Finite Group Databasebeta

← #72118

Subgroups — #72118

Order 384.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C3×#575C211yesmaximal, characteristic, complemented
2#5828C212yesmaximal, complemented
3#5236C212yesmaximal, complemented
4#5126C211yesmaximal, characteristic, complemented
5#4557C211yesmaximal, characteristic, complemented
6#2010—31nomaximal, Sylow
7C2×C3×#41C2212yes—
8C22×C4×D3C2211yescharacteristic
9C3×#137—22no—
10C3×AGL(1,ℤ/8)—22no—
11C3×#135—22no—
12C2×C3×M16C2211yescharacteristic
13#946—22no—
14#938—22no—
15#937—22no—
16#935—22no—
17C4×#67—24no—
18#926C2212yes—
19C2×#208C2211yescharacteristic
20#837—24no—
21(C2×D3)⋊C8—22no—
22#575D311yescharacteristic, complemented
23#525—32no—
24#421—32no—
25#407—31no—
26H(3,ℤ/4)—31no—
27C23×D3D411yescharacteristic
28C3×#41—22no—
29C3×#41—22no—
30C3×#41—22no—
31C3×#41—22no—
32C2×C3×Q8D412yescomplemented
33C2×C3×D4—22no—
34C2×C3×D4D412yescomplemented
35C22×C3×C4—22no—
36C22×C3×C4C2311yescharacteristic
37C2×#67—24no—
38C22×Dic3D411yescharacteristic
39C2×C4×D3—21no—
40C2×C4×D3—22no—
41C2×C4×D3—22no—
42C2×C4×D3—21no—
43C3×Q16—22no—
44C3×SD16—24no—
45C3×D8—22no—
46C3×M16—22no—
47C2×C3×C8D412yescomplemented
48#217—24no—
49#217—22no—
50(C2×D3)⋊C4—24no—
51C4⋊Dic3—22no—
52Dic3⋊C4—24no—
53C4×Dic3D414yescomplemented
54#208—22no—
55C2×C3⋊C8—21no—
56C2×#41C2×D312yes—
57C23×C4—31no—
58#137—22no—
59AGL(1,ℤ/8)—22no—
60#135—22no—
61C2×M16—31no—
62C2×M16C2×D311yescharacteristic
63#123—62no—
64C22⋊Q8—62no—
65#121—62no—
66#120—62no—
67C4×D4—64no—
68#117—32no—
69#104—64no—
70C22⋊C8—62no—
71C23×C3—22no—
72C23×C3C2×D411yescharacteristic
73C22×D3—22no—
74C22×D3—21no—
75C22×D3—42no—
76C3×Q8—22no—
77C3×Q8—22no—
78C3×D4—22no—
79C3×D4—22no—
80C3×D4—42no—
81C3×D4—42no—
82C2×C3×C4—22no—
83C2×C3×C4—42no—
84C2×C3×C4—42no—
85C2×C3×C4C2×D412yes—
86C2×C3×C4C2×D412yes—
87C2×C3×C4C2×D411yescharacteristic
88C2×C3×C4#4111yescharacteristic
89#67—48no—
90C2×Dic3—24no—
91C2×Dic3—22no—
92C2×Dic3—21no—
93C4×D3—41no—
94C4×D3—42no—
95C4×D3—41no—
96C3×C8—22no—
97C3⋊C8—41no—
98C24—31no—
99#41—22no—
100#41—22no—
101#41—22no—
102#41—22no—
103C2×Q8#6712yescomplemented
104C2×D4—22no—
105C2×D4—64no—
106C2×D4#6712yescomplemented
107C22×C4—22no—
108C22×C4—31no—
109C22×C4—62no—
110C22×C4—61no—
111C22×C4—61no—
112C22×C4—62no—
113C22×C4C22×D311yescharacteristic
114Q16—22no—
115SD16—24no—
116D8—22no—
117M16—22no—
118M16—62no—
119C2×C8—61no—
120C2×C8#6712yescomplemented
121C4⋊C4—64no—
122C4⋊C4—62no—
123C22⋊C4—64no—
124C22⋊C4—64no—
125C22⋊C4—62no—
126C42—34no—
127C22×C3—41no—
128C22×C3—44no—
129C22×C3—42no—
130C22×C3#12012yes—
131C22×C3#12111yescharacteristic
132C2×D3—24no—
133C2×D3—22no—
134C2×D3—42no—
135C2×D3—41no—
136C2×D3—41no—
137C3×C4—42no—
138C3×C4#12011yescharacteristic
139C3×C4#12112yes—
140C3×C4#12711yescharacteristic
141Dic3—44no—
142Dic3—41no—
143C23—22no—
144C23—62no—
145C23—61no—
146C23—122no—
147C23D3×D411yescharacteristic
148Q8—22no—
149Q8—22no—
150D4—22no—
151D4—22no—
152D4—42no—
153D4—42no—
154D4—128no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—64no—
159C2×C4—62no—
160C2×C4—61no—
161C2×C4—122no—
162C2×C4—121no—
163C2×C4—121no—
164C2×C4D3×D412yes—
165C2×C4#23711yescharacteristic
166C2×C4C2×#6712yes—
167C2×C4C2×#6711yescharacteristic
168C8—22no—
169C8—121no—
170C2×C3—22no—
171C2×C3—21no—
172C2×C3—82no—
173C2×C3#39211yescharacteristic
174D3—42no—
175C22—41no—
176C22—44no—
177C22—42no—
178C22—64no—
179C22—62no—
180C22—122no—
181C22—121no—
182C22—121no—
183C22#93712yes—
184C22#93811yescharacteristic
185C4—42no—
186C4—124no—
187C4—121no—
188C4#93912yes—
189C4#94011yescharacteristic
190C4#95311yescharacteristic
191C3#201011yescharacteristic, complemented, Sylow
192C2—22no—
193C2—21no—
194C2—82no—
195C2—122no—
196C2#522311yescharacteristic
subgroupG/H
centerC4#940
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#575C2
socleC2×C3#392
hypercenter#575D3
hypocenterC3#2010
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C3×#575C2semidirect
2. #5828C2semidirect
3. #5236C2semidirect
4. #5126C2semidirect
5. #4557C2semidirect
22. #575D3semidirect
32. C2×C3×Q8D4semidirect
34. C2×C3×D4D4semidirect
47. C2×C3×C8D4semidirect
53. C4×Dic3D4semidirect
103. C2×Q8#67semidirect
106. C2×D4#67semidirect
120. C2×C8#67semidirect
191. C3#2010semidirect

Maximal subgroups

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

subgroupindex
C3×#5752
#58282
#52362
#51262
#45572
#20103