Finite Group Databasebeta

← #80751

Subgroups — #80751

Order 384.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#5643C212yesmaximal, complemented
2C3×#467C211yesmaximal, characteristic, complemented
3#5185C212yesmaximal, complemented
4#5115C211yesmaximal, characteristic
5#5102C211yesmaximal, characteristic, complemented
6#5049C212yesmaximal, complemented
7#4890C212yesmaximal, complemented
8#4819C212yesmaximal, complemented
9#4793C212yesmaximal, complemented
10#3262—31nomaximal, Sylow
11C2×C3×Q16C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C3×C22⋊Q8C2212yes—
14#947C2212yes—
15Q8×Dic3C2212yes—
16C2×#215C2212yes—
17#930C2212yes—
18#926C2211yescharacteristic
19C2×#208C2211yescharacteristic
20C2×D24C2211yescharacteristic
21#898—22no—
22#897—22no—
23#893C2212yes—
24#888C2212yes—
25#885—22no—
26#883—22no—
27C3×#107C2211yescharacteristic
28C3×#103C2212yes—
29#835C2212yes—
30#821C2212yes—
31#818C2211yescharacteristic
32Dic3⋊C8C2212yes—
33Dic3⋊C8C2212yes—
34#809C2212yes—
35#807C2212yes—
36#542—32no—
37#477—32no—
38#474—32no—
39#468—32no—
40#467D311yescharacteristic, complemented
41#466—31no—
42#464—32no—
43#463—32no—
44#424—31no—
45C2×C3×Q8C2312yes—
46C22×C3×C4C2311yescharacteristic
47C2×#67—22no—
48C2×D12C2312yes—
49C2×C4×D3—22no—
50C2×Dic6—22no—
51C3×Q16—41no—
52C2×C3×C8—41no—
53C2×C3×C8C2311yescharacteristic
54C2×C3×C8C2311yescharacteristic
55C3×C4⋊C4—22no—
56C3×C4⋊C4C2312yes—
57C3×C22⋊C4—22no—
58#217D412yes—
59#215—42no—
60(C2×D3)⋊C4—22no—
61(C2×D3)⋊C4—22no—
62(C2×D3)⋊C4—22no—
63(C2×D3)⋊C4—22no—
64C4⋊Dic3—22no—
65C4⋊Dic3D411yescharacteristic
66C4⋊Dic3D411yescharacteristic
67C4⋊Dic3C2312yes—
68Dic3⋊C4—22no—
69C4×Dic3—22no—
70C4×Dic3C2312yes—
71#208—41no—
72C2×C3⋊C8C2312yes—
73D24—41no—
74C2×Q16C2×D311yescharacteristic
75C2×SD16—32no—
76C2×D8—31no—
77C2×M16—31no—
78C22×C8C2×D311yescharacteristic
79C4⋊Q8—32no—
80#126—62no—
81#124—32no—
82#124—62no—
83#123—62no—
84C22⋊Q8—62no—
85C22⋊Q8C2×D312yes—
86#121—32no—
87C4×Q8—32no—
88C4×D4—32no—
89#117—31no—
90#107—31no—
91#107—32no—
92#107C2×D311yescharacteristic
93C4⋊C8—32no—
94C4⋊C8—32no—
95#103—32no—
96#103C2×D312yes—
97#102—32no—
98#102—32no—
99C23×C3C2×D411yescharacteristic
100C22×D3—22no—
101C3×Q8—22no—
102C3×Q8—22no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2411yescharacteristic
109#67—82no—
110C2×Dic3—22no—
111C2×Dic3C2×D412yes—
112C2×Dic3C2×D412yes—
113D12—22no—
114D12—22no—
115C4×D3—82no—
116Dic6—82no—
117C3×C8—21no—
118C3×C8—21no—
119C3⋊C8—22no—
120C2×Q8—62no—
121C2×Q8C22×D312yes—
122C2×D4—32no—
123C2×D4—62no—
124C22×C4—62no—
125C22×C4C22×D311yescharacteristic
126Q16—41no—
127SD16—122no—
128D8—121no—
129M16—121no—
130C2×C8—32no—
131C2×C8—41no—
132C2×C8C22×D311yescharacteristic
133C2×C8C22×D311yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—31no—
136C4⋊C4—31no—
137C4⋊C4—32no—
138C4⋊C4—62no—
139C4⋊C4—62no—
140C4⋊C4C22×D312yes—
141C22⋊C4—22no—
142C22⋊C4—32no—
143C22⋊C4—62no—
144C22⋊C4—62no—
145C22⋊C4—62no—
146C22⋊C4—62no—
147C42—32no—
148C42—62no—
149C22×C3—21no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3C22×D411yescharacteristic
153C2×D3—42no—
154C2×D3—42no—
155C2×D3—42no—
156C3×C4—21no—
157C3×C4—42no—
158C3×C4—42no—
159C3×C4H(5,𝔽2)11yescharacteristic
160C3×C42−1+411yescharacteristic
161Dic3—22no—
162Dic3—22no—
163Dic3—42no—
164C23—62no—
165C23D3×D411yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—242no—
169D4—62no—
170D4—62no—
171D4—242no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—32no—
177C2×C4—32no—
178C2×C4—62no—
179C2×C4—242no—
180C2×C4D3×D411yescharacteristic
181C2×C4C23×D311yescharacteristic
182C8—21no—
183C8—21no—
184C8—62no—
185C2×C3—41no—
186C2×C3#53711yescharacteristic
187C2×C3#57611yescharacteristic
188C2×C3#57811yescharacteristic
189D3—82no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—122no—
194C22—122no—
195C22—122no—
196C22C2×D3×D411yescharacteristic
197C4—21no—
198C4—42no—
199C4—42no—
200C4—62no—
201C4—62no—
202C4—122no—
203C4#100411yescharacteristic
204C4#101011yescharacteristic
205C3#326211yescharacteristic, complemented, Sylow
206C2—41no—
207C2—242no—
208C2#559711yescharacteristic
209C2#576611yescharacteristic
210C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#467C2
socleC22×C3C22×D4
prime radical#5115C2
hypercenter#467D3
hypocenterC3#3262
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #5643C2semidirect
2. C3×#467C2semidirect
3. #5185C2semidirect
5. #5102C2semidirect
6. #5049C2semidirect
7. #4890C2semidirect
8. #4819C2semidirect
9. #4793C2semidirect
40. #467D3semidirect
205. C3#3262semidirect

Maximal subgroups

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

subgroupindex
#56432
C3×#4672
#51852
#51152
#51022
#50492
#48902
#48192
#47932
#32623