Finite Group Databasebeta

← #80853

Subgroups — #80853

Order 384.

446 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 119 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#5596C212yesmaximal, complemented
2C3×#469C211yesmaximal, characteristic, complemented
3#5152C211yesmaximal, characteristic, complemented
4#5151C211yesmaximal, characteristic, complemented
5#5135C211yesmaximal, characteristic, complemented
6Dic3×M16C211yesmaximal, characteristic, complemented
7#5039C212yesmaximal, complemented
8#4869C211yesmaximal, characteristic, complemented
9#4855C211yesmaximal, characteristic, complemented
10#4785C212yesmaximal, complemented
11#4760C212yesmaximal, complemented
12#3184—31nomaximal, Sylow
13C2×#237—22no—
14C2×C3×D8C2211yescharacteristic
15C2×C3×M16C2211yescharacteristic
16C3×#121C2212yes—
17#940C2212yes—
18#939—22no—
19D4×Dic3C2212yes—
20C2×#213C2212yes—
21#930C2212yes—
22C2×C4×Dic3C2211yescharacteristic
23C2×#208C2211yescharacteristic
24C2×D24C2211yescharacteristic
25#893C2212yes—
26#888C2212yes—
27#885—22no—
28#67⋊C4—22no—
29C3×#106C2211yescharacteristic
30C3×#102C2212yes—
31#832C2212yes—
32#821C2212yes—
33#817C2211yescharacteristic
34#815C2211yescharacteristic
35#815C2211yescharacteristic
36C8×Dic3C2211yescharacteristic
37C8×Dic3C2211yescharacteristic
38#809C2212yes—
39#808C2212yes—
40#532—32no—
41#498—31no—
42#493—31no—
43#486—32no—
44#469—31no—
45#469—32no—
46#469D311yescharacteristic, complemented
47#442—31no—
48#442—31no—
49#442—32no—
50C4×M16—31no—
51C2×C3×D4—22no—
52C2×C3×D4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C2×#67—22no—
55C2×#67—22no—
56C2×#67—22no—
57C22×Dic3—22no—
58C22×Dic3D412yes—
59#237—48no—
60C2×D12C2312yes—
61C2×C4×D3—22no—
62C2×Dic6—22no—
63C3×D8—22no—
64C3×M16—22no—
65C2×C3×C8C2311yescharacteristic
66C2×C3×C8C2311yescharacteristic
67C3×C4⋊C4C2312yes—
68C3×C22⋊C4—22no—
69#217—22no—
70#213—24no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73C4⋊Dic3C2312yes—
74Dic3⋊C4—22no—
75C4×Dic3D412yes—
76C4×Dic3C2311yescharacteristic
77C4×Dic3C2311yescharacteristic
78#208—22no—
79C2×C3⋊C8C2311yescharacteristic
80C2×C3⋊C8C2311yescharacteristic
81D24—22no—
82C2×#41—62no—
83C2×D8—32no—
84C2×D8—31no—
85C2×D8C2×D311yescharacteristic
86C2×M16—31no—
87C2×M16C2×D311yescharacteristic
88C4⋊Q8—32no—
89#127—32no—
90#124—62no—
91#121—32no—
92#121—62no—
93#121C2×D312yes—
94C4×D4—32no—
95C4×D4—32no—
96C4×D4—62no—
97C2×C42—31no—
98#106—31no—
99#106—32no—
100#106C2×D311yescharacteristic
101#102—32no—
102#102—32no—
103#102—32no—
104#102C2×D312yes—
105#97—31no—
106#97—31no—
107C4×C8—31no—
108C4×C8—31no—
109C23×C3—22no—
110C23×C3C2×D411yescharacteristic
111C22×D3—22no—
112C3×D4—22no—
113C3×D4—22no—
114C3×D4—44no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2411yescharacteristic
120#67—44no—
121#67—44no—
122#67—44no—
123C2×Dic3—24no—
124C2×Dic3—22no—
125C2×Dic3—44no—
126C2×Dic3C2×D412yes—
127C2×Dic3C2×D412yes—
128D12—22no—
129D12—22no—
130C4×D3—44no—
131Dic6—44no—
132C3×C8—21no—
133C3×C8#4112yescomplemented
134C3⋊C8—21no—
135C3⋊C8#4112yescomplemented
136#41—128no—
137C2×Q8—62no—
138C2×D4—22no—
139C2×D4—32no—
140C2×D4—62no—
141C2×D4—62no—
142C2×D4—62no—
143C2×D4C22×D312yes—
144C22×C4—32no—
145C22×C4—62no—
146C22×C4—62no—
147C22×C4C22×D311yescharacteristic
148D8—22no—
149D8—64no—
150D8—62no—
151M16—22no—
152M16—62no—
153C2×C8—31no—
154C2×C8—31no—
155C2×C8C22×D311yescharacteristic
156C2×C8C22×D311yescharacteristic
157C4⋊C4—32no—
158C4⋊C4—62no—
159C4⋊C4C22×D312yes—
160C22⋊C4—22no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C42—32no—
165C42—31no—
166C42—31no—
167C22×C3—21no—
168C22×C3—22no—
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3C22×D411yescharacteristic
173C2×D3—42no—
174C2×D3—42no—
175C2×D3—42no—
176C3×C4—21no—
177C3×C4—42no—
178C3×C4C2×#4111yescharacteristic
179C3×C4C2×#4111yescharacteristic
180Dic3—22no—
181Dic3—42no—
182Dic3AGL(1,ℤ/8)14yes—
183C23—22no—
184C23—62no—
185C23D3×D411yescharacteristic
186Q8—124no—
187D4—22no—
188D4—22no—
189D4—44no—
190D4—62no—
191D4—62no—
192D4—124no—
193D4—124no—
194D4—124no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—64no—
201C2×C4—62no—
202C2×C4—124no—
203C2×C4—124no—
204C2×C4D3×D411yescharacteristic
205C2×C4C23×D311yescharacteristic
206C8—21no—
207C8—32no—
208C8—61no—
209C8#23712yescomplemented
210C2×C3—41no—
211C2×C3—82no—
212C2×C3#53211yescharacteristic
213C2×C3C2×AGL(1,ℤ/8)12yes—
214D3—82no—
215C22—22no—
216C22—21no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—122no—
221C22—122no—
222C22—122no—
223C22C2×D3×D411yescharacteristic
224C4—21no—
225C4—34no—
226C4—42no—
227C4—62no—
228C4—122no—
229C4C2×#23711yescharacteristic
230C4D3×#4111yescharacteristic
231C3#318411yescharacteristic, complemented, Sylow
232C2—41no—
233C2—82no—
234C2—242no—
235C2#559611yescharacteristic
236C2D3×AGL(1,ℤ/8)12yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3184
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #5596C2semidirect
2. C3×#469C2semidirect
3. #5152C2semidirect
4. #5151C2semidirect
5. #5135C2semidirect
6. Dic3×M16C2semidirect
7. #5039C2semidirect
8. #4869C2semidirect
9. #4855C2semidirect
10. #4785C2semidirect
11. #4760C2semidirect
46. #469D3semidirect
133. C3×C8#41semidirect
135. C3⋊C8#41semidirect
209. C8#237semidirect
231. C3#3184semidirect

Maximal subgroups

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

subgroupindex
#55962
C3×#4692
#51522
#51512
#51352
Dic3×M162
#50392
#48692
#48552
#47852
#47602
#31843