Finite Group Databasebeta

← #10692709

Subgroups — #10692709

Order 768.

384 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2987C211yesmaximal, characteristic
2#83337C211yesmaximal, characteristic
3#71823C211yesmaximal, characteristic
4#70823C212yesmaximal
5#67753C212yesmaximal
6#9422—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#446—24no—
10C3×#433—22no—
11C3×#432C2212yes—
12#5093C414yes—
13#5069—22no—
14#5004—22no—
15#4926C2212yes—
16C4×#208C2211yescharacteristic
17#4543—22no—
18#4482—24no—
19#3139—31no—
20#2987D311yescharacteristic
21#1556—32no—
22#1537—31no—
23#1291—32no—
24C22×Dic6—21no—
25C3×C4⋊Q8C2×C412yes—
26C3×#125D412yes—
27C3×C22⋊Q8—22no—
28C3×C4×Q8—22no—
29C3×#117D412yes—
30C2×C3×C42C2311yescharacteristic
31C2×C4⋊Dic3—22no—
32C2×#208—21no—
33#869—22no—
34#869—21no—
35#869C2×C412yes—
36C3×C4⋊C8—24no—
37C3×C4⋊C8D414yes—
38C3×C22⋊C8—22no—
39C3×#97—22no—
40C3×#97—22no—
41C3×C4×C8—22no—
42#830—22no—
43#810—28no—
44C4⋊(C3⋊C8)—22no—
45#803—21no—
46#803—22no—
47C4×C3⋊C8C2×C412yes—
48#533C2×D311yescharacteristic
49C2×C4⋊Q8—31no—
50#491—62no—
51#487—34no—
52#446—24no—
53#433—22no—
54#432—32no—
55#432C2×D312yes—
56#419—62no—
57C4×M16—31no—
58#332—64no—
59#330—62no—
60C2×C3×Q8—22no—
61C22×C3×C4C2×D411yescharacteristic
62C22×C3×C4#4112yes—
63C22×Dic3—41no—
64C2×Dic6—22no—
65C2×Dic6—44no—
66C2×Dic6—41no—
67C2×C3×C8—24no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4—24no—
70C3×C4⋊C4C22⋊C414yes—
71C3×C22⋊C4—22no—
72C3×C42—22no—
73C3×C42C22×C411yescharacteristic
74C3×C42C2×D411yescharacteristic
75C3×C42C2×D412yes—
76C4⋊Dic3—24no—
77C4⋊Dic3—42no—
78#208—42no—
79C2×C3⋊C8—22no—
80C2×C3⋊C8—42no—
81C22×Q8—61no—
82C2×M16—61no—
83C4⋊Q8—32no—
84C4⋊Q8—61no—
85C4⋊Q8—62no—
86C4⋊Q8C4×D312yes—
87#125D1212yes—
88C22⋊Q8—22no—
89C4×Q8—22no—
90#117#6712yes—
91C2×C4⋊C4—62no—
92C2×C42C22×D311yescharacteristic
93C4⋊C8—24no—
94C4⋊C8—62no—
95C4⋊C8#6714yes—
96#103—68no—
97C22⋊C8—22no—
98C22⋊C8—62no—
99#97—22no—
100#97—22no—
101#97—61no—
102#97—62no—
103C4×C8—22no—
104C4×C8—32no—
105C23×C3#12311yescharacteristic
106C3×Q8—82no—
107C2×C3×C4—22no—
108C2×C3×C4—24no—
109C2×C3×C4—42no—
110C2×C3×C4C2×C22⋊C411yescharacteristic
111C2×C3×C4C4×D412yes—
112C2×C3×C4#12011yescharacteristic
113C2×C3×C4#12112yes—
114C2×Dic3—42no—
115C2×Dic3—82no—
116Dic6—44no—
117Dic6—82no—
118C3×C8—44no—
119C3⋊C8—42no—
120C3⋊C8—82no—
121C2×Q8—22no—
122C2×Q8—62no—
123C2×Q8—124no—
124C2×Q8—121no—
125C22×C4—121no—
126C22×C4#23512yes—
127C22×C4C2×#6711yescharacteristic
128M16—122no—
129C2×C8—24no—
130C2×C8—62no—
131C2×C8—122no—
132C4⋊C4—22no—
133C4⋊C4—24no—
134C4⋊C4—64no—
135C4⋊C4—122no—
136C4⋊C4(C2×D3)⋊C414yes—
137C22⋊C4—22no—
138C42—22no—
139C42C2×C4×D311yescharacteristic
140C42C2×D1211yescharacteristic
141C42C2×#6712yes—
142C22×C3—21no—
143C22×C3—22no—
144C22×C3#38611yescharacteristic
145C3×C4—21no—
146C3×C4—22no—
147C3×C4—22no—
148C3×C4—44no—
149C3×C4#45412yes—
150Dic3—82no—
151C23#92911yescharacteristic
152Q8—82no—
153Q8—124no—
154Q8—242no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—42no—
158C2×C4—122no—
159C2×C4—242no—
160C2×C4C2×(C2×D3)⋊C411yescharacteristic
161C2×C4C4×#6712yes—
162C2×C4#93012yes—
163C2×C4#95311yescharacteristic
164C8—44no—
165C8—122no—
166C8—242no—
167C2×C3—41no—
168C2×C3#188811yescharacteristic
169C2×C3#190212yes—
170C22—22no—
171C22—21no—
172C22#521311yescharacteristic
173C4—22no—
174C4—22no—
175C4—21no—
176C4—44no—
177C4—242no—
178C4#524512yes—
179C3#942211yescharacteristic, complemented, Sylow
180C2—41no—
181C2#7166111yescharacteristic
182C2#7202212yes—
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2987C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#2987D3
hypocenterC3#9422
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
179. C3#9422semidirect

Maximal subgroups

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

subgroupindex
C3×#29872
#833372
#718232
#708232
#677532
#94223