Finite Group Databasebeta

← #78977

Subgroups — #78977

Order 384.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#5816C211yesmaximal, characteristic
2C22×#216C211yesmaximal, characteristic, complemented
3C2×#909C211yesmaximal, characteristic
4#5516C211yesmaximal, characteristic, complemented
5C3×#419C211yesmaximal, characteristic, complemented
6#5119C211yesmaximal, characteristic, complemented
7#5002C211yesmaximal, characteristic
8#4798C214yesmaximal, complemented
9#4795C214yesmaximal, complemented
10#3044—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C2×#235C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×M16C2211yescharacteristic
15C3×#117C2211yescharacteristic
16#947C2212yes—
17#946—22no—
18#944C2212yes—
19C2×#216—24no—
20C2×#216C2214yes—
21#929—22no—
22#926C2211yescharacteristic
23C22×C3⋊C8C2211yescharacteristic
24#909—24no—
25C2×Dic12C2212yes—
26C2×#204C2212yes—
27#887C2214yes—
28#877—22no—
29C4×Dic6—22no—
30C3×#103C2214yes—
31Dic3⋊C8C2214yes—
32#810C2212yes—
33#809C2212yes—
34C2×#137—31no—
35C22×Q16—31no—
36#536—31no—
37#519—31no—
38#464—34no—
39#462—34no—
40#424—31no—
41#419D311yescharacteristic, complemented
42#416—31no—
43C2×C3×Q8—21no—
44C2×C3×Q8—22no—
45C2×C3×Q8—22no—
46C2×C3×Q8C2312yes—
47C22×C3×C4—21no—
48C22×C3×C4C2311yescharacteristic
49C2×#67—21no—
50C22×Dic3—21no—
51#235—22no—
52#235—22no—
53C2×D12C2311yescharacteristic
54C2×C4×D3—21no—
55C2×Dic6—21no—
56C2×Dic6—22no—
57C2×Dic6D414yes—
58C2×Dic6C2311yescharacteristic
59C2×Dic6C2312yes—
60C3×M16—22no—
61C2×C3×C8C2312yes—
62C3×C4⋊C4C2312yes—
63C3×C22⋊C4—21no—
64C3×C42—21no—
65#217D412yes—
66#216—28no—
67(C2×D3)⋊C4—24no—
68C4⋊Dic3D412yes—
69Dic3⋊C4—24no—
70Dic3⋊C4—24no—
71C4×Dic3—24no—
72C4×Dic3C2312yes—
73C2×C3⋊C8—22no—
74C2×C3⋊C8C2312yes—
75Dic12—24no—
76#204—24no—
77C2×#41—31no—
78C22×Q8—31no—
79C22×Q8C2×D311yescharacteristic
80#137—64no—
81C2×Q16—34no—
82C2×Q16—32no—
83C2×Q16—64no—
84C2×SD16—32no—
85C2×M16C2×D311yescharacteristic
86C22×C8—31no—
87C4⋊Q8—32no—
88#124—32no—
89#123—62no—
90C22⋊Q8—62no—
91C4×Q8—34no—
92C4×Q8—62no—
93#117—31no—
94#117—62no—
95#117C2×D311yescharacteristic
96C4⋊C8—34no—
97#103—32no—
98#103C2×D314yes—
99#102—32no—
100C23×C3C2×D411yescharacteristic
101C22×D3—21no—
102C3×Q8—22no—
103C3×Q8—22no—
104C3×Q8—42no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—42no—
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D412yes—
110C2×C3×C4#4112yes—
111C2×C3×C4C2411yescharacteristic
112#67—42no—
113C2×Dic3—24no—
114C2×Dic3—22no—
115C2×Dic3—21no—
116C2×Dic3C2×D414yes—
117D12—21no—
118D12—21no—
119C4×D3—42no—
120Dic6—24no—
121Dic6—22no—
122Dic6—21no—
123Dic6—21no—
124Dic6C2×D414yes—
125C3×C8—22no—
126C3⋊C8—22no—
127#41—62no—
128#41—62no—
129C2×Q8—21no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—34no—
133C2×Q8—32no—
134C2×Q8—31no—
135C2×Q8—62no—
136C2×Q8—61no—
137C2×Q8C22×D312yes—
138C2×D4—31no—
139C2×D4—61no—
140C22×C4—21no—
141C22×C4—61no—
142C22×C4—61no—
143C22×C4C22×D311yescharacteristic
144Q16—64no—
145Q16—68no—
146SD16—64no—
147M16—22no—
148C2×C8—32no—
149C2×C8—62no—
150C2×C8C22×D312yes—
151C4⋊C4—32no—
152C4⋊C4—64no—
153C4⋊C4—64no—
154C4⋊C4C22×D312yes—
155C22⋊C4—21no—
156C22⋊C4—32no—
157C22⋊C4—64no—
158C42—21no—
159C42—32no—
160C42—64no—
161C22×C3—22no—
162C22×C3#12112yes—
163C22×C3C22×D411yescharacteristic
164C2×D3—41no—
165C2×D3—42no—
166C3×C4—42no—
167C3×C4—42no—
168C3×C4#12112yes—
169C3×C4C22×D411yescharacteristic
170C3×C4C2×#4111yescharacteristic
171Dic3—24no—
172Dic3—24no—
173Dic3—41no—
174C23—61no—
175C23D3×D411yescharacteristic
176Q8—22no—
177Q8—22no—
178Q8—34no—
179Q8—42no—
180Q8—64no—
181Q8—62no—
182Q8—61no—
183Q8—61no—
184D4—61no—
185D4—61no—
186D4—122no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—34no—
190C2×C4—42no—
191C2×C4—64no—
192C2×C4—62no—
193C2×C4—61no—
194C2×C4—122no—
195C2×C4#23512yes—
196C2×C4D3×D411yescharacteristic
197C2×C4D3×D412yes—
198C2×C4C23×D311yescharacteristic
199C8—22no—
200C8—62no—
201C2×C3—22no—
202C2×C3C2×#12111yescharacteristic
203C2×C3#57812yes—
204D3—81no—
205C22—22no—
206C22—121no—
207C22—122no—
208C22#88412yes—
209C22C2×D3×D411yescharacteristic
210C4—42no—
211C4—42no—
212C4—64no—
213C4—64no—
214C4—121no—
215C4#88412yes—
216C4C2×#23511yescharacteristic
217C4C2×D3×D411yescharacteristic
218C3#304411yescharacteristic, complemented, Sylow
219C2—22no—
220C2—241no—
221C2C2×#88411yescharacteristic
222C2#577912yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#419C2
socleC22×C3C22×D4
prime radicalC2×#235C22
hypercenter#419D3
hypocenterC3#3044
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C22×#216C2semidirect
4. #5516C2semidirect
5. C3×#419C2semidirect
6. #5119C2semidirect
8. #4798C2semidirect
9. #4795C2semidirect
41. #419D3semidirect
218. C3#3044semidirect

Maximal subgroups

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

subgroupindex
#58162
C22×#2162
C2×#9092
#55162
C3×#4192
#51192
#50022
#47982
#47952
#30443