Finite Group Databasebeta

← #82972

Subgroups — #82972

Order 384.

576 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 161 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×C3×#124C211yesmaximal, characteristic
2C2×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×#944C211yesmaximal, characteristic, complemented
4C2×#936C211yesmaximal, characteristic
5#5221C211yesmaximal, characteristic
6#4948C214yesmaximal, complemented
7#4945C214yesmaximal, complemented
8C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
9#4929C211yesmaximal, characteristic, complemented
10#2545—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C3×#124—24no—
15C2×C3×C22⋊C4C2214yes—
16C2×C3×C42C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2214yes—
19C2×#217C2212yes—
20Q8×Dic3—24no—
21#944—24no—
22#936—24no—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×C4⋊Dic3C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2214yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30C3⋊#95C2214yes—
31C3⋊#95C2212yes—
32C2×C4⋊Q8—31no—
33C2×#124—31no—
34C2×#124D311yescharacteristic
35C2×C4×Q8—31no—
36#396—34no—
37#391—31no—
38#390—31no—
39#388—34no—
40C4×C4⋊C4—31no—
41C24×C3C2312yes—
42C2×C3×Q8—22no—
43C2×C3×Q8—22no—
44C2×C3×Q8D414yes—
45C2×C3×D4—22no—
46C2×C3×D4—22no—
47C2×C3×D4—22no—
48C22×C3×C4C2312yes—
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2311yescharacteristic
54C22×Dic3C2311yescharacteristic
55C22×Dic3C2312yes—
56C2×Dic6—22no—
57C2×Dic6—22no—
58C2×Dic6—22no—
59C3×C22⋊C4—28no—
60C3×C42—22no—
61#217—24no—
62#217—24no—
63#217—28no—
64C4⋊Dic3—22no—
65C4⋊Dic3—24no—
66Dic3⋊C4—28no—
67Dic3⋊C4—24no—
68C4×Dic3—22no—
69C4×Dic3—24no—
70C4×Dic3D414yes—
71C22×Q8—31no—
72C22×Q8C2×D311yescharacteristic
73C22×D4C2×D311yescharacteristic
74C4⋊Q8—64no—
75#124—24no—
76#124—64no—
77C4×Q8—64no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—34no—
81C2×C4⋊C4—32no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—34no—
85C2×C22⋊C4C2×D314yes—
86C2×C42—31no—
87C2×C42—32no—
88C2×C42—31no—
89C2×C42C2×D311yescharacteristic
90#95—32no—
91#95—34no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—24no—
95C23×C3—22no—
96C23×C3—24no—
97C23×C3C2411yescharacteristic
98C3×Q8—28no—
99C3×D4—44no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4C2×D412yes—
106C2×C3×C4C2×D414yes—
107C2×C3×C4C2×D412yes—
108C2×C3×C4#4112yes—
109C2×Dic3—22no—
110C2×Dic3—22no—
111C2×Dic3—22no—
112C2×Dic3—22no—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—24no—
117C2×Dic3—22no—
118C2×Dic3C2×D412yes—
119C2×Dic3C2×D412yes—
120C2×Dic3#4114yes—
121Dic6—44no—
122C24C22×D312yes—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—62no—
126C2×Q8—62no—
127C2×Q8—62no—
128C2×Q8#6714yes—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C22×C4—31no—
133C22×C4—32no—
134C22×C4—31no—
135C22×C4—32no—
136C22×C4—32no—
137C22×C4C22×D312yes—
138C22×C4C22×D312yes—
139C22×C4C22×D311yescharacteristic
140C4⋊C4—64no—
141C4⋊C4—64no—
142C4⋊C4—62no—
143C4⋊C4—68no—
144C22⋊C4—28no—
145C22⋊C4—64no—
146C22⋊C4—64no—
147C22⋊C4—68no—
148C42—22no—
149C42—34no—
150C42—64no—
151C42—62no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3—42no—
155C22×C3—44no—
156C22×C3—44no—
157C22×C3C22×D411yescharacteristic
158C22×C3C22×D411yescharacteristic
159C22×C3C2×#4111yescharacteristic
160C22×C3C2×#4112yes—
161C22×C32−1+412yes—
162C3×C4—24no—
163C3×C4—42no—
164C3×C4#12114yes—
165Dic3—22no—
166Dic3—22no—
167Dic3—24no—
168Dic3—42no—
169Dic3—42no—
170C23—22no—
171C23—22no—
172C23—24no—
173C23—24no—
174C23—22no—
175C23C23×D311yescharacteristic
176Q8—28no—
177Q8—124no—
178D4—44no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4—34no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4—64no—
195C2×C4—62no—
196C2×C4D3×D412yes—
197C2×C4#23712yes—
198C2×C4C2×#6714yes—
199C2×C4C2×#6712yes—
200C2×C3—82no—
201C2×C3C2×#12111yescharacteristic
202C2×C3#53211yescharacteristic
203C2×C3#53611yescharacteristic
204C2×C3#54812yes—
205C2×C3#55612yes—
206C22—42no—
207C22—44no—
208C22—42no—
209C22—42no—
210C22—44no—
211C22C2×D3×D411yescharacteristic
212C22C2×#23711yescharacteristic
213C22D3×#4112yes—
214C22#101012yes—
215C22C22×#6711yescharacteristic
216C4—24no—
217C4—42no—
218C4—62no—
219C4—62no—
220C4—64no—
221C4—122no—
222C4—122no—
223C4#93914yes—
224C3#254511yescharacteristic, complemented, Sylow
225C2—82no—
226C2#566811yescharacteristic
227C2#567112yes—
228C2#567511yescharacteristic
229C2C2×#93911yescharacteristic
230C2#583112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radicalC22×C3×D4C22
hypercenterC2×#124D3
hypocenterC3#2545
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic3C2semidirect
3. C2×#944C2semidirect
6. #4948C2semidirect
7. #4945C2semidirect
8. C4×Dic3⋊C4C2semidirect
9. #4929C2semidirect
224. C3#2545semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×Q8×Dic32
C2×#9442
C2×#9362
#52212
#49482
#49452
C4×Dic3⋊C42
#49292
#25453