Finite Group Databasebeta

← #76519

Subgroups — #76519

Order 384.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 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×#946C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic
3C3×#391C211yesmaximal, characteristic, complemented
4#5229C211yesmaximal, characteristic, complemented
5#4975C211yesmaximal, characteristic, complemented
6C4×(C2×D3)⋊C4C211yesmaximal, characteristic
7#4935C211yesmaximal, characteristic, complemented
8#4668C214yesmaximal
9#4657C214yesmaximal, complemented
10#2502—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C4×D3C2211yescharacteristic
13C22×C4×D3C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16#946—24no—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2212yes—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4⋊Dic3C2211yescharacteristic
21C2×C4⋊Dic3C2214yes—
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2214yes—
25C2×C4×Dic3C2211yescharacteristic
26D3×C4⋊C4—24no—
27C3×#95C2214yes—
28C3⋊#95C2212yes—
29C3⋊#95C2214yes—
30C2×C22⋊Q8—31no—
31C22×C4⋊C4—31no—
32#398—34no—
33#391—31no—
34#391D311yescharacteristic, complemented
35#385—34no—
36#384—31no—
37#384—31no—
38C4×C22⋊C4—31no—
39C23×D3C2311yescharacteristic
40C2×C3×Q8—22no—
41C2×C3×Q8—22no—
42C2×C3×Q8—22no—
43C22×C3×C4C2312yes—
44C22×C3×C4C2312yes—
45C22×C3×C4C2312yes—
46C22×C3×C4C2311yescharacteristic
47C22×Dic3C2312yes—
48C22×Dic3C2312yes—
49C22×Dic3C2311yescharacteristic
50C22×Dic3C2312yes—
51C2×C4×D3—22no—
52C2×C4×D3—22no—
53C2×C4×D3—24no—
54C2×C4×D3—44no—
55C2×C4×D3D414yes—
56C3×C4⋊C4—22no—
57C3×C42—22no—
58(C2×D3)⋊C4—24no—
59(C2×D3)⋊C4Q818yescomplemented
60C4⋊Dic3—22no—
61C4⋊Dic3—22no—
62C4⋊Dic3—44no—
63Dic3⋊C4—24no—
64Dic3⋊C4—24no—
65Dic3⋊C4—44no—
66C4×Dic3—22no—
67C22×Q8C2×D311yescharacteristic
68C23×C4—31no—
69C23×C4—32no—
70C22⋊Q8—64no—
71C2×C4⋊C4—31no—
72C2×C4⋊C4—32no—
73C2×C4⋊C4—31no—
74C2×C4⋊C4—32no—
75C2×C4⋊C4—34no—
76C2×C4⋊C4—34no—
77C2×C4⋊C4—64no—
78C2×C4⋊C4C2×D311yescharacteristic
79C2×C22⋊C4—32no—
80C2×C22⋊C4—32no—
81C2×C42—31no—
82C2×C42C2×D311yescharacteristic
83#95—32no—
84#95—34no—
85#95C2×D314yes—
86C23×C3C2411yescharacteristic
87C22×D3—21no—
88C22×D3—21no—
89C22×D3—22no—
90C22×D3C2×D412yes—
91C22×D3C2×Q814yes—
92C3×Q8—44no—
93C2×C3×C4—22no—
94C2×C3×C4—22no—
95C2×C3×C4—22no—
96C2×C3×C4—22no—
97C2×C3×C4—22no—
98C2×C3×C4—22no—
99C2×C3×C4—24no—
100C2×C3×C4C2×D412yes—
101C2×C3×C4C2×Q814yes—
102C2×C3×C4#4112yes—
103C2×C3×C4#4112yes—
104C2×Dic3—22no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—22no—
108C2×Dic3—22no—
109C2×Dic3—22no—
110C2×Dic3—24no—
111C2×Dic3—24no—
112C2×Dic3C2×D412yes—
113C2×Dic3C2×Q814yes—
114C4×D3—28no—
115C4×D3—48no—
116C24—31no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C22×C4—32no—
121C22×C4—31no—
122C22×C4—34no—
123C22×C4—32no—
124C22×C4—32no—
125C22×C4—64no—
126C22×C4—62no—
127C22×C4—62no—
128C22×C4—124no—
129C22×C4C22×D312yes—
130C22×C4C22×D312yes—
131C22×C4C22×D312yes—
132C22×C4C22×D311yescharacteristic
133C4⋊C4—22no—
134C4⋊C4—64no—
135C4⋊C4—64no—
136C4⋊C4—62no—
137C4⋊C4—62no—
138C4⋊C4—124no—
139C4⋊C4—124no—
140C22⋊C4—38no—
141C22⋊C4—64no—
142C42—22no—
143C42—62no—
144C22×C3C22×D411yescharacteristic
145C22×C3C22×Q812yes—
146C22×C3C2×#4111yescharacteristic
147C22×C3C2×#4111yescharacteristic
148C22×C3H(5,𝔽2)12yes—
149C2×D3—22no—
150C2×D3—22no—
151C2×D3—24no—
152C2×D3—24no—
153C2×D3C4⋊Q814yes—
154C3×C4—24no—
155C3×C4—42no—
156C3×C4—42no—
157C3×C4#12314yes—
158Dic3—22no—
159Dic3—24no—
160Dic3—42no—
161Dic3—42no—
162C23—32no—
163C23—34no—
164C23—61no—
165C23—61no—
166C23—62no—
167C23C23×D311yescharacteristic
168Q8—44no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—32no—
177C2×C4—34no—
178C2×C4—62no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—62no—
182C2×C4—68no—
183C2×C4—62no—
184C2×C4—62no—
185C2×C4—64no—
186C2×C4—64no—
187C2×C4—128no—
188C2×C4#23512yes—
189C2×C4D3×D412yes—
190C2×C4#23712yes—
191C2×C4D3×Q814yes—
192C2×C3C2×#12311yescharacteristic
193C2×C3C2×C4⋊Q811yescharacteristic
194C2×C3#53511yescharacteristic
195C2×C3#55412yes—
196C2×C3#55412yes—
197D3—24no—
198C22—34no—
199C22—64no—
200C22—62no—
201C22—62no—
202C22—64no—
203C22C2×#23511yescharacteristic
204C22C2×D3×D411yescharacteristic
205C22C2×#23711yescharacteristic
206C22C2×D3×Q812yes—
207C22#100912yes—
208C4—24no—
209C4—42no—
210C4—42no—
211C4—62no—
212C4—64no—
213C4—122no—
214C4—122no—
215C4#88314yes—
216C3#250211yescharacteristic, complemented, Sylow
217C2—64no—
218C2C2×#88311yescharacteristic
219C2#557912yes—
220C2#563512yes—
221C2#567411yescharacteristic
222C2D3×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#391C2
prime radicalC23×D3C23
hypercenter#391D3
hypocenterC3#2502
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C3×#391C2semidirect
4. #5229C2semidirect
5. #4975C2semidirect
7. #4935C2semidirect
9. #4657C2semidirect
34. #391D3semidirect
59. (C2×D3)⋊C4Q8semidirect
216. C3#2502semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×D3×C4⋊C42
C3×#3912
#52292
#49752
C4×(C2×D3)⋊C42
#49352
#46682
#46572
#25023