Finite Group Databasebeta

← #11780992

Subgroups — #11780992

Order 1152.

363 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 96 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
1Q8×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C3×#5020C212yesmaximal
3#10590387C211yesmaximal, characteristic
4#10589100C211yesmaximal, characteristic
5#10589092C212yesmaximal
6#71117—32nomaximal
7C32×C4×Q8C2211yescharacteristic
8C3×#869C2212yes—
9C2×#3718—21no—
10C2×#3718—21no—
11C4⋊(C3⋊(C3⋊C8))—21no—
12C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C4×C3⋊(C3⋊C8)—21no—
14C4×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#835—22no—
16C3×#808—22no—
17C3×C4⋊(C3⋊C8)C2212yes—
18C3×#475—32no—
19#5092—32no—
20Q8×C3⋊C8—32no—
21Q8×C3⋊C8—61no—
22#5020D312yes—
23#4547—32no—
24#4486—32no—
25#4479—32no—
26C2×C32×Q8D411yescharacteristic
27C2×C3×Dic6—22no—
28C32×C4⋊C4—21no—
29C32×C4⋊C4D411yescharacteristic
30C32×C42—21no—
31C32×C42C2311yescharacteristic
32C2×C3⋊(C3⋊C8)—21no—
33C2×C3⋊(C3⋊C8)D411yescharacteristic
34C2×C3⋊(C3⋊C8)D411yescharacteristic
35C3×C4⋊Dic3—22no—
36C3×C4⋊Dic3—22no—
37C2×C3×C3⋊C8—22no—
38#3718—42no—
39#3718—42no—
40#1670—91noSylow
41C3×C4⋊Q8—32no—
42C3×C4×Q8—21no—
43C3×C4×Q8C2×D312yes—
44C2×#216—62no—
45C2×#216—62no—
46#869C2×D312yes—
47C3×#106—62no—
48C3×C4⋊C8—32no—
49C3×#103—62no—
50#835—22no—
51#808—22no—
52C4⋊(C3⋊C8)—32no—
53C4⋊(C3⋊C8)—61no—
54C4⋊(C3⋊C8)—62no—
55C4⋊(C3⋊C8)—62no—
56C4⋊(C3⋊C8)C2×D312yes—
57C4×C3⋊C8—32no—
58C4×C3⋊C8—61no—
59C4×C3⋊C8—62no—
60C4×C3⋊C8—62no—
61C32×Q8—21no—
62C32×Q8Q1612yescomplemented
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C32×C4C2×D411yescharacteristic
66C2×C32×C4C2×D411yescharacteristic
67C2×C32×C4#4111yescharacteristic
68C2×C3×Dic3—42no—
69C3×Dic6—42no—
70C3×Dic6—42no—
71C3⋊(C3⋊C8)—21no—
72C3⋊(C3⋊C8)—41no—
73C3⋊(C3⋊C8)SD1612yes—
74C3×C3⋊C8—42no—
75#494—91no—
76#475—32no—
77C8×Q8—91no—
78#335—91no—
79#332—92no—
80C2×C3×Q8—21no—
81C2×C3×Q8—62no—
82C2×C3×Q8#6712yes—
83C2×Dic6—22no—
84C2×C3×C8—62no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—21no—
88C3×C4⋊C4—62no—
89C3×C4⋊C4—62no—
90C3×C4⋊C4#6712yes—
91C3×C42—22no—
92C3×C42—22no—
93C3×C42—21no—
94C3×C42C22×D312yes—
95#216—124no—
96#216—124no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99C2×C3⋊C8—22no—
100C2×C3⋊C8—32no—
101C2×C3⋊C8—32no—
102C2×C3⋊C8—61no—
103C2×C3⋊C8—61no—
104C2×C3⋊C8—62no—
105C2×C3⋊C8—62no—
106C22×C32#12111yescharacteristic
107C32×C4—22no—
108C32×C4—21no—
109C32×C4—41no—
110C32×C4C2×SD1611yescharacteristic
111C32×C4C2×Q1611yescharacteristic
112C32×C4#13511yescharacteristic
113C32×C4AGL(1,ℤ/8)11yescharacteristic
114C3×Dic3—82no—
115C2×Q16—181no—
116C2×Q16—181no—
117C4⋊Q8—32no—
118C4×Q8D3211yescharacteristic
119#106—62no—
120C4⋊C8—32no—
121C4⋊C8—91no—
122C4⋊C8—181no—
123#103—62no—
124C4×C8—91no—
125C4×C8—181no—
126C3×Q8—22no—
127C3×Q8—22no—
128C3×Q8—22no—
129C3×Q8—122no—
130C3×Q8—122no—
131C3×Q8#21614yescomplemented
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—122no—
140C2×C3×C4D3×D412yes—
141C2×C3×C4#23712yes—
142C2×C3×C4C2×#6712yes—
143C2×Dic3—42no—
144Dic6—42no—
145Dic6—42no—
146C3×C8—122no—
147C3⋊C8—34no—
148C3⋊C8—42no—
149C3⋊C8—62no—
150C3⋊C8—62no—
151C3⋊C8—121no—
152C3⋊C8—122no—
153C3⋊C8—122no—
154C2×C32#46011yescharacteristic
155C2×C32#46711yescharacteristic
156C2×C32#47311yescharacteristic
157C2×Q8—62no—
158C2×Q8#62511yescharacteristic
159Q16—362no—
160Q16—362no—
161C2×C8—62no—
162C2×C8—91no—
163C2×C8—91no—
164C2×C8—181no—
165C4⋊C4—21no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4#62511yescharacteristic
169C42—21no—
170C42C2×D3211yescharacteristic
171C22×C3—21no—
172C22×C3#93912yes—
173C3×C4—24no—
174C3×C4—22no—
175C3×C4—24no—
176C3×C4—21no—
177C3×C4—21no—
178C3×C4—21no—
179C3×C4—21no—
180C3×C4—41no—
181C3×C4—42no—
182C3×C4—42no—
183C3×C4—242no—
184C3×C4D3×SD1612yes—
185C3×C4#91412yes—
186C3×C4C2×#21612yes—
187C3×C4#95012yes—
188Dic3—82no—
189C32#167011yescharacteristic, complemented, Sylow
190Q8—21no—
191Q8—122no—
192Q8—122no—
193Q8#371812yescomplemented
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—122no—
197C2×C4#380511yescharacteristic
198C2×C4C2×#62511yescharacteristic
199C2×C4#381111yescharacteristic
200C8—92no—
201C8—122no—
202C8—181no—
203C8—361no—
204C2×C3—21no—
205C2×C3—21no—
206C2×C3—21no—
207C2×C3#505112yes—
208C2×C3#516312yes—
209C2×C3#520412yes—
210C22#6398011yescharacteristic
211C4—21no—
212C4—22no—
213C4—41no—
214C4—242no—
215C4#6382211yescharacteristic
216C4C2×#371811yescharacteristic
217C4#6394011yescharacteristic
218C4#6394111yescharacteristic
219C3—21no—
220C3#7111712yescomplemented
221C2#1059118811yescharacteristic
222C2#1059162211yescharacteristic
223C2#1059168311yescharacteristic
subgroupG/H
centerC22#63980
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socle, prime radicalC22×C32#121
hypercenterC4×Q8D32
hypocenterC32#1670
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
62. C32×Q8Q16semidirect
131. C3×Q8#216semidirect
189. C32#1670semidirect
193. Q8#3718semidirect
220. C3#71117semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊C8)2
C3×#50202
#105903872
#105891002
#105890922
#711173