Finite Group Databasebeta

← #78037

Subgroups — #78037

Order 384.

676 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 171 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×#896C212yesmaximal, complemented
2C2×#888C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C211yesmaximal, characteristic
4C2×#878C212yesmaximal, complemented
5C3×#399C211yesmaximal, characteristic, complemented
6#4993C211yesmaximal, characteristic, complemented
7#4975C212yesmaximal, complemented
8#4957C212yesmaximal
9#4949C211yesmaximal, characteristic, complemented
10#4677C211yesmaximal, characteristic, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2542—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C23×Dic3C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×Dic6C2212yes—
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34#896—28no—
35#888—24no—
36#67⋊C4—24no—
37#878—28no—
38C3×#95C2211yescharacteristic
39C3⋊#95C2211yescharacteristic
40C2×C4⋊Q8—31no—
41C2×C22⋊Q8—32no—
42C2×C22⋊Q8—32no—
43C2×C4×D4—31no—
44#399—31no—
45#399—31no—
46#399—31no—
47#399D311yescharacteristic, complemented
48#390—32no—
49#384—32no—
50C4×C4⋊C4—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C2×#67—22no—
58C2×#67—22no—
59C2×#67Q814yescomplemented
60C22×Dic3—24no—
61C22×Dic3—22no—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C2×C4×D3—22no—
68C2×C4×D3—24no—
69C2×Dic6—24no—
70C2×Dic6—24no—
71C2×Dic6—24no—
72C3×C4⋊C4—22no—
73C3×C4⋊C4—24no—
74C3×C22⋊C4—24no—
75C3×C22⋊C4—22no—
76#217—24no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80C4⋊Dic3—24no—
81C4⋊Dic3—22no—
82Dic3⋊C4—24no—
83Dic3⋊C4—22no—
84Dic3⋊C4—24no—
85Dic3⋊C4—24no—
86Dic3⋊C4D418yescomplemented
87C4×Dic3—24no—
88C4×Dic3D414yes—
89C22×Q8—32no—
90C22×D4—31no—
91C23×C4—31no—
92C23×C4—31no—
93C4⋊Q8—64no—
94C22⋊Q8—68no—
95C22⋊Q8—68no—
96C4×D4—64no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4C2×D312yes—
105C2×C4⋊C4C2×D311yescharacteristic
106C2×C22⋊C4—32no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4C2×D311yescharacteristic
111C2×C22⋊C4C2×D312yes—
112C2×C42—31no—
113C2×C42—32no—
114#95—31no—
115#95C2×D311yescharacteristic
116C23×C3—22no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—22no—
120C23×C3C2×Q812yes—
121C23×C3C2411yescharacteristic
122C22×D3—22no—
123C22×D3—21no—
124C22×D3—21no—
125C22×D3—22no—
126C22×D3C2×Q812yes—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4C2×D412yes—
136C2×C3×C4C2×D414yes—
137#67—28no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—44no—
146C2×Dic3C2×D414yes—
147C2×Dic3C2×D412yes—
148C2×Dic3C2×D412yes—
149C2×Dic3C2×D414yes—
150C2×Dic3C2×Q812yes—
151C2×Dic3#4112yes—
152C4×D3—44no—
153Dic6—48no—
154C24—31no—
155C24C22×D311yescharacteristic
156C2×Q8—64no—
157C2×Q8—64no—
158C2×Q8—64no—
159C2×D4—34no—
160C2×D4—62no—
161C2×D4—62no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—32no—
165C22×C4—32no—
166C22×C4—31no—
167C22×C4—62no—
168C22×C4—64no—
169C22×C4—62no—
170C22×C4—64no—
171C22×C4C22×D312yes—
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D312yes—
175C4⋊C4—24no—
176C4⋊C4—22no—
177C4⋊C4—38no—
178C4⋊C4—62no—
179C4⋊C4—64no—
180C4⋊C4—64no—
181C4⋊C4—64no—
182C4⋊C4—64no—
183C4⋊C4—62no—
184C22⋊C4—22no—
185C22⋊C4—24no—
186C22⋊C4—64no—
187C22⋊C4—64no—
188C22⋊C4—62no—
189C22⋊C4—64no—
190C42—34no—
191C42—64no—
192C22×C3—22no—
193C22×C3—24no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—42no—
197C22×C3C22×D411yescharacteristic
198C22×C3C22×D412yes—
199C22×C3C22×Q811yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C32−1+412yes—
202C2×D3—22no—
203C2×D3—24no—
204C2×D3—41no—
205C2×D3—41no—
206C2×D3—42no—
207C3×C4—22no—
208C3×C4—24no—
209C3×C4—42no—
210C3×C4—41no—
211Dic3—24no—
212Dic3—42no—
213Dic3—41no—
214Dic3C22⋊Q814yescomplemented
215Dic3#12714yes—
216C23—22no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—32no—
221C23—62no—
222C23—61no—
223C23—61no—
224C23—62no—
225C23D3×Q812yes—
226C23C23×D311yescharacteristic
227Q8—128no—
228D4—68no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—32no—
238C2×C4—32no—
239C2×C4—34no—
240C2×C4—32no—
241C2×C4—32no—
242C2×C4—34no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—61no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—124no—
251C2×C4—124no—
252C2×C4D3×D412yes—
253C2×C4D3×D414yes—
254C2×C3—42no—
255C2×C3C2×C22⋊Q811yescharacteristic
256C2×C3C2×#12711yescharacteristic
257C2×C3#53611yescharacteristic
258C2×C3#54712yes—
259C2×C3D4×Q812yes—
260D3—42no—
261C22—24no—
262C22—22no—
263C22—41no—
264C22—42no—
265C22—41no—
266C22—64no—
267C22—62no—
268C22—121no—
269C22—122no—
270C22—121no—
271C22C2×D3×D412yes—
272C22C2×D3×D411yescharacteristic
273C22C2×D3×Q811yescharacteristic
274C22D3×#4111yescharacteristic
275C22#101012yes—
276C4—22no—
277C4—24no—
278C4—34no—
279C4—34no—
280C4—41no—
281C4—42no—
282C4—64no—
283C4—121no—
284C4—122no—
285C3#254211yescharacteristic, complemented, Sylow
286C2—42no—
287C2—122no—
288C2D3×C22⋊Q811yescharacteristic
289C2#563212yes—
290C2#564811yescharacteristic
291C2#565512yes—
292C2D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC22×#67C22
hypercenter#399D3
hypocenterC3#2542
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
2. C2×#888C2semidirect
4. C2×#878C2semidirect
5. C3×#399C2semidirect
6. #4993C2semidirect
7. #4975C2semidirect
9. #4949C2semidirect
10. #4677C2semidirect
11. #4649C2semidirect
47. #399D3semidirect
59. C2×#67Q8semidirect
86. Dic3⋊C4D4semidirect
214. Dic3C22⋊Q8semidirect
285. C3#2542semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8882
C2×#67⋊C42
C2×#8782
C3×#3992
#49932
#49752
#49572
#49492
#46772
#46492
#25423