Finite Group Databasebeta

← #76232

Subgroups — #76232

Order 384.

584 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×D12C211yesmaximal, characteristic
3C3×#389C211yesmaximal, characteristic, complemented
4#4993C212yesmaximal, complemented
5#4989C212yesmaximal, complemented
6#4987C211yesmaximal, characteristic, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4987C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4665C212yesmaximal
11#4660C212yesmaximal, complemented
12#2487—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C42C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31D3×C4⋊C4—24no—
32C4×D12—24no—
33C3×#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×D4—31no—
39C22×C4⋊C4—31no—
40#400—32no—
41#399—32no—
42#389—31no—
43#389D311yescharacteristic, complemented
44#386—32no—
45#385—32no—
46#380—31no—
47#380—31no—
48#380—31no—
49C23×D3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C2×D12—22no—
61C2×D12—22no—
62C2×D12Q814yes—
63C2×C4×D3—22no—
64C2×C4×D3—22no—
65C2×C4×D3—24no—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—42no—
70C2×C4×D3—42no—
71C2×C4×D3D414yes—
72C3×C4⋊C4—22no—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—42no—
75C3×C42—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—42no—
80(C2×D3)⋊C4—42no—
81C4⋊Dic3—22no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84Dic3⋊C4—24no—
85Dic3⋊C4—42no—
86C22×D4—31no—
87C23×C4—31no—
88C23×C4—31no—
89C23×C4—32no—
90C4×D4—64no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—64no—
97C2×C4⋊C4C2×D311yescharacteristic
98C2×C4⋊C4C2×D311yescharacteristic
99C2×C4⋊C4C2×D312yes—
100C2×C22⋊C4—31no—
101C2×C22⋊C4—31no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C42C2×D311yescharacteristic
106#95—32no—
107#95—32no—
108#95—32no—
109#95—32no—
110#95C2×D312yes—
111C23×C3C2411yescharacteristic
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3—21no—
116C22×D3—21no—
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—21no—
122C22×D3C2×D412yes—
123C22×D3C2×Q812yes—
124C22×D3C2×Q812yes—
125C22×D3#4112yes—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×Q812yes—
139C2×C3×C4#4112yes—
140C2×C3×C4#4112yes—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3C2×D412yes—
153D12—28no—
154C4×D3—28no—
155C4×D3—48no—
156C4×D3—44no—
157C24—31no—
158C24—31no—
159C2×D4—34no—
160C2×D4—62no—
161C2×D4—62no—
162C22×C4—31no—
163C22×C4—34no—
164C22×C4—31no—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—64no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4—122no—
174C22×C4—122no—
175C22×C4C22×D312yes—
176C22×C4C22×D312yes—
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D311yescharacteristic
179C22×C4C22×D311yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—64no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4—122no—
188C22⋊C4—62no—
189C22⋊C4—62no—
190C22⋊C4—64no—
191C22⋊C4—122no—
192C22⋊C4—122no—
193C42—22no—
194C22×C3C22×D411yescharacteristic
195C22×C3C22×Q811yescharacteristic
196C22×C3C2×#4111yescharacteristic
197C22×C3C2×#4111yescharacteristic
198C22×C3C2×#4111yescharacteristic
199C22×C3H(5,𝔽2)11yescharacteristic
200C22×C3H(5,𝔽2)11yescharacteristic
201C2×D3—22no—
202C2×D3—22no—
203C2×D3—22no—
204C2×D3—22no—
205C2×D3—22no—
206C2×D3—22no—
207C2×D3—22no—
208C2×D3—22no—
209C2×D3—22no—
210C2×D3—41no—
211C2×D3—41no—
212C2×D3—41no—
213C2×D3—41no—
214C2×D3C22⋊Q814yes—
215C3×C4—22no—
216C3×C4—42no—
217C3×C4—41no—
218C3×C4—42no—
219C3×C4#12314yes—
220Dic3—22no—
221Dic3—41no—
222Dic3—42no—
223Dic3—42no—
224C23—32no—
225C23—32no—
226C23—32no—
227C23—32no—
228C23—61no—
229C23—61no—
230C23—61no—
231C23—61no—
232C23—61no—
233C23—61no—
234C23—61no—
235C23—61no—
236C23—61no—
237C23—61no—
238C23C23×D311yescharacteristic
239D4—68no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—32no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—68no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—128no—
265C2×C4—124no—
266C2×C4#23512yes—
267C2×C4D3×D412yes—
268C2×C4D3×Q812yes—
269C2×C4#23912yes—
270C2×C3C2×C22⋊Q811yescharacteristic
271C2×C3C2×#12311yescharacteristic
272C2×C3#53511yescharacteristic
273C2×C3#55311yescharacteristic
274C2×C3#55311yescharacteristic
275C2×C3#55411yescharacteristic
276C2×C3#55411yescharacteristic
277D3—24no—
278D3—42no—
279C22—34no—
280C22—62no—
281C22—62no—
282C22—62no—
283C22—62no—
284C22—62no—
285C22—62no—
286C22—62no—
287C22—62no—
288C22—62no—
289C22—121no—
290C22—121no—
291C22—121no—
292C22—121no—
293C22C2×#23511yescharacteristic
294C22C2×D3×D411yescharacteristic
295C22#100411yescharacteristic
296C22C2×D3×Q811yescharacteristic
297C22C2×#23911yescharacteristic
298C22D3×#4111yescharacteristic
299C22#100911yescharacteristic
300C4—22no—
301C4—42no—
302C4—42no—
303C4—41no—
304C4—62no—
305C4—122no—
306C4—121no—
307C4—122no—
308C4#89414yes—
309C3#248711yescharacteristic, complemented, Sylow
310C2—64no—
311C2—122no—
312C2C2×#89411yescharacteristic
313C2#555811yescharacteristic
314C2#557911yescharacteristic
315C2#560711yescharacteristic
316C2D3×C22⋊Q811yescharacteristic
317C2#569511yescharacteristic
318C2#573011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2487
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊C4C2semidirect
3. C3×#389C2semidirect
4. #4993C2semidirect
5. #4989C2semidirect
6. #4987C2semidirect
7. #4987C2semidirect
8. #4987C2semidirect
9. #4928C2semidirect
11. #4660C2semidirect
43. #389D3semidirect
309. C3#2487semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊C42
C2×C4×D122
C3×#3892
#49932
#49892
#49872
#49282
#46652
#46602
#24873