Finite Group Databasebeta

← #78030

Subgroups — #78030

Order 384.

708 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 161 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, complemented
2C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#884C211yesmaximal, characteristic
4C2×#883C212yesmaximal, complemented
5C3×#399C211yesmaximal, characteristic, complemented
6#4993C211yesmaximal, characteristic, complemented
7#4975C212yesmaximal, complemented
8#4957C212yesmaximal, complemented
9#4949C211yesmaximal, characteristic, complemented
10#4677C211yesmaximal, characteristic, complemented
11#4666C211yesmaximal, characteristic, complemented
12#2501—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×D12C2211yescharacteristic
16C22×C4×D3C2212yes—
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2212yes—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2212yes—
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2212yes—
33C2×C4×Dic3C2212yes—
34#894—28no—
35D3×C4⋊C4—24no—
36#884—24no—
37#883—28no—
38C3×#95C2211yescharacteristic
39C3⋊#95C2211yescharacteristic
40C2×#123—32no—
41C2×#123—32no—
42C2×#121—31no—
43C22×C4⋊C4—31no—
44#399—31no—
45#399—31no—
46#399—31no—
47#399D311yescharacteristic, complemented
48#390—32no—
49#384—32no—
50C4×C22⋊C4—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2312yes—
58C2×#67—24no—
59C2×#67—22no—
60C2×#67—22no—
61C2×#67—24no—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C2×D12—22no—
67C2×D12—24no—
68C2×C4×D3—24no—
69C2×C4×D3—24no—
70C2×C4×D3—24no—
71C2×C4×D3—24no—
72C2×C4×D3D414yes—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—24no—
75C3×C22⋊C4—24no—
76C3×C22⋊C4—22no—
77#217—24no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4D418yescomplemented
82C4⋊Dic3—24no—
83C4⋊Dic3—22no—
84Dic3⋊C4—24no—
85Dic3⋊C4—22no—
86Dic3⋊C4—24no—
87Dic3⋊C4—24no—
88C4×Dic3—24no—
89C22×D4—31no—
90C22×D4—31no—
91C22×D4—31no—
92C23×C4—31no—
93C23×C4—32no—
94#123—68no—
95#123—68no—
96#121—64no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—64no—
104C2×C4⋊C4C2×D311yescharacteristic
105C2×C4⋊C4C2×D312yes—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4C2×D311yescharacteristic
112C2×C22⋊C4C2×D312yes—
113C2×C42—32no—
114#95—31no—
115#95C2×D311yescharacteristic
116C23×C3—22no—
117C23×C3—21no—
118C23×C3—22no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3C2411yescharacteristic
122C22×D3—22no—
123C22×D3—21no—
124C22×D3—21no—
125C22×D3—22no—
126C22×D3—21no—
127C22×D3—22no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3C2×D414yes—
131C22×D3C2×D412yes—
132C2×C3×C4—22no—
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—22no—
140C2×C3×C4C2×D414yes—
141C2×C3×C4C2×D412yes—
142#67—44no—
143#67—44no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3C2×D414yes—
152C2×Dic3C2×D412yes—
153C2×Dic3#4114yes—
154D12—44no—
155C4×D3—28no—
156C4×D3—48no—
157C24—31no—
158C24—31no—
159C24C22×D311yescharacteristic
160C2×D4—62no—
161C2×D4—64no—
162C2×D4—62no—
163C2×D4—64no—
164C2×D4—62no—
165C2×D4—64no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—34no—
169C22×C4—32no—
170C22×C4—31no—
171C22×C4—64no—
172C22×C4—64no—
173C22×C4—64no—
174C22×C4—64no—
175C22×C4C22×D312yes—
176C22×C4C22×D311yescharacteristic
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D312yes—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—64no—
182C4⋊C4—62no—
183C4⋊C4—64no—
184C4⋊C4—64no—
185C4⋊C4—62no—
186C4⋊C4—64no—
187C22⋊C4—22no—
188C22⋊C4—24no—
189C22⋊C4—38no—
190C22⋊C4—64no—
191C22⋊C4—64no—
192C22⋊C4—62no—
193C22⋊C4—64no—
194C42—64no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3C22×D411yescharacteristic
201C22×C3C22×D412yes—
202C22×C3C2×#4112yes—
203C22×C32−1+411yescharacteristic
204C22×C32−1+411yescharacteristic
205C2×D3—22no—
206C2×D3—22no—
207C2×D3—24no—
208C2×D3—24no—
209C2×D3—41no—
210C2×D3—41no—
211C2×D3—41no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3#12714yes—
215C3×C4—22no—
216C3×C4—24no—
217C3×C4—42no—
218C3×C4—41no—
219Dic3—24no—
220Dic3—42no—
221Dic3—41no—
222Dic3#12314yescomplemented
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—34no—
229C23—32no—
230C23—61no—
231C23—62no—
232C23—61no—
233C23—62no—
234C23—61no—
235C23—61no—
236C23—62no—
237C23—61no—
238C23C23×D311yescharacteristic
239D4—124no—
240D4—124no—
241D4—124no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—32no—
251C2×C4—34no—
252C2×C4—34no—
253C2×C4—68no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—128no—
262C2×C4D3×D414yes—
263C2×C4D3×D412yes—
264C2×C3—81no—
265C2×C3C2×#12311yescharacteristic
266C2×C3C2×#12711yescharacteristic
267C2×C3#53611yescharacteristic
268C2×C3#54712yes—
269C2×C3#54712yes—
270D3—24no—
271D3—81no—
272C22—34no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—62no—
279C22—64no—
280C22—62no—
281C22—64no—
282C22—121no—
283C22—122no—
284C22—121no—
285C22—121no—
286C22—122no—
287C22C2×D3×D411yescharacteristic
288C22C2×D3×D412yes—
289C22#100711yescharacteristic
290C22D3×#4112yes—
291C22#101011yescharacteristic
292C4—22no—
293C4—24no—
294C4—34no—
295C4—42no—
296C4—41no—
297C4—64no—
298C4—122no—
299C4—121no—
300C3#250111yescharacteristic, complemented, Sylow
301C2—64no—
302C2—81no—
303C2—241no—
304C2#562811yescharacteristic
305C2#563112yes—
306C2D3×#12311yescharacteristic
307C2#565512yes—
308C2D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC2×#884C2
hypercenter#399D3
hypocenterC3#2501
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#894C2semidirect
2. C2×D3×C4⋊C4C2semidirect
4. C2×#883C2semidirect
5. C3×#399C2semidirect
6. #4993C2semidirect
7. #4975C2semidirect
8. #4957C2semidirect
9. #4949C2semidirect
10. #4677C2semidirect
11. #4666C2semidirect
47. #399D3semidirect
81. (C2×D3)⋊C4D4semidirect
222. Dic3#123semidirect
300. C3#2501semidirect

Maximal subgroups

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

subgroupindex
C2×#8942
C2×D3×C4⋊C42
C2×#8842
C2×#8832
C3×#3992
#49932
#49752
#49572
#49492
#46772
#46662
#25013