Finite Group Databasebeta

← #69165

Subgroups — #69165

Order 384.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#5668C211yesmaximal, characteristic
2#4751C212yesmaximal, complemented
3C3×#355C211yesmaximal, characteristic, complemented
4#4542C211yesmaximal, characteristic, complemented
5#4478C212yesmaximal, complemented
6#2132—31nomaximal, Sylow
7C2×#239C2211yescharacteristic
8C2×#237C411yescharacteristic
9C3×#124C2211yescharacteristic
10#944C2211yescharacteristic
11#940C411yescharacteristic
12D3×M16—22no—
13#884—22no—
14#67⋊C4—22no—
15#875C411yescharacteristic
16C42×D3C411yescharacteristic
17C3×#101C2212yes—
18#836C2212yes—
19#823—22no—
20#532—31no—
21#413—32no—
22#356—32no—
23#355—31no—
24#355D311yescharacteristic, complemented
25C2×C3×Q8C2311yescharacteristic
26C2×C3×D4C2×C411yescharacteristic
27C2×#67—22no—
28C2×#67—21no—
29C22×Dic3—21no—
30#239—41no—
31#239—41no—
32#237—41no—
33#237—41no—
34C2×D12C2×C411yescharacteristic
35C2×D12D412yes—
36C2×C4×D3—21no—
37C2×C4×D3C2×C411yescharacteristic
38C2×C4×D3D412yes—
39C2×Dic6C2×C411yescharacteristic
40C3×M16—22no—
41C3×C22⋊C4—22no—
42C3×C42C2×C411yescharacteristic
43(C2×D3)⋊C4—22no—
44Dic3⋊C4—22no—
45C4×Dic3—21no—
46C4×Dic3C2×C411yescharacteristic
47#208—22no—
48ΓL(1,𝔽25)—42no—
49D3×C8—42no—
50C2×#41—31no—
51C2×#41—31no—
52C2×M16—62no—
53C4⋊Q8—31no—
54#127—31no—
55#124—31no—
56#124C2×D311yescharacteristic
57#121—62no—
58C4×D4—62no—
59C2×C42—31no—
60#101—32no—
61#101C2×D312yes—
62#100—62no—
63C23×C3—21no—
64C22×D3C22⋊C412yes—
65C22×D3C22⋊C411yescharacteristic
66C3×Q8—41no—
67C3×D4—41no—
68C2×C3×C4—21no—
69C2×C3×C4C22×C411yescharacteristic
70C2×C3×C4C2×D412yes—
71#67—44no—
72#67—42no—
73C2×Dic3—21no—
74C2×Dic3—21no—
75C2×Dic3—41no—
76C2×Dic3—41no—
77C2×Dic3C22⋊C411yescharacteristic
78D12—41no—
79D12—42no—
80C4×D3—22no—
81C4×D3—22no—
82C4×D3—21no—
83C4×D3—21no—
84C4×D3—42no—
85Dic6—41no—
86C3×C8—42no—
87C3⋊C8—42no—
88#41—121no—
89#41—121no—
90#41—121no—
91#41—121no—
92C2×Q8—31no—
93C2×Q8C22×D311yescharacteristic
94C2×D4—31no—
95C2×D4—32no—
96C2×D4—62no—
97C2×D4—61no—
98C2×D4C4×D311yescharacteristic
99C22×C4—31no—
100C22×C4—32no—
101C22×C4—61no—
102C22×C4—61no—
103M16—22no—
104M16—62no—
105M16—122no—
106C2×C8—122no—
107C4⋊C4—62no—
108C22⋊C4—22no—
109C22⋊C4—62no—
110C42—31no—
111C42—61no—
112C42C4×D311yescharacteristic
113C22×C3—42no—
114C22×C3—41no—
115C22×C3C2×C22⋊C411yescharacteristic
116C2×D3—21no—
117C2×D3—22no—
118C2×D3—41no—
119C2×D3—42no—
120C3×C4—22no—
121C3×C4—21no—
122C3×C4—41no—
123Dic3—22no—
124Dic3—41no—
125Dic3#9911yescharacteristic
126Dic3#9911yescharacteristic
127C23—21no—
128C23—32no—
129C23—31no—
130Q8—41no—
131Q8—121no—
132D4—41no—
133D4—122no—
134D4—121no—
135D4—124no—
136D4—122no—
137C2×C4—21no—
138C2×C4—31no—
139C2×C4—62no—
140C2×C4—62no—
141C2×C4—61no—
142C2×C4—61no—
143C2×C4—61no—
144C2×C4—61no—
145C2×C4—121no—
146C2×C4—121no—
147C2×C4—122no—
148C2×C4C2×C4×D311yescharacteristic
149C2×C4D3×D412yes—
150C8—42no—
151C8—122no—
152C2×C3—21no—
153C2×C3—81no—
154C2×C3C2×#9911yescharacteristic
155D3—41no—
156D3—42no—
157C22—42no—
158C22—41no—
159C22—61no—
160C22—62no—
161C22—121no—
162C22—122no—
163C22D3×C22⋊C411yescharacteristic
164C4—22no—
165C4—21no—
166C4—31no—
167C4—31no—
168C4—41no—
169C4—62no—
170C4—121no—
171C3#213211yescharacteristic, complemented, Sylow
172C2—21no—
173C2—81no—
174C2—121no—
175C2—122no—
176C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC2×C3×C4C22×C4
FrattiniC2×Q8C22×D3
FittingC3×#355C2
socleC2×C3C2×#99
prime radical#5668C2
cosocleC2×C3×Q8C23
hypercenter#355D3
hypocenterC3#2132
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #4751C2semidirect
3. C3×#355C2semidirect
4. #4542C2semidirect
5. #4478C2semidirect
24. #355D3semidirect
171. C3#2132semidirect

Maximal subgroups

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

subgroupindex
#56682
#47512
C3×#3552
#45422
#44782
#21323