Finite Group Databasebeta

← #71876

Subgroups — #71876

Order 384.

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

normal subgroups only

This group has 306 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×#537C211yesmaximal, characteristic
2C2×#837C212yesmaximal, complemented
3#5232C212yesmaximal, complemented
4#4975C211yesmaximal, characteristic, complemented
5#4530C211yesmaximal, characteristic, complemented
6#2039—31nomaximal, Sylow
7C2×C3×#41C2212yes—
8C3×C22⋊Q8—22no—
9C3×#121—22no—
10C3×#121—22no—
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C4⋊Dic3C2211yescharacteristic
13C2×Dic3⋊C4—22no—
14C2×C4×Dic3C2211yescharacteristic
15C2×#208C2212yes—
16#837—24no—
17#835—22no—
18#832—22no—
19#537D311yescharacteristic
20C2×#104—32no—
21#417—32no—
22#384—31no—
23#343—31no—
24C3×#41—22no—
25C3×#41—22no—
26C3×#41—22no—
27C3×#41—22no—
28C2×C3×Q8D412yes—
29C2×C3×D4—22no—
30C2×C3×D4—22no—
31C2×C3×D4D412yes—
32C22×C3×C4—22no—
33C22×C3×C4D412yes—
34C22×C3×C4C2311yescharacteristic
35C22×Dic3—21no—
36C22×Dic3—21no—
37C3×C4⋊C4—22no—
38C3×C4⋊C4—22no—
39C3×C22⋊C4—24no—
40C4⋊Dic3—22no—
41Dic3⋊C4—44no—
42C4×Dic3—21no—
43C4×Dic3D412yescomplemented
44#208—22no—
45#208—22no—
46C2×C3⋊C8—22no—
47C2×#41C2×D312yes—
48C2×M16—32no—
49C22⋊Q8—22no—
50#121—22no—
51#121—22no—
52C2×C4⋊C4—31no—
53C2×C4⋊C4—62no—
54C2×C4⋊C4C2×D311yescharacteristic
55C2×C42—31no—
56#104—64no—
57#103—62no—
58#102—62no—
59C23×C3—22no—
60C23×C3C2×D411yescharacteristic
61C3×Q8—22no—
62C3×Q8—22no—
63C3×D4—22no—
64C3×D4—22no—
65C3×D4—42no—
66C3×D4—42no—
67C3×D4—82no—
68C2×C3×C4—22no—
69C2×C3×C4—22no—
70C2×C3×C4—42no—
71C2×C3×C4—42no—
72C2×C3×C4—42no—
73C2×C3×C4C2×D411yescharacteristic
74C2×C3×C4C2×D412yes—
75C2×C3×C4#4112yes—
76C2×C3×C4#4111yescharacteristic
77C2×Dic3—22no—
78C2×Dic3—22no—
79C2×Dic3—22no—
80C2×Dic3—42no—
81C2×Dic3—41no—
82C3⋊C8—42no—
83#41—22no—
84#41—22no—
85#41—22no—
86#41—22no—
87C2×Q8#6712yes—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4#6712yes—
91C22×C4—22no—
92C22×C4—61no—
93C22×C4—61no—
94C22×C4#6712yes—
95C22×C4C22×D311yescharacteristic
96M16—62no—
97M16—62no—
98C2×C8—62no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—62no—
102C4⋊C4—124no—
103C22⋊C4—24no—
104C42—32no—
105C42—61no—
106C22×C3—21no—
107C22×C3—21no—
108C22×C3—42no—
109C22×C3—42no—
110C22×C3—42no—
111C22×C3#12011yescharacteristic
112C22×C3#12111yescharacteristic
113C22×C3#12411yescharacteristic
114C3×C4—42no—
115C3×C4—42no—
116C3×C4#12112yes—
117C3×C4#12312yes—
118Dic3—24no—
119Dic3—81no—
120C23—22no—
121C23C2×#6711yescharacteristic
122Q8—22no—
123Q8—22no—
124D4—22no—
125D4—22no—
126D4—42no—
127D4—42no—
128D4—82no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—62no—
135C2×C4—62no—
136C2×C4—62no—
137C2×C4—122no—
138C2×C4—121no—
139C2×C4D3×D411yescharacteristic
140C2×C4#23711yescharacteristic
141C2×C4#23712yes—
142C2×C4C2×#6712yes—
143C8—122no—
144C2×C3—21no—
145C2×C3—21no—
146C2×C3—82no—
147C2×C3#39411yescharacteristic
148C2×C3#45311yescharacteristic
149C2×C3#45611yescharacteristic
150C22—21no—
151C22—21no—
152C22—42no—
153C22—42no—
154C22—42no—
155C22#93611yescharacteristic
156C22#93811yescharacteristic
157C22#95311yescharacteristic
158C4—42no—
159C4—42no—
160C4—64no—
161C4—241no—
162C4#93512yes—
163C4#93912yes—
164C3#203911yescharacteristic, complemented, Sylow
165C2—21no—
166C2—21no—
167C2—82no—
168C2#522211yescharacteristic
169C2#524111yescharacteristic
170C2#524611yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#537C2
socleC22×C3#120
hypercenter#537D3
hypocenterC3#2039
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
2. C2×#837C2semidirect
3. #5232C2semidirect
4. #4975C2semidirect
5. #4530C2semidirect
43. C4×Dic3D4semidirect
164. C3#2039semidirect

Maximal subgroups

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

subgroupindex
C3×#5372
C2×#8372
#52322
#49752
#45302
#20393