Finite Group Databasebeta

← #80463

Subgroups — #80463

Order 384.

336 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 117 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#5732C211yesmaximal, characteristic
2#5570C211yesmaximal, characteristic, complemented
3C3×#462C211yesmaximal, characteristic, complemented
4#5185C212yesmaximal
5Dic3×Q16C212yesmaximal, complemented
6#5079C211yesmaximal, characteristic, complemented
7C4×#215C211yesmaximal, characteristic
8#4846C211yesmaximal, characteristic
9#4830C211yesmaximal, characteristic, complemented
10#4800C212yesmaximal
11#4796C212yesmaximal, complemented
12#3364—31nomaximal, Sylow
13C2×C3×Q16C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16#947C2212yes—
17Q8×Dic3C2212yes—
18Q8×Dic3C2212yes—
19C2×#215C2211yescharacteristic
20#898—22no—
21#897—21no—
22#892—21no—
23#890—22no—
24#890C2212yes—
25C4×D12C2211yescharacteristic
26#869—21no—
27#869C2211yescharacteristic
28C4×Dic6—21no—
29C4×Dic6C2211yescharacteristic
30C3×C4⋊C8C2211yescharacteristic
31C3×#103C2212yes—
32#835C2211yescharacteristic
33#835C2212yes—
34#835C2212yes—
35#821C2212yes—
36#818C2212yes—
37Dic3⋊C8C2212yes—
38C8×Dic3C2212yes—
39#809C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#557—31no—
43#556—31no—
44#488—32no—
45#487—31no—
46#479—32no—
47#474—31no—
48#464—32no—
49#462D311yescharacteristic, complemented
50C8×Q8—31no—
51C4×Q16—32no—
52C4×SD16—31no—
53C2×C3×Q8C2312yes—
54C2×C3×Q8C2311yescharacteristic
55C2×D12C2311yescharacteristic
56C2×C4×D3—21no—
57C2×Dic6—21no—
58C2×Dic6D411yescharacteristic
59C3×Q16—24no—
60C2×C3×C8C2312yes—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4C2311yescharacteristic
65C3×C42—21no—
66C3×C42C2311yescharacteristic
67#215—22no—
68(C2×D3)⋊C4—21no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4—22no—
71C4⋊Dic3—21no—
72C4⋊Dic3—22no—
73C4⋊Dic3—22no—
74C4⋊Dic3—22no—
75C4⋊Dic3D411yescharacteristic
76C4⋊Dic3C2311yescharacteristic
77C4⋊Dic3C2312yes—
78Dic3⋊C4—22no—
79Dic3⋊C4D412yes—
80C4×Dic3—22no—
81C4×Dic3—22no—
82C4×Dic3C2312yes—
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2311yescharacteristic
85C2×Q16C2×D312yes—
86C2×SD16—31no—
87C4⋊Q8—31no—
88C4⋊Q8—61no—
89C4⋊Q8C2×D311yescharacteristic
90#126—62no—
91#125—32no—
92#125—62no—
93#124—32no—
94C22⋊Q8—61no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—32no—
98C4×Q8—61no—
99C4×Q8C2×D311yescharacteristic
100C4×D4—31no—
101#117—61no—
102#107—32no—
103#106—31no—
104C4⋊C8—32no—
105C4⋊C8C2×D311yescharacteristic
106#103—31no—
107#103—32no—
108#103—32no—
109#103C2×D312yes—
110#102—32no—
111#102—31no—
112C4×C8—32no—
113C4×C8—31no—
114C22×D3—21no—
115C3×Q8—21no—
116C3×Q8—22no—
117C3×Q8—22no—
118C3×Q8#4112yes—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2411yescharacteristic
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3C2×D412yes—
128C2×Dic3C2×D412yes—
129D12—21no—
130D12—21no—
131C4×D3—42no—
132Dic6—22no—
133Dic6—42no—
134C3×C8—22no—
135C3⋊C8—21no—
136C3⋊C8—21no—
137C2×Q8—31no—
138C2×Q8—61no—
139C2×Q8C22×D312yes—
140C2×Q8C22×D311yescharacteristic
141C2×D4—31no—
142C22×C4—61no—
143Q16—24no—
144SD16—62no—
145C2×C8—31no—
146C2×C8—31no—
147C2×C8C22×D312yes—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—31no—
152C4⋊C4—32no—
153C4⋊C4—32no—
154C4⋊C4—31no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—61no—
160C4⋊C4C22×D311yescharacteristic
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—61no—
164C42—21no—
165C42—32no—
166C42—62no—
167C42—62no—
168C42C22×D311yescharacteristic
169C22×C3C22×D411yescharacteristic
170C2×D3—41no—
171C2×D3—41no—
172C2×D3—41no—
173C3×C4—22no—
174C3×C4—21no—
175C3×C4—42no—
176C3×C4—41no—
177C3×C4#13512yes—
178C3×C4C2×#4111yescharacteristic
179C3×C42−1+411yescharacteristic
180Dic3—22no—
181Dic3—22no—
182Dic3—42no—
183Dic3—41no—
184C23—61no—
185Q8—21no—
186Q8—22no—
187Q8—22no—
188Q8—62no—
189Q8—122no—
190Q8#23712yes—
191D4—61no—
192D4—61no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—61no—
199C2×C4—62no—
200C2×C4—122no—
201C2×C4D3×D411yescharacteristic
202C2×C4D3×D411yescharacteristic
203C2×C4C23×D311yescharacteristic
204C8—22no—
205C8—61no—
206C8—61no—
207C2×C3#54811yescharacteristic
208C2×C3C2×#13511yescharacteristic
209C2×C3#57811yescharacteristic
210D3—81no—
211C22—121no—
212C22—121no—
213C22—121no—
214C22C2×D3×D411yescharacteristic
215C4—21no—
216C4—22no—
217C4—41no—
218C4—42no—
219C4—62no—
220C4—62no—
221C4—121no—
222C4—122no—
223C4#91912yes—
224C4C2×#23711yescharacteristic
225C4#101011yescharacteristic
226C3#336411yescharacteristic, complemented, Sylow
227C2—241no—
228C2#559811yescharacteristic
229C2C2×#91911yescharacteristic
230C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#462C2
socleC22×C3C22×D4
prime radicalC2×D12C23
hypercenter#462D3
hypocenterC3#3364
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #5570C2semidirect
3. C3×#462C2semidirect
5. Dic3×Q16C2semidirect
6. #5079C2semidirect
9. #4830C2semidirect
11. #4796C2semidirect
49. #462D3semidirect
226. C3#3364semidirect

Maximal subgroups

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

subgroupindex
#57322
#55702
C3×#4622
#51852
Dic3×Q162
#50792
C4×#2152
#48462
#48302
#48002
#47962
#33643