Finite Group Databasebeta

← #80702

Subgroups — #80702

Order 384.

436 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 121 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#5596C212yesmaximal, complemented
2C3×#466C211yesmaximal, characteristic, complemented
3#5150C212yesmaximal, complemented
4#5115C211yesmaximal, characteristic, complemented
5C2×Dic3⋊C8C211yesmaximal, characteristic, complemented
6#5038C212yesmaximal, complemented
7#4890C212yesmaximal, complemented
8#4786C212yesmaximal, complemented
9#4759C212yesmaximal, complemented
10#3245—31nomaximal, Sylow
11C2×#237—22no—
12C2×C3×D8C2211yescharacteristic
13C22×C3×C8C2211yescharacteristic
14C3×#121C2212yes—
15#940C2212yes—
16#939—22no—
17D4×Dic3C2212yes—
18C2×#213C2212yes—
19#930C2212yes—
20C2×C4×Dic3C2211yescharacteristic
21C22×C3⋊C8C2211yescharacteristic
22C2×D24C2211yescharacteristic
23#893C2212yes—
24#888C2212yes—
25#885—22no—
26#67⋊C4—22no—
27C3×#107C2211yescharacteristic
28C3×#102C2212yes—
29#832C2212yes—
30#821C2212yes—
31#818C2211yescharacteristic
32Dic3⋊C8C2212yes—
33Dic3⋊C8C2212yes—
34#809C2212yes—
35#807C2212yes—
36#532—32no—
37#474—32no—
38#474—32no—
39#466—31no—
40#466—32no—
41#466D311yescharacteristic, complemented
42#459—32no—
43#459—32no—
44C2×C4⋊C8—31no—
45C2×C3×D4—22no—
46C2×C3×D4C2312yes—
47C22×C3×C4C2311yescharacteristic
48C2×#67—22no—
49C2×#67—22no—
50C2×#67—22no—
51C22×Dic3—22no—
52C22×Dic3D411yescharacteristic
53C22×Dic3D411yescharacteristic
54#237—48no—
55C2×D12C2312yes—
56C2×C4×D3—22no—
57C2×Dic6—22no—
58C3×D8—41no—
59C2×C3×C8—41no—
60C2×C3×C8C2311yescharacteristic
61C2×C3×C8C2311yescharacteristic
62C3×C4⋊C4C2312yes—
63C3×C22⋊C4—22no—
64#217—22no—
65#213—42no—
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4—22no—
68C4⋊Dic3C2312yes—
69Dic3⋊C4—22no—
70C4×Dic3D412yes—
71C4×Dic3C2312yes—
72C2×C3⋊C8—41no—
73C2×C3⋊C8C2312yes—
74D24—41no—
75C2×#41—62no—
76C2×D8—32no—
77C2×D8—31no—
78C2×D8C2×D311yescharacteristic
79C22×C8—31no—
80C22×C8C2×D311yescharacteristic
81C4⋊Q8—32no—
82#127—32no—
83#124—62no—
84#121—32no—
85#121—62no—
86#121C2×D312yes—
87C4×D4—32no—
88C4×D4—32no—
89C4×D4—62no—
90C2×C42—31no—
91#107—31no—
92#107—32no—
93#107C2×D311yescharacteristic
94C4⋊C8—32no—
95C4⋊C8—32no—
96#102—32no—
97#102—32no—
98#102—32no—
99#102C2×D312yes—
100C23×C3—22no—
101C23×C3C2×D411yescharacteristic
102C22×D3—22no—
103C3×D4—22no—
104C3×D4—22no—
105C3×D4—44no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—22no—
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2411yescharacteristic
111#67—44no—
112#67—44no—
113#67—44no—
114C2×Dic3—22no—
115C2×Dic3—41no—
116C2×Dic3—44no—
117C2×Dic3D814yescomplemented
118C2×Dic3C2×D412yes—
119C2×Dic3C2×D412yes—
120D12—22no—
121D12—22no—
122C4×D3—44no—
123Dic6—44no—
124C3×C8—21no—
125C3×C8—21no—
126C3⋊C8—22no—
127#41—128no—
128C2×Q8—62no—
129C2×D4—22no—
130C2×D4—32no—
131C2×D4—62no—
132C2×D4—62no—
133C2×D4—62no—
134C2×D4C22×D312yes—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—62no—
138C22×C4—62no—
139C22×C4C22×D311yescharacteristic
140D8—41no—
141D8—121no—
142D8—122no—
143C2×C8—32no—
144C2×C8—41no—
145C2×C8—121no—
146C2×C8C22×D311yescharacteristic
147C2×C8C22×D311yescharacteristic
148C4⋊C4—32no—
149C4⋊C4—62no—
150C4⋊C4C22×D312yes—
151C22⋊C4—22no—
152C22⋊C4—62no—
153C22⋊C4—62no—
154C22⋊C4—62no—
155C42—32no—
156C42—32no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3—42no—
162C22×C3C2×D812yes—
163C22×C3C22×D411yescharacteristic
164C2×D3—42no—
165C2×D3—42no—
166C2×D3—42no—
167C3×C4—21no—
168C3×C4—42no—
169C3×C4H(5,𝔽2)11yescharacteristic
170C3×C42−1+411yescharacteristic
171Dic3—22no—
172Dic3—42no—
173Dic3C2×D814yes—
174C23—22no—
175C23—62no—
176C23D3×D411yescharacteristic
177Q8—124no—
178D4—22no—
179D4—22no—
180D4—44no—
181D4—62no—
182D4—62no—
183D4—124no—
184D4—124no—
185D4—124no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—34no—
190C2×C4—32no—
191C2×C4—32no—
192C2×C4—62no—
193C2×C4—124no—
194C2×C4—124no—
195C2×C4—121no—
196C2×C4D3×D411yescharacteristic
197C2×C4C23×D311yescharacteristic
198C8—21no—
199C8—21no—
200C8—62no—
201C2×C3—22no—
202C2×C3—82no—
203C2×C3#53711yescharacteristic
204C2×C3C22×D811yescharacteristic
205C2×C3#57511yescharacteristic
206D3—82no—
207C22—21no—
208C22—21no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—122no—
213C22—122no—
214C22—122no—
215C22D3×D812yes—
216C22C2×D3×D411yescharacteristic
217C4—21no—
218C4—34no—
219C4—42no—
220C4—62no—
221C4—122no—
222C4#100411yescharacteristic
223C4#101011yescharacteristic
224C3#324511yescharacteristic, complemented, Sylow
225C2—22no—
226C2—82no—
227C2—242no—
228C2#559711yescharacteristic
229C2C2×D3×D811yescharacteristic
230C2#576811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3245
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #5596C2semidirect
2. C3×#466C2semidirect
3. #5150C2semidirect
4. #5115C2semidirect
5. C2×Dic3⋊C8C2semidirect
6. #5038C2semidirect
7. #4890C2semidirect
8. #4786C2semidirect
9. #4759C2semidirect
41. #466D3semidirect
117. C2×Dic3D8semidirect
224. C3#3245semidirect

Maximal subgroups

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

subgroupindex
#55962
C3×#4662
#51502
#51152
C2×Dic3⋊C82
#50382
#48902
#47862
#47592
#32453