Finite Group Databasebeta

← #69694

Subgroups — #69694

Order 384.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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×#942C212yesmaximal, complemented
2#5510C211yesmaximal, characteristic
3C3×#391C211yesmaximal, characteristic, complemented
4C2×#805C212yesmaximal, complemented
5#4530C211yesmaximal, characteristic, complemented
6#2007—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8C2×#235C2212yes—
9C2×C3×C4⋊C4C2211yescharacteristic
10C2×C3×C42C2211yescharacteristic
11C2×#216—22no—
12#942—24no—
13#942—24no—
14C2×#215—22no—
15#930—22no—
16C2×#208C2212yes—
17#897—22no—
18#895—22no—
19C3×#95—22no—
20#805—24no—
21C2×#137—32no—
22#537—31no—
23C2×#104—32no—
24#391D311yescharacteristic, complemented
25#343—31no—
26C2×C3×Q8—21no—
27C2×C3×Q8—22no—
28C2×C3×Q8—22no—
29C2×C3×Q8D412yescomplemented
30C22×C3×C4—21no—
31C22×C3×C4—21no—
32C22×C3×C4D412yes—
33C22×C3×C4C2311yescharacteristic
34C2×#67—22no—
35#235—22no—
36#235—22no—
37#235—22no—
38#235—22no—
39C2×D12—22no—
40C2×D12D412yes—
41C2×C4×D3—22no—
42C2×Dic6D412yes—
43C3×C4⋊C4—22no—
44C3×C42—22no—
45#216—28no—
46#215—28no—
47(C2×D3)⋊C4—24no—
48C4⋊Dic3—22no—
49#208—22no—
50#208D414yescomplemented
51C2×C3⋊C8—22no—
52C2×#41—32no—
53C22×Q8C2×D311yescharacteristic
54#137—64no—
55#137—64no—
56C2×Q16—62no—
57C2×SD16—62no—
58C2×M16—32no—
59C22⋊Q8—62no—
60#121—62no—
61#121—62no—
62C2×C4⋊C4C2×D311yescharacteristic
63C2×C42C2×D311yescharacteristic
64#104—64no—
65#95—22no—
66C23×C3C2×D411yescharacteristic
67C22×D3—22no—
68C3×Q8—24no—
69C3×Q8—42no—
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—41no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4—41no—
77C2×C3×C4—42no—
78C2×C3×C4C2×D411yescharacteristic
79C2×C3×C4C2×D412yes—
80C2×C3×C4C2×D412yes—
81C2×C3×C4#4111yescharacteristic
82#67—42no—
83#67—42no—
84C2×Dic3—22no—
85D12—22no—
86D12—22no—
87D12—82no—
88C4×D3—42no—
89C4×D3—42no—
90Dic6—22no—
91Dic6—22no—
92C3⋊C8—24no—
93#41—62no—
94#41—62no—
95#41—62no—
96#41—62no—
97C2×Q8—21no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—32no—
101C2×Q8D1212yescomplemented
102C2×D4—32no—
103C2×D4—62no—
104C2×D4—62no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—62no—
108C22×C4#6712yes—
109C22×C4C22×D311yescharacteristic
110Q16—68no—
111SD16—68no—
112M16—34no—
113M16—62no—
114C2×C8—62no—
115C4⋊C4—22no—
116C4⋊C4—62no—
117C22⋊C4—64no—
118C42—22no—
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—42no—
129C3×C4—42no—
130C3×C4#12012yes—
131C3×C4#12112yes—
132Dic3—42no—
133C23—62no—
134C23C2×#6711yescharacteristic
135Q8—24no—
136Q8—42no—
137Q8—62no—
138Q8—62no—
139D4—62no—
140D4—62no—
141D4—122no—
142D4—122no—
143D4—242no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—62no—
153C2×C4—122no—
154C2×C4—122no—
155C2×C4C2×D1211yescharacteristic
156C2×C4#23511yescharacteristic
157C2×C4D3×D412yes—
158C2×C4D3×D412yes—
159C8—64no—
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#93011yescharacteristic
172C22#93711yescharacteristic
173C22#94011yescharacteristic
174C4—42no—
175C4—42no—
176C4—42no—
177C4—122no—
178C4#88212yes—
179C4#88412yes—
180C3#200711yescharacteristic, complemented, Sylow
181C2—21no—
182C2—21no—
183C2—242no—
184C2#496011yescharacteristic
185C2#506111yescharacteristic
186C2#509511yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#391C2
socleC22×C3#120
prime radical#5510C2
hypercenter#391D3
hypocenterC3#2007
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#942C2semidirect
3. C3×#391C2semidirect
4. C2×#805C2semidirect
5. #4530C2semidirect
24. #391D3semidirect
29. C2×C3×Q8D4semidirect
50. #208D4semidirect
101. C2×Q8D12semidirect
180. C3#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#9422
#55102
C3×#3912
C2×#8052
#45302
#20073