Finite Group Databasebeta

← #78614

Subgroups — #78614

Order 384.

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

normal subgroups only

This group has 424 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#5791C212yesmaximal, complemented
2C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
3#5116C211yesmaximal, characteristic, complemented
4#5115C211yesmaximal, characteristic, complemented
5#5114C212yesmaximal, complemented
6#4737C212yesmaximal, complemented
7#4734C212yesmaximal, complemented
8#4733C212yesmaximal, complemented
9#4723C212yesmaximal, complemented
10#3191—31nomaximal, Sylow
11C23×C3×C4C2211yescharacteristic
12C2×#235C2212yes—
13C22×C3×C8C2211yescharacteristic
14C22×C3×C8C2211yescharacteristic
15#953—22no—
16#930C2212yes—
17#930C2212yes—
18#929—22no—
19C4×#67—22no—
20C4×#67—22no—
21#926C2212yes—
22#924C2212yes—
23#924C2212yes—
24C2×Dic12C2211yescharacteristic, complemented
25C2×D24C2211yescharacteristic, complemented
26C2×#204C2212yescomplemented
27C3×C22⋊C8C2212yes—
28C3×C22⋊C8C2212yes—
29#821C2212yescomplemented
30#821C2212yescomplemented
31#818C2211yescharacteristic, complemented
32#818C2211yescharacteristic, complemented
33#817C2212yescomplemented
34#816C2212yescomplemented
35#816C2212yescomplemented
36#525—32no—
37#485—32no—
38#482—32no—
39#467—31no—
40#466—31no—
41#465—32no—
42#452—32no—
43#449—32no—
44C2×C22⋊C8D311yescharacteristic, complemented
45C24×C3D411yescharacteristic
46C22×C3×C4—41no—
47C22×C3×C4—41no—
48C22×C3×C4D411yescharacteristic
49C22×C3×C4D412yes—
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C2×#67—22no—
53C2×#67—22no—
54C2×#67—22no—
55#235—42no—
56#235—42no—
57C2×D12C2312yes—
58C2×C4×D3—22no—
59C2×Dic6C2312yes—
60C2×C3×C8—41no—
61C2×C3×C8—41no—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2312yes—
64C2×C3×C8C2311yescharacteristic
65#217—22no—
66#217—22no—
67#217—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70C4⋊Dic3C2312yes—
71C4⋊Dic3C2312yes—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74C4×Dic3—22no—
75Dic12—41no—
76D24—41no—
77#204—42no—
78C2×#41—32no—
79C23×C4C2×D311yescharacteristic
80C2×Q16—31no—
81C2×SD16—32no—
82C2×D8—31no—
83C22×C8C2×D311yescharacteristic
84C22×C8C2×D311yescharacteristic
85#123—62no—
86C22⋊Q8—32no—
87C22⋊Q8—32no—
88#121—32no—
89#121—32no—
90#120—62no—
91C4×D4—62no—
92C4×D4—62no—
93#117—32no—
94#107—31no—
95#107—31no—
96#106—32no—
97#103—32no—
98#103—32no—
99#102—32no—
100#102—32no—
101C22⋊C8C2×D312yes—
102C22⋊C8C2×D312yes—
103C23×C3—22no—
104C23×C3—41no—
105C23×C3—41no—
106C23×C3C2×D412yes—
107C23×C3C2×D411yescharacteristic
108C22×D3—22no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—41no—
114C2×C3×C4—42no—
115C2×C3×C4C2×D412yes—
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2411yescharacteristic
118#67—44no—
119#67—44no—
120#67—44no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124D12—22no—
125D12—22no—
126C4×D3—44no—
127Dic6—22no—
128Dic6—22no—
129C3×C8—22no—
130C3×C8—21no—
131C3×C8—21no—
132C24D1211yescharacteristic
133#41—122no—
134#41—122no—
135C2×Q8—32no—
136C2×D4—32no—
137C2×D4—62no—
138C2×D4—62no—
139C2×D4—62no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—62no—
143C22×C4D1211yescharacteristic
144C22×C4D1212yes—
145C22×C4C22×D312yes—
146C22×C4C22×D311yescharacteristic
147Q16—121no—
148SD16—122no—
149D8—121no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8C22×D311yescharacteristic
153C2×C8C22×D312yes—
154C2×C8C22×D311yescharacteristic
155C4⋊C4—32no—
156C4⋊C4—32no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C42—62no—
165C22×C3—21no—
166C22×C3—21no—
167C22×C3—22no—
168C22×C3—22no—
169C22×C3—22no—
170C22×C3—44no—
171C22×C3#13512yes—
172C22×C3C22×D411yescharacteristic
173C2×D3—42no—
174C2×D3—42no—
175C2×D3—42no—
176C3×C4—22no—
177C3×C4—21no—
178C3×C4H(5,𝔽2)11yescharacteristic
179C3×C4H(5,𝔽2)11yescharacteristic
180Dic3—42no—
181Dic3—42no—
182Dic3—42no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—62no—
187C23C2×D1212yes—
188C23C2×D1211yescharacteristic
189Q8—62no—
190Q8—62no—
191D4—62no—
192D4—62no—
193D4—124no—
194D4—124no—
195D4—124no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—62no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—124no—
206C2×C4C2×D1212yes—
207C2×C4C2×D1211yescharacteristic
208C2×C4C23×D311yescharacteristic
209C8—21no—
210C8—22no—
211C8—21no—
212C2×C3—22no—
213C2×C3—42no—
214C2×C3#53411yescharacteristic
215C2×C3C2×#13511yescharacteristic
216C2×C3#57511yescharacteristic
217D3—82no—
218C22—21no—
219C22—21no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—44no—
224C22—122no—
225C22—122no—
226C22—122no—
227C22#90412yes—
228C22C22×D1211yescharacteristic
229C4—22no—
230C4—21no—
231C4—122no—
232C4—122no—
233C4—122no—
234C4#100411yescharacteristic
235C4#100411yescharacteristic
236C3#319111yescharacteristic, complemented, Sylow
237C2—22no—
238C2—42no—
239C2—242no—
240C2#549311yescharacteristic
241C2C2×#90411yescharacteristic
242C2#574811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C22⋊C8C2
socleC22×C3C22×D4
hypercenterC2×C22⋊C8D3
hypocenterC3#3191
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. #5791C2semidirect
2. C2×C3×C22⋊C8C2semidirect
3. #5116C2semidirect
4. #5115C2semidirect
5. #5114C2semidirect
6. #4737C2semidirect
7. #4734C2semidirect
8. #4733C2semidirect
9. #4723C2semidirect
24. C2×Dic12C22semidirect
25. C2×D24C22semidirect
26. C2×#204C22semidirect
29. #821C22semidirect
30. #821C22semidirect
31. #818C22semidirect
32. #818C22semidirect
33. #817C22semidirect
34. #816C22semidirect
35. #816C22semidirect
44. C2×C22⋊C8D3semidirect
236. C3#3191semidirect

Maximal subgroups

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

subgroupindex
#57912
C2×C3×C22⋊C82
#51162
#51152
#51142
#47372
#47342
#47332
#47232
#31913