Finite Group Databasebeta

← #11776229

Subgroups — #11776229

Order 1152.

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

normal subgroups only

This group has 647 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#10596196C211yesmaximal, characteristic
2#10590256C212yesmaximal, complemented
3C3×#4546C212yesmaximal, complemented
4#10589091C212yesmaximal, complemented
5#68482—32nomaximal
6C2×#3832C412yes—
7C32×C4⋊Q8C2211yescharacteristic
8#64156C412yes—
9#64104—22no—
10#64102—22no—
11#64083C2211yescharacteristic
12C42×C3⋊D3C2211yescharacteristic
13C3×C4⋊(C3⋊C8)C2214yes—
14#63557—22no—
15#63553—22no—
16#5725—32no—
17#5725—61no—
18#4833—34no—
19C3×#333—32no—
20#4546D312yescomplemented
21#4485—34no—
22C2×C32×Q8C2×C412yes—
23#3832—44no—
24C2×C4⋊(C3⋊D3)—22no—
25C2×C4⋊(C3⋊D3)—22no—
26C2×C4⋊(C3⋊D3)C2×C412yes—
27C2×C4×C3⋊D3—22no—
28C2×C4×C3⋊D3C2×C411yescharacteristic
29C2×C4×C3⋊D3D412yes—
30C32×C4⋊C4—22no—
31C32×C42C2311yescharacteristic
32#3752—24no—
33C4×C3⋊Dic3C2×C411yescharacteristic
34C4×C3⋊Dic3D412yes—
35C2×C3×C3⋊C8—24no—
36#1543—91noSylow
37C2×#239—34no—
38C2×#239—62no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8C2×D312yes—
41#947—34no—
42#947—62no—
43#895—64no—
44#895—64no—
45#893—64no—
46#893—64no—
47#874—32no—
48#874—61no—
49C42×D3—32no—
50C42×D3—61no—
51C3×C4⋊C8—34no—
52(C2×D3)⋊C8—64no—
53C8×Dic3—64no—
54C4⋊(C3⋊C8)C2×D314yes—
55C22×C3⋊D3—22no—
56C22×C3⋊D3C22⋊C411yescharacteristic
57C32×Q8—42no—
58C2×C32×C4—22no—
59C2×C32×C4C22×C411yescharacteristic
60C2×C32×C4C2×D412yes—
61C2×C3⋊Dic3C22⋊C412yes—
62C2×C3⋊Dic3C22⋊C411yescharacteristic
63C4⋊(C3⋊D3)—42no—
64C4⋊(C3⋊D3)—44no—
65C4⋊(C3⋊D3)—44no—
66C4×C3⋊D3—22no—
67C4×C3⋊D3—24no—
68C4×C3⋊D3—44no—
69C3×C3⋊C8—44no—
70#532—91no—
71#431—92no—
72#333—32no—
73#331—92no—
74C2×C3×Q8—22no—
75C2×C3×Q8C4×D314yes—
76#239—128no—
77#239—128no—
78C2×D12—34no—
79C2×D12—62no—
80C2×D12—64no—
81C2×D12—64no—
82C2×D12—64no—
83C2×D12—64no—
84C2×C4×D3—32no—
85C2×C4×D3—34no—
86C2×C4×D3—61no—
87C2×C4×D3—62no—
88C2×C4×D3—64no—
89C2×C4×D3—64no—
90C2×C3×C8—64no—
91C3×C4⋊C4—24no—
92C3×C4⋊C4—24no—
93C3×C42—21no—
94C3×C42C22×D312yes—
95(C2×D3)⋊C4—68no—
96(C2×D3)⋊C4—68no—
97C4×Dic3—32no—
98C4×Dic3—34no—
99C4×Dic3—62no—
100C4×Dic3—61no—
101C2×C3⋊C8—24no—
102C22×C32C2×C22⋊C411yescharacteristic
103C2×C3⋊D3—21no—
104C2×C3⋊D3—22no—
105C2×C3⋊D3—42no—
106C2×C3⋊D3—44no—
107C32×C4—21no—
108C32×C4—42no—
109C32×C4#13514yes—
110C3⋊Dic3—22no—
111C3⋊Dic3#10012yes—
112C2×#41—92no—
113C4⋊Q8D3211yescharacteristic
114#127—91no—
115#124—92no—
116#121—182no—
117C4×D4—182no—
118C2×C42—91no—
119C4⋊C8—34no—
120C22⋊C8—182no—
121C4×C8—182no—
122C22×D3—32no—
123C22×D3—61no—
124C22×D3—64no—
125C22×D3—64no—
126C3×Q8—44no—
127C3×Q8—44no—
128C2×C3×C4—24no—
129C2×C3×C4—24no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4C2×C4×D312yes—
133C2×C3×C4D3×D414yes—
134C2×Dic3—32no—
135C2×Dic3—34no—
136C2×Dic3—61no—
137C2×Dic3—62no—
138D12—124no—
139D12—128no—
140D12—124no—
141D12—128no—
142D12—128no—
143D12—124no—
144D12—124no—
145C4×D3—64no—
146C4×D3—64no—
147C4×D3—64no—
148C4×D3—64no—
149C4×D3—124no—
150C4×D3—128no—
151C4×D3—128no—
152C3×C8—124no—
153C3⋊C8—44no—
154C2×C32C2×#10011yescharacteristic
155C2×C32#41612yes—
156C3⋊D3—41no—
157C3⋊D3—82no—
158#41—364no—
159C2×Q8#62412yes—
160C2×D4—92no—
161C2×D4—182no—
162C2×D4—182no—
163C22×C4—91no—
164C22×C4—92no—
165C22×C4—182no—
166C2×C8—64no—
167C4⋊C4—22no—
168C22⋊C4—184no—
169C42—91no—
170C42—92no—
171C42C2×D3211yescharacteristic
172C22×C3—21no—
173C22×C3D3×C22⋊C412yes—
174C2×D3—62no—
175C2×D3—62no—
176C2×D3—64no—
177C2×D3—122no—
178C2×D3—124no—
179C2×D3—128no—
180C2×D3—124no—
181C2×D3—128no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—24no—
185C3×C4—44no—
186C3×C4—44no—
187C3×C4#91614yes—
188C3×C4#91914yes—
189Dic3—34no—
190Dic3—62no—
191Dic3—64no—
192Dic3—122no—
193C32#154311yescharacteristic, complemented, Sylow
194C23—91no—
195C23—182no—
196Q8—42no—
197D4—362no—
198D4—364no—
199D4—364no—
200C2×C4—22no—
201C2×C4—91no—
202C2×C4—92no—
203C2×C4—182no—
204C2×C4—184no—
205C2×C4—364no—
206C2×C4C2×#62411yescharacteristic
207C2×C4#381112yes—
208C8—124no—
209C2×C3—21no—
210C2×C3—22no—
211C2×C3D3×#10012yes—
212C2×C3#480414yes—
213D3—122no—
214D3—122no—
215D3—244no—
216D3—244no—
217C22—181no—
218C22—182no—
219C22—362no—
220C22—364no—
221C22#6397411yescharacteristic
222C4—21no—
223C4—42no—
224C4—92no—
225C4—182no—
226C4#6395014yes—
227C3—21no—
228C3#6848212yescomplemented
229C2—361no—
230C2—722no—
231C2#1059150311yescharacteristic
232C2#1059156012yes—
subgroupG/H
centerC22#63974
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32C2×C22⋊C4
prime radical#10596196C2
cosocleC32×C42C23
hypercenterC4⋊Q8D32
hypocenterC32#1543
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10590256C2semidirect
3. C3×#4546C2semidirect
4. #10589091C2semidirect
20. #4546D3semidirect
193. C32#1543semidirect
228. C3#68482semidirect

Maximal subgroups

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

subgroupindex
#105961962
#105902562
C3×#45462
#105890912
#684823