Finite Group Databasebeta

← #10706708

Subgroups — #10706708

Order 768.

268 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 69 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#81910C211yesmaximal, characteristic
2#69303C211yesmaximal, characteristic
3#69299C211yesmaximal, characteristic
4C3×#1366C211yesmaximal, characteristic, complemented
5#67800C211yesmaximal, characteristic, complemented
6#67572C211yesmaximal, characteristic, complemented
7#67572C211yesmaximal, characteristic, complemented
8#11990—31nomaximal, Sylow
9#5698C412yes—
10C3×#499C2211yescharacteristic
11#5069C412yes—
12D3×M32—21no—
13#4887—21no—
14#4873—21no—
15#4867—21no—
16#4853—21no—
17#4699C2211yescharacteristic
18#4689C2211yescharacteristic
19C3×#364C2211yescharacteristic
20C3×#364C2211yescharacteristic
21#4514—21no—
22#4512—21no—
23#4511—21no—
24#4463C2211yescharacteristic
25#4463C2211yescharacteristic
26#3177—31no—
27#2166—31no—
28#2153—31no—
29#1366—31no—
30#1366D311yescharacteristic, complemented
31#1358—31no—
32#1358—31no—
33C3×#125C2×C412yes—
34C2×Dic12—21no—
35C2×Dic12—21no—
36C2×ΓL(1,𝔽25)D411yescharacteristic
37C2×D3×C8D411yescharacteristic
38#898—42no—
39#897—22no—
40#890—42no—
41#887—22no—
42#872C2×C411yescharacteristic
43#869C2×C412yes—
44C3×M32—21no—
45C3×M32—21no—
46C3×#107—21no—
47C3×#106—21no—
48C3×C4×C8C2311yescharacteristic
49#815D411yescharacteristic
50C8×Dic3D411yescharacteristic
51#812—21no—
52#812—21no—
53#810—22no—
54#810—22no—
55#803C2×C411yescharacteristic
56#798—41no—
57D3×C16—41no—
58#541—32no—
59C2×M32—61no—
60#499C2×D311yescharacteristic
61#494—31no—
62#491—32no—
63#470—61no—
64#467—61no—
65#441—61no—
66C4×Q16—61no—
67#405—31no—
68#364—31no—
69#364—31no—
70#364—61no—
71#364C2×D311yescharacteristic
72#364C2×D311yescharacteristic
73#349—61no—
74#347—61no—
75C2×C4×D3C22⋊C411yescharacteristic
76C2×Dic6—22no—
77C2×C3×C8C2×D411yescharacteristic
78C2×C3×C8C2×D411yescharacteristic
79C3×C4⋊C4—22no—
80C3×C4⋊C4—42no—
81C3×C42C22×C411yescharacteristic
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—42no—
84C4⋊Dic3—42no—
85C4⋊Dic3—42no—
86Dic3⋊C4—21no—
87Dic3⋊C4—42no—
88C4×Dic3—42no—
89C4×Dic3C22⋊C411yescharacteristic
90C2×C3⋊C8C22⋊C411yescharacteristic
91C2×C3⋊C8C22⋊C411yescharacteristic
92Dic12—41no—
93Dic12—41no—
94Dic12—41no—
95Dic12—41no—
96ΓL(1,𝔽25)—21no—
97ΓL(1,𝔽25)—21no—
98D3×C8—21no—
99D3×C8—21no—
100C3×C16—41no—
101C3×C16—41no—
102C3⋊C16—41no—
103C3⋊C16—41no—
104C2×Q16—61no—
105C2×Q16—61no—
106C2×M16—31no—
107C22×C8—31no—
108C4⋊Q8—32no—
109#126—122no—
110#125—122no—
111#125C4×D312yes—
112C22⋊Q8—62no—
113C4×Q8—62no—
114#117—31no—
115M32—21no—
116M32—21no—
117M32—61no—
118M32—61no—
119M32—121no—
120C2×C16—121no—
121#107—21no—
122#106—21no—
123#103—62no—
124#103—62no—
125#97—31no—
126#97—31no—
127C4×C8—31no—
128C4×C8C22×D311yescharacteristic
129C22×D3#10011yescharacteristic
130C2×C3×C4—21no—
131C2×C3×C4—42no—
132C2×C3×C4C2×C22⋊C411yescharacteristic
133C2×Dic3—21no—
134C2×Dic3—42no—
135C2×Dic3#10011yescharacteristic
136C4×D3—21no—
137C4×D3—21no—
138Dic6—42no—
139Dic6—42no—
140C3×C8#13511yescharacteristic
141C3×C8#13511yescharacteristic
142C3×C8AGL(1,ℤ/8)11yescharacteristic
143C3×C8#13711yescharacteristic
144C3⋊C8—21no—
145C3⋊C8—21no—
146C2×Q8—62no—
147C22×C4—31no—
148Q16—121no—
149Q16—121no—
150Q16—121no—
151Q16—121no—
152M16—61no—
153M16—61no—
154C2×C8—31no—
155C2×C8—31no—
156C2×C8—61no—
157C2×C8—61no—
158C2×C8D3×D411yescharacteristic
159C2×C8D3×D411yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—61no—
163C4⋊C4—122no—
164C4⋊C4—122no—
165C4⋊C4—122no—
166C22⋊C4—61no—
167C22⋊C4—122no—
168C42—31no—
169C42—122no—
170C42C2×C4×D311yescharacteristic
171C16—41no—
172C16—41no—
173C16—121no—
174C16—121no—
175C22×C3C2×#10011yescharacteristic
176C2×D3—21no—
177C2×D3—41no—
178C3×C4—41no—
179C3×C4—82no—
180C3×C4#41611yescharacteristic
181C3×C4#41711yescharacteristic
182Dic3—21no—
183Dic3—41no—
184Dic3—82no—
185C23—31no—
186Q8—122no—
187Q8—122no—
188C2×C4—21no—
189C2×C4—31no—
190C2×C4—42no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—122no—
195C2×C4D3×C22⋊C411yescharacteristic
196C8—61no—
197C8—61no—
198C8#91411yescharacteristic
199C8#91611yescharacteristic
200C8#91811yescharacteristic
201C8#91911yescharacteristic
202C2×C3—21no—
203C2×C3#155111yescharacteristic
204D3—41no—
205C22—61no—
206C22—121no—
207C22D3×#10011yescharacteristic
208C4—41no—
209C4—61no—
210C4—82no—
211C4—121no—
212C4—242no—
213C4#480311yescharacteristic
214C4#480411yescharacteristic
215C3#1199011yescharacteristic, complemented, Sylow
216C2—21no—
217C2—121no—
218C2#6848511yescharacteristic
subgroupG/H
centerC2#68485
commutatorC3×C42C22×C4
FrattiniC4×C8C22×D3
FittingC3×#1366C2
socleC2×C3#1551
prime radicalC22×D3#100
cosocleC3×C4×C8C23
hypercenter#1366D3
hypocenterC3#11990
LCSG ▷ C3×C42 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
4. C3×#1366C2semidirect
5. #67800C2semidirect
6. #67572C2semidirect
7. #67572C2semidirect
30. #1366D3semidirect
215. C3#11990semidirect

Maximal subgroups

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

subgroupindex
#819102
#693032
#692992
C3×#13662
#678002
#675722
#119903