Finite Group Databasebeta

← #69716

Subgroups — #69716

Order 384.

236 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 53 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#5483C211yesmaximal, characteristic
2C3×#399C211yesmaximal, characteristic
3#4970C212yesmaximal, complemented
4#4525C211yesmaximal, characteristic
5#4452C212yesmaximal, complemented
6#1614—31nomaximal, Sylow
7C2×C3×C4⋊C4—21no—
8C2×C3×C4⋊C4C2211yescharacteristic
9C2×C3×C22⋊C4—21no—
10C2×C3×C22⋊C4C2211yescharacteristic
11C2×#217—21no—
12#924—21no—
13#924C2212yes—
14#878—22no—
15#878—22no—
16C3×#95C2211yescharacteristic
17#830C2212yes—
18#810—22no—
19#807—22no—
20#543—31no—
21#484—32no—
22#399D311yescharacteristic
23#328—32no—
24#323—31no—
25C24×C3D411yescharacteristic
26C22×C3×C4—21no—
27C22×C3×C4—21no—
28C22×C3×C4D411yescharacteristic
29C22×C3×C4C2311yescharacteristic
30C22×Dic3—21no—
31C22×Dic3—21no—
32C2×Dic6—21no—
33C2×Dic6D412yes—
34C3×C4⋊C4—22no—
35C3×C4⋊C4—42no—
36C3×C22⋊C4—22no—
37C3×C22⋊C4—42no—
38#217—22no—
39#217—22no—
40C4⋊Dic3—21no—
41C4⋊Dic3D412yes—
42Dic3⋊C4—22no—
43Dic3⋊C4—22no—
44C2×C3⋊C8—22no—
45C22⋊Q8—32no—
46C22⋊Q8—62no—
47C22⋊Q8—62no—
48C22⋊Q8—61no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4C2×D311yescharacteristic
51C2×C22⋊C4—21no—
52C2×C22⋊C4—61no—
53C2×C22⋊C4C2×D311yescharacteristic
54#107—62no—
55#103—62no—
56C22⋊C8—32no—
57#95C2×D311yescharacteristic
58C23×C3—21no—
59C23×C3—21no—
60C23×C3—41no—
61C23×C3—41no—
62C23×C3Q1612yes—
63C23×C3C2×D411yescharacteristic
64C2×C3×C4—22no—
65C2×C3×C4—21no—
66C2×C3×C4—41no—
67C2×C3×C4—41no—
68C2×C3×C4—41no—
69C2×C3×C4—41no—
70C2×C3×C4—41no—
71C2×C3×C4—41no—
72C2×C3×C4—41no—
73C2×C3×C4C2×D412yes—
74C2×Dic3—22no—
75C2×Dic3—22no—
76C2×Dic3—81no—
77C2×Dic3—81no—
78Dic6—42no—
79Dic6—81no—
80C3⋊C8—42no—
81C24#6711yescharacteristic
82C2×Q8—32no—
83C2×Q8—61no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—61no—
87C22×C4—61no—
88C22×C4#6711yescharacteristic
89C22×C4C22×D311yescharacteristic
90C2×C8—62no—
91C4⋊C4—22no—
92C4⋊C4—32no—
93C4⋊C4—42no—
94C4⋊C4—62no—
95C4⋊C4—61no—
96C4⋊C4—62no—
97C22⋊C4—22no—
98C22⋊C4—42no—
99C22⋊C4—62no—
100C22⋊C4—62no—
101C22×C3—21no—
102C22×C3—21no—
103C22×C3—22no—
104C22×C3—42no—
105C22×C3—41no—
106C22×C3—41no—
107C22×C3—81no—
108C22×C3#12011yescharacteristic
109C22×C3C2×Q1611yescharacteristic
110C22×C3#13711yescharacteristic
111C3×C4—22no—
112C3×C4—42no—
113C3×C4—41no—
114C3×C4—81no—
115Dic3—42no—
116Dic3—42no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23#21612yes—
122C23C2×#6711yescharacteristic
123Q8—122no—
124Q8—241no—
125C2×C4—22no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—62no—
135C2×C4—62no—
136C2×C4—241no—
137C2×C4—241no—
138C2×C4D3×D412yes—
139C8—122no—
140C2×C3—21no—
141C2×C3—21no—
142C2×C3—42no—
143C2×C3#45111yescharacteristic
144C2×C3#45311yescharacteristic
145C2×C3#45811yescharacteristic
146C22—21no—
147C22—21no—
148C22—22no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—81no—
153C22#93711yescharacteristic
154C22C2×#21611yescharacteristic
155C22#95111yescharacteristic
156C4—22no—
157C4—42no—
158C4—41no—
159C4—81no—
160C4—122no—
161C4—122no—
162C3#161411yescharacteristic, complemented, Sylow
163C2—21no—
164C2—21no—
165C2—42no—
166C2#505011yescharacteristic
167C2#505311yescharacteristic
168C2#507711yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#399C2
socleC22×C3#120
prime radicalC24×C3D4
hypercenter#399D3
hypocenterC3#1614
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
3. #4970C2semidirect
5. #4452C2semidirect
162. C3#1614semidirect

Maximal subgroups

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

subgroupindex
#54832
C3×#3992
#49702
#45252
#44522
#16143