Finite Group Databasebeta

← #10695164

Subgroups — #10695164

Order 768.

412 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 123 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#78463C211yesmaximal, characteristic
2C2×C3×#322C211yesmaximal, characteristic, complemented
3#69791C211yesmaximal, characteristic, complemented
4#69738C212yesmaximal, complemented
5#69735C212yesmaximal
6#9863—31nomaximal, Sylow
7#5477C2211yescharacteristic
8#5470C2211yescharacteristic
9C2×C3×C4×C8C2212yes—
10C2×C3×C4×C8C2211yescharacteristic
11#5113—22no—
12#4924C2212yes—
13#4697—22no—
14#4694C412yes—
15#4681C412yes—
16C3×#322—22no—
17#4552—22no—
18#4459—24no—
19#4456—24no—
20#3131—31no—
21#1837—31no—
22#1493—32no—
23#1492—32no—
24C2×#322D311yescharacteristic, complemented
25C2×#235—21no—
26C22×C3×C8—22no—
27C22×C3×C8D412yes—
28C2×C3×C42C2311yescharacteristic
29#930—22no—
30#926—21no—
31#924—22no—
32C2×#208—22no—
33#875—21no—
34#874C2×C411yescharacteristic
35C4×D12—22no—
36#870—21no—
37#869C2×C411yescharacteristic
38#869C2×C412yes—
39C4×Dic6—22no—
40C3×C4×C8—22no—
41C3×C4×C8C2×C412yes—
42C3×C4×C8D414yes—
43C3×C4×C8D412yes—
44#821—24no—
45#816—24no—
46C4⋊(C3⋊C8)—22no—
47C4⋊(C3⋊C8)—22no—
48#533—31no—
49#532—31no—
50#488—62no—
51#487—32no—
52#486—32no—
53#423—32no—
54#416—62no—
55C2×C4×C8C2×D312yes—
56C2×C4×C8C2×D311yescharacteristic
57#341—62no—
58Q8⋊C8—64no—
59D4⋊C8—64no—
60#322—22no—
61C22×C3×C4C2×D411yescharacteristic
62C22×C3×C4#4112yes—
63C2×#67—41no—
64#235—44no—
65C2×D12—21no—
66C2×D12—22no—
67C2×C4×D3—41no—
68C2×Dic6—21no—
69C2×Dic6—22no—
70C2×C3×C8—24no—
71C2×C3×C8—24no—
72C2×C3×C8—44no—
73C2×C3×C8C22⋊C414yes—
74C3×C42C22×C411yescharacteristic
75C3×C42C2×D412yes—
76C3×C42C2×D411yescharacteristic
77#217—41no—
78(C2×D3)⋊C4—42no—
79C4⋊Dic3—22no—
80C4⋊Dic3—22no—
81C4⋊Dic3—22no—
82Dic3⋊C4—42no—
83C4×Dic3—41no—
84#208—44no—
85C2×C3⋊C8—42no—
86C2×#41—61no—
87C2×M16—62no—
88C22×C8—22no—
89C22×C8D1212yes—
90C4⋊Q8—31no—
91C4⋊Q8—32no—
92#127—31no—
93#125—61no—
94#124—61no—
95C22⋊Q8—62no—
96#121—62no—
97C4×Q8—62no—
98C4×D4—62no—
99#117—61no—
100C2×C42C22×D311yescharacteristic
101C4⋊C8—62no—
102C4⋊C8—62no—
103#103—64no—
104#102—64no—
105C4×C8—22no—
106C4×C8C4×D312yes—
107C4×C8D1214yes—
108C4×C8#6712yes—
109C23×C3#12411yescharacteristic
110C22×D3—41no—
111C2×C3×C4—24no—
112C2×C3×C4C2×C22⋊C411yescharacteristic
113C2×C3×C4C4×D412yes—
114C2×C3×C4#12112yes—
115C2×C3×C4#12711yescharacteristic
116C2×C3×C4#13514yes—
117#67—82no—
118C2×Dic3—42no—
119C2×Dic3—41no—
120D12—42no—
121D12—44no—
122C4×D3—82no—
123Dic6—42no—
124Dic6—44no—
125C3×C8—24no—
126C3×C8—44no—
127C3⋊C8—82no—
128#41—124no—
129C2×Q8—62no—
130C2×Q8—61no—
131C2×D4—62no—
132C2×D4—61no—
133C2×D4—121no—
134C22×C4—121no—
135C22×C4C2×D1211yescharacteristic
136C22×C4#23512yes—
137M16—124no—
138C2×C8—24no—
139C2×C8—24no—
140C2×C8—44no—
141C2×C8—122no—
142C2×C8(C2×D3)⋊C414yes—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—122no—
147C22⋊C4—121no—
148C22⋊C4—122no—
149C42—121no—
150C42C2×C4×D311yescharacteristic
151C42C2×D1212yes—
152C42C2×#6711yescharacteristic
153C22×C3—22no—
154C22×C3#39011yescharacteristic
155C22×C3#48812yes—
156C2×D3—81no—
157C2×D3—82no—
158C3×C4—22no—
159C3×C4#41612yes—
160C3×C4#44414yes—
161C3×C4#49512yes—
162Dic3—81no—
163Dic3—82no—
164C23—121no—
165C23#87511yescharacteristic
166Q8—124no—
167Q8—122no—
168D4—124no—
169D4—122no—
170D4—242no—
171C2×C4—24no—
172C2×C4—122no—
173C2×C4—121no—
174C2×C4—242no—
175C2×C4C4×D1212yes—
176C2×C4#87411yescharacteristic
177C2×C4#90414yes—
178C2×C4C2×(C2×D3)⋊C411yescharacteristic
179C2×C4#93012yes—
180C8—24no—
181C8—44no—
182C8—242no—
183C2×C3—22no—
184C2×C3#195811yescharacteristic
185C2×C3#197512yes—
186D3—161no—
187C22—22no—
188C22—241no—
189C22—242no—
190C22#469712yes—
191C22#493911yescharacteristic
192C4—22no—
193C4—241no—
194C4—242no—
195C4#469612yes—
196C4#471714yes—
197C4#511312yes—
198C3#986311yescharacteristic, complemented, Sylow
199C2—22no—
200C2—481no—
201C2#6977311yescharacteristic
202C2#6984512yes—
subgroupG/H
centerC2×C4#874
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#322C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenterC2×#322D3
hypocenterC3#9863
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322

Complements

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

NH ≅ G/Ntype
2. C2×C3×#322C2semidirect
3. #69791C2semidirect
4. #69738C2semidirect
24. C2×#322D3semidirect
198. C3#9863semidirect

Maximal subgroups

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

subgroupindex
#784632
C2×C3×#3222
#697912
#697382
#697352
#98633