Finite Group Databasebeta

← #11781785

Subgroups — #11781785

Order 1152.

393 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 82 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#10592537C211yesmaximal, characteristic
2#10590387C211yesmaximal, characteristic, complemented
3#10590313C212yesmaximal
4C3×#4482C212yesmaximal, complemented
5#10589098C211yesmaximal, characteristic, complemented
6#68449—32nomaximal
7C2×#3743—21no—
8C4×C4⋊(C3⋊D3)C2211yescharacteristic
9C3×#869C2212yes—
10C2×#3718—21no—
11C2×#3718—21no—
12C32×C4⋊C8C2211yescharacteristic
13#63647—21no—
14C4×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×C4⋊(C3⋊C8)C2212yes—
16#63575—22no—
17#63564—22no—
18#5092—32no—
19#5087—32no—
20#4846—32no—
21#4838—32no—
22#4838—61no—
23C3×#332—32no—
24#4482D312yescomplemented
25#4480—32no—
26C2×C4⋊(C3⋊D3)D411yescharacteristic
27C2×C4×C3⋊D3—21no—
28C2×C3×Dic6—22no—
29C2×C32×C8—21no—
30C32×C42C2311yescharacteristic
31#3752—21no—
32C4⋊(C3⋊Dic3)D411yescharacteristic
33C2×C3⋊(C3⋊C8)D411yescharacteristic
34C2×C3⋊(C3⋊C8)D411yescharacteristic
35#3743—42no—
36C3×C4⋊Dic3—22no—
37C3×C4⋊Dic3—22no—
38C2×C3×C3⋊C8—22no—
39#3718—42no—
40#3718—42no—
41#1689—91noSylow
42C3×C4⋊Q8—32no—
43C2×#216—62no—
44C2×#216—62no—
45C2×ΓL(1,𝔽25)—62no—
46C2×ΓL(1,𝔽25)—62no—
47C4×D12—32no—
48C4×D12—61no—
49#869C2×D312yes—
50C3×C4⋊C8—21no—
51C3×C4⋊C8—32no—
52C3×C4⋊C8C2×D312yes—
53#821—62no—
54(C2×D3)⋊C8—62no—
55(C2×D3)⋊C8—62no—
56#818—62no—
57#809—62no—
58#807—62no—
59C4⋊(C3⋊C8)C2×D312yes—
60C4×C3⋊C8—32no—
61C4×C3⋊C8—61no—
62C22×C3⋊D3—21no—
63C2×C32×C4C2×D411yescharacteristic
64C2×C32×C4C2×D411yescharacteristic
65C2×C32×C4#4111yescharacteristic
66C2×C3⋊Dic3—21no—
67C4⋊(C3⋊D3)—22no—
68C4×C3⋊D3—41no—
69C4×C3⋊D3—41no—
70C2×C3×Dic3—42no—
71C3×Dic6—42no—
72C3×Dic6—42no—
73C32×C8—41no—
74C3⋊(C3⋊C8)—21no—
75C3⋊(C3⋊C8)D812yes—
76C3×C3⋊C8—42no—
77#494—91no—
78#474—92no—
79#436—91no—
80#334—91no—
81#332—32no—
82C2×C3×Q8—62no—
83C2×D12—32no—
84C2×D12—61no—
85C2×C4×D3—62no—
86C2×C4×D3—62no—
87C2×Dic6—22no—
88C2×C3×C8—22no—
89C2×C3×C8—22no—
90C2×C3×C8—62no—
91C3×C4⋊C4—62no—
92C3×C4⋊C4—62no—
93C3×C42—21no—
94C3×C42C22×D312yes—
95#216—124no—
96#216—124no—
97(C2×D3)⋊C4—62no—
98(C2×D3)⋊C4—62no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101C4⋊Dic3—32no—
102C4⋊Dic3—61no—
103C2×C3⋊C8—22no—
104C2×C3⋊C8—32no—
105C2×C3⋊C8—32no—
106C2×C3⋊C8—61no—
107C2×C3⋊C8—61no—
108ΓL(1,𝔽25)—124no—
109ΓL(1,𝔽25)—124no—
110C22×C32#12111yescharacteristic
111C2×C3⋊D3—41no—
112C2×C3⋊D3—41no—
113C2×C3⋊D3—41no—
114C32×C4—21no—
115C32×C4C2×D811yescharacteristic
116C32×C4AGL(1,ℤ/8)11yescharacteristic
117C32×C4#13712yes—
118C3⋊Dic3—41no—
119C3×Dic3—82no—
120C2×Q16—181no—
121C2×Q16—181no—
122C2×M16—181no—
123C4⋊Q8—32no—
124C4×D4—91no—
125#107—182no—
126C4⋊C8—32no—
127C4⋊C8D3211yescharacteristic
128#102—182no—
129C22⋊C8—181no—
130C4×C8—91no—
131C22×D3—62no—
132C22×D3—62no—
133C3×Q8—122no—
134C3×Q8—122no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—122no—
139C2×C3×C4#23512yes—
140C2×C3×C4D3×D412yes—
141C2×C3×C4D3×D412yes—
142C2×Dic3—42no—
143C2×Dic3—62no—
144C2×Dic3—62no—
145D12—62no—
146D12—62no—
147D12—122no—
148C4×D3—122no—
149C4×D3—122no—
150C4×D3—122no—
151C4×D3—122no—
152Dic6—42no—
153Dic6—42no—
154C3×C8—42no—
155C3×C8—42no—
156C3×C8—122no—
157C3⋊C8—34no—
158C3⋊C8—42no—
159C3⋊C8—62no—
160C3⋊C8—62no—
161C3⋊C8—121no—
162C2×C32#45911yescharacteristic
163C2×C32#47011yescharacteristic
164C2×C32#47311yescharacteristic
165C3⋊D3—81no—
166C2×Q8—62no—
167C2×D4—91no—
168C22×C4—181no—
169Q16—362no—
170Q16—362no—
171M16—362no—
172C2×C8—21no—
173C2×C8—62no—
174C2×C8—91no—
175C2×C8—91no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—91no—
179C22⋊C4—181no—
180C42C2×D3211yescharacteristic
181C22×C3—21no—
182C22×C3#88412yes—
183C2×D3—122no—
184C2×D3—122no—
185C2×D3—122no—
186C2×D3—122no—
187C2×D3—122no—
188C2×D3—122no—
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—21no—
192C3×C4—21no—
193C3×C4—41no—
194C3×C4—242no—
195C3×C4D3×D812yes—
196C3×C4#91412yes—
197C3×C4#91512yes—
198C3×C4#91812yes—
199Dic3—82no—
200Dic3—122no—
201Dic3—122no—
202C32#168911yescharacteristic, complemented, Sylow
203C23—181no—
204Q8—122no—
205Q8—122no—
206D4—182no—
207C2×C4—122no—
208C2×C4—181no—
209C2×C4—361no—
210C2×C4—361no—
211C2×C4#379811yescharacteristic
212C2×C4#380211yescharacteristic
213C2×C4#381111yescharacteristic
214C8—41no—
215C8—92no—
216C8—122no—
217C8—181no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3#475512yes—
222C2×C3#478612yes—
223C2×C3#481612yes—
224D3—242no—
225D3—242no—
226C22—361no—
227C22—361no—
228C22—361no—
229C22#6391211yescharacteristic
230C4—21no—
231C4—242no—
232C4—361no—
233C4#6381112yes—
234C4#6392611yescharacteristic
235C4#6394111yescharacteristic
236C3—21no—
237C3#6844912yescomplemented
238C2—721no—
239C2#1059055111yescharacteristic
240C2#1059085611yescharacteristic
241C2#1059099311yescharacteristic
subgroupG/H
centerC22#63912
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊C8D32
hypocenterC32#1689
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #10590387C2semidirect
4. C3×#4482C2semidirect
5. #10589098C2semidirect
24. #4482D3semidirect
202. C32#1689semidirect
237. C3#68449semidirect

Maximal subgroups

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

subgroupindex
#105925372
#105903872
#105903132
C3×#44822
#105890982
#684493