Finite Group Databasebeta

← #79358

Subgroups — #79358

Order 384.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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
2C2×#869C211yesmaximal, characteristic, complemented
3C3×#432C211yesmaximal, characteristic, complemented
4#4849C212yesmaximal, complemented
5#4841C212yesmaximal, complemented
6#4739C212yesmaximal, complemented
7#4738C212yesmaximal
8#4715C212yesmaximal, complemented
9#4703C212yesmaximal, complemented
10#3218—31nomaximal, Sylow
11C22×Dic6C2212yes—
12C2×C3×C42C2211yescharacteristic
13#930C2211yescharacteristic
14#929—21no—
15C4×#67—21no—
16C2×C4⋊Dic3—21no—
17C2×C4⋊Dic3—21no—
18#924C2211yescharacteristic
19C2×Dic12C2214yes—
20C2×#204C2214yes—
21#876—21no—
22#875C2211yescharacteristic
23C4×D12C2211yescharacteristic
24#872—21no—
25#870C2211yescharacteristic
26#869—21no—
27#869—21no—
28#869C2212yes—
29#869C2212yes—
30C4×Dic6C2211yescharacteristic
31C3×C4⋊C8C2212yes—
32C3×C22⋊C8C2212yes—
33C3×#97C2212yes—
34#816C2214yes—
35#816C2214yes—
36C2×C4⋊Q8—31no—
37#529—31no—
38#497—32no—
39#491—32no—
40#462—32no—
41#461—32no—
42#451—32no—
43#448—32no—
44#432D311yescharacteristic, complemented
45C22×C3×C4D411yescharacteristic
46C22×C3×C4D411yescharacteristic
47C22×C3×C4C2311yescharacteristic
48C2×#67—21no—
49C22×Dic3—22no—
50C2×D12C2311yescharacteristic
51C2×C4×D3—21no—
52C2×Dic6—22no—
53C2×Dic6—44no—
54C2×Dic6C2314yes—
55C2×Dic6C2311yescharacteristic
56C2×C3×C8C2314yes—
57C3×C42D411yescharacteristic
58C3×C42D411yescharacteristic
59C3×C42C2311yescharacteristic
60C3×C42C2311yescharacteristic
61#217—21no—
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—21no—
64(C2×D3)⋊C4—21no—
65(C2×D3)⋊C4—21no—
66C4⋊Dic3—22no—
67C4⋊Dic3—22no—
68C4⋊Dic3—22no—
69C4⋊Dic3—22no—
70C4⋊Dic3C2311yescharacteristic
71C4⋊Dic3C2311yescharacteristic
72Dic3⋊C4—21no—
73Dic3⋊C4—21no—
74Dic3⋊C4—21no—
75Dic3⋊C4—21no—
76C4×Dic3—21no—
77Dic12—44no—
78#204—44no—
79C22×Q8—32no—
80C2×Q16—34no—
81C2×SD16—34no—
82C4⋊Q8—32no—
83C4⋊Q8—32no—
84C4⋊Q8—61no—
85C4⋊Q8—61no—
86#126—61no—
87#125—31no—
88#124—31no—
89#123—61no—
90C22⋊Q8—31no—
91#121—31no—
92C4×Q8—31no—
93C4×D4—31no—
94C4×D4—61no—
95#117—61no—
96C2×C4⋊C4—61no—
97C2×C4⋊C4—61no—
98C2×C42C2×D311yescharacteristic
99C4⋊C8C2×D312yes—
100#103—34no—
101#103—34no—
102C22⋊C8C2×D312yes—
103#97C2×D312yes—
104C23×C3C2×D411yescharacteristic
105C22×D3—21no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2411yescharacteristic
115#67—42no—
116C2×Dic3—21no—
117C2×Dic3—21no—
118C2×Dic3—21no—
119C2×Dic3—24no—
120C2×Dic3—44no—
121D12—22no—
122C4×D3—42no—
123Dic6—24no—
124Dic6—22no—
125Dic6—24no—
126Dic6—44no—
127C3×C8—24no—
128C2×Q8—34no—
129C2×Q8—31no—
130C2×Q8—62no—
131C2×Q8—124no—
132C2×D4—31no—
133C2×D4—61no—
134C22×C4—62no—
135C22×C4—61no—
136C22×C4D1211yescharacteristic
137C22×C4D1211yescharacteristic
138C22×C4C22×D311yescharacteristic
139Q16—124no—
140SD16—124no—
141C2×C8C22×D314yes—
142C4⋊C4—31no—
143C4⋊C4—31no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4—62no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C22⋊C4—61no—
153C22⋊C4—61no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C42—61no—
158C42D1211yescharacteristic
159C42D1211yescharacteristic
160C42C22×D311yescharacteristic
161C42C22×D311yescharacteristic
162C22×C3—21no—
163C22×C3—21no—
164C22×C3#13712yes—
165C22×C3C22×D411yescharacteristic
166C2×D3—41no—
167C2×D3—41no—
168C2×D3—41no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—21no—
172C3×C4—21no—
173C3×C4#13712yes—
174C3×C4H(5,𝔽2)12yes—
175Dic3—44no—
176Dic3—41no—
177Dic3—41no—
178Dic3—41no—
179C23—61no—
180C23C2×D1211yescharacteristic
181Q8—64no—
182Q8—62no—
183Q8—64no—
184Q8—124no—
185D4—62no—
186D4—122no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—64no—
194C2×C4—124no—
195C2×C4—122no—
196C2×C4C2×D1211yescharacteristic
197C2×C4C2×D1211yescharacteristic
198C2×C4C2×D1211yescharacteristic
199C2×C4C2×D1211yescharacteristic
200C2×C4C2×D1211yescharacteristic
201C2×C4C23×D311yescharacteristic
202C8—24no—
203C2×C3—22no—
204C2×C3#53511yescharacteristic
205C2×C3C2×#13711yescharacteristic
206C2×C3C2×#13711yescharacteristic
207D3—81no—
208C22—21no—
209C22—21no—
210C22—121no—
211C22—121no—
212C22—121no—
213C22#90912yes—
214C22C22×D1211yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—121no—
220C4—121no—
221C4—124no—
222C4—121no—
223C4#90912yes—
224C4#100912yes—
225C3#321811yescharacteristic, complemented, Sylow
226C2—22no—
227C2—241no—
228C2#552511yescharacteristic
229C2C2×#90911yescharacteristic
230C2C2×#90911yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#432C2
socleC22×C3C22×D4
prime radical#930C22
hypercenter#432D3
hypocenterC3#3218
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. C2×#869C2semidirect
3. C3×#432C2semidirect
4. #4849C2semidirect
5. #4841C2semidirect
6. #4739C2semidirect
8. #4715C2semidirect
9. #4703C2semidirect
44. #432D3semidirect
225. C3#3218semidirect

Maximal subgroups

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

subgroupindex
#54792
C2×#8692
C3×#4322
#48492
#48412
#47392
#47382
#47152
#47032
#32183