Finite Group Databasebeta

← #69696

Subgroups — #69696

Order 384.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#5493C211yesmaximal, characteristic, complemented
2C3×#392C211yesmaximal, characteristic, complemented
3#5217C212yesmaximal, complemented
4#4525C211yesmaximal, characteristic, complemented
5#4451C212yesmaximal, complemented
6#1595—31nomaximal, Sylow
7C22×C3×D4—21no—
8C22×C3×D4C2211yescharacteristic, complemented
9C22×#67—21no—
10C2×C3×C22⋊C4—21no—
11C2×C3×C22⋊C4C2211yescharacteristic
12C2×#213—22no—
13#930—21no—
14#930C2212yes—
15#886—22no—
16#882—22no—
17C3×#95C2211yescharacteristic, complemented
18#832—22no—
19#830C2212yescomplemented
20#534—31no—
21#447—32no—
22#392D311yescharacteristic, complemented
23#327—32no—
24#323—31no—
25C24×C3—21no—
26C24×C3D411yescharacteristic
27C23×D3—21no—
28C2×C3×D4—22no—
29C2×C3×D4—42no—
30C2×C3×D4—42no—
31C2×C3×D4—42no—
32C2×C3×D4—42no—
33C22×C3×C4—21no—
34C22×C3×C4D411yescharacteristic
35C22×C3×C4C2311yescharacteristic
36C2×#67—22no—
37C2×#67—22no—
38C2×#67—42no—
39C2×#67—42no—
40C22×Dic3—21no—
41C2×D12—21no—
42C2×D12D412yescomplemented
43C3×C22⋊C4—22no—
44C3×C22⋊C4—42no—
45#213—44no—
46(C2×D3)⋊C4—22no—
47(C2×D3)⋊C4—22no—
48C4⋊Dic3—21no—
49C4⋊Dic3D412yescomplemented
50C2×C3⋊C8—22no—
51C22×D4—21no—
52C22×D4—61no—
53C22×D4C2×D311yescharacteristic, complemented
54C2×D8—62no—
55#123—62no—
56#121—32no—
57#121—61no—
58#120—62no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4C2×D311yescharacteristic
61#102—62no—
62C22⋊C8—32no—
63#95C2×D311yescharacteristic, complemented
64C23×C3—21no—
65C23×C3—21no—
66C23×C3—22no—
67C23×C3—41no—
68C23×C3—41no—
69C23×C3—41no—
70C23×C3—41no—
71C23×C3—41no—
72C23×C3—41no—
73C23×C3—41no—
74C23×C3—41no—
75C23×C3D812yescomplemented
76C23×C3C2×D411yescharacteristic
77C22×D3—22no—
78C22×D3—42no—
79C22×D3—81no—
80C22×D3—81no—
81C3×D4—24no—
82C3×D4—42no—
83C3×D4—48no—
84C2×C3×C4—22no—
85C2×C3×C4—21no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4C2×D412yes—
91#67—44no—
92#67—42no—
93#67—42no—
94C2×Dic3—22no—
95C2×Dic3—42no—
96D12—42no—
97D12—81no—
98C3⋊C8—42no—
99C24—21no—
100C24—61no—
101C24#6711yescharacteristic
102C2×D4—22no—
103C2×D4—32no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—61no—
109C2×D4—62no—
110C2×D4—62no—
111C2×D4—122no—
112C2×D4—122no—
113C22×C4—21no—
114C22×C4—61no—
115C22×C4#6711yescharacteristic
116C22×C4C22×D311yescharacteristic
117D8—124no—
118C2×C8—62no—
119C4⋊C4—32no—
120C4⋊C4—61no—
121C22⋊C4—22no—
122C22⋊C4—42no—
123C22⋊C4—62no—
124C22⋊C4—62no—
125C22×C3—21no—
126C22×C3—21no—
127C22×C3—22no—
128C22×C3—42no—
129C22×C3—42no—
130C22×C3—42no—
131C22×C3—41no—
132C22×C3—41no—
133C22×C3—42no—
134C22×C3—81no—
135C22×C3—81no—
136C22×C3—81no—
137C22×C3—81no—
138C22×C3—81no—
139C22×C3#12011yescharacteristic
140C22×C3C2×D811yescharacteristic
141C22×C3AGL(1,ℤ/8)11yescharacteristic
142C2×D3—42no—
143C2×D3—42no—
144C2×D3—42no—
145C2×D3—84no—
146C3×C4—22no—
147C3×C4—42no—
148C3×C4—41no—
149Dic3—42no—
150C23—21no—
151C23—21no—
152C23—22no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—62no—
162C23—122no—
163C23—241no—
164C23—241no—
165C23#21312yescomplemented
166C23C2×#6711yescharacteristic
167D4—24no—
168D4—42no—
169D4—48no—
170D4—122no—
171D4—124no—
172D4—122no—
173D4—122no—
174D4—241no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—62no—
182C2×C4—122no—
183C2×C4D3×D412yes—
184C8—122no—
185C2×C3—21no—
186C2×C3—21no—
187C2×C3—42no—
188C2×C3—82no—
189C2×C3#44711yescharacteristic
190C2×C3#45311yescharacteristic
191C2×C3B(4,𝔽2)11yescharacteristic
192D3—82no—
193C22—21no—
194C22—21no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—122no—
208C22—122no—
209C22—122no—
210C22—244no—
211C22C2×#21311yescharacteristic
212C22#93711yescharacteristic
213C22#94911yescharacteristic
214C4—22no—
215C4—42no—
216C4—41no—
217C4—122no—
218C3#159511yescharacteristic, complemented, Sylow
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—242no—
224C2#503211yescharacteristic
225C2#503611yescharacteristic
226C2#507711yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#392C2
socleC22×C3#120
hypercenter#392D3
hypocenterC3#1595
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #5493C2semidirect
2. C3×#392C2semidirect
3. #5217C2semidirect
4. #4525C2semidirect
5. #4451C2semidirect
8. C22×C3×D4C22semidirect
17. C3×#95C22semidirect
19. #830C22semidirect
22. #392D3semidirect
42. C2×D12D4semidirect
49. C4⋊Dic3D4semidirect
53. C22×D4C2×D3semidirect
63. #95C2×D3semidirect
75. C23×C3D8semidirect
165. C23#213semidirect
218. C3#1595semidirect

Maximal subgroups

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

subgroupindex
#54932
C3×#3922
#52172
#45252
#44512
#15953