Finite Group Databasebeta

← #76231

Subgroups — #76231

Order 384.

624 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#894C212yesmaximal
2C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C4×D12C211yesmaximal, characteristic
4C3×#389C211yesmaximal, characteristic, complemented
5#4987C211yesmaximal, characteristic, complemented
6#4987C211yesmaximal, characteristic, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4982C212yesmaximal, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4664C212yesmaximal, complemented
12#2486—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3C2212yes—
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
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—
31#894—28no—
32D3×C4⋊C4—24no—
33C4×D12—24no—
34C3×#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#123—32no—
40C2×C4×D4—31no—
41C22×C4⋊C4—31no—
42#398—32no—
43#397—32no—
44#389—31no—
45#389D311yescharacteristic, complemented
46#381—32no—
47#380—31no—
48#380—31no—
49#380—31no—
50C23×D3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C2×D12—22no—
62C2×D12—22no—
63C2×D12D414yes—
64C2×C4×D3—22no—
65C2×C4×D3—22no—
66C2×C4×D3—24no—
67C2×C4×D3—24no—
68C2×C4×D3—24no—
69C2×C4×D3—22no—
70C2×C4×D3—22no—
71C2×C4×D3—22no—
72C2×C4×D3D414yes—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—24no—
75C3×C4⋊C4—22no—
76C3×C42—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—22no—
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—22no—
85Dic3⋊C4—24no—
86Dic3⋊C4—24no—
87C22×D4—31no—
88C23×C4—31no—
89C23×C4—32no—
90C23×C4—31no—
91#123—68no—
92C4×D4—64no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—64no—
99C2×C4⋊C4C2×D311yescharacteristic
100C2×C4⋊C4C2×D312yes—
101C2×C4⋊C4C2×D311yescharacteristic
102C2×C22⋊C4—31no—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C42C2×D311yescharacteristic
108#95—32no—
109#95—32no—
110#95—32no—
111#95—32no—
112#95C2×D312yes—
113C23×C3C2411yescharacteristic
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×D3—21no—
123C22×D3—21no—
124C22×D3C2×D412yes—
125C22×D3C2×D412yes—
126C22×D3C2×D412yes—
127C22×D3#4112yes—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4C2×D412yes—
140C2×C3×C4C2×D412yes—
141C2×C3×C4#4112yes—
142C2×C3×C4#4112yes—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3C2×D412yes—
155D12—28no—
156C4×D3—28no—
157C4×D3—48no—
158C4×D3—44no—
159C24—31no—
160C24—31no—
161C2×D4—34no—
162C2×D4—62no—
163C2×D4—62no—
164C22×C4—31no—
165C22×C4—34no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—32no—
169C22×C4—64no—
170C22×C4—64no—
171C22×C4—64no—
172C22×C4—62no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D312yes—
179C22×C4C22×D312yes—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—64no—
186C4⋊C4—62no—
187C4⋊C4—62no—
188C4⋊C4—64no—
189C4⋊C4—62no—
190C22⋊C4—62no—
191C22⋊C4—62no—
192C22⋊C4—64no—
193C22⋊C4—64no—
194C22⋊C4—64no—
195C42—22no—
196C22×C3C22×D411yescharacteristic
197C22×C3C22×D411yescharacteristic
198C22×C3C2×#4111yescharacteristic
199C22×C3C2×#4111yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C32−1+411yescharacteristic
202C22×C32−1+411yescharacteristic
203C2×D3—22no—
204C2×D3—22no—
205C2×D3—22no—
206C2×D3—22no—
207C2×D3—22no—
208C2×D3—22no—
209C2×D3—22no—
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—41no—
213C2×D3—41no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3#12114yes—
217C3×C4—22no—
218C3×C4—42no—
219C3×C4—41no—
220C3×C4—42no—
221C3×C4#12314yescomplemented
222Dic3—22no—
223Dic3—42no—
224Dic3—41no—
225Dic3—42no—
226C23—32no—
227C23—32no—
228C23—32no—
229C23—32no—
230C23—61no—
231C23—61no—
232C23—61no—
233C23—61no—
234C23—61no—
235C23—61no—
236C23—61no—
237C23—61no—
238C23—61no—
239C23—61no—
240C23C23×D311yescharacteristic
241D4—68no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—32no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—68no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—128no—
267C2×C4—124no—
268C2×C4#23512yes—
269C2×C4D3×D412yes—
270C2×C4D3×D412yes—
271C2×C4#23912yes—
272C2×C3C2×#12111yescharacteristic
273C2×C3C2×#12311yescharacteristic
274C2×C3#53611yescharacteristic
275C2×C3#54711yescharacteristic
276C2×C3#54711yescharacteristic
277C2×C3#55211yescharacteristic
278C2×C3#55211yescharacteristic
279D3—24no—
280D3—42no—
281C22—34no—
282C22—62no—
283C22—62no—
284C22—62no—
285C22—62no—
286C22—62no—
287C22—62no—
288C22—62no—
289C22—62no—
290C22—62no—
291C22—121no—
292C22—121no—
293C22—121no—
294C22—121no—
295C22C2×#23511yescharacteristic
296C22C2×D3×D411yescharacteristic
297C22C2×D3×D411yescharacteristic
298C22C2×#23911yescharacteristic
299C22#100711yescharacteristic
300C22D3×#4111yescharacteristic
301C22#101011yescharacteristic
302C4—22no—
303C4—42no—
304C4—42no—
305C4—41no—
306C4—62no—
307C4—122no—
308C4—121no—
309C4—122no—
310C4#89414yescomplemented
311C3#248611yescharacteristic, complemented, Sylow
312C2—64no—
313C2—122no—
314C2C2×#89411yescharacteristic
315C2#555111yescharacteristic
316C2#558111yescharacteristic
317C2D3×#12111yescharacteristic
318C2#562811yescharacteristic
319C2#569411yescharacteristic
320C2#572611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC22×D12C22
hypercenter#389D3
hypocenterC3#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×D3×C4⋊C4C2semidirect
4. C3×#389C2semidirect
5. #4987C2semidirect
6. #4987C2semidirect
7. #4987C2semidirect
8. #4982C2semidirect
9. #4928C2semidirect
10. #4675C2semidirect
11. #4664C2semidirect
45. #389D3semidirect
221. C3×C4#123semidirect
310. C4#894semidirect
311. C3#2486semidirect

Maximal subgroups

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

subgroupindex
C2×#8942
C2×D3×C4⋊C42
C2×C4×D122
C3×#3892
#49872
#49822
#49282
#46752
#46642
#24863