Finite Group Databasebeta

← #80870

Subgroups — #80870

Order 384.

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

normal subgroups only

This group has 418 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#5606C212yesmaximal, complemented
2C3×#469C211yesmaximal, characteristic, complemented
3#5158C212yesmaximal, complemented
4#5137C211yesmaximal, characteristic, complemented
5#5126C211yesmaximal, characteristic, complemented
6#5042C212yesmaximal, complemented
7#4868C212yesmaximal, complemented
8#4783C212yesmaximal, complemented
9#4779C212yesmaximal, complemented
10#3197—31nomaximal, Sylow
11C2×#237C2212yes—
12C22×C4×D3C2211yescharacteristic
13C2×C3×D8C2211yescharacteristic, complemented
14C2×C3×M16C2211yescharacteristic
15C3×#121C2212yes—
16#938C2212yes—
17#937—22no—
18D4×Dic3—22no—
19C2×#214C2212yescomplemented
20#924C2212yes—
21C2×#208C2211yescharacteristic
22C2×Dic12C2211yescharacteristic, complemented
23#897C2212yes—
24#892C2212yes—
25#883—22no—
26#67⋊C4—22no—
27C3×#106C2211yescharacteristic, complemented
28C3×#102C2212yescomplemented
29#832C2212yescomplemented
30(C2×D3)⋊C8C2212yes—
31(C2×D3)⋊C8C2212yes—
32#817C2211yescharacteristic, complemented
33#816C2212yescomplemented
34#810C2212yescomplemented
35#807C2212yescomplemented
36#525—32no—
37#485—32no—
38#482—32no—
39#470—31no—
40#469D311yescharacteristic, complemented
41#468—32no—
42#452—32no—
43#449—32no—
44#407—31no—
45C23×D3D411yescharacteristic
46C2×C3×D4—22no—
47C2×C3×D4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C2×#67—22no—
50C2×#67—22no—
51C22×Dic3—22no—
52C22×Dic3D411yescharacteristic
53#237—42no—
54#237—42no—
55C2×C4×D3—41no—
56C2×C4×D3—41no—
57C2×C4×D3D412yes—
58C2×C4×D3C2312yes—
59C2×Dic6C2312yes—
60C3×D8—41no—
61C3×M16—41no—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4C2312yes—
65C3×C22⋊C4—22no—
66#217—22no—
67#217—22no—
68#214—42no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71C4⋊Dic3—22no—
72C4⋊Dic3C2312yes—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75#208—41no—
76C2×C3⋊C8C2312yes—
77Dic12—41no—
78C2×#41—32no—
79C23×C4—31no—
80C2×Q16—31no—
81C2×SD16—32no—
82C2×D8C2×D311yescharacteristic, complemented
83C2×M16—31no—
84C2×M16C2×D311yescharacteristic
85#123—62no—
86C22⋊Q8—32no—
87C22⋊Q8—32no—
88#121—32no—
89#121C2×D312yes—
90#120—62no—
91C4×D4—62no—
92C4×D4—62no—
93#117—32no—
94#107—32no—
95#106—31no—
96#106C2×D311yescharacteristic, complemented
97#103—32no—
98#103—32no—
99#102—32no—
100#102C2×D312yescomplemented
101C22⋊C8—32no—
102C22⋊C8—32no—
103C23×C3—22no—
104C23×C3C2×D411yescharacteristic
105C22×D3—41no—
106C22×D3—42no—
107C22×D3C2×D412yes—
108C3×D4—22no—
109C3×D4—22no—
110C3×D4—44no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4C2411yescharacteristic
116#67—44no—
117#67—44no—
118C2×Dic3—22no—
119C2×Dic3—22no—
120C2×Dic3—41no—
121C2×Dic3—44no—
122C2×Dic3C2×D412yes—
123C4×D3—22no—
124C4×D3—22no—
125C4×D3—42no—
126Dic6—22no—
127Dic6—22no—
128C3×C8—21no—
129C3×C8—21no—
130C3⋊C8—22no—
131C24—31no—
132#41—122no—
133#41—122no—
134C2×Q8—32no—
135C2×D4—22no—
136C2×D4—62no—
137C2×D4—62no—
138C2×D4C22×D312yes—
139C22×C4—31no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—62no—
143C22×C4—121no—
144C22×C4—121no—
145C22×C4C22×D311yescharacteristic
146Q16—121no—
147SD16—122no—
148D8—41no—
149M16—41no—
150M16—121no—
151C2×C8—32no—
152C2×C8C22×D311yescharacteristic
153C2×C8C22×D311yescharacteristic
154C4⋊C4—32no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4C22×D312yes—
158C22⋊C4—22no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C42—62no—
164C22×C3—21no—
165C22×C3—22no—
166C22×C3—42no—
167C22×C3—42no—
168C22×C3—42no—
169C22×C3C22×D411yescharacteristic
170C2×D3—22no—
171C2×D3—22no—
172C2×D3—22no—
173C2×D3—44no—
174C3×C4—21no—
175C3×C4—42no—
176C3×C4H(5,𝔽2)11yescharacteristic
177C3×C4H(5,𝔽2)11yescharacteristic
178Dic3—22no—
179Dic3—42no—
180Dic3—42no—
181C23—22no—
182C23—32no—
183C23—121no—
184C23—122no—
185C23D3×D411yescharacteristic
186Q8—62no—
187Q8—62no—
188D4—22no—
189D4—22no—
190D4—44no—
191D4—124no—
192D4—124no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—32no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—122no—
202C2×C4—121no—
203C2×C4—124no—
204C2×C4D3×D411yescharacteristic
205C2×C4C23×D311yescharacteristic
206C8—21no—
207C8—21no—
208C8—62no—
209C2×C3—41no—
210C2×C3—82no—
211C2×C3#53411yescharacteristic
212C2×C3#57512yes—
213D3—42no—
214C22—22no—
215C22—21no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—62no—
220C22—62no—
221C22—62no—
222C22—124no—
223C22C2×D3×D411yescharacteristic
224C4—21no—
225C4—42no—
226C4—62no—
227C4—122no—
228C4—122no—
229C4#100411yescharacteristic
230C4#100911yescharacteristic
231C3#319711yescharacteristic, complemented, Sylow
232C2—41no—
233C2—82no—
234C2—122no—
235C2#560511yescharacteristic
236C2#577312yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3197
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. #5606C2semidirect
2. C3×#469C2semidirect
3. #5158C2semidirect
4. #5137C2semidirect
5. #5126C2semidirect
6. #5042C2semidirect
7. #4868C2semidirect
8. #4783C2semidirect
9. #4779C2semidirect
13. C2×C3×D8C22semidirect
19. C2×#214C22semidirect
22. C2×Dic12C22semidirect
27. C3×#106C22semidirect
28. C3×#102C22semidirect
29. #832C22semidirect
32. #817C22semidirect
33. #816C22semidirect
34. #810C22semidirect
35. #807C22semidirect
40. #469D3semidirect
82. C2×D8C2×D3semidirect
96. #106C2×D3semidirect
100. #102C2×D3semidirect
231. C3#3197semidirect

Maximal subgroups

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

subgroupindex
#56062
C3×#4692
#51582
#51372
#51262
#50422
#48682
#47832
#47792
#31973