Finite Group Databasebeta

← #10692719

Subgroups — #10692719

Order 768.

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

normal subgroups only

This group has 436 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, complemented
2#83336C211yesmaximal, characteristic
3#71823C211yesmaximal, characteristic, complemented
4#70818C212yesmaximal
5#67753C212yesmaximal, complemented
6#9424—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×#874C2211yescharacteristic
9C3×#446—24no—
10C3×#433—22no—
11C3×#432C2212yes—
12#5088C414yes—
13#5067—22no—
14#5001—22no—
15#4926C2212yes—
16C4×#208C2211yescharacteristic
17#4543—22no—
18#4485—24no—
19#3138—31no—
20#2987D311yescharacteristic, complemented
21#1555—32no—
22#1537—31no—
23#1291—32no—
24C22×D12—21no—
25C22×D12—22no—
26C3×C4⋊Q8C2×C412yes—
27C3×#125D412yes—
28C3×C22⋊Q8—22no—
29C3×C4×Q8—22no—
30C3×#117D412yes—
31C2×C3×C42C2311yescharacteristic
32C2×#208—21no—
33#874—22no—
34#874—21no—
35#874C2×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#809—28no—
44C4⋊(C3⋊C8)—22no—
45#803—21no—
46#803—22no—
47C4×C3⋊C8C2×C412yes—
48#533C2×D311yescharacteristic
49C2×#127—31no—
50#490—62no—
51#486—34no—
52#446—24no—
53#433—22no—
54#432—32no—
55#432C2×D312yes—
56#418—62no—
57C4×M16—31no—
58#331—64no—
59#330—62no—
60C23×D3—41no—
61C2×C3×Q8—22no—
62C22×C3×C4C2×D411yescharacteristic
63C22×C3×C4#4112yes—
64C2×D12—22no—
65C2×D12—24no—
66C2×D12—44no—
67C2×D12—41no—
68C2×D12—42no—
69C2×D12—84no—
70C2×C3×C8—24no—
71C3×C4⋊C4—22no—
72C3×C4⋊C4—24no—
73C3×C4⋊C4C22⋊C414yes—
74C3×C22⋊C4—22no—
75C3×C42—22no—
76C3×C42C22×C411yescharacteristic
77C3×C42C2×D411yescharacteristic
78C3×C42C2×D412yes—
79#208—42no—
80C2×C3⋊C8—22no—
81C2×C3⋊C8—42no—
82C22×D4—62no—
83C22×D4—61no—
84C2×M16—61no—
85C4⋊Q8C4×D312yes—
86#127—32no—
87#127—61no—
88#127—62no—
89#125D1212yes—
90C22⋊Q8—22no—
91C4×Q8—22no—
92#117#6712yes—
93C2×C42C22×D311yescharacteristic
94C4⋊C8—24no—
95C4⋊C8—62no—
96C4⋊C8#6714yes—
97#102—68no—
98C22⋊C8—22no—
99C22⋊C8—62no—
100#97—22no—
101#97—22no—
102#97—61no—
103#97—62no—
104C4×C8—22no—
105C4×C8—32no—
106C23×C3#12311yescharacteristic
107C22×D3—42no—
108C22×D3—82no—
109C22×D3—84no—
110C3×Q8—82no—
111C2×C3×C4—22no—
112C2×C3×C4—24no—
113C2×C3×C4—42no—
114C2×C3×C4C2×C22⋊C411yescharacteristic
115C2×C3×C4C4×D412yes—
116C2×C3×C4#12011yescharacteristic
117C2×C3×C4#12112yes—
118D12—44no—
119D12—84no—
120D12—84no—
121D12—82no—
122C3×C8—44no—
123C3⋊C8—42no—
124C3⋊C8—82no—
125C24—121no—
126C2×Q8—22no—
127C2×D4—62no—
128C2×D4—64no—
129C2×D4—124no—
130C2×D4—122no—
131C2×D4—121no—
132C2×D4—244no—
133C22×C4#23512yes—
134C22×C4C2×#6711yescharacteristic
135M16—122no—
136C2×C8—24no—
137C2×C8—62no—
138C2×C8—122no—
139C4⋊C4—22no—
140C4⋊C4—24no—
141C4⋊C4(C2×D3)⋊C414yes—
142C22⋊C4—22no—
143C42—22no—
144C42C2×C4×D311yescharacteristic
145C42C2×D1211yescharacteristic
146C42C2×#6712yes—
147C22×C3—21no—
148C22×C3—22no—
149C22×C3#38611yescharacteristic
150C2×D3—82no—
151C2×D3—84no—
152C2×D3—164no—
153C3×C4—21no—
154C3×C4—22no—
155C3×C4—22no—
156C3×C4—44no—
157C3×C4#45412yes—
158C23—122no—
159C23—242no—
160C23—244no—
161C23#92911yescharacteristic
162Q8—82no—
163D4—124no—
164D4—244no—
165D4—244no—
166D4—242no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—42no—
170C2×C4C2×(C2×D3)⋊C411yescharacteristic
171C2×C4C4×#6712yes—
172C2×C4#93012yes—
173C2×C4#95311yescharacteristic
174C8—44no—
175C8—122no—
176C8—242no—
177C2×C3—41no—
178C2×C3#188811yescharacteristic
179C2×C3#190312yes—
180D3—162no—
181C22—22no—
182C22—21no—
183C22—244no—
184C22—242no—
185C22—484no—
186C22#521311yescharacteristic
187C4—22no—
188C4—22no—
189C4—21no—
190C4—44no—
191C4#524512yes—
192C3#942411yescharacteristic, complemented, Sylow
193C2—41no—
194C2—482no—
195C2#7166111yescharacteristic
196C2#7202312yes—
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2987C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#2987D3
hypocenterC3#9424
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
1. C3×#2987C2semidirect
3. #71823C2semidirect
5. #67753C2semidirect
20. #2987D3semidirect
192. C3#9424semidirect

Maximal subgroups

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

subgroupindex
C3×#29872
#833362
#718232
#708182
#677532
#94243