Finite Group Databasebeta

← #79324

Subgroups — #79324

Order 384.

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

normal subgroups only

This group has 328 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#5479C211yesmaximal, characteristic
2#5470C211yesmaximal, characteristic, complemented
3C3×#432C211yesmaximal, characteristic, complemented
4#4847C212yesmaximal, complemented
5#4829C212yesmaximal, complemented
6#4737C212yesmaximal
7#4723C212yesmaximal, complemented
8#4714C211yesmaximal, characteristic, complemented
9#4714C211yesmaximal, characteristic, complemented
10#4702C211yesmaximal, characteristic, complemented
11#4702C211yesmaximal, characteristic, complemented
12#3240—31nomaximal, Sylow
13C2×C3×C42C2211yescharacteristic
14#930C2211yescharacteristic
15#929—21no—
16C4×#67—21no—
17#926C2212yes—
18#924—21no—
19#924—21no—
20#924C2211yescharacteristic
21#876—21no—
22#875C2211yescharacteristic
23C4×D12C2211yescharacteristic
24#872—21no—
25#870C2211yescharacteristic
26#870C2212yes—
27#869C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6—21no—
30C4×Dic6—21no—
31C4×Dic6C2211yescharacteristic
32C3×C4⋊C8C2212yes—
33C3×C22⋊C8C2212yes—
34C3×#97C2211yescharacteristic
35C3×#97C2211yescharacteristic
36#821C2212yes—
37#821C2212yes—
38#818C2212yes—
39#818C2212yes—
40#817C2212yes—
41#817C2212yes—
42#816C2212yes—
43#816C2212yes—
44#533—31no—
45#529—31no—
46#501—31no—
47#501—31no—
48#489—31no—
49#489—31no—
50#485—32no—
51#482—32no—
52#479—32no—
53#478—32no—
54#432D311yescharacteristic, complemented
55C22×C3×C4D411yescharacteristic
56C22×C3×C4D411yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C2×#67—21no—
59C2×D12C2311yescharacteristic
60C2×C4×D3—21no—
61C2×Dic6—21no—
62C2×Dic6—21no—
63C2×Dic6C2311yescharacteristic
64C2×C3×C8C2312yes—
65C2×C3×C8C2312yes—
66C3×C42D411yescharacteristic
67C3×C42D411yescharacteristic
68C3×C42C2311yescharacteristic
69C3×C42C2311yescharacteristic
70#217—22no—
71#217—21no—
72(C2×D3)⋊C4—21no—
73(C2×D3)⋊C4—21no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—21no—
76C4⋊Dic3—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3C2312yes—
79C4⋊Dic3C2312yes—
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84Dic3⋊C4—21no—
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88C4×Dic3—22no—
89C4×Dic3—21no—
90C4⋊Q8—31no—
91C4⋊Q8—31no—
92#126—61no—
93#125—31no—
94#125—32no—
95#124—31no—
96#123—61no—
97C22⋊Q8—31no—
98C22⋊Q8—61no—
99C22⋊Q8—61no—
100#121—31no—
101C4×Q8—31no—
102C4×Q8—61no—
103C4×Q8—61no—
104C4×D4—31no—
105C4×D4—61no—
106#117—32no—
107#117—61no—
108C2×C42C2×D311yescharacteristic
109#107—32no—
110#107—32no—
111#106—32no—
112#106—32no—
113C4⋊C8C2×D312yes—
114#103—32no—
115#103—32no—
116#102—32no—
117#102—32no—
118C22⋊C8C2×D312yes—
119#97C2×D311yescharacteristic
120#97C2×D311yescharacteristic
121C23×C3C2×D411yescharacteristic
122C22×D3—21no—
123C2×C3×C4—22no—
124C2×C3×C4—41no—
125C2×C3×C4—41no—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2411yescharacteristic
132#67—42no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138D12—22no—
139C4×D3—42no—
140Dic6—22no—
141Dic6—42no—
142Dic6—42no—
143C3×C8—22no—
144C3×C8—22no—
145C2×Q8—31no—
146C2×Q8—61no—
147C2×Q8—61no—
148C2×D4—31no—
149C2×D4—61no—
150C22×C4—61no—
151C22×C4D1211yescharacteristic
152C22×C4D1211yescharacteristic
153C22×C4C22×D311yescharacteristic
154C2×C8C22×D312yes—
155C2×C8C22×D312yes—
156C4⋊C4—32no—
157C4⋊C4—32no—
158C4⋊C4—31no—
159C4⋊C4—31no—
160C4⋊C4—61no—
161C4⋊C4—61no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4—61no—
167C4⋊C4—61no—
168C22⋊C4—62no—
169C22⋊C4—61no—
170C22⋊C4—61no—
171C22⋊C4—61no—
172C22⋊C4—61no—
173C22⋊C4—61no—
174C42—62no—
175C42—61no—
176C42D1211yescharacteristic
177C42D1211yescharacteristic
178C42C22×D311yescharacteristic
179C42C22×D311yescharacteristic
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3C22×D411yescharacteristic
184C2×D3—41no—
185C2×D3—41no—
186C2×D3—41no—
187C3×C4—21no—
188C3×C4—21no—
189C3×C4—21no—
190C3×C4—21no—
191C3×C4—21no—
192C3×C42−1+412yes—
193Dic3—41no—
194Dic3—42no—
195Dic3—42no—
196Dic3—41no—
197Dic3—41no—
198C23—61no—
199C23C2×D1211yescharacteristic
200Q8—62no—
201Q8—122no—
202Q8—122no—
203D4—62no—
204D4—122no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—122no—
214C2×C4C2×D1211yescharacteristic
215C2×C4C2×D1211yescharacteristic
216C2×C4C2×D1211yescharacteristic
217C2×C4C2×D1211yescharacteristic
218C2×C4C2×D1211yescharacteristic
219C2×C4C23×D311yescharacteristic
220C8—22no—
221C8—22no—
222C2×C3—41no—
223C2×C3#53611yescharacteristic
224C2×C3#57511yescharacteristic
225C2×C3#57511yescharacteristic
226D3—81no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—121no—
231C22—121no—
232C22—121no—
233C22C22×D1211yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—121no—
240C4—121no—
241C4—122no—
242C4—122no—
243C4—121no—
244C4#101012yes—
245C3#324011yescharacteristic, complemented, Sylow
246C2—41no—
247C2—241no—
248C2#552611yescharacteristic
249C2#574811yescharacteristic
250C2#574811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#432C2
socleC22×C3C22×D4
prime radical#930C22
hypercenter#432D3
hypocenterC3#3240
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #5470C2semidirect
3. C3×#432C2semidirect
4. #4847C2semidirect
5. #4829C2semidirect
7. #4723C2semidirect
8. #4714C2semidirect
9. #4714C2semidirect
10. #4702C2semidirect
11. #4702C2semidirect
54. #432D3semidirect
245. C3#3240semidirect

Maximal subgroups

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

subgroupindex
#54792
#54702
C3×#4322
#48472
#48292
#47372
#47232
#47142
#47022
#32403