Finite Group Databasebeta

← #79322

Subgroups — #79322

Order 384.

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

normal subgroups only

This group has 362 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#4850C212yesmaximal, complemented
5#4842C212yesmaximal, complemented
6#4736C212yesmaximal
7#4721C212yesmaximal, complemented
8#4714C212yesmaximal, complemented
9#4702C212yesmaximal, complemented
10#3238—31nomaximal, Sylow
11C22×Dic6—21no—
12C2×C3×C42C2211yescharacteristic
13#930C2211yescharacteristic
14#929—21no—
15C4×#67—21no—
16C2×C4⋊Dic3—21no—
17C2×C4⋊Dic3C2212yes—
18#924C2211yescharacteristic
19#876—21no—
20#875C2211yescharacteristic
21C4×D12C2211yescharacteristic
22#872—21no—
23#870C2211yescharacteristic
24#869—21no—
25#869—21no—
26#869C2212yes—
27#869C2212yes—
28C4×Dic6C2211yescharacteristic
29C3×C4⋊C8C2212yes—
30C3×C22⋊C8C2212yes—
31C3×#97C2212yes—
32#821C2214yes—
33#818C2214yes—
34#817C2214yes—
35#816C2214yes—
36C2×C4⋊Q8—31no—
37#529—31no—
38#501—32no—
39#489—32no—
40#483—32no—
41#480—32no—
42#477—32no—
43#476—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—22no—
54C2×Dic6—44no—
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⋊Dic3C2314yes—
70C4⋊Dic3C2311yescharacteristic
71C4⋊Dic3C2311yescharacteristic
72Dic3⋊C4—21no—
73Dic3⋊C4—21no—
74Dic3⋊C4—21no—
75Dic3⋊C4—21no—
76C4×Dic3—21no—
77C22×Q8—61no—
78C4⋊Q8—32no—
79C4⋊Q8—32no—
80C4⋊Q8—61no—
81C4⋊Q8—61no—
82#126—61no—
83#125—31no—
84#124—31no—
85#123—61no—
86C22⋊Q8—31no—
87#121—31no—
88C4×Q8—31no—
89C4×D4—31no—
90C4×D4—61no—
91#117—61no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—61no—
94C2×C42C2×D311yescharacteristic
95#107—34no—
96#106—34no—
97C4⋊C8C2×D312yes—
98#103—34no—
99#102—34no—
100C22⋊C8C2×D312yes—
101#97C2×D312yes—
102C23×C3C2×D411yescharacteristic
103C22×D3—21no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2411yescharacteristic
113#67—42no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—24no—
118C2×Dic3—44no—
119D12—22no—
120C4×D3—42no—
121Dic6—22no—
122Dic6—44no—
123Dic6—44no—
124C3×C8—24no—
125C2×Q8—31no—
126C2×Q8—62no—
127C2×Q8—62no—
128C2×Q8—124no—
129C2×D4—31no—
130C2×D4—61no—
131C22×C4—62no—
132C22×C4—61no—
133C22×C4D1211yescharacteristic
134C22×C4D1211yescharacteristic
135C22×C4C22×D311yescharacteristic
136C2×C8C22×D314yes—
137C4⋊C4—34no—
138C4⋊C4—31no—
139C4⋊C4—31no—
140C4⋊C4—62no—
141C4⋊C4—62no—
142C4⋊C4—62no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4—61no—
152C42—61no—
153C42D1211yescharacteristic
154C42D1211yescharacteristic
155C42C22×D311yescharacteristic
156C42C22×D311yescharacteristic
157C22×C3—21no—
158C22×C3—21no—
159C22×C3AGL(1,ℤ/8)12yes—
160C22×C3C22×D411yescharacteristic
161C2×D3—41no—
162C2×D3—41no—
163C2×D3—41no—
164C3×C4—21no—
165C3×C4—21no—
166C3×C4—21no—
167C3×C4—21no—
168C3×C4#13712yes—
169C3×C42−1+412yes—
170Dic3—41no—
171Dic3—44no—
172Dic3—41no—
173Dic3—41no—
174C23—61no—
175C23C2×D1211yescharacteristic
176Q8—62no—
177Q8—124no—
178Q8—124no—
179D4—62no—
180D4—122no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—64no—
188C2×C4—124no—
189C2×C4—122no—
190C2×C4C2×D1211yescharacteristic
191C2×C4C2×D1211yescharacteristic
192C2×C4C2×D1211yescharacteristic
193C2×C4C2×D1211yescharacteristic
194C2×C4C2×D1211yescharacteristic
195C2×C4C23×D311yescharacteristic
196C8—24no—
197C2×C3—22no—
198C2×C3#53611yescharacteristic
199C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C3C2×#13711yescharacteristic
201D3—81no—
202C22—21no—
203C22—21no—
204C22—121no—
205C22—121no—
206C22—121no—
207C22#90812yes—
208C22C22×D1211yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—124no—
214C4—121no—
215C4—121no—
216C4—121no—
217C4#90912yes—
218C4#101012yes—
219C3#323811yescharacteristic, complemented, Sylow
220C2—22no—
221C2—241no—
222C2#552611yescharacteristic
223C2C2×#90811yescharacteristic
224C2C2×#90911yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#432C2
socleC22×C3C22×D4
prime radical#930C22
hypercenter#432D3
hypocenterC3#3238
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. #4850C2semidirect
5. #4842C2semidirect
7. #4721C2semidirect
8. #4714C2semidirect
9. #4702C2semidirect
44. #432D3semidirect
219. C3#3238semidirect

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
#48502
#48422
#47362
#47212
#47142
#47022
#32383