Finite Group Databasebeta

← #13167375

Subgroups — #13167375

Order 1728.

376 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 55 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#11724337C211yesmaximal, characteristic
2#11723652C212yesmaximal, complemented
3C3×C8×#157C211yesmaximal, characteristic, complemented
4#11722787C211yesmaximal, characteristic, complemented
5#11722785C212yesmaximal, complemented
6#10593299—31nomaximal
7#89042C2212yes—
8C3×C4×#157C2211yescharacteristic
9C3×#3787C2211yescharacteristic
10#88943C411yescharacteristic, complemented
11#88942C412yescomplemented
12#88941C411yescharacteristic, complemented
13C3×#3743—22no—
14C3×C8×C3⋊D3C2211yescharacteristic
15#88809C2212yes—
16#88808—22no—
17#64223—31no—
18#63805—32no—
19#63804—62no—
20C8×#157D311yescharacteristic, complemented
21#63733—31no—
22#63731—32no—
23C2×C3×#157—21no—
24C3×C4×C3⋊D3—22no—
25C3×C4×C3⋊D3C2311yescharacteristic
26#6296C2×C412yes—
27#6295—22no—
28#6295C2×C412yes—
29C3×#622C2×C411yescharacteristic
30C3×#622C2×C411yescharacteristic
31C33×C8D411yescharacteristic
32C3×C3⋊(C3⋊C8)—22no—
33C3×C3⋊(C3⋊C8)D411yescharacteristic
34#4700—91nomaximal
35#3798—32no—
36#3798—62no—
37C4×#157C2×D311yescharacteristic
38#3787C2×D311yescharacteristic
39#3776—31no—
40#3775—32no—
41#3774—31no—
42#3743—22no—
43C8×C3⋊D3C2×D311yescharacteristic
44C3×ΓL(1,𝔽25)—62no—
45C3×ΓL(1,𝔽25)—62no—
46C3×ΓL(1,𝔽25)—122no—
47C3×D3×C8—62no—
48#3712—32no—
49#3711—62no—
50#3710—62no—
51D3×C3⋊C8—62no—
52D3×C3⋊C8—62no—
53C2×C3×C3⋊D3—22no—
54C2×C3×C3⋊D3#4111yescharacteristic
55C3×#157—22no—
56C33×C4C2×D411yescharacteristic
57C3×C3⋊Dic3—22no—
58C3×C3⋊Dic3C22×C411yescharacteristic
59#907—182no—
60#904—91no—
61#901—92no—
62C3×C4×C8—91no—
63#819—91no—
64#805—92no—
65C2×#157—21no—
66C4×C3⋊D3—22no—
67C4×C3⋊D3C22×D311yescharacteristic
68C3×C4×D3—62no—
69C3×C4×D3—62no—
70C3×C4×D3—62no—
71C3×C4×D3—122no—
72#627—32no—
73#627—62no—
74#626—62no—
75#626—62no—
76#626—62no—
77#625—32no—
78#625—62no—
79#622C4×D311yescharacteristic
80#622C4×D311yescharacteristic
81C32×C8—22no—
82C32×C8—42no—
83C32×C8D1211yescharacteristic
84C3⋊(C3⋊C8)—22no—
85C3⋊(C3⋊C8)D1211yescharacteristic
86C3×C3⋊C8—62no—
87C3×C3⋊C8—62no—
88C3×C3⋊C8—62no—
89C3×C3⋊C8—122no—
90#443—271noSylow
91C2×C33C4×D411yescharacteristic
92C3×C3⋊D3—21no—
93C3×C3⋊D3—42no—
94#235—92no—
95#235—182no—
96C3×M16—182no—
97C2×C3×C8—91no—
98C2×C3×C8—91no—
99C3×C42—91no—
100#208—92no—
101C2×C3⋊C8—182no—
102Dic12—91no—
103D24—91no—
104#204—92no—
105ΓL(1,𝔽25)—62no—
106ΓL(1,𝔽25)—62no—
107ΓL(1,𝔽25)—122no—
108ΓL(1,𝔽25)—182no—
109D3×C8—62no—
110D3×C8—182no—
111D3×C8—182no—
112C2×C3⋊D3—22no—
113C2×C3⋊D3#23511yescharacteristic
114C2×C3×D3—62no—
115C2×C3×D3—62no—
116C2×C3×D3—62no—
117C2×C3×D3—122no—
118#157—22no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4C2×D1211yescharacteristic
122C3⋊Dic3—22no—
123C3⋊Dic3C2×C4×D311yescharacteristic
124C3×Dic3—62no—
125C3×Dic3—62no—
126C3×Dic3—62no—
127C3×Dic3—122no—
128#135—271no—
129#131—272no—
130#108—271no—
131#104—272no—
132C4×C8—91no—
133C33#44311yescharacteristic, complemented, Sylow
134C2×C3×C4—91no—
135C2×C3×C4—182no—
136C2×C3×C4—181no—
137#67—182no—
138#67—182no—
139#67—182no—
140D12—92no—
141D12—182no—
142C4×D3—62no—
143C4×D3—62no—
144C4×D3—62no—
145C4×D3—122no—
146Dic6—92no—
147Dic6—182no—
148C3×C8—22no—
149C3×C8—42no—
150C3×C8—91no—
151C3×C8—91no—
152C3×C8—91no—
153C3×C8—182no—
154C3×C8GO+(4,𝔽2)11yescharacteristic
155C3⋊C8—62no—
156C3⋊C8—62no—
157C3⋊C8—62no—
158C3⋊C8—122no—
159C2×C32—22no—
160C2×C32—42no—
161C2×C32C4×D1211yescharacteristic
162C3⋊D3—21no—
163C3⋊D3—42no—
164C3×D3—122no—
165C3×D3—122no—
166C3×D3—122no—
167C3×D3—242no—
168#41—272no—
169Q16—271no—
170SD16—272no—
171D8—271no—
172M16—182no—
173M16—272no—
174C2×C8—91no—
175C2×C8—91no—
176C2×C8—542no—
177C42—91no—
178C22×C3—91no—
179C22×C3—182no—
180C2×D3—62no—
181C2×D3—62no—
182C2×D3—62no—
183C2×D3—122no—
184C3×C4—22no—
185C3×C4—42no—
186C3×C4—91no—
187C3×C4—182no—
188C3×C4—182no—
189C3×C4C2×GO+(4,𝔽2)11yescharacteristic
190Dic3—62no—
191Dic3—62no—
192Dic3—62no—
193Dic3—122no—
194C32—22no—
195C32—42no—
196C32#470011yescharacteristic, complemented
197Q8—272no—
198D4—272no—
199D4—542no—
200C2×C4—91no—
201C2×C4—181no—
202C2×C4—182no—
203C8—91no—
204C8—91no—
205C8—91no—
206C8—182no—
207C8#157⋊D311yescharacteristic
208C2×C3—22no—
209C2×C3—42no—
210C2×C3—181no—
211C2×C3—362no—
212C2×C3C4×GO+(4,𝔽2)11yescharacteristic
213D3—122no—
214D3—122no—
215D3—122no—
216D3—242no—
217C22—91no—
218C22—182no—
219C4—91no—
220C4—182no—
221C4—182no—
222C4C2×#157⋊D311yescharacteristic
223C3—22no—
224C3—42no—
225C3#1059329911yescharacteristic, complemented
226C2—181no—
227C2—362no—
228C2C4×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC4C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC33×C8D4
socleC2×C33C4×D4
prime radical#11724337C2
cosocleC3×C4×C3⋊D3C23
hypercenterC8#157⋊D3
hypocenterC33#443
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #11723652C2semidirect
3. C3×C8×#157C2semidirect
4. #11722787C2semidirect
5. #11722785C2semidirect
10. #88943C4semidirect
11. #88942C4semidirect
12. #88941C4semidirect
20. C8×#157D3semidirect
133. C33#443semidirect
196. C32#4700semidirect
225. C3#10593299semidirect

Maximal subgroups

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

subgroupindex
#117243372
#117236522
C3×C8×#1572
#117227872
#117227852
#105932993
#47009