Finite Group Databasebeta

← #13181155

Subgroups — #13181155

Order 1728.

864 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 119 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#11724624C212yesmaximal, complemented
2C3×#64289C211yesmaximal, characteristic, complemented
3#11724341C213yesmaximal, complemented
4#11724338C216yesmaximal, complemented
5#11724337C213yesmaximal, complemented
6#10596520—31nomaximal
7C3×#3832C2211yescharacteristic
8C3×Q8×C3⋊D3—22no—
9#89042C2216yes—
10#89041—26no—
11#89040C2216yes—
12C3×#3788C2213yes—
13C3×#3787C2213yes—
14#88943C2213yes—
15#88943C2216yes—
16#88942C2216yes—
17#88941C2211yescharacteristic
18#64316—32no—
19#64315—62no—
20#64289D311yescharacteristic, complemented
21#64228—33no—
22#64226—36no—
23#64223—33no—
24C33×Q8D411yescharacteristic
25C3×C4⋊(C3⋊D3)D413yes—
26C3×C4×C3⋊D3—26no—
27C3×C4×C3⋊D3C2313yes—
28C3×#634—26no—
29#6296C2316yes—
30#6295—26no—
31#6295C2312yes—
32#6294—26no—
33C3×#622C2311yescharacteristic
34C3×#622C2313yes—
35#5753—91nomaximal
36#3832C2×D311yescharacteristic
37Q8×C3⋊D3—22no—
38C3×#239—62no—
39C3×D3×Q8—62no—
40C3×D3×Q8—62no—
41C3×D3×Q8—122no—
42#3799—66no—
43#3798—36no—
44#3798—66no—
45#3797—36no—
46#3796—66no—
47#3795—66no—
48D3×Dic6—66no—
49#3788C2×D313yes—
50#3787C2×D313yes—
51#3776—33no—
52#3776—36no—
53#3775—36no—
54#3774—31no—
55C2×C3×C3⋊D3—22no—
56C2×C3×C3⋊D3C2×D413yes—
57C33×C4C2×D413yes—
58C3×C3⋊Dic3—26no—
59C3×C3⋊Dic3C2411yescharacteristic
60#1010—92no—
61#1007—182no—
62C3×#131—91no—
63#909—96no—
64C2×Dic12—93no—
65#904—93no—
66C32×Q8—22no—
67C32×Q8—42no—
68C32×Q8D1211yescharacteristic
69C4⋊(C3⋊D3)D1213yes—
70C4×C3⋊D3—26no—
71C4×C3⋊D3C22×D313yes—
72#634—26no—
73C3×D12—66no—
74C3×C4×D3—66no—
75C3×C4×D3—66no—
76C3×C4×D3—66no—
77C3×C4×D3—126no—
78C3×Dic6—66no—
79C3×Dic6—66no—
80C3×Dic6—126no—
81#627—36no—
82#627—66no—
83#626—62no—
84#626—66no—
85#626—66no—
86#625—32no—
87#625—66no—
88#624—66no—
89D3×Dic3—612no—
90#622C22×D311yescharacteristic
91#622C22×D313yes—
92#578—271noSylow
93C2×C33C22×D411yescharacteristic
94C3×C3⋊D3—23no—
95C3×C3⋊D3—42no—
96C3×#41—91no—
97C2×C3×Q8—182no—
98#239—62no—
99#239—186no—
100D3×Q8—62no—
101D3×Q8—62no—
102D3×Q8—122no—
103D3×Q8—186no—
104#237—186no—
105#235—96no—
106#235—186no—
107#235—186no—
108C2×Dic6—96no—
109C3×M16—93no—
110C2×C3×C8—93no—
111Dic12—93no—
112Dic12—96no—
113D24—91no—
114#204—96no—
115C2×C3⋊D3—22no—
116C2×C3⋊D3C2×D1213yes—
117C2×C3×D3—66no—
118C2×C3×D3—62no—
119C2×C3×D3—62no—
120C2×C3×D3—122no—
121C32×C4—26no—
122C32×C4—46no—
123C32×C4C2×D1213yes—
124C3⋊Dic3—26no—
125C3⋊Dic3C23×D311yescharacteristic
126C3×Dic3—62no—
127C3×Dic3—66no—
128C3×Dic3—66no—
129C3×Dic3—126no—
1302−1+4—272no—
131#137—276no—
132#135—273no—
133C2×Q16—273no—
134#131—91no—
135C33#57811yescharacteristic, complemented, Sylow
136C3×Q8—22no—
137C3×Q8—42no—
138C3×Q8—186no—
139C3×Q8GO+(4,𝔽2)11yescharacteristic
140C3×D4—93no—
141C2×C3×C4—93no—
142C2×C3×C4—186no—
143#67—182no—
144#67—186no—
145#67—186no—
146C2×Dic3—186no—
147D12—66no—
148D12—92no—
149D12—186no—
150C4×D3—66no—
151C4×D3—66no—
152C4×D3—66no—
153C4×D3—126no—
154C4×D3—1812no—
155Dic6—66no—
156Dic6—66no—
157Dic6—96no—
158Dic6—126no—
159Dic6—186no—
160C3×C8—93no—
161C3×C8—91no—
162C2×C32—22no—
163C2×C32—42no—
164C2×C32C22×D1211yescharacteristic
165C3⋊D3—23no—
166C3⋊D3—42no—
167C3×D3—126no—
168C3×D3—122no—
169C3×D3—124no—
170C3×D3—122no—
171#41—91no—
172#41—276no—
173#41—546no—
174C2×Q8—182no—
175C2×Q8—276no—
176Q16—273no—
177Q16—276no—
178SD16—276no—
179D8—271no—
180M16—93no—
181C2×C8—93no—
182C22×C3—93no—
183C22×C3—182no—
184C2×D3—66no—
185C2×D3—62no—
186C2×D3—62no—
187C2×D3—122no—
188C3×C4—26no—
189C3×C4—46no—
190C3×C4—91no—
191C3×C4—186no—
192C3×C4C2×GO+(4,𝔽2)13yes—
193Dic3—62no—
194Dic3—66no—
195Dic3—66no—
196Dic3—126no—
197C32—22no—
198C32—42no—
199C32#575311yescharacteristic, complemented
200Q8—186no—
201Q8—276no—
202Q8#157⋊D311yescharacteristic
203D4—93no—
204D4—272no—
205D4—546no—
206C2×C4—93no—
207C2×C4—186no—
208C2×C4—546no—
209C8—91no—
210C8—93no—
211C2×C3—22no—
212C2×C3—42no—
213C2×C3—183no—
214C2×C3—362no—
215C2×C3C22×GO+(4,𝔽2)11yescharacteristic
216D3—126no—
217D3—122no—
218D3—124no—
219D3—122no—
220C22—93no—
221C22—182no—
222C4—91no—
223C4—186no—
224C4C2×#157⋊D313yes—
225C3—22no—
226C3—42no—
227C3#1059652011yescharacteristic, complemented
228C2—183no—
229C2—362no—
230C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FittingC33×Q8D4
socleC2×C33C22×D4
hypercenterQ8#157⋊D3
hypocenterC33#578
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #11724624C2semidirect
2. C3×#64289C2semidirect
3. #11724341C2semidirect
4. #11724338C2semidirect
5. #11724337C2semidirect
20. #64289D3semidirect
135. C33#578semidirect
199. C32#5753semidirect
227. C3#10596520semidirect

Maximal subgroups

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

subgroupindex
#117246242
C3×#642892
#117243412
#117243382
#117243372
#105965203
#57539