Finite Group Databasebeta

← #73673

Subgroups — #73673

Order 384.

512 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#875C211yesmaximal, characteristic
2C2×C4×Dic6C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4#4993C212yesmaximal, complemented
5#4990C212yesmaximal, complemented
6#4939C211yesmaximal, characteristic
7#4939C211yesmaximal, characteristic
8#4929C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4669C212yesmaximal, complemented
12#2448—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
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×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2212yes—
30C2×C4×Dic3C2212yes—
31#875—24no—
32C4×Dic6—24no—
33C3×#95C2211yescharacteristic
34C3×#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#124—31no—
39C2×C4×Q8—31no—
40#399—32no—
41#397—32no—
42#391—31no—
43#390—31no—
44#390—31no—
45#389—31no—
46#388—32no—
47#388—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×D12—22no—
63C2×Dic6—22no—
64C2×Dic6—22no—
65C2×Dic6D414yes—
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—24no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—42no—
76(C2×D3)⋊C4—42no—
77C4⋊Dic3—22no—
78C4⋊Dic3—22no—
79Dic3⋊C4—24no—
80Dic3⋊C4—42no—
81C4×Dic3—24no—
82C22×Q8—31no—
83C22×D4—31no—
84#124—64no—
85C4×Q8—64no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4C2×D311yescharacteristic
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C42—32no—
99C2×C42C2×D311yescharacteristic
100C2×C42C2×D311yescharacteristic
101C2×C42C2×D311yescharacteristic
102#95—32no—
103#95—32no—
104#95—32no—
105#95C2×D311yescharacteristic
106#95C2×D311yescharacteristic
107C23×C3C2411yescharacteristic
108C22×D3—22no—
109C22×D3—22no—
110C22×D3—22no—
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—22no—
115C2×C3×C4—21no—
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×C4C2×D412yes—
132C2×C3×C4#4112yes—
133C2×C3×C4#4112yes—
134C2×C3×C4#4112yes—
135C2×C3×C4#4112yes—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3C2×D414yes—
145D12—44no—
146Dic6—28no—
147C24—32no—
148C2×Q8—34no—
149C2×Q8—62no—
150C2×Q8—62no—
151C2×D4—62no—
152C2×D4—62no—
153C2×D4—62no—
154C22×C4—32no—
155C22×C4—32no—
156C22×C4—32no—
157C22×C4C22×D311yescharacteristic
158C22×C4C22×D311yescharacteristic
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C22×C4C22×D311yescharacteristic
162C22×C4C22×D311yescharacteristic
163C22×C4C22×D311yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—64no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—122no—
170C22⋊C4—64no—
171C22⋊C4—64no—
172C22⋊C4—64no—
173C22⋊C4—64no—
174C22⋊C4—122no—
175C22⋊C4—122no—
176C42—22no—
177C42—22no—
178C42—22no—
179C42—64no—
180C22×C3C22×D411yescharacteristic
181C22×C3C2×#4111yescharacteristic
182C22×C3C2×#4111yescharacteristic
183C22×C3C2×#4111yescharacteristic
184C22×C3C2×#4111yescharacteristic
185C22×C3H(5,𝔽2)11yescharacteristic
186C22×C32−1+411yescharacteristic
187C2×D3—42no—
188C2×D3—42no—
189C2×D3—42no—
190C2×D3—42no—
191C2×D3—42no—
192C2×D3—42no—
193C2×D3—42no—
194C3×C4—22no—
195C3×C4—22no—
196C3×C4—41no—
197C3×C4—41no—
198C3×C4—41no—
199C3×C4—41no—
200C3×C4#12314yes—
201Dic3—24no—
202Dic3—42no—
203Dic3—42no—
204C23—62no—
205C23—62no—
206C23—62no—
207C23—62no—
208C23—62no—
209C23—62no—
210C23—62no—
211C23C23×D311yescharacteristic
212Q8—68no—
213D4—124no—
214C2×C4—21no—
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—34no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4#23512yes—
240C2×C4#23512yes—
241C2×C4#23512yes—
242C2×C4D3×D412yes—
243C2×C4#23912yes—
244C2×C3C2×#12311yescharacteristic
245C2×C3#52911yescharacteristic
246C2×C3#54211yescharacteristic
247C2×C3#54811yescharacteristic
248C2×C3#55011yescharacteristic
249C2×C3#55611yescharacteristic
250C2×C3#55911yescharacteristic
251D3—82no—
252C22—122no—
253C22—122no—
254C22—122no—
255C22—122no—
256C22—122no—
257C22—122no—
258C22—122no—
259C22C2×#23511yescharacteristic
260C22C2×#23511yescharacteristic
261C22C2×#23511yescharacteristic
262C22C2×D3×D411yescharacteristic
263C22C2×#23911yescharacteristic
264C22#100911yescharacteristic
265C22#101011yescharacteristic
266C4—22no—
267C4—22no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—64no—
273C4—122no—
274C4—122no—
275C4#89414yes—
276C3#244811yescharacteristic, complemented, Sylow
277C2—242no—
278C2#547911yescharacteristic
279C2C2×#89411yescharacteristic
280C2#553411yescharacteristic
281C2#555211yescharacteristic
282C2#555311yescharacteristic
283C2#558411yescharacteristic
284C2#558511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C4⋊C4C2
prime radicalC22×D12C22
hypercenterC4×C4⋊C4D3
hypocenterC3#2448
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×C4×Dic6C2semidirect
3. C3×C4×C4⋊C4C2semidirect
4. #4993C2semidirect
5. #4990C2semidirect
8. #4929C2semidirect
9. #4928C2semidirect
10. #4675C2semidirect
11. #4669C2semidirect
48. C4×C4⋊C4D3semidirect
276. C3#2448semidirect

Maximal subgroups

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

subgroupindex
C2×#8752
C2×C4×Dic62
C3×C4×C4⋊C42
#49932
#49902
#49392
#49292
#49282
#46752
#46692
#24483