Finite Group Databasebeta

← #10695087

Subgroups — #10695087

Order 768.

422 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#78475C211yesmaximal, characteristic
2C3×#1744C211yesmaximal, characteristic
3#69738C212yesmaximal
4#69731C211yesmaximal, characteristic
5#69396C212yesmaximal
6#9848—31nomaximal, Sylow
7#5470C2212yes—
8C2×C3×C4×C8C2211yescharacteristic
9C2×C3×C4×C8C2211yescharacteristic
10C3×C43C2211yescharacteristic
11#5116—22no—
12#4924C2212yes—
13#4699C411yescharacteristic, complemented
14C4×Dic12—22no—
15#4682C411yescharacteristic, complemented
16#4681C412yescomplemented
17C3×#336—22no—
18#4523—24no—
19#4458—22no—
20#4456—24no—
21#3170—31no—
22#1994—32no—
23#1744D311yescharacteristic
24#1567—31no—
25#1493—32no—
26C22×C3×C8—21no—
27C22×C3×C8D412yes—
28C2×C3×C42—21no—
29C2×C3×C42—21no—
30C2×C3×C42D412yes—
31C2×C3×C42C2311yescharacteristic
32#926—22no—
33#924—22no—
34#924—22no—
35C2×#208—22no—
36C2×Dic12—22no—
37#870—22no—
38#869C2×C412yes—
39#869C2×C412yes—
40C4×Dic6—22no—
41C4×Dic6—22no—
42C3×C4×C8—21no—
43C3×C4×C8C2×C411yescharacteristic
44C3×C4×C8C2×C411yescharacteristic
45C3×C4×C8D412yes—
46C3×C4×C8D412yes—
47#818—22no—
48#816—24no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)—22no—
51#533—32no—
52#500—31no—
53#494—31no—
54#487—32no—
55#467—62no—
56C4×Q16—62no—
57#423—32no—
58C2×C4×C8C2×D311yescharacteristic
59C2×C4×C8C2×D311yescharacteristic
60C43C2×D311yescharacteristic
61#339—64no—
62#336—22no—
63#335—62no—
64Q8⋊C8—64no—
65C22×C3×C4—22no—
66C22×C3×C4C2×D411yescharacteristic
67C22×C3×C4C2×D411yescharacteristic
68C22×C3×C4#4111yescharacteristic
69C2×Dic6—22no—
70C2×Dic6—22no—
71C2×C3×C8—22no—
72C2×C3×C8—42no—
73C2×C3×C8—42no—
74C2×C3×C8C22⋊C412yes—
75C2×C3×C8C22⋊C412yes—
76C3×C42—22no—
77C3×C42—42no—
78C3×C42—42no—
79C3×C42Q1614yescomplemented
80C3×C42C22×C411yescharacteristic
81C3×C42C2×D411yescharacteristic
82C3×C42C2×D411yescharacteristic
83C3×C42#4111yescharacteristic
84#217—42no—
85C4⋊Dic3—22no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89Dic3⋊C4—42no—
90Dic3⋊C4—42no—
91C4×Dic3—42no—
92#208—44no—
93C2×C3⋊C8—42no—
94Dic12—44no—
95C2×Q16—62no—
96C2×M16—62no—
97C22×C8—21no—
98C22×C8#6712yes—
99C4⋊Q8—32no—
100C4⋊Q8—32no—
101#125—62no—
102C22⋊Q8—62no—
103C22⋊Q8—62no—
104C4×Q8—62no—
105C4×Q8—62no—
106#117—62no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42D1212yes—
110C2×C42C22×D311yescharacteristic
111#107—62no—
112C4⋊C8—62no—
113C4⋊C8—62no—
114#103—64no—
115C4×C8—21no—
116C4×C8C4×D311yescharacteristic
117C4×C8C4×D311yescharacteristic
118C4×C8D1212yes—
119C4×C8D1212yes—
120C23×C3#12111yescharacteristic
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—24no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4C2×C22⋊C411yescharacteristic
127C2×C3×C4C4×D411yescharacteristic
128C2×C3×C4C4×D411yescharacteristic
129C2×C3×C4#12111yescharacteristic
130C2×C3×C4#12411yescharacteristic
131C2×C3×C4#12711yescharacteristic
132C2×C3×C4C2×Q1612yes—
133C2×C3×C4#13512yes—
134C2×Dic3—42no—
135C2×Dic3—42no—
136Dic6—44no—
137Dic6—44no—
138C3×C8—22no—
139C3×C8—22no—
140C3×C8—42no—
141C3⋊C8—82no—
142C2×Q8—62no—
143C2×Q8—62no—
144C22×C4—22no—
145C22×C4C2×D1211yescharacteristic
146C22×C4#23511yescharacteristic
147C22×C4C2×#6711yescharacteristic
148Q16—124no—
149M16—124no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—122no—
154C2×C8(C2×D3)⋊C412yes—
155C2×C8(C2×D3)⋊C412yes—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—122no—
161C4⋊C4—122no—
162C22⋊C4—122no—
163C42—22no—
164C42—42no—
165C42—42no—
166C42—122no—
167C42Dic1214yescomplemented
168C42C2×C4×D311yescharacteristic
169C42C2×D1211yescharacteristic
170C42C2×D1211yescharacteristic
171C42#23511yescharacteristic
172C22×C3—21no—
173C22×C3—21no—
174C22×C3#39011yescharacteristic
175C22×C3#46712yes—
176C3×C4—21no—
177C3×C4—21no—
178C3×C4—44no—
179C3×C4#42112yes—
180C3×C4C4×Q1612yes—
181C3×C4#44412yes—
182C3×C4#49411yescharacteristic
183C3×C4#49511yescharacteristic
184Dic3—82no—
185Dic3—82no—
186C23#93011yescharacteristic
187Q8—124no—
188Q8—124no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4C4×D1211yescharacteristic
197C2×C4C4×D1211yescharacteristic
198C2×C4#87411yescharacteristic
199C2×C4#87511yescharacteristic
200C2×C4#90412yes—
201C2×C4C2×Dic1212yes—
202C2×C4C2×(C2×D3)⋊C411yescharacteristic
203C2×C4#93011yescharacteristic
204C8—22no—
205C8—22no—
206C8—42no—
207C8—242no—
208C2×C3—22no—
209C2×C3#196311yescharacteristic
210C2×C3#196911yescharacteristic
211C2×C3#197511yescharacteristic
212C22—21no—
213C22—21no—
214C22#493911yescharacteristic
215C22#511612yes—
216C4—21no—
217C4—21no—
218C4—44no—
219C4—242no—
220C4—242no—
221C4#469611yescharacteristic
222C4C4×Dic1212yes—
223C4#469911yescharacteristic
224C4#471712yes—
225C4#500512yes—
226C3#984811yescharacteristic, complemented, Sylow
227C2—22no—
228C2#6944011yescharacteristic
229C2#6977511yescharacteristic
230C2#6984511yescharacteristic
subgroupG/H
centerC2×C4#874
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1744C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1744D3
hypocenterC3#9848
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1744

Complements

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

NH ≅ G/Ntype
13. #4699C4semidirect
15. #4682C4semidirect
16. #4681C4semidirect
79. C3×C42Q16semidirect
167. C42Dic12semidirect
226. C3#9848semidirect

Maximal subgroups

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

subgroupindex
#784752
C3×#17442
#697382
#697312
#693962
#98483