Finite Group Databasebeta

← #80850

Subgroups — #80850

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#5619C212yesmaximal, complemented
2C3×#469C211yesmaximal, characteristic, complemented
3#5152C211yesmaximal, characteristic, complemented
4#5151C211yesmaximal, characteristic, complemented
5#5135C211yesmaximal, characteristic, complemented
6#5122C211yesmaximal, characteristic, complemented
7#5038C212yesmaximal, complemented
8#4869C211yesmaximal, characteristic, complemented
9#4858C211yesmaximal, characteristic, complemented
10#4764C212yesmaximal, complemented
11#4756C212yesmaximal, complemented
12#3176—31nomaximal, Sylow
13C2×C3×D8C2211yescharacteristic
14C2×C3×M16C2211yescharacteristic
15C3×#121C2212yes—
16#940C2212yes—
17#939—22no—
18#938—22no—
19D4×Dic3C2212yes—
20C2×#213C2212yes—
21#930C2212yes—
22#926C2211yescharacteristic
23C22×C3⋊C8C2211yescharacteristic
24C2×D24C2211yescharacteristic
25#893C2212yes—
26#890C2212yes—
27#886—22no—
28#883—22no—
29C3×#106C2211yescharacteristic
30C3×#102C2212yes—
31#832C2212yes—
32#821C2212yes—
33#817C2211yescharacteristic
34#815C2211yescharacteristic
35#815C2211yescharacteristic
36C8×Dic3C2211yescharacteristic
37C8×Dic3C2211yescharacteristic
38#809C2212yes—
39#807C2212yes—
40#540—32no—
41#499—31no—
42#493—31no—
43#490—32no—
44#469—31no—
45#469D311yescharacteristic, complemented
46#466—32no—
47#442—31no—
48#442—31no—
49C4×D8—32no—
50#405—31no—
51C2×C3×D4—22no—
52C2×C3×D4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C2×#67—22no—
55C2×#67—22no—
56C2×#67—22no—
57C22×Dic3—22no—
58C2×D12C2312yes—
59C2×C4×D3—22no—
60C3×D8—22no—
61C3×M16—22no—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4C2312yes—
65C3×C22⋊C4—22no—
66#217—22no—
67#217D412yes—
68#213—24no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71C4⋊Dic3—22no—
72C4⋊Dic3D412yes—
73C4⋊Dic3C2312yes—
74Dic3⋊C4—22no—
75C4×Dic3C2311yescharacteristic
76C4×Dic3C2311yescharacteristic
77C2×C3⋊C8—22no—
78C2×C3⋊C8C2311yescharacteristic
79C2×C3⋊C8C2311yescharacteristic
80D24—22no—
81C2×D8—32no—
82C2×D8—31no—
83C2×D8C2×D311yescharacteristic
84C2×M16C2×D311yescharacteristic
85C22×C8—31no—
86#127—32no—
87#125—32no—
88#123—62no—
89#123—62no—
90#121—32no—
91#121—62no—
92#121—62no—
93#121C2×D312yes—
94C4×D4—32no—
95C4×D4—32no—
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#97—31no—
106C4×C8—31no—
107C4×C8—31no—
108C23×C3—22no—
109C23×C3C2×D411yescharacteristic
110C22×D3—22no—
111C3×D4—22no—
112C3×D4—22no—
113C3×D4—82no—
114C2×C3×C4—21no—
115C2×C3×C4—21no—
116C2×C3×C4—22no—
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2411yescharacteristic
119#67—82no—
120#67—82no—
121#67—82no—
122C2×Dic3—22no—
123C2×Dic3—82no—
124C2×Dic3C2×D412yes—
125C2×Dic3C2×D412yes—
126D12—22no—
127D12—22no—
128C4×D3—82no—
129C3×C8—21no—
130C3×C8#4112yes—
131C3⋊C8—21no—
132C3⋊C8#4112yes—
133C2×D4—22no—
134C2×D4—32no—
135C2×D4—62no—
136C2×D4—62no—
137C2×D4—62no—
138C2×D4C22×D312yes—
139C22×C4—62no—
140C22×C4—62no—
141C22×C4C22×D311yescharacteristic
142D8—22no—
143D8—64no—
144D8—62no—
145M16—22no—
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—
162C42—31no—
163C22×C3—21no—
164C22×C3—22no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3—42no—
168C22×C3C22×D411yescharacteristic
169C2×D3—42no—
170C2×D3—42no—
171C2×D3—42no—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4C2×#4111yescharacteristic
175C3×C4C2×#4111yescharacteristic
176Dic3—22no—
177Dic3—22no—
178Dic3—42no—
179C23—22no—
180C23—62no—
181C23D3×D411yescharacteristic
182D4—22no—
183D4—22no—
184D4—62no—
185D4—62no—
186D4—82no—
187D4—242no—
188D4—242no—
189D4—242no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—32no—
194C2×C4—32no—
195C2×C4—62no—
196C2×C4—242no—
197C2×C4—242no—
198C2×C4D3×D411yescharacteristic
199C2×C4C23×D311yescharacteristic
200C8—21no—
201C8—32no—
202C8—61no—
203C8#23712yes—
204C2×C3—41no—
205C2×C3—82no—
206C2×C3#53211yescharacteristic
207C2×C3#57612yes—
208D3—82no—
209C22—22no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—122no—
215C22—122no—
216C22—122no—
217C22C2×D3×D411yescharacteristic
218C4—21no—
219C4—42no—
220C4—62no—
221C4—62no—
222C4—122no—
223C4C2×#23711yescharacteristic
224C4D3×#4111yescharacteristic
225C3#317611yescharacteristic, complemented, Sylow
226C2—41no—
227C2—82no—
228C2—242no—
229C2#559611yescharacteristic
230C2#577412yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3176
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #5619C2semidirect
2. C3×#469C2semidirect
3. #5152C2semidirect
4. #5151C2semidirect
5. #5135C2semidirect
6. #5122C2semidirect
7. #5038C2semidirect
8. #4869C2semidirect
9. #4858C2semidirect
10. #4764C2semidirect
11. #4756C2semidirect
45. #469D3semidirect
225. C3#3176semidirect

Maximal subgroups

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

subgroupindex
#56192
C3×#4692
#51522
#51512
#51352
#51222
#50382
#48692
#48582
#47642
#47562
#31763