Finite Group Databasebeta

← #10692799

Subgroups — #10692799

Order 768.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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#83336C211yesmaximal, characteristic
2#83332C211yesmaximal, characteristic, complemented
3C3×#1561C211yesmaximal, characteristic, complemented
4#70502C212yesmaximal
5#67710C212yesmaximal, complemented
6#9438—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×#874C2211yescharacteristic
9C3×#423C2212yes—
10#5088—22no—
11#5067C414yes—
12#5024—22no—
13Q8×C3⋊C8—22no—
14#5001—22no—
15#5000—22no—
16#4925C2212yes—
17C4×#208C2211yescharacteristic
18C3×#332—22no—
19#4485—22no—
20#4485—22no—
21#3138—31no—
22#2986—31no—
23#1561D311yescharacteristic, complemented
24#1545—32no—
25#1299—32no—
26C22×D12—21no—
27C22×D12—22no—
28C2×C3×M16—22no—
29C3×C4⋊Q8D412yes—
30C3×#125C2×C412yes—
31C3×C22⋊Q8—22no—
32C3×C4×Q8—22no—
33C3×#117D412yes—
34C2×C3×C42C2311yescharacteristic
35C2×#208—21no—
36#874—22no—
37#874—21no—
38#874C2×C412yes—
39C3×C4⋊C8—22no—
40C3×C4⋊C8—22no—
41#830—22no—
42#809—28no—
43C4⋊(C3⋊C8)—24no—
44C4⋊(C3⋊C8)D412yes—
45C4⋊(C3⋊C8)D412yes—
46#803—22no—
47#803C2×C412yes—
48C4×C3⋊C8—22no—
49C4×C3⋊C8—21no—
50C4×C3⋊C8—22no—
51#533C2×D311yescharacteristic
52C2×#127—31no—
53#490—34no—
54#486—62no—
55#446—62no—
56C8×Q8—62no—
57#433—62no—
58#431—32no—
59#423C2×D312yes—
60#418—62no—
61C4×M16—31no—
62#332—22no—
63#331—62no—
64#331—62no—
65C23×D3—41no—
66C2×C3×Q8—22no—
67C22×C3×C4C2×D411yescharacteristic
68C22×C3×C4#4112yes—
69C2×D12—22no—
70C2×D12—24no—
71C2×D12—44no—
72C2×D12—41no—
73C2×D12—42no—
74C2×D12—82no—
75C2×D12—82no—
76C3×M16—82no—
77C2×C3×C8—42no—
78C3×C4⋊C4—24no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4C22⋊C414yes—
81C3×C22⋊C4—22no—
82C3×C42—22no—
83C3×C42C22×C411yescharacteristic
84C3×C42C2×D412yes—
85C3×C42C2×D411yescharacteristic
86#208—42no—
87C2×C3⋊C8—22no—
88C2×C3⋊C8—22no—
89C2×C3⋊C8—22no—
90C22×D4—61no—
91C22×D4—62no—
92C2×M16—22no—
93C2×M16—61no—
94C4⋊Q8#6712yes—
95#127—32no—
96#127—62no—
97#127—61no—
98#125C4×D312yes—
99C22⋊Q8—22no—
100C4×Q8—22no—
101#117D1212yes—
102C2×C42C22×D311yescharacteristic
103C4⋊C8—22no—
104C4⋊C8—22no—
105C4⋊C8—32no—
106C4⋊C8—32no—
107C4⋊C8—64no—
108#102—68no—
109C22⋊C8—62no—
110#97—32no—
111#97—62no—
112C4×C8—62no—
113C4×C8—62no—
114C4×C8—61no—
115C23×C3#12311yescharacteristic
116C22×D3—42no—
117C22×D3—82no—
118C22×D3—84no—
119C3×Q8—42no—
120C3×Q8—42no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—24no—
125C2×C3×C4C2×C22⋊C411yescharacteristic
126C2×C3×C4C4×D412yes—
127C2×C3×C4#12011yescharacteristic
128C2×C3×C4#12112yes—
129D12—44no—
130D12—84no—
131D12—82no—
132D12—82no—
133D12—82no—
134C3×C8—82no—
135C3⋊C8—42no—
136C3⋊C8—42no—
137C3⋊C8—42no—
138C24—121no—
139C2×Q8—22no—
140C2×D4—64no—
141C2×D4—62no—
142C2×D4—124no—
143C2×D4—121no—
144C2×D4—122no—
145C2×D4—242no—
146C2×D4—242no—
147C22×C4C2×D1211yescharacteristic
148C22×C4#23712yes—
149M16—82no—
150M16—122no—
151C2×C8—42no—
152C2×C8—62no—
153C2×C8—62no—
154C2×C8—62no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4(C2×D3)⋊C414yes—
158C22⋊C4—22no—
159C42—22no—
160C42C2×C4×D311yescharacteristic
161C42D3×D412yes—
162C42C2×#6711yescharacteristic
163C22×C3—21no—
164C22×C3—22no—
165C22×C3#38611yescharacteristic
166C2×D3—82no—
167C2×D3—84no—
168C2×D3—164no—
169C3×C4—21no—
170C3×C4—22no—
171C3×C4—44no—
172C3×C4#45412yes—
173C3×C4#47112yes—
174C3×C4#47212yes—
175C23—122no—
176C23—242no—
177C23—244no—
178C23#88611yescharacteristic
179Q8—42no—
180Q8—42no—
181D4—124no—
182D4—244no—
183D4—242no—
184D4—242no—
185D4—242no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4#67⋊C412yes—
191C2×C4#88211yescharacteristic
192C2×C4C2×(C2×D3)⋊C411yescharacteristic
193C2×C4#93912yes—
194C8—82no—
195C8—122no—
196C8—122no—
197C8—122no—
198C2×C3—41no—
199C2×C3#188811yescharacteristic
200C2×C3#190712yes—
201D3—162no—
202C22—22no—
203C22—21no—
204C22—244no—
205C22—242no—
206C22—484no—
207C22#495811yescharacteristic
208C4—22no—
209C4—21no—
210C4—44no—
211C4#475312yes—
212C4#519912yes—
213C4#520312yes—
214C3#943811yescharacteristic, complemented, Sylow
215C2—41no—
216C2—482no—
217C2#7010411yescharacteristic
218C2#7030312yes—
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1561C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1561D3
hypocenterC3#9438
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
2. #83332C2semidirect
3. C3×#1561C2semidirect
5. #67710C2semidirect
23. #1561D3semidirect
214. C3#9438semidirect

Maximal subgroups

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

subgroupindex
#833362
#833322
C3×#15612
#705022
#677102
#94383