Finite Group Databasebeta

← #80767

Subgroups — #80767

Order 384.

378 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 109 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#5628C212yesmaximal
2C3×#467C211yesmaximal, characteristic, complemented
3#5186C212yesmaximal, complemented
4#5116C211yesmaximal, characteristic, complemented
5#5110C211yesmaximal, characteristic
6#5052C212yesmaximal, complemented
7#4888C212yesmaximal, complemented
8#4814C212yesmaximal, complemented
9#4809C212yesmaximal, complemented
10#3202—31nomaximal, Sylow
11C2×D3×Q8C2212yes—
12C2×#235C2211yescharacteristic
13C2×C3×Q16C2211yescharacteristic
14C22×C3×C8C2211yescharacteristic
15C3×C22⋊Q8C2212yes—
16#946C2212yes—
17C2×#216C2212yes—
18#924C2212yes—
19C2×#208C2211yescharacteristic
20C2×Dic12C2211yescharacteristic
21#897C2212yes—
22#894—22no—
23#892C2212yes—
24#888—22no—
25#885—22no—
26#883—22no—
27C3×#107C2211yescharacteristic
28C3×#103C2212yes—
29#835C2212yes—
30(C2×D3)⋊C8C2212yes—
31(C2×D3)⋊C8C2212yes—
32#818C2211yescharacteristic
33#816C2212yes—
34#810C2212yes—
35#808C2212yes—
36#536—32no—
37#485—32no—
38#485—32no—
39#470—32no—
40#467—31no—
41#467D311yescharacteristic, complemented
42#451—32no—
43#451—32no—
44#408—31no—
45C2×C3×Q8C2312yes—
46C22×C3×C4C2311yescharacteristic
47C2×#67D412yes—
48D3×Q8—42no—
49D3×Q8—42no—
50#235—41no—
51#235—41no—
52C2×D12D411yescharacteristic
53C2×C4×D3—22no—
54C2×C4×D3C2312yes—
55C2×Dic6—22no—
56C2×Dic6D411yescharacteristic
57C2×Dic6C2312yes—
58C3×Q16—41no—
59C2×C3×C8—41no—
60C2×C3×C8C2311yescharacteristic
61C2×C3×C8C2311yescharacteristic
62C3×C4⋊C4—22no—
63C3×C4⋊C4C2312yes—
64C3×C22⋊C4—22no—
65#217—22no—
66#216—42no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3C2312yes—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75#208—41no—
76C2×C3⋊C8C2312yes—
77Dic12—41no—
78C2×#41—31no—
79C22×Q8—32no—
80C2×Q16—32no—
81C2×Q16—31no—
82C2×Q16C2×D311yescharacteristic
83C2×M16—31no—
84C22×C8C2×D311yescharacteristic
85C4⋊Q8—62no—
86#124—62no—
87#123—62no—
88#123—62no—
89C22⋊Q8—32no—
90C22⋊Q8—32no—
91C22⋊Q8—32no—
92C22⋊Q8C2×D312yes—
93#117—32no—
94#107—31no—
95#107C2×D311yescharacteristic
96#106—32no—
97#103—32no—
98#103—32no—
99#103—32no—
100#103C2×D312yes—
101C22⋊C8—32no—
102C22⋊C8—32no—
103C23×C3C2×D411yescharacteristic
104C22×D3C2×D412yes—
105C3×Q8—22no—
106C3×Q8—22no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2411yescharacteristic
113#67—42no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3C2×D412yes—
117D12—41no—
118C4×D3—22no—
119C4×D3—22no—
120C4×D3—44no—
121Dic6—22no—
122Dic6—22no—
123Dic6—41no—
124Dic6—44no—
125C3×C8—21no—
126C3×C8—21no—
127C3⋊C8—22no—
128#41—121no—
129#41—121no—
130C2×Q8—31no—
131C2×Q8—32no—
132C2×Q8—62no—
133C2×Q8—122no—
134C2×Q8—122no—
135C2×Q8C22×D312yes—
136C2×D4—32no—
137C2×D4—31no—
138C22×C4—32no—
139C22×C4—62no—
140C22×C4C22×D311yescharacteristic
141Q16—41no—
142Q16—121no—
143Q16—122no—
144M16—121no—
145C2×C8—32no—
146C2×C8—41no—
147C2×C8C22×D311yescharacteristic
148C2×C8C22×D311yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—32no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4C22×D312yes—
155C22⋊C4—22no—
156C22⋊C4—62no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C22⋊C4—62no—
160C42—62no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—21no—
164C22×C3C22×D411yescharacteristic
165C2×D3—22no—
166C2×D3—22no—
167C2×D3—22no—
168C3×C4—21no—
169C3×C4—42no—
170C3×C4—42no—
171C3×C4H(5,𝔽2)11yescharacteristic
172C3×C4H(5,𝔽2)11yescharacteristic
173Dic3—22no—
174Dic3—42no—
175Dic3—42no—
176C23—32no—
177C23D3×D411yescharacteristic
178Q8—22no—
179Q8—22no—
180Q8—62no—
181Q8—62no—
182Q8—121no—
183Q8—124no—
184D4—121no—
185D4—122no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—32no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—124no—
196C2×C4D3×D411yescharacteristic
197C2×C4C23×D311yescharacteristic
198C8—21no—
199C8—21no—
200C8—62no—
201C2×C3—41no—
202C2×C3#53411yescharacteristic
203C2×C3#57811yescharacteristic
204C2×C3#57811yescharacteristic
205D3—42no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—62no—
210C22—62no—
211C22—62no—
212C22C2×D3×D411yescharacteristic
213C4—21no—
214C4—42no—
215C4—42no—
216C4—62no—
217C4—122no—
218C4—122no—
219C4#100411yescharacteristic
220C4#100911yescharacteristic
221C3#320211yescharacteristic, complemented, Sylow
222C2—41no—
223C2—122no—
224C2#560511yescharacteristic
225C2#576611yescharacteristic
226C2#577111yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#467C2
socleC22×C3C22×D4
prime radicalC2×#235C22
hypercenter#467D3
hypocenterC3#3202
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. C3×#467C2semidirect
3. #5186C2semidirect
4. #5116C2semidirect
6. #5052C2semidirect
7. #4888C2semidirect
8. #4814C2semidirect
9. #4809C2semidirect
41. #467D3semidirect
221. C3#3202semidirect

Maximal subgroups

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

subgroupindex
#56282
C3×#4672
#51862
#51162
#51102
#50522
#48882
#48142
#48092
#32023