Finite Group Databasebeta

← #76215

Subgroups — #76215

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×#895C211yesmaximal, characteristic
2C2×#894C212yesmaximal, complemented
3C2×C4×D12C211yesmaximal, characteristic, complemented
4C3×#389C211yesmaximal, characteristic, complemented
5#4989C212yesmaximal, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4975C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4672C212yesmaximal, complemented
12#2422—31nomaximal, Sylow
13C22×D12C2212yes—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C42C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#895—24no—
33#894—28no—
34C4×D12—24no—
35C3×#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C2×#123—32no—
40C2×#121—31no—
41C2×C4×D4—31no—
42#397—32no—
43#394—32no—
44#390—31no—
45#389D311yescharacteristic, complemented
46#385—32no—
47#384—31no—
48#380—31no—
49C4×C22⋊C4—31no—
50C23×D3C2312yes—
51C23×D3C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C2×D12—22no—
62C2×D12—22no—
63C2×D12—24no—
64C2×D12—24no—
65C2×D12—42no—
66C2×D12D414yes—
67C2×C4×D3—24no—
68C2×C4×D3—24no—
69C2×C4×D3—22no—
70C2×C4×D3—22no—
71C2×C4×D3—22no—
72C2×C4×D3—42no—
73C3×C4⋊C4—24no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4—22no—
76C3×C42—22no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4D414yescomplemented
84C4⋊Dic3—22no—
85Dic3⋊C4—24no—
86Dic3⋊C4—42no—
87C4×Dic3—22no—
88C22×D4—31no—
89C22×D4—32no—
90C23×C4—31no—
91C23×C4—32no—
92#123—68no—
93#121—64no—
94C4×D4—64no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4C2×D311yescharacteristic
99C2×C4⋊C4C2×D312yes—
100C2×C4⋊C4C2×D311yescharacteristic
101C2×C22⋊C4—31no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C42—31no—
109C2×C42C2×D311yescharacteristic
110#95—31no—
111#95—31no—
112#95—32no—
113#95C2×D312yes—
114C23×C3C2411yescharacteristic
115C22×D3—22no—
116C22×D3—21no—
117C22×D3—21no—
118C22×D3—22no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—21no—
125C22×D3—21no—
126C22×D3—22no—
127C22×D3C2×D412yes—
128C22×D3C2×D414yes—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4C2×D412yes—
141C2×C3×C4C2×D412yes—
142C2×C3×C4#4112yes—
143C2×C3×C4#4112yes—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3C2×D412yes—
155C2×Dic3#4112yes—
156D12—28no—
157D12—48no—
158C4×D3—48no—
159C4×D3—44no—
160C24—31no—
161C24—32no—
162C2×D4—34no—
163C2×D4—64no—
164C2×D4—64no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—122no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—32no—
172C22×C4—64no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4—64no—
176C22×C4—62no—
177C22×C4—122no—
178C22×C4C22×D311yescharacteristic
179C22×C4C22×D312yes—
180C22×C4C22×D312yes—
181C22×C4C22×D311yescharacteristic
182C22×C4C22×D311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—24no—
186C4⋊C4—64no—
187C4⋊C4—62no—
188C4⋊C4—122no—
189C22⋊C4—34no—
190C22⋊C4—64no—
191C22⋊C4—62no—
192C22⋊C4—64no—
193C22⋊C4—64no—
194C22⋊C4—64no—
195C22⋊C4—64no—
196C42—22no—
197C42—62no—
198C22×C3C22×D411yescharacteristic
199C22×C3C22×D411yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C3C2×#4111yescharacteristic
202C22×C3C2×#4111yescharacteristic
203C22×C3H(5,𝔽2)11yescharacteristic
204C22×C32−1+411yescharacteristic
205C2×D3—24no—
206C2×D3—22no—
207C2×D3—24no—
208C2×D3—22no—
209C2×D3—22no—
210C2×D3—24no—
211C2×D3—41no—
212C2×D3—42no—
213C2×D3—41no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—41no—
217C2×D3—41no—
218C2×D3—42no—
219C3×C4—22no—
220C3×C4—42no—
221C3×C4—42no—
222C3×C4—41no—
223C3×C4#12314yescomplemented
224Dic3—22no—
225Dic3—22no—
226Dic3—41no—
227Dic3—42no—
228C23—32no—
229C23—34no—
230C23—61no—
231C23—62no—
232C23—62no—
233C23—61no—
234C23—61no—
235C23—62no—
236C23—61no—
237C23—62no—
238C23—61no—
239C23—62no—
240C23—62no—
241C23—61no—
242C23C23×D311yescharacteristic
243D4—68no—
244D4—128no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—128no—
269C2×C4—124no—
270C2×C4#23512yes—
271C2×C4D3×D412yes—
272C2×C4D3×D412yes—
273C2×C4#23912yes—
274C2×C3C2×#12311yescharacteristic
275C2×C3#53211yescharacteristic
276C2×C3#54211yescharacteristic
277C2×C3D4211yescharacteristic
278C2×C3#54711yescharacteristic
279C2×C3#54711yescharacteristic
280C2×C3#55311yescharacteristic
281D3—44no—
282D3—42no—
283C22—62no—
284C22—64no—
285C22—62no—
286C22—64no—
287C22—62no—
288C22—64no—
289C22—121no—
290C22—122no—
291C22—122no—
292C22—121no—
293C22—122no—
294C22—121no—
295C22—122no—
296C22—121no—
297C22C2×#23511yescharacteristic
298C22C2×D3×D411yescharacteristic
299C22C2×D3×D411yescharacteristic
300C22C2×#23911yescharacteristic
301C22#100711yescharacteristic
302C22D3×#4111yescharacteristic
303C22#100911yescharacteristic
304C4—22no—
305C4—42no—
306C4—42no—
307C4—41no—
308C4—62no—
309C4—62no—
310C4—121no—
311C4—122no—
312C4#89414yescomplemented
313C3#242211yescharacteristic, complemented, Sylow
314C2—122no—
315C2—124no—
316C2C2×#89411yescharacteristic
317C2#554811yescharacteristic
318C2#558211yescharacteristic
319C2#560911yescharacteristic
320C2#563111yescharacteristic
321C2#569511yescharacteristic
322C2#572611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC2×#895C2
hypercenter#389D3
hypocenterC3#2422
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#894C2semidirect
3. C2×C4×D12C2semidirect
4. C3×#389C2semidirect
5. #4989C2semidirect
6. #4988C2semidirect
7. #4987C2semidirect
8. #4975C2semidirect
9. C4×(C2×D3)⋊C4C2semidirect
10. #4675C2semidirect
11. #4672C2semidirect
45. #389D3semidirect
83. (C2×D3)⋊C4D4semidirect
223. C3×C4#123semidirect
312. C4#894semidirect
313. C3#2422semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8942
C2×C4×D122
C3×#3892
#49892
#49882
#49872
#49752
C4×(C2×D3)⋊C42
#46752
#46722
#24223