Finite Group Databasebeta

← #77764

Subgroups — #77764

Order 384.

624 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 151 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×#894C212yesmaximal
2C2×#893C211yesmaximal, characteristic
3C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#398C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4981C211yesmaximal, characteristic, complemented
7#4976C212yesmaximal, complemented
8#4674C211yesmaximal, characteristic, complemented
9#4667C212yesmaximal, complemented
10#4667C211yesmaximal, characteristic, 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×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#894—28no—
31#893—24no—
32D3×C4⋊C4—24no—
33C3×#95C2212yes—
34C3×#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C2×#123—32no—
40C2×C4×D4—31no—
41C22×C4⋊C4—31no—
42#398—31no—
43#398D311yescharacteristic, complemented
44#397—31no—
45#397—31no—
46#389—32no—
47#381—32no—
48#380—32no—
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—24no—
65C2×C4×D3—24no—
66C2×C4×D3—24no—
67C2×C4×D3—24no—
68C2×C4×D3—22no—
69C2×C4×D3—24no—
70C2×C4×D3D414yes—
71C3×C4⋊C4—22no—
72C3×C4⋊C4—22no—
73C3×C4⋊C4—24no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—22no—
79C4⋊Dic3—22no—
80Dic3⋊C4—24no—
81Dic3⋊C4—24no—
82Dic3⋊C4—24no—
83C4×Dic3—22no—
84C22×D4—31no—
85C23×C4—31no—
86C23×C4—32no—
87C23×C4—31no—
88#123—68no—
89C4×D4—64no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—64no—
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C4⋊C4C2×D311yescharacteristic
97C2×C4⋊C4C2×D312yes—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C42—31no—
104#95—32no—
105#95—31no—
106#95—32no—
107#95—32no—
108#95C2×D312yes—
109#95C2×D311yescharacteristic
110C23×C3C2411yescharacteristic
111C22×D3—22no—
112C22×D3—22no—
113C22×D3—21no—
114C22×D3—22no—
115C22×D3—21no—
116C22×D3—22no—
117C22×D3C2×D412yes—
118C22×D3C2×D412yes—
119C22×D3C2×D412yes—
120C22×D3#4112yes—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D412yes—
134C2×Dic3—21no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3C2×D412yes—
143C2×Dic3#4114yes—
144D12—28no—
145C4×D3—28no—
146C4×D3—48no—
147C4×D3—44no—
148C24—31no—
149C24—31no—
150C2×D4—34no—
151C2×D4—62no—
152C2×D4—62no—
153C22×C4—31no—
154C22×C4—34no—
155C22×C4—31no—
156C22×C4—32no—
157C22×C4—32no—
158C22×C4—64no—
159C22×C4—62no—
160C22×C4—64no—
161C22×C4—64no—
162C22×C4—64no—
163C22×C4—64no—
164C22×C4C22×D311yescharacteristic
165C22×C4C22×D312yes—
166C22×C4C22×D312yes—
167C22×C4C22×D311yescharacteristic
168C22×C4C22×D311yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—24no—
172C4⋊C4—64no—
173C4⋊C4—64no—
174C4⋊C4—62no—
175C4⋊C4—64no—
176C22⋊C4—62no—
177C22⋊C4—64no—
178C22⋊C4—64no—
179C22⋊C4—64no—
180C22⋊C4—62no—
181C42—62no—
182C22×C3C22×D411yescharacteristic
183C22×C3C22×D411yescharacteristic
184C22×C3C2×#4111yescharacteristic
185C22×C3C2×#4112yes—
186C22×C32−1+412yes—
187C2×D3—22no—
188C2×D3—22no—
189C2×D3—24no—
190C2×D3—24no—
191C2×D3—22no—
192C2×D3—24no—
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—42no—
196C2×D3#12114yes—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—42no—
200C3×C4—41no—
201C3×C4—42no—
202Dic3—42no—
203Dic3—41no—
204Dic3—42no—
205Dic3#12314yescomplemented
206C23—32no—
207C23—32no—
208C23—32no—
209C23—32no—
210C23—62no—
211C23—62no—
212C23—61no—
213C23—62no—
214C23—62no—
215C23—61no—
216C23C23×D311yescharacteristic
217D4—68no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—32no—
230C2×C4—34no—
231C2×C4—68no—
232C2×C4—61no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—124no—
241C2×C4—128no—
242C2×C4D3×D412yes—
243C2×C4D3×D412yes—
244C2×C3C2×#12111yescharacteristic
245C2×C3C2×#12311yescharacteristic
246C2×C3#53611yescharacteristic
247C2×C3#54712yes—
248C2×C3#55212yes—
249D3—24no—
250D3—42no—
251C22—34no—
252C22—62no—
253C22—64no—
254C22—62no—
255C22—64no—
256C22—62no—
257C22—64no—
258C22—121no—
259C22—122no—
260C22—121no—
261C22C2×D3×D411yescharacteristic
262C22C2×D3×D411yescharacteristic
263C22#100712yes—
264C22D3×#4112yes—
265C22D3×#4111yescharacteristic
266C4—22no—
267C4—22no—
268C4—34no—
269C4—41no—
270C4—42no—
271C4—42no—
272C4—121no—
273C4—122no—
274C4—122no—
275C3#248611yescharacteristic, complemented, Sylow
276C2—64no—
277C2—122no—
278C2D3×#12111yescharacteristic
279C2#563112yes—
280C2D3×#12311yescharacteristic
281C2#569312yes—
282C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#398C2
prime radicalC22×D12C22
hypercenter#398D3
hypocenterC3#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C2×D3×C4⋊C4C2semidirect
4. C3×#398C2semidirect
5. #4992C2semidirect
6. #4981C2semidirect
7. #4976C2semidirect
8. #4674C2semidirect
9. #4667C2semidirect
10. #4667C2semidirect
11. #4664C2semidirect
43. #398D3semidirect
205. Dic3#123semidirect
275. 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×#8932
C2×D3×C4⋊C42
C3×#3982
#49922
#49812
#49762
#46742
#46672
#46642
#24863