Finite Group Databasebeta

← #81706

Subgroups — #81706

Order 384.

386 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 113 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#5719C211yesmaximal, characteristic, complemented
2#5697C211yesmaximal, characteristic, complemented
3C3×#490C211yesmaximal, characteristic, complemented
4#5073C211yesmaximal, characteristic, complemented
5#5065C211yesmaximal, characteristic, complemented
6#4785C212yesmaximal, complemented
7#4782C212yesmaximal, complemented
8#4775C212yesmaximal, complemented
9#4756C212yesmaximal, complemented
10#4713C211yesmaximal, characteristic, complemented
11#4708C211yesmaximal, characteristic, complemented
12#3176—31nomaximal, Sylow
13C3×#127C2211yescharacteristic
14C3×#125C2211yescharacteristic
15#939—22no—
16#938C2212yes—
17#935—22no—
18D4×Dic3C2212yes—
19C2×#213C2212yes—
20C2×#213C2212yes—
21C2×ΓL(1,𝔽25)C2211yescharacteristic
22C2×D3×C8C2211yescharacteristic
23#895—22no—
24#895C2212yes—
25#894—22no—
26#893C2212yes—
27#874C2211yescharacteristic
28#872C2211yescharacteristic
29#870C2211yescharacteristic
30C3×#102C2212yes—
31C3×#102C2212yes—
32C3×#97C2211yescharacteristic
33#821C2212yes—
34#821C2212yes—
35#815C2211yescharacteristic
36C8×Dic3C2211yescharacteristic
37#808C2212yes—
38#807C2212yes—
39C4×C3⋊C8C2211yescharacteristic
40#540—31no—
41#540—31no—
42#499—31no—
43#493—31no—
44#490—31no—
45#490D311yescharacteristic, complemented
46#469—32no—
47#466—32no—
48#442—32no—
49C4×D8—32no—
50#405—31no—
51C2×C3×D4—22no—
52C2×C3×D4C2312yes—
53C2×#67—22no—
54C22×Dic3—22no—
55C2×D12—22no—
56C2×D12—22no—
57C2×D12C2312yes—
58C2×C4×D3—22no—
59C2×C4×D3C2311yescharacteristic
60C2×C3×C8C2311yescharacteristic
61C2×C3×C8C2311yescharacteristic
62C3×C4⋊C4—22no—
63C3×C4⋊C4C2312yes—
64C3×C42C2311yescharacteristic
65#217—22no—
66#217—22no—
67#213—24no—
68#213—24no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4D412yes—
72C4⋊Dic3C2312yes—
73Dic3⋊C4—22no—
74Dic3⋊C4D412yes—
75C4×Dic3C2311yescharacteristic
76C2×C3⋊C8C2311yescharacteristic
77C2×C3⋊C8C2311yescharacteristic
78ΓL(1,𝔽25)—22no—
79D3×C8—22no—
80C2×D8—32no—
81C2×D8—32no—
82C2×M16—31no—
83C22×C8—31no—
84#127—31no—
85#127C2×D311yescharacteristic
86#125—31no—
87#125C2×D311yescharacteristic
88#123—62no—
89#123—62no—
90#121—32no—
91#121—32no—
92#121—62no—
93#121—62no—
94C4×D4—32no—
95C4×D4—32no—
96#117—31no—
97#107—32no—
98#106—32no—
99#102—32no—
100#102—32no—
101#102C2×D312yes—
102#102C2×D312yes—
103#97—31no—
104#97C2×D311yescharacteristic
105C4×C8—31no—
106C4×C8—31no—
107C23×C3—22no—
108C22×D3—22no—
109C22×D3C2×D411yescharacteristic
110C3×D4—22no—
111C3×D4—22no—
112C3×D4—82no—
113C2×C3×C4—22no—
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2411yescharacteristic
116#67—82no—
117C2×Dic3—22no—
118C2×Dic3—82no—
119C2×Dic3C2×D412yes—
120C2×Dic3C2×D411yescharacteristic
121D12—22no—
122D12—22no—
123D12—82no—
124D12—82no—
125C4×D3—21no—
126C4×D3—21no—
127C4×D3—82no—
128C3×C8—21no—
129C3×C8—21no—
130C3⋊C8#4112yes—
131C3⋊C8#4112yes—
132C2×D4—22no—
133C2×D4—32no—
134C2×D4—62no—
135C2×D4—62no—
136C2×D4—62no—
137C2×D4C22×D312yes—
138C22×C4—31no—
139C22×C4—62no—
140C22×C4—62no—
141D8—64no—
142D8—64no—
143M16—62no—
144C2×C8—31no—
145C2×C8—31no—
146C2×C8—62no—
147C2×C8C22×D311yescharacteristic
148C2×C8C22×D311yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—32no—
151C4⋊C4—32no—
152C4⋊C4—62no—
153C4⋊C4C22×D312yes—
154C22⋊C4—32no—
155C22⋊C4—62no—
156C22⋊C4—62no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C42—31no—
160C42C22×D311yescharacteristic
161C22×C3—42no—
162C22×C3—42no—
163C22×C3—42no—
164C22×C3C22×D411yescharacteristic
165C2×D3—21no—
166C2×D3—22no—
167C2×D3—42no—
168C2×D3—42no—
169C2×D3—42no—
170C3×C4—22no—
171C3×C4—42no—
172C3×C4C2×#4111yescharacteristic
173C3×C4C2×#4111yescharacteristic
174Dic3—21no—
175Dic3—22no—
176Dic3—42no—
177C23—22no—
178C23—31no—
179C23—62no—
180D4—22no—
181D4—22no—
182D4—62no—
183D4—62no—
184D4—82no—
185D4—242no—
186D4—242no—
187D4—242no—
188C2×C4—22no—
189C2×C4—32no—
190C2×C4—31no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—62no—
194C2×C4—242no—
195C2×C4—242no—
196C2×C4D3×D412yes—
197C2×C4C23×D311yescharacteristic
198C8—21no—
199C8—21no—
200C8—32no—
201C8—32no—
202C2×C3—82no—
203C2×C3#53211yescharacteristic
204C2×C3#57612yes—
205D3—41no—
206D3—82no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—61no—
211C22—62no—
212C22—122no—
213C22—122no—
214C22—122no—
215C22C2×D3×D411yescharacteristic
216C4—22no—
217C4—42no—
218C4—61no—
219C4—62no—
220C4—122no—
221C4D3×#4111yescharacteristic
222C4D3×#4111yescharacteristic
223C3#317611yescharacteristic, complemented, Sylow
224C2—82no—
225C2—121no—
226C2—242no—
227C2#566811yescharacteristic
228C2#577412yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#490C2
socleC22×C3C22×D4
hypercenter#490D3
hypocenterC3#3176
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #5719C2semidirect
2. #5697C2semidirect
3. C3×#490C2semidirect
4. #5073C2semidirect
5. #5065C2semidirect
6. #4785C2semidirect
7. #4782C2semidirect
8. #4775C2semidirect
9. #4756C2semidirect
10. #4713C2semidirect
11. #4708C2semidirect
45. #490D3semidirect
223. C3#3176semidirect

Maximal subgroups

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

subgroupindex
#57192
#56972
C3×#4902
#50732
#50652
#47852
#47822
#47752
#47562
#47132
#47082
#31763