Finite Group Databasebeta

← #79860

Subgroups — #79860

Order 384.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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#5562C212yesmaximal, complemented
2C3×#442C211yesmaximal, characteristic, complemented
3#5159C211yesmaximal, characteristic, complemented
4#5151C211yesmaximal, characteristic, complemented
5C4×#213C212yesmaximal, complemented
6#4865C211yesmaximal, characteristic, complemented
7#4858C211yesmaximal, characteristic, complemented
8#4775C212yesmaximal, complemented
9#4764C212yesmaximal, complemented
10#4711C211yesmaximal, characteristic, complemented
11#4708C211yesmaximal, characteristic, complemented
12#3176—31nomaximal, Sylow
13C2×C3×D8C2211yescharacteristic
14C3×C4×D4C2212yes—
15#940C2212yes—
16#938C2212yes—
17C2×#213C2212yes—
18#930—22no—
19#929—22no—
20C2×D24C2211yescharacteristic
21C2×ΓL(1,𝔽25)C2211yescharacteristic
22C2×D3×C8C2211yescharacteristic
23#895C2212yes—
24#890C2212yes—
25#884—22no—
26#883—22no—
27C4×D12C2212yes—
28#872C2211yescharacteristic
29C3×#106C2211yescharacteristic
30C3×#102C2212yes—
31C3×#97C2211yescharacteristic
32#832C2212yes—
33#821C2212yes—
34#817C2211yescharacteristic
35#815C2211yescharacteristic
36C8×Dic3C2211yescharacteristic
37#809C2212yes—
38#807C2212yes—
39C4×C3⋊C8C2211yescharacteristic
40#540—32no—
41#499—31no—
42#493—31no—
43#490—32no—
44#469—31no—
45#469—31no—
46#466—32no—
47#442—31no—
48#442D311yescharacteristic, complemented
49C4×D8—32no—
50#405—31no—
51C2×C3×D4C2312yes—
52C22×C3×C4—22no—
53C2×#67—22no—
54C2×#67—22no—
55C2×#67—22no—
56C2×D12—22no—
57C2×D12C2312yes—
58C2×C4×D3—22no—
59C2×C4×D3C2311yescharacteristic
60C3×D8—22no—
61C2×C3×C8C2311yescharacteristic
62C2×C3×C8C2311yescharacteristic
63C3×C4⋊C4C2312yes—
64C3×C22⋊C4—22no—
65C3×C42C2311yescharacteristic
66#217—22no—
67#213—24no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4D412yes—
71C4⋊Dic3—22no—
72C4⋊Dic3C2312yes—
73Dic3⋊C4—22no—
74Dic3⋊C4D412yes—
75C4×Dic3C2311yescharacteristic
76C2×C3⋊C8C2311yescharacteristic
77C2×C3⋊C8C2311yescharacteristic
78D24—22no—
79ΓL(1,𝔽25)—22no—
80D3×C8—22no—
81C2×D8—32no—
82C2×D8—31no—
83C2×D8C2×D311yescharacteristic
84C2×M16—31no—
85C22×C8—31no—
86#127—32no—
87#125—32no—
88#123—62no—
89#123—62no—
90#121—32no—
91#121—32no—
92#121—62no—
93#121—62no—
94C4×D4—32no—
95C4×D4C2×D312yes—
96#117—31no—
97#107—32no—
98#106—31no—
99#106C2×D311yescharacteristic
100#102—32no—
101#102—32no—
102#102—32no—
103#102C2×D312yes—
104#97—31no—
105#97C2×D311yescharacteristic
106C4×C8—31no—
107C4×C8—31no—
108C23×C3—22no—
109C22×D3—22no—
110C22×D3C2×D411yescharacteristic
111C3×D4—22no—
112C3×D4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—82no—
115C2×C3×C4C2×D412yes—
116C2×C3×C4C2411yescharacteristic
117#67—82no—
118#67—82no—
119#67—82no—
120C2×Dic3—22no—
121C2×Dic3C2×D412yes—
122C2×Dic3C2×D411yescharacteristic
123D12—22no—
124D12—22no—
125D12—82no—
126C4×D3—21no—
127C4×D3—21no—
128C4×D3—82no—
129C3×C8—21no—
130C3×C8#4112yes—
131C3⋊C8—21no—
132C3⋊C8#4112yes—
133C2×D4—32no—
134C2×D4—62no—
135C2×D4—62no—
136C2×D4—62no—
137C2×D4—62no—
138C2×D4C22×D312yes—
139C22×C4—22no—
140C22×C4—31no—
141C22×C4—62no—
142D8—22no—
143D8—64no—
144D8—62no—
145M16—62no—
146C2×C8—31no—
147C2×C8—31no—
148C2×C8—62no—
149C2×C8C22×D311yescharacteristic
150C2×C8C22×D311yescharacteristic
151C4⋊C4—32no—
152C4⋊C4—32no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4C22×D312yes—
156C22⋊C4—22no—
157C22⋊C4—32no—
158C22⋊C4—62no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C42—31no—
162C42C22×D311yescharacteristic
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—42no—
166C22×C3C22×D411yescharacteristic
167C2×D3—21no—
168C2×D3—22no—
169C2×D3—42no—
170C2×D3—42no—
171C2×D3—42no—
172C3×C4—22no—
173C3×C4—42no—
174C3×C4C2×#4111yescharacteristic
175C3×C4C2×#4111yescharacteristic
176Dic3—21no—
177Dic3—22no—
178Dic3—42no—
179C23—22no—
180C23—31no—
181C23—62no—
182D4—22no—
183D4—22no—
184D4—62no—
185D4—62no—
186D4—242no—
187D4—242no—
188D4—242no—
189D4—242no—
190C2×C4—22no—
191C2×C4—32no—
192C2×C4—31no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—62no—
196C2×C4—82no—
197C2×C4—242no—
198C2×C4D3×D412yes—
199C2×C4C23×D311yescharacteristic
200C8—21no—
201C8—32no—
202C8—61no—
203C8#23512yes—
204C2×C3—82no—
205C2×C3#53211yescharacteristic
206C2×C3#57612yes—
207D3—41no—
208D3—82no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—62no—
213C22—61no—
214C22—122no—
215C22—122no—
216C22—122no—
217C22C2×D3×D411yescharacteristic
218C4—22no—
219C4—42no—
220C4—62no—
221C4—61no—
222C4—122no—
223C4C2×#23511yescharacteristic
224C4D3×#4111yescharacteristic
225C3#317611yescharacteristic, complemented, Sylow
226C2—82no—
227C2—121no—
228C2—242no—
229C2#554811yescharacteristic
230C2#577412yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
hypercenter#442D3
hypocenterC3#3176
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #5562C2semidirect
2. C3×#442C2semidirect
3. #5159C2semidirect
4. #5151C2semidirect
5. C4×#213C2semidirect
6. #4865C2semidirect
7. #4858C2semidirect
8. #4775C2semidirect
9. #4764C2semidirect
10. #4711C2semidirect
11. #4708C2semidirect
48. #442D3semidirect
225. C3#3176semidirect

Maximal subgroups

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

subgroupindex
#55622
C3×#4422
#51592
#51512
C4×#2132
#48652
#48582
#47752
#47642
#47112
#47082
#31763