Finite Group Databasebeta

← #78401

Subgroups — #78401

Order 384.

578 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#895C211yesmaximal, characteristic, complemented
2C2×#894C211yesmaximal, characteristic, complemented
3C3×#400C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4991C211yesmaximal, characteristic, complemented
6#4676C212yesmaximal, complemented
7#4672C212yesmaximal, complemented
8#4672C212yesmaximal, complemented
9#4671C211yesmaximal, characteristic, complemented
10#4671C212yesmaximal, complemented
11#4667C211yesmaximal, characteristic, complemented
12#2813—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×D12C2212yes—
15C22×D12C2211yescharacteristic
16C22×D12C2212yes—
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32#895—24no—
33#894—24no—
34C3×#95C2211yescharacteristic
35C3×#95C2212yes—
36C3×#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×#123—31no—
39C2×#121—31no—
40#400D311yescharacteristic, complemented
41#399—31no—
42#396—32no—
43#394—31no—
44#394—32no—
45#394—32no—
46#392—31no—
47#392—32no—
48#380—31no—
49C23×D3C2311yescharacteristic
50C23×D3C2312yes—
51C23×D3C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×D12—22no—
61C2×D12—24no—
62C2×D12—22no—
63C2×D12—42no—
64C2×D12—42no—
65C2×D12—42no—
66C2×D12—42no—
67C2×D12—42no—
68C2×D12—42no—
69C2×D12—42no—
70C2×C4×D3—42no—
71C2×C4×D3D414yescomplemented
72C3×C4⋊C4—22no—
73C3×C4⋊C4—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—42no—
79(C2×D3)⋊C4—42no—
80(C2×D3)⋊C4—42no—
81(C2×D3)⋊C4—42no—
82(C2×D3)⋊C4—42no—
83Dic3⋊C4—22no—
84Dic3⋊C4—42no—
85Dic3⋊C4—42no—
86C22×D4—32no—
87C22×D4—31no—
88C22×D4—31no—
89C22×D4—32no—
90C23×C4—31no—
91#123—64no—
92#121—64no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4C2×D311yescharacteristic
97C2×C4⋊C4C2×D311yescharacteristic
98C2×C22⋊C4—32no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—31no—
101C2×C22⋊C4—31no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107#95—31no—
108#95C2×D311yescharacteristic
109#95C2×D312yes—
110#95C2×D312yes—
111C23×C3C2411yescharacteristic
112C22×D3—21no—
113C22×D3—22no—
114C22×D3—22no—
115C22×D3—21no—
116C22×D3—21no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—21no—
126C22×D3—21no—
127C22×D3—22no—
128C22×D3C2×D412yes—
129C2×C3×C4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4C2×D412yes—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3C2×D412yes—
149D12—44no—
150D12—44no—
151D12—48no—
152D12—48no—
153C4×D3—44no—
154C24—31no—
155C24—31no—
156C24—32no—
157C2×D4—64no—
158C2×D4—62no—
159C2×D4—62no—
160C2×D4—122no—
161C2×D4—122no—
162C2×D4—122no—
163C2×D4—122no—
164C2×D4—122no—
165C2×D4—122no—
166C2×D4—122no—
167C22×C4—31no—
168C22×C4—34no—
169C22×C4—32no—
170C22×C4—31no—
171C22×C4—122no—
172C22×C4C22×D312yes—
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D312yes—
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C4⋊C4—122no—
182C22⋊C4—64no—
183C22⋊C4—62no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22⋊C4—122no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C22×C3C22×D411yescharacteristic
192C22×C3H(5,𝔽2)12yes—
193C22×C3H(5,𝔽2)12yes—
194C22×C3H(5,𝔽2)11yescharacteristic
195C22×C32−1+411yescharacteristic
196C2×D3—22no—
197C2×D3—22no—
198C2×D3—22no—
199C2×D3—42no—
200C2×D3—42no—
201C2×D3—42no—
202C2×D3—41no—
203C2×D3—42no—
204C2×D3—41no—
205C2×D3—41no—
206C2×D3—42no—
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C3×C4—22no—
214C3×C4—41no—
215C3×C4—42no—
216C3×C4—41no—
217C3×C4—42no—
218Dic3—22no—
219Dic3—41no—
220Dic3—42no—
221C23—32no—
222C23—61no—
223C23—61no—
224C23—62no—
225C23—62no—
226C23—61no—
227C23—62no—
228C23—61no—
229C23—62no—
230C23—62no—
231C23—62no—
232C23—62no—
233C23—62no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—61no—
238C23C23×D311yescharacteristic
239D4—124no—
240D4—124no—
241D4—128no—
242D4—128no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—32no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—124no—
263C2×C4D3×D412yes—
264C2×C3#53411yescharacteristic
265C2×C3#53511yescharacteristic
266C2×C3#53711yescharacteristic
267C2×C3#56012yes—
268C2×C3#56212yes—
269D3—42no—
270D3—82no—
271D3—81no—
272C22—62no—
273C22—62no—
274C22—62no—
275C22—122no—
276C22—122no—
277C22—122no—
278C22—121no—
279C22—122no—
280C22—121no—
281C22—122no—
282C22—122no—
283C22—122no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22—121no—
288C22—122no—
289C22C2×D3×D411yescharacteristic
290C22#100711yescharacteristic
291C22#100912yes—
292C22#100912yes—
293C22#100911yescharacteristic
294C4—22no—
295C4—41no—
296C4—42no—
297C4—42no—
298C4—41no—
299C4—62no—
300C4—121no—
301C4—122no—
302C3#281311yescharacteristic, complemented, Sylow
303C2—122no—
304C2—242no—
305C2—241no—
306C2#562911yescharacteristic
307C2#565311yescharacteristic
308C2#567411yescharacteristic
309C2#570012yes—
310C2#571112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
hypercenter#400D3
hypocenterC3#2813
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#895C2semidirect
2. C2×#894C2semidirect
3. C3×#400C2semidirect
4. #4993C2semidirect
5. #4991C2semidirect
6. #4676C2semidirect
7. #4672C2semidirect
8. #4672C2semidirect
9. #4671C2semidirect
10. #4671C2semidirect
11. #4667C2semidirect
40. #400D3semidirect
71. C2×C4×D3D4semidirect
302. C3#2813semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8942
C3×#4002
#49932
#49912
#46762
#46722
#46712
#46672
#28133