Finite Group Databasebeta

← #68359

Subgroups — #68359

Order 384.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#5672C211yesmaximal, characteristic
2#5055C211yesmaximal, characteristic
3#4703C211yesmaximal, characteristic, complemented
4#4702C211yesmaximal, characteristic, complemented
5C3×#329C211yesmaximal, characteristic
6#4465C211yesmaximal, characteristic, complemented
7#4465C211yesmaximal, characteristic, complemented
8#1664—31nomaximal, Sylow
9C3×#124C2211yescharacteristic
10#944—21no—
11#935—21no—
12#879—21no—
13#879—21no—
14#878—21no—
15#878—21no—
16#870C2211yescharacteristic
17#870C2211yescharacteristic
18#869C2211yescharacteristic
19C3×#97C2211yescharacteristic
20C3×#97C2211yescharacteristic
21#835—21no—
22#832—21no—
23#818—21no—
24#817—21no—
25#816—21no—
26#816—21no—
27#803C2211yescharacteristic
28#563—31no—
29#501—31no—
30#491—31no—
31#489—31no—
32#330—31no—
33#330—31no—
34#329D311yescharacteristic
35C2×C3×Q8D411yescharacteristic
36C2×C3×D4D411yescharacteristic
37C22×Dic3—21no—
38C2×Dic6—21no—
39C2×Dic6—21no—
40C2×C3×C8—21no—
41C2×C3×C8—21no—
42C3×C22⋊C4—21no—
43C3×C22⋊C4—21no—
44C3×C42C2311yescharacteristic
45#217—21no—
46#217—21no—
47#217—21no—
48C4⋊Dic3—21no—
49C4⋊Dic3—21no—
50C4⋊Dic3D411yescharacteristic
51C4⋊Dic3D411yescharacteristic
52C4⋊Dic3D411yescharacteristic
53C4⋊Dic3D411yescharacteristic
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56Dic3⋊C4—21no—
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60C4×Dic3—21no—
61C2×C3⋊C8—21no—
62C4⋊Q8—31no—
63C4⋊Q8—61no—
64#126—61no—
65#126—61no—
66#125—31no—
67#125—31no—
68#124C2×D311yescharacteristic
69#123—61no—
70C22⋊Q8—61no—
71C22⋊Q8—61no—
72#107—61no—
73#106—61no—
74#103—61no—
75#103—61no—
76#103—61no—
77#102—61no—
78#97—31no—
79#97C2×D311yescharacteristic
80#97C2×D311yescharacteristic
81C23×C3—21no—
82C3×Q8—41no—
83C3×D4—41no—
84C2×C3×C4—21no—
85C2×C3×C4C2×D411yescharacteristic
86C2×C3×C4C2×D411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88C2×Dic3—21no—
89C2×Dic3—21no—
90C2×Dic3—21no—
91C2×Dic3—21no—
92C2×Dic3—41no—
93C2×Dic3—81no—
94Dic6—42no—
95Dic6—81no—
96C3×C8—22no—
97C3×C8—41no—
98C3⋊C8—41no—
99C2×Q8—61no—
100C2×Q8—61no—
101C2×Q8D1211yescharacteristic
102C2×D4D1211yescharacteristic
103C22×C4—61no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—61no—
107C4⋊C4—31no—
108C4⋊C4—31no—
109C4⋊C4—31no—
110C4⋊C4—31no—
111C4⋊C4—61no—
112C4⋊C4—61no—
113C4⋊C4—61no—
114C4⋊C4—61no—
115C4⋊C4—61no—
116C4⋊C4—61no—
117C4⋊C4—61no—
118C4⋊C4—61no—
119C22⋊C4—21no—
120C22⋊C4—21no—
121C22⋊C4—61no—
122C22⋊C4—61no—
123C22⋊C4—61no—
124C42—61no—
125C42C22×D311yescharacteristic
126C22×C3—41no—
127C22×C3—41no—
128C22×C3—41no—
129C22×C3#12011yescharacteristic
130C3×C4—21no—
131C3×C4—21no—
132C3×C4—21no—
133C3×C4—41no—
134Dic3—41no—
135Dic3—41no—
136Dic3—41no—
137Dic3—41no—
138Dic3—81no—
139C23—21no—
140Q8—41no—
141Q8—122no—
142Q8—241no—
143D4—41no—
144C2×C4—21no—
145C2×C4—61no—
146C2×C4—61no—
147C2×C4—61no—
148C2×C4—61no—
149C2×C4—121no—
150C2×C4—241no—
151C2×C4C2×D1211yescharacteristic
152C2×C4D3×D411yescharacteristic
153C2×C4D3×D411yescharacteristic
154C8—22no—
155C8—41no—
156C8—121no—
157C2×C3—81no—
158C2×C3#45411yescharacteristic
159C2×C3#45511yescharacteristic
160C2×C3#45611yescharacteristic
161C22—41no—
162C22—41no—
163C22—41no—
164C22#88211yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—41no—
169C4—121no—
170C4—121no—
171C4—121no—
172C4—121no—
173C4—241no—
174C3#166411yescharacteristic, complemented, Sylow
175C2—81no—
176C2#474611yescharacteristic
177C2#482611yescharacteristic
178C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3#120
prime radicalC2×C3×D4D4
hypercenter#329D3
hypocenterC3#1664
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
3. #4703C2semidirect
4. #4702C2semidirect
6. #4465C2semidirect
7. #4465C2semidirect
174. C3#1664semidirect

Maximal subgroups

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

subgroupindex
#56722
#50552
#47032
#47022
C3×#3292
#44652
#16643