Finite Group Databasebeta

← #78041

Subgroups — #78041

Order 384.

564 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 147 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×#887C211yesmaximal, characteristic, complemented
2C2×#67⋊C4C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4989C212yesmaximal, complemented
6#4976C212yesmaximal, complemented
7#4957C212yesmaximal
8#4947C211yesmaximal, characteristic, complemented
9#4946C212yesmaximal, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4650C211yesmaximal, characteristic, complemented
12#2580—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2212yes—
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×Dic3⋊C4C2212yes—
29C2×C4×Dic3C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31#887—24no—
32#67⋊C4—24no—
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×Q8—31no—
39C2×C4×D4—31no—
40#399—31no—
41#399D311yescharacteristic, complemented
42#393—31no—
43#393—31no—
44#390—32no—
45#389—32no—
46#385—32no—
47#385—32no—
48#382—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C2×#67—22no—
56C2×#67—22no—
57C2×#67Q814yescomplemented
58C22×Dic3—22no—
59C22×Dic3—42no—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2312yes—
64C22×Dic3C2311yescharacteristic
65C2×C4×D3—22no—
66C2×C4×D3—42no—
67C2×Dic6—22no—
68C2×Dic6—24no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4—42no—
71C3×C22⋊C4—22no—
72C3×C22⋊C4—42no—
73#217—24no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—42no—
77Dic3⋊C4—22no—
78Dic3⋊C4—24no—
79Dic3⋊C4—24no—
80Dic3⋊C4—42no—
81C4×Dic3—24no—
82C4×Dic3D414yes—
83C22×Q8—31no—
84C22×D4—31no—
85C23×C4—31no—
86C23×C4—31no—
87C4×Q8—64no—
88C4×D4—64no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C4⋊C4C2×D312yes—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D311yescharacteristic
100C2×C22⋊C4C2×D312yes—
101C2×C42—31no—
102C2×C42—32no—
103#95—32no—
104#95—31no—
105#95—32no—
106#95—32no—
107#95C2×D311yescharacteristic
108C23×C3—21no—
109C23×C3—22no—
110C23×C3—21no—
111C23×C3—22no—
112C23×C3C2×Q812yes—
113C23×C3C2411yescharacteristic
114C22×D3—21no—
115C22×D3—22no—
116C22×D3—21no—
117C22×D3—22no—
118C22×D3C2×Q812yes—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4C2×D412yes—
129#67—28no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—44no—
139C2×Dic3C2×D412yes—
140C2×Dic3C2×D412yes—
141C2×Dic3C2×Q812yes—
142C2×Dic3#4114yes—
143C2×Dic3#4112yes—
144C4×D3—44no—
145Dic6—28no—
146C24—31no—
147C24C22×D311yescharacteristic
148C2×Q8—64no—
149C2×Q8—62no—
150C2×D4—34no—
151C2×D4—62no—
152C2×D4—62no—
153C22×C4—31no—
154C22×C4—32no—
155C22×C4—31no—
156C22×C4—32no—
157C22×C4—31no—
158C22×C4—62no—
159C22×C4—62no—
160C22×C4—122no—
161C22×C4—122no—
162C22×C4C22×D312yes—
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D312yes—
165C22×C4C22×D311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—62no—
169C4⋊C4—64no—
170C4⋊C4—64no—
171C4⋊C4—122no—
172C22⋊C4—22no—
173C22⋊C4—42no—
174C22⋊C4—64no—
175C22⋊C4—64no—
176C22⋊C4—62no—
177C22⋊C4—122no—
178C42—34no—
179C42—64no—
180C22×C3—22no—
181C22×C3—24no—
182C22×C3—41no—
183C22×C3—41no—
184C22×C3—42no—
185C22×C3C22×D411yescharacteristic
186C22×C3C22×Q811yescharacteristic
187C22×C3C2×#4111yescharacteristic
188C22×C3C2×#4112yes—
189C22×C3H(5,𝔽2)12yes—
190C2×D3—22no—
191C2×D3—24no—
192C2×D3—41no—
193C2×D3—41no—
194C2×D3—42no—
195C3×C4—22no—
196C3×C4—41no—
197C3×C4—42no—
198C3×C4—42no—
199Dic3—24no—
200Dic3—22no—
201Dic3—41no—
202Dic3—42no—
203Dic3C22⋊Q814yes—
204C23—21no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—32no—
209C23—61no—
210C23—62no—
211C23—62no—
212C23—61no—
213C23D3×Q812yes—
214C23C23×D311yescharacteristic
215Q8—68no—
216D4—68no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—32no—
230C2×C4—34no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—61no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—124no—
240C2×C4—124no—
241C2×C4D3×D412yes—
242C2×C3—42no—
243C2×C3C2×C22⋊Q811yescharacteristic
244C2×C3#52511yescharacteristic
245C2×C3#53511yescharacteristic
246C2×C3#55412yes—
247C2×C3#55912yes—
248D3—42no—
249C22—24no—
250C22—22no—
251C22—41no—
252C22—42no—
253C22—41no—
254C22—64no—
255C22—62no—
256C22—121no—
257C22—122no—
258C22—121no—
259C22C2×D3×D411yescharacteristic
260C22#100412yes—
261C22C2×D3×Q811yescharacteristic
262C22D3×#4112yes—
263C22D3×#4111yescharacteristic
264C4—22no—
265C4—34no—
266C4—42no—
267C4—41no—
268C4—42no—
269C4—64no—
270C4—62no—
271C4—122no—
272C4—121no—
273C3#258011yescharacteristic, complemented, Sylow
274C2—42no—
275C2—122no—
276C2D3×C22⋊Q811yescharacteristic
277C2#563612yes—
278C2#565611yescharacteristic
279C2#566412yes—
280C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC22×#67C22
hypercenter#399D3
hypocenterC3#2580
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#887C2semidirect
3. C3×#399C2semidirect
4. #4993C2semidirect
5. #4989C2semidirect
6. #4976C2semidirect
8. #4947C2semidirect
9. #4946C2semidirect
10. #4673C2semidirect
11. #4650C2semidirect
41. #399D3semidirect
57. C2×#67Q8semidirect
273. C3#2580semidirect

Maximal subgroups

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

subgroupindex
C2×#8872
C2×#67⋊C42
C3×#3992
#49932
#49892
#49762
#49572
#49472
#49462
#46732
#46502
#25803