Finite Group Databasebeta

← #76349

Subgroups — #76349

Order 384.

676 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 185 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×D4×Dic3C211yesmaximal, characteristic
2C2×#888C211yesmaximal, characteristic, complemented
3C2×#878C214yesmaximal, complemented
4C3×#390C211yesmaximal, characteristic
5#5221C211yesmaximal, characteristic
6#4975C211yesmaximal, characteristic, complemented
7#4949C214yesmaximal, complemented
8#4935C211yesmaximal, characteristic, complemented
9C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
10#2542—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×Dic3C2212yes—
13C22×Dic6C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2214yes—
16C2×C3×C42C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2214yes—
19D4×Dic3—24no—
20C2×C4⋊Dic3C2214yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2214yes—
25C2×Dic3⋊C4C2212yes—
26C2×C4×Dic3C2211yescharacteristic
27C2×C4×Dic3C2211yescharacteristic
28#888—24no—
29#878—216no—
30C3⋊#95C2212yes—
31C2×C4⋊Q8—31no—
32C2×C22⋊Q8—34no—
33C2×C4×D4—31no—
34#399—34no—
35#390—31no—
36#390D311yescharacteristic
37#384—31no—
38#384—31no—
39C4×C4⋊C4—31no—
40C24×C3C2312yes—
41C2×C3×D4—22no—
42C2×C3×D4—22no—
43C2×C3×D4Q814yescomplemented
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×Dic3—28no—
48C22×Dic3—24no—
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2311yescharacteristic
52C22×Dic3C2312yes—
53C22×Dic3C2311yescharacteristic
54C2×Dic6—28no—
55C2×Dic6—24no—
56C3×C4⋊C4—22no—
57C3×C22⋊C4—28no—
58C3×C42—22no—
59#217—28no—
60#217—24no—
61C4⋊Dic3—28no—
62C4⋊Dic3—22no—
63C4⋊Dic3—22no—
64Dic3⋊C4—28no—
65Dic3⋊C4—24no—
66Dic3⋊C4D418yescomplemented
67C4×Dic3—22no—
68C4×Dic3D414yes—
69C22×Q8—32no—
70C22×D4C2×D311yescharacteristic
71C23×C4—32no—
72C4⋊Q8—64no—
73C22⋊Q8—616no—
74C4×D4—64no—
75C2×C4⋊C4—32no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4—34no—
78C2×C4⋊C4—34no—
79C2×C4⋊C4—31no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4C2×D311yescharacteristic
82C2×C22⋊C4—32no—
83C2×C22⋊C4—34no—
84C2×C22⋊C4C2×D314yes—
85C2×C42—31no—
86C2×C42—31no—
87C2×C42C2×D311yescharacteristic
88#95—32no—
89C23×C3—24no—
90C23×C3—22no—
91C23×C3—22no—
92C23×C3—24no—
93C23×C3C2×Q814yes—
94C23×C3C2411yescharacteristic
95C3×D4—28no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—24no—
99C2×C3×C4—22no—
100C2×C3×C4C2×D414yes—
101C2×C3×C4C2×D412yes—
102C2×C3×C4C2×Q812yes—
103C2×C3×C4#4112yes—
104C2×Dic3—22no—
105C2×Dic3—22no—
106C2×Dic3—22no—
107C2×Dic3—22no—
108C2×Dic3—24no—
109C2×Dic3—24no—
110C2×Dic3—22no—
111C2×Dic3—48no—
112C2×Dic3C2×D412yes—
113C2×Dic3C2×D414yes—
114C2×Dic3C2×D412yes—
115C2×Dic3C2×D414yes—
116Dic6—48no—
117C24C22×D312yes—
118C2×Q8—68no—
119C2×Q8—64no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4Dic614yescomplemented
123C22×C4—31no—
124C22×C4—31no—
125C22×C4—32no—
126C22×C4—32no—
127C22×C4—32no—
128C22×C4—64no—
129C22×C4—68no—
130C22×C4C22×D312yes—
131C22×C4C22×D312yes—
132C22×C4C22×D311yescharacteristic
133C4⋊C4—22no—
134C4⋊C4—38no—
135C4⋊C4—68no—
136C4⋊C4—64no—
137C4⋊C4—62no—
138C4⋊C4—68no—
139C4⋊C4—62no—
140C22⋊C4—28no—
141C22⋊C4—64no—
142C22⋊C4—68no—
143C42—22no—
144C42—34no—
145C42—62no—
146C22×C3—24no—
147C22×C3—28no—
148C22×C3—42no—
149C22×C3—42no—
150C22×C3—44no—
151C22×C3C22×D411yescharacteristic
152C22×C3C22×D412yes—
153C22×C3C22×Q811yescharacteristic
154C22×C3C2×#4111yescharacteristic
155C22×C32−1+412yes—
156C3×C4—24no—
157C3×C4—42no—
158C3×C4C22⋊Q814yescomplemented
159Dic3—22no—
160Dic3—24no—
161Dic3—42no—
162Dic3—42no—
163Dic3#12714yes—
164C23—24no—
165C23—22no—
166C23—22no—
167C23—24no—
168C23C2×Dic614yes—
169C23C23×D311yescharacteristic
170Q8—128no—
171D4—28no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—32no—
177C2×C4—34no—
178C2×C4—32no—
179C2×C4—34no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—62no—
184C2×C4—64no—
185C2×C4—64no—
186C2×C4—62no—
187C2×C4—128no—
188C2×C4C2×Dic612yes—
189C2×C4D3×D414yes—
190C2×C4D3×D412yes—
191C2×C4#23712yes—
192C2×C3—44no—
193C2×C3C2×C22⋊Q811yescharacteristic
194C2×C3C2×#12711yescharacteristic
195C2×C3#53611yescharacteristic
196C2×C3#54712yes—
197C2×C3D4×Q812yes—
198C22—24no—
199C22—28no—
200C22—42no—
201C22—42no—
202C22—44no—
203C22C22×Dic611yescharacteristic
204C22C2×D3×D411yescharacteristic
205C22C2×D3×D412yes—
206C22C2×#23711yescharacteristic
207C22#101012yes—
208C4—24no—
209C4—34no—
210C4—42no—
211C4—62no—
212C4—64no—
213C4—122no—
214C4—122no—
215C4#87814yescomplemented
216C3#254211yescharacteristic, complemented, Sylow
217C2—44no—
218C2C2×#87811yescharacteristic
219C2D4×Dic612yes—
220C2#561112yes—
221C2#567511yescharacteristic
222C2D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC22×C3×D4C22
hypercenter#390D3
hypocenterC3#2542
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#888C2semidirect
3. C2×#878C2semidirect
6. #4975C2semidirect
7. #4949C2semidirect
8. #4935C2semidirect
9. C4×Dic3⋊C4C2semidirect
43. C2×C3×D4Q8semidirect
66. Dic3⋊C4D4semidirect
122. C2×D4Dic6semidirect
158. C3×C4C22⋊Q8semidirect
215. C4#878semidirect
216. C3#2542semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic32
C2×#8882
C2×#8782
C3×#3902
#52212
#49752
#49492
#49352
C4×Dic3⋊C42
#25423