×C412yes—
73C22×C3×C4C22×C414yes—
74C22×C3×C4C2×D414yes—
75C22×C3×C4C2×D414yes—
76C22×C3×C4C2×D414yes—
77C22×Dic3—28no—
78C3×C4⋊C4—28no—
79C3×C4⋊C4C22⋊C4116yes—
80C3×C22⋊C4—216no—
81#217—216no—
82C4⋊Dic3—84no—
83C4×Dic3—84no—
84C2×#41—28no—
85C2×#41—48no—
86C22×Q8C2×Dic312yes—
87C22×D4C2×Dic312yes—
88C22×D4C2×Dic314yes—
89C22×D4C22×D314yes—
90C23×C4C2×Dic314yes—
91C23×C4C22×D311yescharacteristic
92C23×C4C22×D312yes—
93C22⋊Q8—216no—
94#121—28no—
95#121—28no—
96#121—216no—
97C2×C4⋊C4—64no—
98C2×C4⋊C4C2×Dic314yes—
99C2×C4⋊C4C2×Dic314yes—
100C2×C4⋊C4#6718yes—
101C2×C22⋊C4—38no—
102C2×C22⋊C4C2×Dic318yes—
103C2×C42—64no—
104#99—1216no—
105#95—64no—
106#95—64no—
107#95—68no—
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×C4C2×C22⋊C418yes—
130C2×C3×C4C2×C22⋊C414yes—
131C2×Dic3—48no—
132C2×Dic3—48no—
133C2×Dic3—48no—
134C24C22×Dic314yes—
135C24C23×D311yescharacteristic
136#41—432no—
137C2×Q8—24no—
138C2×Q8—44no—
139C2×D4—216no—
140C2×D4—28no—
141C2×D4—24no—
142C2×D4—28no—
143C2×D4—44no—
144C2×D4—84no—
145C22×C4—24no—
146C22×C4—28no—
147C22×C4—48no—
148C22×C4—68no—
149C22×C4—82no—
150C22×C4C22×Dic314yes—
151C22×C4C22×Dic312yes—
152C22×C4C22×Dic314yes—
153C22×C4C2×#6714yes—
154C22×C4C2×#6714yes—
155C22×C4C2×#6714yes—
156C4⋊C4—28no—
157C4⋊C4—244no—
158C4⋊C4#217116yes—
159C22⋊C4—216no—
160C22⋊C4—616no—
161C42—244no—
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—
177Dic3—88no—
178C23—24no—
179C23—28no—
180C23—41no—
181C23—41no—
182C23—44no—
183C23—44no—
184C23#100414yes—
185C23C23×Dic311yescharacteristic
186C23C22×#6712yes—
187Q8—48no—
188D4—48no—
189D4—416no—
190D4—88no—
191C2×C4—24no—
192C2×C4—28no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—416no—
199C2×C4—128no—
200C2×C4—128no—
201C2×C4—128no—
202C2×C4C2×#21718yes—
203C2×C4C2×#21714yes—
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—
219C22#579812yes—
220C22#580514yes—
221C22C22×#21711yescharacteristic
222C4—24no—
223C4—28no—
224C4—44no—
225C4—248no—
226C3#2179611yescharacteristic, complemented, Sylow
227C2—24no—
228C2—22no—
229C2—84no—
230C2#8404411yescharacteristic
231C2#8410711yescharacteristic
232C2#8411911yescharacteristic
subgroupG/H
centerC22C22×#217
commutatorC23×C3C23×C4
FrattiniC24C23×D3
Fitting, prime radicalC2×C3×#537C2
socleC22×C3C22×C22⋊C4
cosocleC24×C3C24
hypercenterC2×#537D3
hypocenterC3#21796
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
3. #72139C2semidirect
4. #71686C2semidirect
8. C2×C3×C22⋊Q8C4semidirect
9. C2×C3×#121C4semidirect
47. C2×C22⋊Q8Dic3semidirect
48. C2×#121Dic3semidirect
226. C3#21796semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5372
#821552
#721392
#716862
#217963