Finite Group Databasebeta

← #73585

Subgroups — #73585

Order 384.

652 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#930C211yesmaximal, characteristic
2C2×#883C212yesmaximal, complemented
3C2×C4×D12C211yesmaximal, characteristic, complemented
4C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
5#5212C211yesmaximal, characteristic, complemented
6#4961C212yesmaximal, complemented
7#4939C211yesmaximal, characteristic, complemented
8#4935C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4674C212yesmaximal, complemented
11#4668C212yesmaximal, complemented
12#2422—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2211yescharacteristic
16C22×C4×D3C2212yes—
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
21C2×#217C2212yes—
22#930—24no—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2212yes—
32C2×Dic3⋊C4C2212yes—
33#883—28no—
34C4×D12—24no—
35C3×#95C2211yescharacteristic
36C3×#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#123—32no—
40C2×#121—31no—
41C2×C4×D4—31no—
42#397—32no—
43#394—32no—
44#390—31no—
45#389—31no—
46#385—32no—
47#384—31no—
48#380—31no—
49C4×C22⋊C4D311yescharacteristic, complemented
50C24×C3C2311yescharacteristic
51C23×D3C2312yes—
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4—22no—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C2×#67—24no—
62C2×#67—24no—
63C2×#67—42no—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C2×D12—22no—
68C2×D12—22no—
69C2×D12D414yes—
70C2×C4×D3—24no—
71C2×C4×D3—24no—
72C2×C4×D3—42no—
73C3×C22⋊C4—22no—
74C3×C22⋊C4D414yescomplemented
75C3×C42—22no—
76C3×C42—22no—
77#217—24no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—24no—
82C4⋊Dic3—22no—
83C4⋊Dic3—24no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86C4⋊Dic3—42no—
87Dic3⋊C4—24no—
88C22×D4—32no—
89C22×D4—31no—
90C23×C4—32no—
91C23×C4C2×D311yescharacteristic
92#123—68no—
93#121—64no—
94C4×D4—64no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4C2×D311yescharacteristic
107C2×C22⋊C4C2×D311yescharacteristic
108C2×C42C2×D311yescharacteristic
109C2×C42C2×D311yescharacteristic
110#95—32no—
111#95—32no—
112#95C2×D311yescharacteristic
113#95C2×D311yescharacteristic
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3C2×D412yes—
121C23×C3C2411yescharacteristic
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3C2×D414yes—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—44no—
142C2×C3×C4C2×D412yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4C2×D412yes—
145C2×C3×C4#4112yes—
146C2×C3×C4#4112yes—
147C2×C3×C4#4112yes—
148#67—48no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158D12—28no—
159C4×D3—48no—
160C24—32no—
161C24C22×D311yescharacteristic
162C2×D4—34no—
163C2×D4—64no—
164C2×D4—64no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—122no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—32no—
172C22×C4—32no—
173C22×C4—32no—
174C22×C4—64no—
175C22×C4—64no—
176C22×C4—122no—
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D311yescharacteristic
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C22×C4C22×D311yescharacteristic
183C4⋊C4—64no—
184C4⋊C4—64no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4—62no—
188C4⋊C4—122no—
189C22⋊C4—22no—
190C22⋊C4—64no—
191C22⋊C4—64no—
192C22⋊C4—64no—
193C22⋊C4—64no—
194C22⋊C4—64no—
195C22⋊C4D1214yescomplemented
196C42—22no—
197C42—22no—
198C22×C3—22no—
199C22×C3—22no—
200C22×C3—22no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3C22×D411yescharacteristic
206C22×C3C22×D411yescharacteristic
207C22×C3C2×#4111yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3H(5,𝔽2)11yescharacteristic
211C22×C32−1+411yescharacteristic
212C2×D3—24no—
213C2×D3—24no—
214C2×D3—24no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—42no—
219C3×C4—22no—
220C3×C4—22no—
221C3×C4—22no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4#12314yescomplemented
225Dic3—42no—
226Dic3—42no—
227Dic3—42no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—34no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23C2×D1212yes—
242C23C23×D311yescharacteristic
243D4—68no—
244D4—128no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—44no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—128no—
268C2×C4C2×D1212yes—
269C2×C4C2×D1212yes—
270C2×C4#23512yes—
271C2×C4#23512yes—
272C2×C4D3×D412yes—
273C2×C4#23712yes—
274C2×C3—42no—
275C2×C3C2×#12311yescharacteristic
276C2×C3#53211yescharacteristic
277C2×C3#54211yescharacteristic
278C2×C3D4211yescharacteristic
279C2×C3#54711yescharacteristic
280C2×C3#54711yescharacteristic
281C2×C3#55311yescharacteristic
282D3—44no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—64no—
291C22—64no—
292C22—64no—
293C22—122no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22C22×D1211yescharacteristic
298C22C2×#23511yescharacteristic
299C22C2×#23511yescharacteristic
300C22C2×D3×D411yescharacteristic
301C22C2×#23711yescharacteristic
302C22#100911yescharacteristic
303C22#101011yescharacteristic
304C4—22no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—122no—
310C4—122no—
311C4—122no—
312C4#88314yescomplemented
313C3#242211yescharacteristic, complemented, Sylow
314C2—42no—
315C2—124no—
316C2#547711yescharacteristic
317C2C2×#88311yescharacteristic
318C2#553411yescharacteristic
319C2D4×D1211yescharacteristic
320C2#555111yescharacteristic
321C2#555511yescharacteristic
322C2#556511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C22⋊C4C2
prime radicalC2×#930C2
hypercenterC4×C22⋊C4D3
hypocenterC3#2422
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#883C2semidirect
3. C2×C4×D12C2semidirect
4. C3×C4×C22⋊C4C2semidirect
5. #5212C2semidirect
6. #4961C2semidirect
7. #4939C2semidirect
8. #4935C2semidirect
9. #4928C2semidirect
10. #4674C2semidirect
11. #4668C2semidirect
49. C4×C22⋊C4D3semidirect
74. C3×C22⋊C4D4semidirect
195. C22⋊C4D12semidirect
224. C3×C4#123semidirect
312. C4#883semidirect
313. C3#2422semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×#8832
C2×C4×D122
C3×C4×C22⋊C42
#52122
#49612
#49392
#49352
#49282
#46742
#46682
#24223