Finite Group Databasebeta

← #10692791

Subgroups — #10692791

Order 768.

400 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#2986C211yesmaximal, characteristic
2#83337C211yesmaximal, characteristic
3#71834C211yesmaximal, characteristic
4#70780C212yesmaximal
5#67743C212yesmaximal
6#9436—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#446—22no—
10C3×C8×Q8—22no—
11C3×#433—22no—
12C3×#431C2212yes—
13#5093—22no—
14#5069C414yes—
15#5004—22no—
16#4924C2212yes—
17C4×#208C2211yescharacteristic
18#4547—22no—
19#4482—22no—
20#4482—22no—
21#3139—31no—
22#2986D311yescharacteristic
23#1561—31no—
24#1549—32no—
25#1299—32no—
26C22×Dic6—21no—
27C3×C4⋊Q8D412yes—
28C3×#125C2×C412yes—
29C3×C22⋊Q8—22no—
30C3×C4×Q8—22no—
31C3×#117D412yes—
32C2×C3×C42C2311yescharacteristic
33C2×C4⋊Dic3—22no—
34C2×#208—21no—
35C2×#208—22no—
36#869—22no—
37#869—21no—
38#869C2×C412yes—
39C3×C4⋊C8—24no—
40C3×C4⋊C8D412yes—
41C3×C4⋊C8D412yes—
42C3×C22⋊C8—22no—
43C3×#97—22no—
44C3×C4×C8—22no—
45C3×C4×C8—22no—
46#810—28no—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)—22no—
49#803C2×C412yes—
50C4×C3⋊C8—21no—
51#533C2×D311yescharacteristic
52C2×C4⋊Q8—31no—
53#491—34no—
54#487—62no—
55#446—22no—
56C8×Q8—22no—
57#433—22no—
58#431C2×D312yes—
59#423—32no—
60#419—62no—
61C4×M16—31no—
62#332—62no—
63#332—62no—
64#332—62no—
65C2×C3×Q8—22no—
66C22×C3×C4C2×D411yescharacteristic
67C22×C3×C4#4112yes—
68C22×Dic3—41no—
69C2×Dic6—22no—
70C2×Dic6—44no—
71C2×Dic6—41no—
72C2×C3×C8—22no—
73C2×C3×C8—22no—
74C3×C4⋊C4—24no—
75C3×C4⋊C4—22no—
76C3×C4⋊C4C22⋊C414yes—
77C3×C22⋊C4—22no—
78C3×C42—22no—
79C3×C42C22×C411yescharacteristic
80C3×C42C2×D411yescharacteristic
81C3×C42C2×D412yes—
82C4⋊Dic3—24no—
83C4⋊Dic3—42no—
84#208—42no—
85#208—82no—
86C2×C3⋊C8—22no—
87C2×C3⋊C8—42no—
88C22×Q8—61no—
89C2×M16—61no—
90C2×M16—62no—
91C4⋊Q8—32no—
92C4⋊Q8—61no—
93C4⋊Q8—62no—
94C4⋊Q8D1212yes—
95#125C4×D312yes—
96C22⋊Q8—22no—
97C4×Q8—22no—
98#117#6712yes—
99C2×C4⋊C4—62no—
100C2×C42C22×D311yescharacteristic
101C4⋊C8—24no—
102C4⋊C8—62no—
103C4⋊C8—62no—
104C4⋊C8#6712yes—
105C4⋊C8#6712yes—
106#103—68no—
107C22⋊C8—22no—
108#97—22no—
109#97—32no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—61no—
113C23×C3#12311yescharacteristic
114C3×Q8—42no—
115C3×Q8—42no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—24no—
120C2×C3×C4C2×C22⋊C411yescharacteristic
121C2×C3×C4C4×D412yes—
122C2×C3×C4#12011yescharacteristic
123C2×C3×C4#12112yes—
124C2×Dic3—42no—
125C2×Dic3—82no—
126Dic6—44no—
127Dic6—82no—
128C3×C8—42no—
129C3×C8—42no—
130C3⋊C8—42no—
131C3⋊C8—82no—
132C2×Q8—22no—
133C2×Q8—62no—
134C2×Q8—124no—
135C2×Q8—121no—
136C22×C4—121no—
137C22×C4#23512yes—
138C22×C4C2×#6711yescharacteristic
139M16—122no—
140M16—242no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—62no—
144C2×C8—122no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—64no—
148C4⋊C4—122no—
149C4⋊C4(C2×D3)⋊C414yes—
150C22⋊C4—22no—
151C42—22no—
152C42C2×C4×D311yescharacteristic
153C42C2×D1211yescharacteristic
154C42C2×#6712yes—
155C22×C3—21no—
156C22×C3—22no—
157C22×C3#38611yescharacteristic
158C3×C4—21no—
159C3×C4—22no—
160C3×C4—44no—
161C3×C4#45412yes—
162C3×C4#47112yes—
163C3×C4#47312yes—
164Dic3—82no—
165C23#92911yescharacteristic
166Q8—42no—
167Q8—42no—
168Q8—124no—
169Q8—242no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—122no—
175C2×C4—242no—
176C2×C4C2×(C2×D3)⋊C411yescharacteristic
177C2×C4C4×#6712yes—
178C2×C4#93012yes—
179C2×C4#95311yescharacteristic
180C8—42no—
181C8—42no—
182C8—122no—
183C8—242no—
184C2×C3—41no—
185C2×C3#188811yescharacteristic
186C2×C3#190812yes—
187C22—22no—
188C22—21no—
189C22#521311yescharacteristic
190C4—22no—
191C4—21no—
192C4—44no—
193C4—242no—
194C4#514112yes—
195C4#514312yes—
196C4#524512yes—
197C3#943611yescharacteristic, complemented, Sylow
198C2—41no—
199C2#7166111yescharacteristic
200C2#7201912yes—
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2986C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#2986D3
hypocenterC3#9436
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
197. C3#9436semidirect

Maximal subgroups

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

subgroupindex
C3×#29862
#833372
#718342
#707802
#677432
#94363