Finite Group Databasebeta

← #13165182

Subgroups — #13165182

Order 1728.

376 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C3×#64223C211yesmaximal, characteristic
2#11722821C212yesmaximal, complemented
3#11722787C211yesmaximal, characteristic, complemented
4#11722762C212yesmaximal, complemented
5#11722703C211yesmaximal, characteristic, complemented
6#10593297—31nomaximal
7C3×#3798C2212yes—
8C4×C3⋊#157C2211yescharacteristic
9C3×#3787C2211yescharacteristic
10C3×#3776C411yescharacteristic, complemented
11C3×#3775C412yescomplemented
12C3×#3774C411yescharacteristic, complemented
13#88809C2212yes—
14#88786—22no—
15C3⋊D3×C3⋊C8C2211yescharacteristic
16C3×D3×C3⋊C8—22no—
17#64223D311yescharacteristic
18#63814—62no—
19#63803—32no—
20C8×#157—31no—
21#63733—31no—
22#63731—32no—
23C2×C3⋊#157—21no—
24C3×C4×C3⋊D3C2311yescharacteristic
25C32×C4×D3—22no—
26C3×#627C2×C412yes—
27C3×#626—22no—
28C3×#625C2×C412yes—
29C3×#622C2×C411yescharacteristic
30C3×#622C2×C411yescharacteristic
31C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
32C3×C3⋊(C3⋊C8)—22no—
33C32×C3⋊C8D411yescharacteristic
34#5196—91nomaximal
35C3×#235—62no—
36#3798C2×D312yes—
37C4×#157—31no—
38#3787C2×D311yescharacteristic
39#3776Dic311yescharacteristic, complemented
40#3775Dic312yescomplemented
41#3774Dic311yescharacteristic, complemented
42#3748—62no—
43C8×C3⋊D3—31no—
44C2×C3×C3⋊C8—62no—
45C3×D3×C8—62no—
46#3712—32no—
47#3711—62no—
48#3711—62no—
49#3711—62no—
50#3711—122no—
51D3×C3⋊C8—22no—
52D3×C3⋊C8—62no—
53C2×C3×C3⋊D3#4111yescharacteristic
54C2×C32×D3—22no—
55C3⋊#157—22no—
56C33×C4C2×D411yescharacteristic
57C3×C3⋊Dic3C22×C411yescharacteristic
58C32×Dic3—22no—
59C3×#135—91no—
60#948—92no—
61#901—182no—
62#837—92no—
63#819—91no—
64C8×Dic3—91no—
65C2×#157—61no—
66C2×C32×C4—62no—
67C4×C3⋊D3C22×D311yescharacteristic
68C3×#67—62no—
69C3×#67—62no—
70C3×D12—62no—
71C3×C4×D3—22no—
72C3×C4×D3—22no—
73C3×C4×D3—42no—
74C3×C4×D3—62no—
75C3×Dic6—62no—
76#627C2×Dic312yes—
77#626—22no—
78#625C2×Dic312yes—
79#622C2×Dic311yescharacteristic
80#622C2×Dic311yescharacteristic
81C32×C8—31no—
82C3⋊(C3⋊C8)—22no—
83C3⋊(C3⋊C8)—31no—
84C3⋊(C3⋊C8)—62no—
85C3⋊(C3⋊C8)—122no—
86C3×C3⋊C8—22no—
87C3×C3⋊C8—62no—
88C3×C3⋊C8—62no—
89C3×C3⋊C8—122no—
90#443—271noSylow
91C2×C33C4×D411yescharacteristic
92C3×C3⋊D3—21no—
93C32×D3—42no—
94C3×#41—92no—
95#235—62no—
96C3×Q16—91no—
97C3×SD16—92no—
98C3×D8—91no—
99C2×C3×C8—91no—
100C2×C3×C8—182no—
101C4×Dic3—91no—
102#208—92no—
103#208—182no—
104#208—182no—
105#208—362no—
106C2×C3⋊C8—62no—
107C2×C3⋊C8—91no—
108ΓL(1,𝔽25)—182no—
109ΓL(1,𝔽25)—182no—
110D3×C8—62no—
111D3×C8—182no—
112C22×C32—62no—
113C2×C3⋊D3#23711yescharacteristic
114C2×C3×D3—22no—
115C2×C3×D3—22no—
116C2×C3×D3—42no—
117C2×C3×D3—62no—
118#157—62no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—62no—
122C32×C4D3×D411yescharacteristic
123C3⋊Dic3C22×Dic311yescharacteristic
124C3×Dic3—22no—
125C3×Dic3—22no—
126C3×Dic3—42no—
127C3×Dic3—62no—
128#135—91no—
129#131—272no—
130#108—271no—
131#104—272no—
132C4×C8—271no—
133C33#44311yescharacteristic, complemented, Sylow
134C3×Q8—92no—
135C3×D4—92no—
136C3×D4—182no—
137C2×C3×C4—62no—
138C2×C3×C4—62no—
139C2×C3×C4—91no—
140C2×C3×C4—122no—
141#67—62no—
142#67—62no—
143C2×Dic3—181no—
144D12—62no—
145C4×D3—22no—
146C4×D3—62no—
147Dic6—62no—
148C3×C8—62no—
149C3×C8—91no—
150C3×C8—91no—
151C3×C8—182no—
152C3⋊C8—62no—
153C3⋊C8—62no—
154C3⋊C8—91no—
155C3⋊C8—122no—
156C3⋊C8—182no—
157C3⋊C8—362no—
158C3⋊C8GO+(4,𝔽2)11yescharacteristic
159C2×C32—22no—
160C2×C32—42no—
161C2×C32—122no—
162C2×C32D4×Dic311yescharacteristic
163C3⋊D3—21no—
164C3×D3—42no—
165C3×D3—42no—
166C3×D3—82no—
167C3×D3—122no—
168#41—92no—
169Q16—91no—
170SD16—92no—
171D8—91no—
172M16—272no—
173M16—542no—
174C2×C8—91no—
175C2×C8—182no—
176C2×C8—271no—
177C42—271no—
178C22×C3—62no—
179C22×C3—62no—
180C22×C3—91no—
181C22×C3—122no—
182C2×D3—22no—
183C2×D3—62no—
184C3×C4—22no—
185C3×C4—42no—
186C3×C4—62no—
187C3×C4—62no—
188C3×C4—91no—
189C3×C4—122no—
190C3×C4C2×GO+(4,𝔽2)11yescharacteristic
191Dic3—22no—
192Dic3—62no—
193Dic3—182no—
194C32—22no—
195C32—42no—
196C32#519611yescharacteristic, complemented
197Q8—92no—
198D4—92no—
199D4—182no—
200C2×C4—62no—
201C2×C4—91no—
202C2×C4—541no—
203C8—31no—
204C8—91no—
205C8—91no—
206C8—182no—
207C8—271no—
208C2×C3—22no—
209C2×C3—42no—
210C2×C3—122no—
211C2×C3—122no—
212C2×C3—181no—
213C2×C3—242no—
214C2×C3C4×GO+(4,𝔽2)11yescharacteristic
215D3—42no—
216D3—122no—
217C22—62no—
218C22—91no—
219C4—62no—
220C4—91no—
221C4—542no—
222C4D3×GO+(4,𝔽2)11yescharacteristic
223C3—22no—
224C3—42no—
225C3#1059329711yescharacteristic, complemented
226C2—122no—
227C2—181no—
228C2Dic3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
FittingC33×C4C2×D4
socleC2×C33C4×D4
prime radicalC3×#64223C2
cosocleC3×C4×C3⋊D3C23
hypocenterC33#443
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #11722821C2semidirect
3. #11722787C2semidirect
4. #11722762C2semidirect
5. #11722703C2semidirect
10. C3×#3776C4semidirect
11. C3×#3775C4semidirect
12. C3×#3774C4semidirect
39. #3776Dic3semidirect
40. #3775Dic3semidirect
41. #3774Dic3semidirect
133. C33#443semidirect
196. C32#5196semidirect
225. C3#10593297semidirect

Maximal subgroups

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

subgroupindex
C3×#642232
#117228212
#117227872
#117227622
#117227032
#105932973
#51969