Finite Group Databasebeta

← #76641

Subgroups — #76641

Order 384.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#944C211yesmaximal, characteristic, complemented
3C2×#897C211yesmaximal, characteristic
4C2×#872C211yesmaximal, characteristic
5C3×#391C211yesmaximal, characteristic, complemented
6#4977C211yesmaximal, characteristic, complemented
7#4929C211yesmaximal, characteristic, complemented
8#4675C214yesmaximal
9#4657C214yesmaximal, complemented
10#2621—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C4×D3C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16#946—24no—
17#944—24no—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2214yes—
26C2×C4×Dic3C2211yescharacteristic
27#897—24no—
28#872—24no—
29C3×#95C2214yes—
30C3⋊#95C2214yes—
31C3⋊#95C2214yes—
32C2×C4⋊Q8—31no—
33C2×C22⋊Q8—31no—
34C2×C22⋊Q8—31no—
35C2×#117—31no—
36#398—34no—
37#397—34no—
38#391—31no—
39#391D311yescharacteristic, complemented
40#382—31no—
41C23×D3C2311yescharacteristic
42C2×C3×Q8—22no—
43C2×C3×Q8—22no—
44C2×C3×Q8—22no—
45C22×C3×C4C2312yes—
46C22×C3×C4C2312yes—
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×Dic3C2312yes—
50C22×Dic3C2312yes—
51C22×Dic3C2312yes—
52C22×Dic3C2311yescharacteristic
53C2×C4×D3—22no—
54C2×C4×D3—22no—
55C2×C4×D3D414yes—
56C2×Dic6—22no—
57C2×Dic6—22no—
58C2×Dic6—22no—
59C3×C4⋊C4—22no—
60C3×C42—22no—
61(C2×D3)⋊C4—24no—
62(C2×D3)⋊C4—24no—
63(C2×D3)⋊C4—24no—
64C4⋊Dic3—22no—
65C4⋊Dic3—24no—
66Dic3⋊C4—24no—
67Dic3⋊C4—28no—
68Dic3⋊C4—24no—
69C4×Dic3D414yes—
70C22×Q8—31no—
71C22×Q8C2×D311yescharacteristic
72C23×C4—31no—
73C4⋊Q8—64no—
74C22⋊Q8—64no—
75C22⋊Q8—64no—
76#117—64no—
77C2×C4⋊C4—32no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—34no—
82C2×C4⋊C4C2×D311yescharacteristic
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C42—31no—
87C2×C42C2×D311yescharacteristic
88#95—34no—
89#95—34no—
90#95C2×D314yes—
91C23×C3C2411yescharacteristic
92C22×D3—21no—
93C22×D3—21no—
94C22×D3—22no—
95C22×D3—22no—
96C22×D3C2×D412yes—
97C3×Q8—44no—
98C2×C3×C4—22no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—24no—
106C2×C3×C4C2×D412yes—
107C2×C3×C4C2×D412yes—
108C2×C3×C4#4112yes—
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—24no—
118C2×Dic3C2×D414yes—
119C2×Dic3C2×D412yes—
120C4×D3—28no—
121Dic6—44no—
122C24—31no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—62no—
127C2×Q8—62no—
128C2×Q8—62no—
129C22×C4—32no—
130C22×C4—31no—
131C22×C4—34no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—62no—
135C22×C4—62no—
136C22×C4C22×D312yes—
137C22×C4C22×D312yes—
138C22×C4C22×D312yes—
139C22×C4C22×D311yescharacteristic
140C4⋊C4—22no—
141C4⋊C4—64no—
142C4⋊C4—64no—
143C4⋊C4—64no—
144C4⋊C4—68no—
145C4⋊C4—62no—
146C22⋊C4—64no—
147C22⋊C4—64no—
148C22⋊C4—64no—
149C42—22no—
150C42—34no—
151C22×C3C22×D411yescharacteristic
152C22×C3C22×D411yescharacteristic
153C22×C3C2×#4111yescharacteristic
154C22×C32−1+412yes—
155C22×C32−1+412yes—
156C2×D3—22no—
157C2×D3—22no—
158C2×D3—22no—
159C2×D3—42no—
160C2×D3—42no—
161C3×C4—42no—
162C3×C4—42no—
163C3×C4—42no—
164C3×C4#12114yes—
165Dic3—22no—
166Dic3—24no—
167Dic3—42no—
168Dic3—42no—
169C23—32no—
170C23—61no—
171C23—61no—
172C23—62no—
173C23—62no—
174C23C23×D311yescharacteristic
175Q8—44no—
176Q8—124no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—34no—
186C2×C4—32no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—68no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—62no—
195C2×C4—62no—
196C2×C4—64no—
197C2×C4#23512yes—
198C2×C4D3×D412yes—
199C2×C4D3×D412yes—
200C2×C3C2×#12111yescharacteristic
201C2×C3#53611yescharacteristic
202C2×C3#53611yescharacteristic
203C2×C3#53612yes—
204C2×C3#54112yes—
205D3—42no—
206C22—62no—
207C22—62no—
208C22—62no—
209C22—122no—
210C22—122no—
211C22C2×#23511yescharacteristic
212C22C2×D3×D411yescharacteristic
213C22C2×D3×D411yescharacteristic
214C22#100712yes—
215C22#101012yes—
216C4—42no—
217C4—42no—
218C4—42no—
219C4—62no—
220C4—64no—
221C4—122no—
222C4—122no—
223C4#88414yes—
224C3#262111yescharacteristic, complemented, Sylow
225C2—122no—
226C2C2×#88411yescharacteristic
227C2#558312yes—
228C2#562812yes—
229C2#567511yescharacteristic
230C2#572411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#391C2
prime radicalC23×D3C23
hypercenter#391D3
hypocenterC3#2621
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#944C2semidirect
5. C3×#391C2semidirect
6. #4977C2semidirect
7. #4929C2semidirect
9. #4657C2semidirect
39. #391D3semidirect
224. C3#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
C2×#9442
C2×#8972
C2×#8722
C3×#3912
#49772
#49292
#46752
#46572
#26213