Finite Group Databasebeta

← #10695070

Subgroups — #10695070

Order 768.

478 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 139 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×C42×C8C211yesmaximal, characteristic
3#69738C212yesmaximal
4#69731C211yesmaximal, characteristic
5#69396C212yesmaximal
6#9845—31nomaximal, Sylow
7#5470C2212yes—
8C2×C3×C4×C8—22no—
9C2×C3×C4×C8C2211yescharacteristic
10C2×C3×C4×C8C2211yescharacteristic
11C3×C43C2211yescharacteristic
12#5116—22no—
13#4924C2212yes—
14#4699C411yescharacteristic, complemented
15C4×Dic12—22no—
16#4682C411yescharacteristic, complemented
17#4681C412yescomplemented
18#4523—24no—
19#4458—22no—
20#4456—24no—
21#3170—31no—
22#1994—32no—
23C42×C8D311yescharacteristic
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—28no—
43C3×C4×C8—21no—
44C3×C4×C8C2×C411yescharacteristic
45C3×C4×C8C2×C411yescharacteristic
46C3×C4×C8D412yes—
47C3×C4×C8D412yes—
48#818—22no—
49#816—24no—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)—22no—
52#533—32no—
53#500—31no—
54#494—31no—
55#487—32no—
56#467—62no—
57C4×Q16—62no—
58#423—32no—
59C2×C4×C8—22no—
60C2×C4×C8C2×D311yescharacteristic
61C2×C4×C8C2×D311yescharacteristic
62C43C2×D311yescharacteristic
63#339—64no—
64#335—62no—
65Q8⋊C8—64no—
66C22×C3×C4—22no—
67C22×C3×C4C2×D411yescharacteristic
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4#4111yescharacteristic
70C2×Dic6—22no—
71C2×Dic6—22no—
72C2×C3×C8—24no—
73C2×C3×C8—22no—
74C2×C3×C8—24no—
75C2×C3×C8C22⋊C412yes—
76C2×C3×C8C22⋊C412yes—
77C3×C42—24no—
78C3×C42—24no—
79C3×C42—22no—
80C3×C42Q1614yescomplemented
81C3×C42C22×C411yescharacteristic
82C3×C42C2×D411yescharacteristic
83C3×C42C2×D411yescharacteristic
84C3×C42#4111yescharacteristic
85#217—42no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90Dic3⋊C4—42no—
91Dic3⋊C4—42no—
92C4×Dic3—42no—
93#208—44no—
94C2×C3⋊C8—42no—
95Dic12—44no—
96C2×Q16—62no—
97C2×M16—62no—
98C22×C8—21no—
99C22×C8#6712yes—
100C4⋊Q8—32no—
101C4⋊Q8—32no—
102#125—62no—
103C22⋊Q8—62no—
104C22⋊Q8—62no—
105C4×Q8—62no—
106C4×Q8—62no—
107#117—62no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42D1212yes—
111C2×C42C22×D311yescharacteristic
112#107—62no—
113C4⋊C8—62no—
114C4⋊C8—62no—
115#103—64no—
116C4×C8—28no—
117C4×C8—21no—
118C4×C8C4×D311yescharacteristic
119C4×C8C4×D311yescharacteristic
120C4×C8D1212yes—
121C4×C8D1212yes—
122C23×C3#12111yescharacteristic
123C2×C3×C4—24no—
124C2×C3×C4—24no—
125C2×C3×C4—24no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4C2×C22⋊C411yescharacteristic
129C2×C3×C4C4×D411yescharacteristic
130C2×C3×C4C4×D411yescharacteristic
131C2×C3×C4#12111yescharacteristic
132C2×C3×C4#12411yescharacteristic
133C2×C3×C4#12711yescharacteristic
134C2×C3×C4C2×Q1612yes—
135C2×C3×C4#13512yes—
136C2×Dic3—42no—
137C2×Dic3—42no—
138Dic6—44no—
139Dic6—44no—
140C3×C8—24no—
141C3×C8—22no—
142C3×C8#10414yes—
143C3⋊C8—82no—
144C2×Q8—62no—
145C2×Q8—62no—
146C22×C4—22no—
147C22×C4C2×D1211yescharacteristic
148C22×C4#23511yescharacteristic
149C22×C4C2×#6711yescharacteristic
150Q16—124no—
151M16—124no—
152C2×C8—24no—
153C2×C8—22no—
154C2×C8—24no—
155C2×C8—122no—
156C2×C8(C2×D3)⋊C412yes—
157C2×C8(C2×D3)⋊C412yes—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C22⋊C4—122no—
165C42—24no—
166C42—24no—
167C42—22no—
168C42—122no—
169C42Dic1214yescomplemented
170C42C2×C4×D311yescharacteristic
171C42C2×D1211yescharacteristic
172C42C2×D1211yescharacteristic
173C42#23511yescharacteristic
174C22×C3—21no—
175C22×C3—21no—
176C22×C3#39011yescharacteristic
177C22×C3#46712yes—
178C3×C4—28no—
179C3×C4—21no—
180C3×C4—21no—
181C3×C4C2×#10412yes—
182C3×C4C4×Q1612yes—
183C3×C4#44312yes—
184C3×C4#49411yescharacteristic
185C3×C4#49511yescharacteristic
186Dic3—82no—
187Dic3—82no—
188C23#93011yescharacteristic
189Q8—124no—
190Q8—124no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4C4×D1211yescharacteristic
199C2×C4C4×D1211yescharacteristic
200C2×C4#87411yescharacteristic
201C2×C4#87511yescharacteristic
202C2×C4#90412yes—
203C2×C4C2×Dic1212yes—
204C2×C4C2×(C2×D3)⋊C411yescharacteristic
205C2×C4#93011yescharacteristic
206C8—24no—
207C8—22no—
208C8—242no—
209C8#80514yes—
210C2×C3—22no—
211C2×C3#196211yescharacteristic
212C2×C3#196911yescharacteristic
213C2×C3#197411yescharacteristic
214C22—21no—
215C22—21no—
216C22#493911yescharacteristic
217C22#511612yes—
218C4—28no—
219C4—21no—
220C4—21no—
221C4—242no—
222C4—242no—
223C4#469611yescharacteristic
224C4C4×Dic1212yes—
225C4#469911yescharacteristic
226C4#470012yes—
227C4C2×#80512yes—
228C3#984511yescharacteristic, complemented, Sylow
229C2—22no—
230C2#6939511yescharacteristic
231C2#6977511yescharacteristic
232C2#6979511yescharacteristic
subgroupG/H
centerC2×C4#874
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C42×C8C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenterC42×C8D3
hypocenterC3#9845
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8

Complements

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

NH ≅ G/Ntype
14. #4699C4semidirect
16. #4682C4semidirect
17. #4681C4semidirect
80. C3×C42Q16semidirect
169. C42Dic12semidirect
228. C3#9845semidirect

Maximal subgroups

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

subgroupindex
#784752
C3×C42×C82
#697382
#697312
#693962
#98453