Finite Group Databasebeta

← #11723334

Subgroups — #11723334

Order 864.

460 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 102 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#89020C211yesmaximal, characteristic
2C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#3743C211yesmaximal, characteristic, complemented
4C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
5#88788C211yesmaximal, characteristic, complemented
6C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
7D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#64116—31nomaximal
9#63805—34nomaximal
10C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C3×C4×C3⋊D3C2211yescharacteristic
12C32×C4×D3C2211yescharacteristic
13#6293C411yescharacteristic
14#6292C411yescharacteristic
15#6291C411yescharacteristic
16#6290C411yescharacteristic
17C33×C8C2211yescharacteristic
18C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
19C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C32×C3⋊C8C2211yescharacteristic
21#3828—31no—
22#3798—34no—
23C32×M16—31no—
24C2×C3⋊(C3⋊C8)—31no—
25#3743—31no—
26#3743D311yescharacteristic, complemented
27C8×C3⋊D3—31no—
28C8×C3⋊D3—31no—
29C8×C3⋊D3—34no—
30C8×C3⋊D3—64no—
31C3×ΓL(1,𝔽25)—34no—
32C3×ΓL(1,𝔽25)D314yescomplemented
33#3712—34no—
34D3×C3⋊C8—34no—
35D3×C3⋊C8—34no—
36C2×C3⋊(C3⋊D3)C2×C411yescharacteristic
37C2×C3×C3⋊D3C2×C411yescharacteristic
38C2×C32×D3C2×C411yescharacteristic
39C33×C4C2311yescharacteristic
40C3⋊(C3⋊Dic3)C2×C411yescharacteristic
41C3×C3⋊Dic3C2×C411yescharacteristic
42C32×Dic3C2×C411yescharacteristic
43#907—94no—
44#907—91no—
45C2×C32×C4—31no—
46#638—31no—
47#638—31no—
48C4⋊(C3⋊D3)—31no—
49C4×C3⋊D3—31no—
50C4×C3⋊D3—34no—
51C4×C3⋊D3—64no—
52C4×C3⋊D3C2×D311yescharacteristic
53#634—31no—
54C3×C4×D3—34no—
55C3×C4×D3C2×D314yes—
56#627—34no—
57#626—34no—
58#626—34no—
59#625—34no—
60C32×C8—24no—
61C32×C8—31no—
62C32×C8C2×D314yes—
63C32×C8C2×D311yescharacteristic
64C3⋊(C3⋊C8)—31no—
65C3⋊(C3⋊C8)—34no—
66C3⋊(C3⋊C8)—64no—
67C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×C3⋊C8—34no—
69C3×C3⋊C8C2×D314yes—
70C2×C33C22×C411yescharacteristic
71C3⋊(C3⋊D3)—21no—
72C3×C3⋊D3—21no—
73C32×D3—21no—
74#235—94no—
75#235—91no—
76C3×M16—34no—
77C3×M16—91no—
78C2×C3⋊C8—94no—
79C2×C3⋊C8—91no—
80ΓL(1,𝔽25)—34no—
81ΓL(1,𝔽25)—94no—
82ΓL(1,𝔽25)—91no—
83ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
84D3×C8—94no—
85D3×C8—91no—
86D3×C8—94no—
87D3×C8—91no—
88D3×C8—184no—
89C22×C32—31no—
90C2×C3⋊D3—31no—
91C2×C3⋊D3—34no—
92C2×C3⋊D3—64no—
93C2×C3⋊D3C4×D311yescharacteristic
94C2×C3×D3—34no—
95C2×C3×D3C4×D314yes—
96C32×C4—24no—
97C32×C4—31no—
98C32×C4C22×D314yes—
99C32×C4C22×D311yescharacteristic
100C3⋊Dic3—31no—
101C3⋊Dic3—34no—
102C3⋊Dic3—64no—
103C3⋊Dic3C4×D311yescharacteristic
104C3×Dic3—34no—
105C3×Dic3C4×D314yes—
106#131—271noSylow
107C33#13111yescharacteristic, complemented, Sylow
108C2×C3×C4—34no—
109C2×C3×C4—91no—
110#67—94no—
111#67—94no—
112#67—91no—
113#67—91no—
114D12—94no—
115D12—91no—
116C4×D3—34no—
117C4×D3—94no—
118C4×D3—91no—
119C4×D3—184no—
120C4×D3C2×C3⋊D311yescharacteristic
121Dic6—94no—
122Dic6—91no—
123C3×C8—24no—
124C3×C8—34no—
125C3×C8—91no—
126C3×C8D3214yes—
127C3×C8C2×C3⋊D311yescharacteristic
128C3⋊C8—34no—
129C3⋊C8—94no—
130C3⋊C8—91no—
131C3⋊C8—184no—
132C3⋊C8C2×C3⋊D311yescharacteristic
133C2×C32—24no—
134C2×C32—61no—
135C2×C32C2×C4×D314yes—
136C2×C32C2×C4×D311yescharacteristic
137C3⋊D3—21no—
138C3⋊D3—68no—
139C3⋊D3—64no—
140C3⋊D3—61no—
141C3×D3—24no—
142C3×D3—64no—
143#41—271no—
144M16—31no—
145M16—91no—
146M16—271no—
147C2×C8—271no—
148C2×C8—271no—
149C2×C8—271no—
150C22×C3—34no—
151C22×C3—91no—
152C2×D3—34no—
153C2×D3—94no—
154C2×D3—91no—
155C2×D3—184no—
156C2×D3C4×C3⋊D311yescharacteristic
157C3×C4—24no—
158C3×C4—34no—
159C3×C4—91no—
160C3×C4C2×D3214yes—
161C3×C4C22×C3⋊D311yescharacteristic
162Dic3—34no—
163Dic3—94no—
164Dic3—91no—
165Dic3—184no—
166Dic3C4×C3⋊D311yescharacteristic
167C32—24no—
168C32#90714yescomplemented
169C32#90711yescharacteristic, complemented
170Q8—271no—
171D4—271no—
172D4—271no—
173D4—271no—
174C2×C4—31no—
175C2×C4—91no—
176C2×C4—271no—
177C8—31no—
178C8—91no—
179C8—271no—
180C8D3×C3⋊D311yescharacteristic
181C2×C3—24no—
182C2×C3—64no—
183C2×C3—181no—
184C2×C3C4×D3214yes—
185C2×C3C2×C4×C3⋊D311yescharacteristic
186D3—21no—
187D3—64no—
188D3—188no—
189D3—184no—
190D3—181no—
191C22—31no—
192C22—91no—
193C22—271no—
194C4—31no—
195C4—91no—
196C4—271no—
197C4C2×D3×C3⋊D311yescharacteristic
198C3—24no—
199C3#6380514yescomplemented
200C3#6411611yescharacteristic, complemented
201C2—61no—
202C2—181no—
203C2—541no—
204C2C4×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC4C2×D3×C3⋊D3
commutator, socleC2×C33C22×C4
FittingC33×C8C22
prime radical#89020C2
cosocleC33×C4C23
hypercenterC8D3×C3⋊D3
hypocenterC33#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C8×C3⋊(C3⋊D3)C2semidirect
3. C3×#3743C2semidirect
4. C32×ΓL(1,𝔽25)C2semidirect
5. #88788C2semidirect
6. C3⋊D3×C3⋊C8C2semidirect
7. D3×C3⋊(C3⋊C8)C2semidirect
26. #3743D3semidirect
32. C3×ΓL(1,𝔽25)D3semidirect
83. ΓL(1,𝔽25)C3⋊D3semidirect
107. C33#131semidirect
168. C32#907semidirect
169. C32#907semidirect
199. C3#63805semidirect
200. C3#64116semidirect

Maximal subgroups

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

subgroupindex
#890202
C8×C3⋊(C3⋊D3)2
C3×#37432
C32×ΓL(1,𝔽25)2
#887882
C3⋊D3×C3⋊C82
D3×C3⋊(C3⋊C8)2
#641163
#638053