Finite Group Databasebeta

← #74455

Subgroups — #74455

Order 384.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×D3×C22⋊C4C211yesmaximal, characteristic
3C3×#385C211yesmaximal, characteristic, complemented
4#5210C211yesmaximal, characteristic, complemented
5#4992C212yesmaximal, complemented
6#4989C212yesmaximal, complemented
7#4955C211yesmaximal, characteristic
8#4943C211yesmaximal, characteristic, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4664C212yesmaximal, complemented
12#2483—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2212yescomplemented
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×(C2×D3)⋊C4—24no—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yes—
30C2×Dic3⋊C4C2212yescomplemented
31D3×C22⋊C4—24no—
32C3×#95C2212yescomplemented
33C3⋊#95C2212yescomplemented
34C3⋊#95C2212yescomplemented
35C3⋊#95C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C22×C22⋊C4—31no—
39C22×C22⋊C4—31no—
40#397—32no—
41#397—32no—
42#385—31no—
43#385—32no—
44#385D311yescharacteristic, complemented
45#381—31no—
46#381—31no—
47#381—32no—
48#379—31no—
49C24×C3C2311yescharacteristic
50C23×D3—22no—
51C23×D3—22no—
52C23×D3—24no—
53C23×D3—42no—
54C23×D3D414yes—
55C23×D3C2311yescharacteristic
56C23×D3C2311yescharacteristic
57C22×C3×C4—22no—
58C22×C3×C4—42no—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2312yes—
63C22×Dic3—22no—
64C22×Dic3—22no—
65C22×Dic3—22no—
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70C2×C4×D3—22no—
71C2×C4×D3—22no—
72C2×C4×D3—24no—
73C2×C4×D3—42no—
74C3×C4⋊C4—42no—
75C3×C22⋊C4—22no—
76C3×C22⋊C4—22no—
77#217—22no—
78#217—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—42no—
85Dic3⋊C4—42no—
86C25—31no—
87C23×C4—31no—
88C23×C4—31no—
89C23×C4—31no—
90C23×C4C2×D311yescharacteristic
91C2×C4⋊C4—32no—
92C2×C4⋊C4C2×D312yescomplemented
93C2×C22⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—64no—
102C2×C22⋊C4—64no—
103C2×C22⋊C4C2×D311yescharacteristic
104C2×C22⋊C4C2×D311yescharacteristic
105#95—32no—
106#95—32no—
107#95—32no—
108#95—32no—
109#95—32no—
110#95C2×D312yescomplemented
111C23×C3—22no—
112C23×C3—22no—
113C23×C3C2×D412yes—
114C23×C3#4112yes—
115C23×C3#4112yes—
116C23×C3C2411yescharacteristic
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—22no—
120C22×D3—28no—
121C22×D3—21no—
122C22×D3—21no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—28no—
126C22×D3—44no—
127C22×D3—48no—
128C22×D3—48no—
129C22×D3C2×D412yes—
130C22×D3C2×D412yes—
131C22×D3#4114yes—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—44no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—44no—
154C4×D3—44no—
155C4×D3—44no—
156C24—31no—
157C24—31no—
158C24—34no—
159C24—64no—
160C24—62no—
161C24—62no—
162C24—122no—
163C24C22×D311yescharacteristic
164C22×C4—22no—
165C22×C4—32no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—42no—
170C22×C4—62no—
171C22×C4—62no—
172C22×C4—64no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4—122no—
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D312yes—
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D312yes—
181C4⋊C4—42no—
182C4⋊C4—122no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—62no—
186C22⋊C4—62no—
187C22⋊C4—64no—
188C22⋊C4—64no—
189C22⋊C4—64no—
190C22⋊C4—62no—
191C22⋊C4—62no—
192C22⋊C4—122no—
193C22×C3—22no—
194C22×C3—22no—
195C22×C3—24no—
196C22×C3—24no—
197C22×C3#12314yes—
198C22×C3C22×D411yescharacteristic
199C22×C3C2×#4111yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C3C2×#4112yes—
202C22×C3H(5,𝔽2)12yes—
203C2×D3—22no—
204C2×D3—22no—
205C2×D3—24no—
206C2×D3—24no—
207C2×D3—22no—
208C2×D3—22no—
209C2×D3—22no—
210C2×D3—416no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3#12314yes—
214C3×C4—41no—
215C3×C4—41no—
216C3×C4—42no—
217C3×C4—42no—
218Dic3—41no—
219Dic3—42no—
220Dic3—41no—
221Dic3—42no—
222C23—22no—
223C23—22no—
224C23—32no—
225C23—34no—
226C23—32no—
227C23—68no—
228C23—61no—
229C23—62no—
230C23—61no—
231C23—68no—
232C23—61no—
233C23—62no—
234C23—61no—
235C23—62no—
236C23—124no—
237C23—128no—
238C23—128no—
239C23#23512yes—
240C23D3×D412yes—
241C23#23912yes—
242C23C23×D311yescharacteristic
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—124no—
265C2×C4—124no—
266C2×C4—124no—
267C2×C3—24no—
268C2×C3C2×#12311yescharacteristic
269C2×C3C2×#12311yescharacteristic
270C2×C3#53411yescharacteristic
271C2×C3#55112yes—
272C2×C3#55112yes—
273D3—24no—
274D3—42no—
275C22—24no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—34no—
280C22—64no—
281C22—62no—
282C22—64no—
283C22—62no—
284C22—62no—
285C22—62no—
286C22—62no—
287C22—1216no—
288C22—122no—
289C22—122no—
290C22#89414yes—
291C22C2×#23511yescharacteristic
292C22C2×D3×D411yescharacteristic
293C22#100412yes—
294C22C2×#23911yescharacteristic
295C22D3×#4112yes—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—41no—
300C4—122no—
301C4—121no—
302C4—122no—
303C4—121no—
304C3#248311yescharacteristic, complemented, Sylow
305C2—24no—
306C2—64no—
307C2—122no—
308C2C2×#89411yescharacteristic
309C2#555612yes—
310C2#558911yescharacteristic
311C2#563712yes—
312C2D3×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC24×D3C22
hypercenter#385D3
hypocenterC3#2483
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C3×#385C2semidirect
4. #5210C2semidirect
5. #4992C2semidirect
6. #4989C2semidirect
8. #4943C2semidirect
9. #4943C2semidirect
10. #4675C2semidirect
11. #4664C2semidirect
18. C2×C3×C4⋊C4C22semidirect
30. C2×Dic3⋊C4C22semidirect
32. C3×#95C22semidirect
33. C3⋊#95C22semidirect
34. C3⋊#95C22semidirect
35. C3⋊#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
44. #385D3semidirect
92. C2×C4⋊C4C2×D3semidirect
110. #95C2×D3semidirect
304. C3#2483semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D3)⋊C42
C2×D3×C22⋊C42
C3×#3852
#52102
#49922
#49892
#49552
#49432
#46752
#46642
#24833