Finite Group Databasebeta

← #11785483

Subgroups — #11785483

Order 1152.

780 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#64116C211yesmaximal, characteristic
2#10592705C212yesmaximal
3#10592697C212yesmaximal, complemented
4C32×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
5#10589423C211yesmaximal, characteristic, complemented
6#68710—34nomaximal
7C2×#3828C2211yescharacteristic
8C2×C32×M16C2211yescharacteristic
9C32×#117C2212yes—
10#64135C2212yes—
11C22×C3⋊(C3⋊C8)C2211yescharacteristic
12#64116—22no—
13#64116—22no—
14C2×#3743C412yescomplemented
15C2×C8×C3⋊D3C412yescomplemented
16#63635—22no—
17#63634—22no—
18#63633—24no—
19C2×#907—34no—
20#5006—38no—
21#4998—38no—
22C3×H(3,ℤ/4)D314yescomplemented
23#4557—34no—
24C22×C32×C4C2311yescharacteristic
25C2×#638D412yes—
26#3828—22no—
27#3828—22no—
28C2×C4⋊(C3⋊D3)Q811yescharacteristic
29C2×C4×C3⋊D3C2×C412yes—
30C2×#634Q811yescharacteristic
31C32×M16—22no—
32C2×C32×C8C2×C412yes—
33C32×C4⋊C4—22no—
34C32×C22⋊C4—22no—
35C32×C42—22no—
36#3757—22no—
37C4⋊(C3⋊Dic3)—22no—
38C4×C3⋊Dic3—22no—
39C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
40C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
41C2×C3⋊(C3⋊C8)D412yes—
42C2×C3⋊(C3⋊C8)D412yes—
43#3743—24no—
44C8×C3⋊D3—22no—
45C8×C3⋊D3—22no—
46#1812—91noSylow
47C2×#235—34no—
48C2×C3×M16C2×D314yes—
49C3×#117C2×D318yes—
50#926—38no—
51C22×C3⋊C8—34no—
52#907—68no—
53#907—68no—
54C2×ΓL(1,𝔽25)—38no—
55C2×D3×C8—38no—
56#808—68no—
57#807—68no—
58#806—616no—
59C23×C32C2×D411yescharacteristic
60C22×C3⋊D3C4⋊C412yes—
61C2×C32×C4—42no—
62C2×C32×C4C22×C411yescharacteristic
63C2×C32×C4C2×D412yes—
64C2×C32×C4C2×Q811yescharacteristic
65C2×C32×C4#4112yes—
66#638—24no—
67C2×C3⋊Dic3—42no—
68C2×C3⋊Dic3C4⋊C412yes—
69C4⋊(C3⋊D3)—22no—
70C4×C3⋊D3—22no—
71C4×C3⋊D3—22no—
72#634—22no—
73C32×C8—22no—
74C3⋊(C3⋊C8)C22⋊C411yescharacteristic
75C3⋊(C3⋊C8)C22⋊C411yescharacteristic
76C3⋊(C3⋊C8)C22⋊C412yes—
77C2×#131—91no—
78#427—92no—
79#410—92no—
80H(3,ℤ/4)—91no—
81H(3,ℤ/4)C3⋊D311yescharacteristic, complemented
82C22×C3×C4C22×D314yes—
83C2×#67—38no—
84#235—68no—
85#235—68no—
86C2×D12—34no—
87C2×C4×D3—38no—
88C2×Dic6—34no—
89C3×M16—28no—
90C2×C3×C8C4×D318yes—
91C3×C4⋊C4—28no—
92C3×C22⋊C4—28no—
93C3×C42—28no—
94#217—68no—
95C4⋊Dic3—68no—
96C4×Dic3—68no—
97C2×C3⋊C8—34no—
98C2×C3⋊C8—34no—
99C2×C3⋊C8—38no—
100C2×C3⋊C8—38no—
101ΓL(1,𝔽25)—616no—
102D3×C8—68no—
103D3×C8—68no—
104C22×C32—41no—
105C22×C32C2×C4⋊C411yescharacteristic
106C22×C32#12112yes—
107C2×C3⋊D3—22no—
108C2×C3⋊D3—42no—
109C32×C4—82no—
110C32×C4C2×C22⋊C411yescharacteristic
111C32×C4#11711yescharacteristic
112C32×C4C22⋊Q812yes—
113C3⋊Dic3—22no—
114C3⋊Dic3—82no—
115C2×#41—91no—
116#131—182no—
117#131—182no—
118C2×M16—92no—
119C2×M16C2×C3⋊D311yescharacteristic
120C22×C8—91no—
121C22×C8—92no—
122#117—92no—
123#117C2×C3⋊D312yes—
124#107—182no—
125#106—182no—
126#99—184no—
127C23×C3D3×D414yes—
128C22×D3—38no—
129C2×C3×C4—48no—
130C2×C3×C4C2×C4×D314yes—
131C2×C3×C4#23518yes—
132C2×C3×C4D3×D418yes—
133C2×C3×C4D3×Q814yes—
134#67—616no—
135C2×Dic3—38no—
136C2×Dic3—128no—
137D12—68no—
138C4×D3—68no—
139C4×D3—68no—
140Dic6—68no—
141C3×C8—28no—
142C3⋊C8—34no—
143C3⋊C8—34no—
144C3⋊C8—38no—
145C2×C32—22no—
146C2×C32—21no—
147C2×C32#38011yescharacteristic
148C3⋊D3—42no—
149#41—182no—
150#41—182no—
151C2×Q8—91no—
152C2×D4—92no—
153C2×D4—91no—
154C22×C4—92no—
155C22×C4C22×C3⋊D311yescharacteristic
156M16—22no—
157M16—184no—
158C2×C8—91no—
159C2×C8—91no—
160C2×C8—92no—
161C2×C8—92no—
162C2×C8—182no—
163C2×C8—182no—
164C2×C8C4×C3⋊D312yes—
165C4⋊C4—22no—
166C4⋊C4—182no—
167C22⋊C4—22no—
168C22⋊C4—182no—
169C42—22no—
170C42—182no—
171C22×C3—44no—
172C22×C3#88418yes—
173C22×C3D3×C4⋊C414yes—
174C2×D3—68no—
175C2×D3—128no—
176C3×C4—88no—
177C3×C4#87214yes—
178C3×C4D3×C22⋊C414yes—
179C3×C4#89618yes—
180Dic3—68no—
181Dic3—248no—
182C32#181211yescharacteristic, complemented, Sylow
183C23—92no—
184C23D4×C3⋊D311yescharacteristic
185Q8—182no—
186D4—184no—
187D4—182no—
188C2×C4—42no—
189C2×C4—92no—
190C2×C4—182no—
191C2×C4—182no—
192C2×C4—362no—
193C2×C4C2×C4×C3⋊D311yescharacteristic
194C2×C4#382812yes—
195C2×C4D4×C3⋊D312yes—
196C2×C4Q8×C3⋊D311yescharacteristic
197C8—22no—
198C8—91no—
199C8—91no—
200C8—92no—
201C2×C3—28no—
202C2×C3—24no—
203C2×C3#466714yes—
204D3—128no—
205C22—41no—
206C22—182no—
207C22—362no—
208C22#6409312yes—
209C22C4⋊C4×C3⋊D311yescharacteristic
210C4—82no—
211C4—182no—
212C4—722no—
213C4#6408111yescharacteristic
214C4C22⋊C4×C3⋊D311yescharacteristic
215C4#6410512yes—
216C3#6871014yescomplemented
217C2—22no—
218C2—21no—
219C2—362no—
220C2#1059236611yescharacteristic
subgroupG/H
centerC4#64081
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×H(3,ℤ/4)C2
socleC2×C32#380
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterH(3,ℤ/4)C3⋊D3
hypocenterC32#1812
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×M16 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
3. #10592697C2semidirect
4. C32×H(3,ℤ/4)C2semidirect
5. #10589423C2semidirect
14. C2×#3743C4semidirect
15. C2×C8×C3⋊D3C4semidirect
22. C3×H(3,ℤ/4)D3semidirect
81. H(3,ℤ/4)C3⋊D3semidirect
182. C32#1812semidirect
216. C3#68710semidirect

Maximal subgroups

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

subgroupindex
C2×#641162
#105927052
#105926972
C32×H(3,ℤ/4)2
#105894232
#687103