Finite Group Databasebeta

← #70057

Subgroups — #70057

Order 384.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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#5830C211yesmaximal, characteristic
2C3×#410C211yesmaximal, characteristic, complemented
3#5237C211yesmaximal, characteristic, complemented
4#4471C214yesmaximal, complemented
5#2122—31nomaximal, Sylow
6C2×C3×#41C2211yescharacteristic
7C22×D12C411yescharacteristic, complemented
8C3×#117—21no—
9#947—21no—
10#940—21no—
11#937C412yescomplemented
12#937C2212yes—
13#930—21no—
14#930—21no—
15#926C411yescharacteristic, complemented
16C2×#208—21no—
17C3×#99—21no—
18C3×#99C2212yes—
19#836—21no—
20#833C2212yes—
21#535—31no—
22#413—31no—
23#410D311yescharacteristic, complemented
24AΣL(1,𝔽16)—34no—
25C23×D3C2×C412yes—
26C3×#41—41no—
27C3×#41—41no—
28C2×C3×Q8D411yescharacteristic
29C2×C3×D4C2×C411yescharacteristic
30C2×C3×D4D411yescharacteristic
31C2×C3×D4C2311yescharacteristic
32C22×C3×C4C2×C411yescharacteristic
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C2×#67—22no—
36C2×#67—22no—
37C2×D12—21no—
38C2×D12—21no—
39C2×D12—41no—
40C2×D12—41no—
41C3×C4⋊C4—41no—
42C3×C22⋊C4—22no—
43C3×C42—41no—
44#217C2×C412yes—
45(C2×D3)⋊C4—22no—
46(C2×D3)⋊C4—22no—
47C4⋊Dic3—21no—
48C4×Dic3—21no—
49#208—22no—
50#208—41no—
51C2×C3⋊C8—41no—
52C2×#41C2×D311yescharacteristic
53C22×D4—31no—
54C2×M16—61no—
55#127—61no—
56#124—61no—
57#121—61no—
58#121—61no—
59#120—32no—
60#120—32no—
61#117—21no—
62#117—31no—
63#101—61no—
64#100—32no—
65#99—21no—
66#99C2×D312yes—
67C23×C3C22⋊C411yescharacteristic
68C23×C3C22×C411yescharacteristic
69C23×C3C2×D411yescharacteristic
70C22×D3—22no—
71C22×D3—42no—
72C22×D3—82no—
73C3×Q8—41no—
74C3×D4—41no—
75C3×D4—41no—
76C3×D4—41no—
77C2×C3×C4—21no—
78C2×C3×C4—21no—
79C2×C3×C4—21no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—42no—
84C2×C3×C4C22⋊C411yescharacteristic
85C2×C3×C4C22⋊C411yescharacteristic
86C2×C3×C4C22⋊C411yescharacteristic
87C2×C3×C4C2×D411yescharacteristic
88#67—82no—
89#67—82no—
90C2×Dic3—22no—
91D12—42no—
92D12—41no—
93D12—41no—
94C3⋊C8—42no—
95C24—32no—
96#41—41no—
97#41—41no—
98C2×Q8D1211yescharacteristic
99C2×D4—62no—
100C2×D4—62no—
101C2×D4—61no—
102C2×D4—61no—
103C2×D4—121no—
104C2×D4—121no—
105C2×D4C4×D311yescharacteristic
106C2×D4#6711yescharacteristic
107C2×D4C22×D311yescharacteristic
108C22×C4C4×D311yescharacteristic
109C22×C4D1211yescharacteristic
110C22×C4#6711yescharacteristic
111M16—62no—
112M16—121no—
113C2×C8—121no—
114C4⋊C4—41no—
115C4⋊C4—61no—
116C22⋊C4—22no—
117C22⋊C4—32no—
118C22⋊C4—62no—
119C22⋊C4—62no—
120C42—41no—
121C42—61no—
122C22×C3—21no—
123C22×C3—21no—
124C22×C3—21no—
125C22×C3—41no—
126C22×C3—41no—
127C22×C3—41no—
128C22×C3C2×C22⋊C411yescharacteristic
129C2×D3—42no—
130C2×D3—42no—
131C2×D3—42no—
132C2×D3—84no—
133C3×C4—21no—
134C3×C4—21no—
135C3×C4—21no—
136C3×C4—82no—
137C3×C4#9911yescharacteristic
138C3×C4#9911yescharacteristic
139Dic3—42no—
140C23—62no—
141C23—122no—
142C23—242no—
143C23(C2×D3)⋊C411yescharacteristic
144C23C2×C4×D311yescharacteristic
145C23C2×#6711yescharacteristic
146Q8—41no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—122no—
151D4—121no—
152D4—121no—
153D4—242no—
154D4—242no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—62no—
163C2×C4(C2×D3)⋊C411yescharacteristic
164C2×C4(C2×D3)⋊C411yescharacteristic
165C2×C4(C2×D3)⋊C411yescharacteristic
166C2×C4C2×D1211yescharacteristic
167C8—122no—
168C2×C3—21no—
169C2×C3—41no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3C2×#9911yescharacteristic
173D3—82no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—122no—
181C22—122no—
182C22—122no—
183C22—244no—
184C22C2×(C2×D3)⋊C411yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—82no—
189C4—122no—
190C4#80611yescharacteristic
191C4#80611yescharacteristic
192C3#212211yescharacteristic, complemented, Sylow
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—242no—
198C2C2×#80611yescharacteristic
subgroupG/H
centerC2C2×#806
commutatorC23×C3C22×C4
FrattiniC2×D4C22×D3
FittingC3×#410C2
socleC2×C3C2×#99
prime radical#5830C2
cosocleC2×C3×D4C23
hypercenter#410D3
hypocenterC3#2122
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. C3×#410C2semidirect
3. #5237C2semidirect
4. #4471C2semidirect
7. C22×D12C4semidirect
11. #937C4semidirect
15. #926C4semidirect
23. #410D3semidirect
192. C3#2122semidirect

Maximal subgroups

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

subgroupindex
#58302
C3×#4102
#52372
#44712
#21223