Finite Group Databasebeta

← #69009

Subgroups — #69009

Order 384.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#909C212yesmaximal, complemented
2#5510C211yesmaximal, characteristic
3C2×#825C212yesmaximal, complemented
4#4973C211yesmaximal, characteristic, complemented
5C3×#343C211yesmaximal, characteristic, complemented
6#2007—31nomaximal, Sylow
7C2×#235C2212yes—
8C22×Dic6C2211yescharacteristic
9C2×C3×M16C2212yes—
10C2×C3×C4⋊C4C2211yescharacteristic
11#930—22no—
12C2×C4×Dic3C2211yescharacteristic
13#909—24no—
14#909—24no—
15C2×Dic12—22no—
16C2×#204—22no—
17#897—22no—
18#895—22no—
19C3⋊#95—22no—
20#825—24no—
21C2×#137—32no—
22#537—31no—
23C2×#104—32no—
24#391—31no—
25#343D311yescharacteristic, complemented
26C22×C3×C4D412yes—
27C22×C3×C4C2311yescharacteristic
28C2×#67—22no—
29C22×Dic3—21no—
30C22×Dic3—21no—
31#235—22no—
32#235—22no—
33#235—22no—
34#235—22no—
35C2×D12—22no—
36C2×D12D412yes—
37C2×C4×D3—22no—
38C2×Dic6—22no—
39C2×Dic6—22no—
40C2×Dic6—21no—
41C2×Dic6D412yes—
42C2×Dic6D412yescomplemented
43C3×M16—22no—
44C3×M16D414yescomplemented
45C2×C3×C8—22no—
46C3×C4⋊C4—22no—
47(C2×D3)⋊C4—24no—
48C4⋊Dic3—22no—
49C4×Dic3—22no—
50Dic12—28no—
51#204—28no—
52C2×#41—32no—
53C22×Q8—31no—
54#137—64no—
55#137—64no—
56C2×Q16—62no—
57C2×SD16—62no—
58C2×M16C2×D312yes—
59C22⋊Q8—62no—
60#121—62no—
61#121—62no—
62C2×C4⋊C4C2×D311yescharacteristic
63C2×C42—31no—
64#104—64no—
65#95—62no—
66C23×C3C2×D411yescharacteristic
67C22×D3—22no—
68C2×C3×C4—22no—
69C2×C3×C4—42no—
70C2×C3×C4C2×D412yes—
71C2×C3×C4C2×D411yescharacteristic
72C2×C3×C4C2×D412yes—
73C2×C3×C4#4111yescharacteristic
74#67—42no—
75#67—42no—
76C2×Dic3—22no—
77C2×Dic3—22no—
78C2×Dic3—22no—
79C2×Dic3—41no—
80C2×Dic3—41no—
81C2×Dic3—41no—
82C2×Dic3—41no—
83D12—22no—
84D12—22no—
85D12—82no—
86C4×D3—42no—
87C4×D3—42no—
88Dic6—24no—
89Dic6—22no—
90Dic6—22no—
91Dic6—42no—
92C3×C8—24no—
93#41—62no—
94#41—62no—
95#41—62no—
96#41—62no—
97C2×Q8—32no—
98C2×Q8—32no—
99C2×Q8—61no—
100C2×Q8—62no—
101C2×Q8—62no—
102C2×D4—32no—
103C2×D4—62no—
104C2×D4—62no—
105C22×C4—61no—
106C22×C4—61no—
107C22×C4—62no—
108C22×C4D1212yes—
109C22×C4C22×D311yescharacteristic
110Q16—68no—
111SD16—68no—
112M16—22no—
113M16D1214yescomplemented
114C2×C8—22no—
115C4⋊C4—22no—
116C4⋊C4—62no—
117C22⋊C4—64no—
118C42—62no—
119C22×C3—21no—
120C22×C3—21no—
121C22×C3#12011yescharacteristic
122C22×C3#12111yescharacteristic
123C22×C3#12711yescharacteristic
124C2×D3—42no—
125C2×D3—42no—
126C2×D3—42no—
127C3×C4—42no—
128C3×C4#12012yes—
129C3×C4#12112yes—
130Dic3—42no—
131Dic3—42no—
132Dic3—42no—
133C23—62no—
134C23C2×D1211yescharacteristic
135Q8—64no—
136Q8—62no—
137Q8—62no—
138Q8—122no—
139D4—62no—
140D4—62no—
141D4—122no—
142D4—122no—
143D4—242no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—62no—
147C2×C4—62no—
148C2×C4—62no—
149C2×C4—121no—
150C2×C4—121no—
151C2×C4—121no—
152C2×C4—121no—
153C2×C4—122no—
154C2×C4—122no—
155C2×C4C2×D1212yes—
156C2×C4D3×D411yescharacteristic
157C2×C4D3×D412yes—
158C2×C4#23911yescharacteristic
159C8—24no—
160C2×C3—21no—
161C2×C3—21no—
162C2×C3#39211yescharacteristic
163C2×C3#45511yescharacteristic
164C2×C3#45611yescharacteristic
165D3—82no—
166C22—21no—
167C22—21no—
168C22—122no—
169C22—122no—
170C22—122no—
171C22#87411yescharacteristic
172C22#88211yescharacteristic
173C22#89511yescharacteristic
174C4—42no—
175C4—122no—
176C4—122no—
177C4—122no—
178C4#88212yes—
179C4#89512yes—
180C3#200711yescharacteristic, complemented, Sylow
181C2—21no—
182C2—21no—
183C2—242no—
184C2#467111yescharacteristic
185C2#474611yescharacteristic
186C2#475211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#343C2
socleC22×C3#120
prime radical#5510C2
hypercenter#343D3
hypocenterC3#2007
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#909C2semidirect
3. C2×#825C2semidirect
4. #4973C2semidirect
5. C3×#343C2semidirect
25. #343D3semidirect
42. C2×Dic6D4semidirect
44. C3×M16D4semidirect
113. M16D12semidirect
180. C3#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
#55102
C2×#8252
#49732
C3×#3432
#20073