Finite Group Databasebeta

← #73677

Subgroups — #73677

Order 384.

548 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D12C211yesmaximal, characteristic
2C2×#870C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4#4993C212yesmaximal, complemented
5#4989C212yesmaximal, complemented
6#4939C211yesmaximal, characteristic
7#4939C211yesmaximal, characteristic
8#4928C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4676C212yesmaximal, complemented
11#4667C212yesmaximal, complemented
12#2454—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C42C2211yescharacteristic
18C2×C3×C42C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C4×D12—24no—
33#870—24no—
34C3×#95C2211yescharacteristic
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×C4×D4—31no—
40#399—32no—
41#396—32no—
42#390—31no—
43#390—31no—
44#389—31no—
45#389—31no—
46#385—32no—
47#380—32no—
48C4×C4⋊C4D311yescharacteristic, complemented
49C23×D3C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C2×D12—22no—
61C2×D12—22no—
62C2×D12Q814yes—
63C2×C4×D3—24no—
64C2×C4×D3—24no—
65C2×C4×D3—42no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68C3×C42—22no—
69C3×C42—22no—
70C3×C42—22no—
71(C2×D3)⋊C4—24no—
72(C2×D3)⋊C4—24no—
73(C2×D3)⋊C4—42no—
74(C2×D3)⋊C4—42no—
75(C2×D3)⋊C4—42no—
76C4⋊Dic3—22no—
77C4⋊Dic3—22no—
78C4⋊Dic3D414yes—
79Dic3⋊C4—24no—
80Dic3⋊C4—24no—
81Dic3⋊C4—42no—
82Dic3⋊C4—42no—
83C22×D4—31no—
84C23×C4—32no—
85#125—64no—
86C4×D4—64no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C42C2×D311yescharacteristic
102C2×C42C2×D311yescharacteristic
103C2×C42C2×D311yescharacteristic
104#95—32no—
105#95—32no—
106#95C2×D311yescharacteristic
107#95C2×D311yescharacteristic
108C23×C3C2411yescharacteristic
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3C2×Q814yes—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×Q812yes—
134C2×C3×C4#4112yes—
135C2×C3×C4#4112yes—
136C2×C3×C4#4112yes—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3C2×D414yes—
146D12—28no—
147C4×D3—48no—
148C24—32no—
149C2×D4—34no—
150C2×D4—62no—
151C2×D4—62no—
152C22×C4—32no—
153C22×C4—32no—
154C22×C4—32no—
155C22×C4—64no—
156C22×C4—64no—
157C22×C4—122no—
158C22×C4C22×D311yescharacteristic
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C22×C4C22×D311yescharacteristic
162C22×C4C22×D311yescharacteristic
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D311yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—34no—
168C4⋊C4—64no—
169C4⋊C4—64no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C4⋊C4—122no—
174C22⋊C4—64no—
175C22⋊C4—64no—
176C22⋊C4—122no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C42—22no—
180C42—22no—
181C42—22no—
182C22×C3C22×D411yescharacteristic
183C22×C3C22×Q811yescharacteristic
184C22×C3C2×#4111yescharacteristic
185C22×C3C2×#4111yescharacteristic
186C22×C3C2×#4111yescharacteristic
187C22×C3H(5,𝔽2)11yescharacteristic
188C22×C3H(5,𝔽2)11yescharacteristic
189C2×D3—24no—
190C2×D3—24no—
191C2×D3—24no—
192C2×D3—42no—
193C2×D3—42no—
194C2×D3—42no—
195C2×D3—42no—
196C3×C4—22no—
197C3×C4—22no—
198C3×C4—41no—
199C3×C4—41no—
200C3×C4—41no—
201C3×C4—41no—
202C3×C4C22⋊Q814yes—
203Dic3—24no—
204Dic3—42no—
205Dic3—42no—
206C23—34no—
207C23—62no—
208C23—62no—
209C23—62no—
210C23—62no—
211C23—62no—
212C23—62no—
213C23C23×D311yescharacteristic
214D4—68no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—34no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—128no—
241C2×C4#23512yes—
242C2×C4#23512yes—
243C2×C4#23512yes—
244C2×C4D3×D412yes—
245C2×C4D3×Q812yes—
246C2×C3C2×C22⋊Q811yescharacteristic
247C2×C3#52911yescharacteristic
248C2×C3#54011yescharacteristic
249C2×C3#54611yescharacteristic
250C2×C3#55011yescharacteristic
251C2×C3#55411yescharacteristic
252C2×C3#55411yescharacteristic
253D3—44no—
254C22—64no—
255C22—64no—
256C22—64no—
257C22—122no—
258C22—122no—
259C22—122no—
260C22—122no—
261C22C2×#23511yescharacteristic
262C22C2×#23511yescharacteristic
263C22C2×#23511yescharacteristic
264C22C2×D3×D411yescharacteristic
265C22C2×D3×Q811yescharacteristic
266C22#100911yescharacteristic
267C22#100911yescharacteristic
268C4—22no—
269C4—22no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—64no—
275C4—122no—
276C4—122no—
277C4#89614yes—
278C3#245411yescharacteristic, complemented, Sylow
279C2—124no—
280C2#547911yescharacteristic
281C2C2×#89611yescharacteristic
282C2#553511yescharacteristic
283C2#555011yescharacteristic
284C2#555311yescharacteristic
285C2#557911yescharacteristic
286C2#557911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C4⋊C4C2
prime radicalC22×D12C22
hypercenterC4×C4⋊C4D3
hypocenterC3#2454
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#870C2semidirect
3. C3×C4×C4⋊C4C2semidirect
4. #4993C2semidirect
5. #4989C2semidirect
8. #4928C2semidirect
9. #4928C2semidirect
10. #4676C2semidirect
11. #4667C2semidirect
48. C4×C4⋊C4D3semidirect
278. C3#2454semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D122
C2×#8702
C3×C4×C4⋊C42
#49932
#49892
#49392
#49282
#46762
#46672
#24543