Finite Group Databasebeta

← #10713736

Subgroups — #10713736

Order 768.

316 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 115 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#78542C211yesmaximal, characteristic
2C3×#2110C211yesmaximal, characteristic
3#69885C211yesmaximal, characteristic
4#67814C212yesmaximal
5#67810C212yesmaximal
6#13270—31nomaximal, Sylow
7#5517C411yescharacteristic
8C2×C3×M32C2212yes—
9C3×#405C2211yescharacteristic
10#5119C411yescharacteristic
11#5118C411yescharacteristic
12#5101C2211yescharacteristic
13C2×C8×Dic3C2211yescharacteristic
14C2×#812C2212yes—
15#4999C411yescharacteristic
16#4701—22no—
17C8×Dic6—22no—
18C3×#347—22no—
19#4463—22no—
20#2960—31no—
21#2153—31no—
22#2110D311yescharacteristic
23#1378—32no—
24#1376—32no—
25C2×C3×M16C2×C411yescharacteristic
26C22×C3×C8C2311yescharacteristic
27C3×#117C2×C411yescharacteristic
28#926C2×C411yescharacteristic
29C2×C4×Dic3C2×C411yescharacteristic
30C2×#208C2×C411yescharacteristic
31C22×C3⋊C8C2×C411yescharacteristic
32#890—21no—
33#888—21no—
34#878—22no—
35C4×Dic6—22no—
36C3×M32—22no—
37C3×M32Q814yescomplemented
38C2×C3×C16—22no—
39C3×#97D412yes—
40C3×C4×C8D412yes—
41#815—21no—
42Dic3⋊C8—22no—
43Dic3⋊C8—22no—
44C8×Dic3—21no—
45C8×Dic3—21no—
46C8×Dic3—21no—
47#812—22no—
48#812—22no—
49C2×C3⋊C16—22no—
50C4⋊(C3⋊C8)—22no—
51#533—31no—
52C2×M32—32no—
53C2×M32C2×D312yes—
54#446—62no—
55C8×Q8—62no—
56#424—31no—
57#424—31no—
58#423—31no—
59#405—31no—
60#405C2×D311yescharacteristic
61C2×C4×C8—31no—
62#364—62no—
63#347—22no—
64C22×C3×C4C22×C411yescharacteristic
65C22×Dic3—21no—
66C2×Dic6—22no—
67C3×M16—22no—
68C2×C3×C8C22⋊C412yes—
69C2×C3×C8C2×D411yescharacteristic
70C2×C3×C8C2×D411yescharacteristic
71C2×C3×C8C2×Q812yes—
72C2×C3×C8#4112yes—
73C3×C4⋊C4—21no—
74C3×C22⋊C4—21no—
75C3×C42C22⋊C412yes—
76#217M1612yes—
77C4⋊Dic3—22no—
78C4⋊Dic3—21no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81C4×Dic3—21no—
82C4×Dic3—21no—
83C4×Dic3M1611yescharacteristic
84C4×Dic3M1611yescharacteristic
85#208—22no—
86C2×C3⋊C8—21no—
87C2×C3⋊C8—21no—
88C2×C3⋊C8—22no—
89C3×C16—24no—
90C3⋊C16—42no—
91C2×M16—31no—
92C2×M16C4×D311yescharacteristic
93C22×C8—31no—
94C22×C8C22×D311yescharacteristic
95C4⋊Q8—61no—
96#125—61no—
97C22⋊Q8—62no—
98C4×Q8—62no—
99#117—31no—
100#117C4×D311yescharacteristic
101C2×C42—31no—
102M32—22no—
103M32—62no—
104M32—62no—
105M32Dic614yescomplemented
106C2×C16—22no—
107C2×C16—62no—
108C4⋊C8—62no—
109C4⋊C8—62no—
110C4⋊C8—62no—
111#97—61no—
112#97#6712yes—
113C4×C8—61no—
114C4×C8—61no—
115C4×C8—61no—
116C4×C8D1212yes—
117C23×C3C2×M1611yescharacteristic
118C2×C3×C4—22no—
119C2×C3×C4C2×C22⋊C411yescharacteristic
120C2×C3×C4C4×Q812yes—
121C2×C3×C4C2×M1611yescharacteristic
122C2×C3×C4#13112yes—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—42no—
127Dic6—44no—
128C3×C8—22no—
129C3×C8C22⋊Q812yes—
130C3×C8#12411yescharacteristic
131C3×C8C4⋊Q811yescharacteristic
132C3⋊C8—41no—
133C3⋊C8—41no—
134C2×Q8—62no—
135C22×C4—61no—
136C22×C4C2×C4×D311yescharacteristic
137M16—22no—
138M16—62no—
139C2×C8—62no—
140C2×C8—61no—
141C2×C8—61no—
142C2×C8(C2×D3)⋊C412yes—
143C2×C8C2×Dic612yes—
144C2×C8C2×D1211yescharacteristic
145C2×C8#23512yes—
146C2×C8C2×#6711yescharacteristic
147C4⋊C4—21no—
148C4⋊C4—62no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—61no—
152C22⋊C4—21no—
153C22⋊C4—32no—
154C42—31no—
155C42—31no—
156C42—61no—
157C42—61no—
158C42(C2×D3)⋊C412yes—
159C16—24no—
160C16—122no—
161C22×C3—41no—
162C22×C3#40711yescharacteristic
163C22×C3#44612yes—
164C3×C4—42no—
165C3×C4#39111yescharacteristic
166C3×C4#40811yescharacteristic
167C3×C4#44612yes—
168Dic3—22no—
169Dic3—42no—
170Dic3—41no—
171C23D3×M1611yescharacteristic
172Q8—124no—
173C2×C4—22no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—62no—
177C2×C4—122no—
178C2×C4C4×Dic612yes—
179C2×C4D3×M1611yescharacteristic
180C2×C4#90712yes—
181C2×C4C2×(C2×D3)⋊C411yescharacteristic
182C8—22no—
183C8—121no—
184C8—121no—
185C8#86911yescharacteristic
186C8#87511yescharacteristic
187C8#92412yes—
188C2×C3—21no—
189C2×C3—22no—
190C2×C3#198511yescharacteristic
191C22—41no—
192C22#470112yes—
193C22#512611yescharacteristic
194C4—42no—
195C4—62no—
196C4—121no—
197C4—122no—
198C4#470112yes—
199C4#492911yescharacteristic
200C4#512911yescharacteristic
201C3#1327011yescharacteristic, complemented, Sylow
202C2—21no—
203C2—22no—
204C2#6983811yescharacteristic
subgroupG/H
centerC8#875
commutatorC22×C3×C4C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2110C2
socleC2×C3#1985
prime radicalC23×C3C2×M16
cosocleC22×C3×C8C23
hypercenter#2110D3
hypocenterC3#13270
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2110

Complements

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

NH ≅ G/Ntype
37. C3×M32Q8semidirect
105. M32Dic6semidirect
201. C3#13270semidirect

Maximal subgroups

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

subgroupindex
#785422
C3×#21102
#698852
#678142
#678102
#132703