Finite Group Databasebeta

← #13146637

Subgroups — #13146637

Order 1728.

362 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 37 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
1D4×#1088C211yesmaximal, characteristic, complemented
2#11722305C211yesmaximal, characteristic, complemented
3#11721190C212yesmaximal, complemented
4#11720695C212yesmaximal, complemented
5#11720693C211yesmaximal, characteristic, complemented
6C22×#1088—21no—
7C2×#1088⋊C2—21no—
8C2×#1088⋊C2—21no—
9D4×#272C2211yescharacteristic
10#88655C2212yes—
11C22⋊#1088—21no—
12C4⋊#1088C2211yescharacteristic
13C4×#1088C2211yescharacteristic
14#3753⋊C3—22no—
15#88436—22no—
16#88434C2212yes—
17D3×#213—62no—
18C22×#272—21no—
19C2×((C3⋊D3)⋊C3)⋊C2—22no—
20C2×#1088—21no—
21C2×#1088—42no—
22C2×#1088D411yescharacteristic
23C2×#1088D411yescharacteristic
24#1088⋊C2—42no—
25#1088⋊C2—42no—
26D4×H(3,𝔽3)D411yescharacteristic, complemented
27#6235—21no—
28C4⋊#272D411yescharacteristic, complemented
29C4×#272C2311yescharacteristic
30C4⋊((C3⋊D3)⋊C3)—22no—
31#6198—22no—
32(C3⋊(C3⋊C8))⋊C3—22no—
33#5015—91nomaximal
34C3×D3×D4—62no—
35D3×D12—62no—
36#3753—22no—
37C3×#213—62no—
38#3714—62no—
39#3713—62no—
40D3×C3⋊C8—62no—
41C22×H(3,𝔽3)—21no—
42C2×#272—21no—
43C2×#272—41no—
44C2×#272—41no—
45C2×#272C2×D411yescharacteristic
46C2×(C3⋊D3)⋊C3—42no—
47((C3⋊D3)⋊C3)⋊C2—42no—
48((C3⋊D3)⋊C3)⋊C2—42no—
49#1088—21no—
50#1088—41no—
51#1088D812yescomplemented
52C4×H(3,𝔽3)C2×D411yescharacteristic
53#1084#4111yescharacteristic
54C3×C4×D4—91no—
55C2×#213—92no—
56D3×D8—182no—
57#874—91no—
58#809—92no—
59C4⋊(C3⋊C8)—91no—
60C22×C3×D3—122no—
61C2×D32—122no—
62C32×D4—22no—
63C4⋊(C3⋊D3)—22no—
64C3×#67—122no—
65C3×D12—62no—
66C3×D12—62no—
67C3×C4×D3—62no—
68#626—122no—
69C3⋊(C3⋊C8)—22no—
70C3×C3⋊C8—62no—
71#459—271noSylow
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)#12111yescharacteristic
74#272—41no—
75#272C2×D811yescharacteristic
76#272AGL(1,ℤ/8)11yescharacteristic
77(C3⋊D3)⋊C3—82no—
78C2×C3×D4—91no—
79C22×C3×C4—181no—
80D3×D4—62no—
81D3×D4—182no—
82C2×D12—92no—
83C2×D12—181no—
84C2×D12—181no—
85C3×D8—182no—
86C3×C4⋊C4—91no—
87C3×C22⋊C4—181no—
88C3×C42—91no—
89#213—62no—
90#213—182no—
91#213—182no—
92#213—182no—
93C2×C3⋊C8—92no—
94D24—182no—
95D3×C8—182no—
96C22×C32—42no—
97C2×C3⋊D3—42no—
98C2×C3×D3—62no—
99C2×C3×D3—122no—
100C2×C3×D3—122no—
101C2×C3×D3—242no—
102C2×C3×D3—242no—
103D32—242no—
104D32—242no—
105C32×C4—22no—
106C3×Dic3—62no—
107C2×D8—272no—
108#127—271no—
109C4×D4—91no—
110C4⋊C8—271no—
111#102—272no—
112H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
113C23×C3—181no—
114C22×D3—122no—
115C22×D3—182no—
116C22×D3—362no—
117C3×D4—62no—
118C3×D4—91no—
119C3×D4—182no—
120C3×D4—181no—
121C3×D4GO+(4,𝔽2)11yescharacteristic
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—181no—
126C2×C3×C4—362no—
127#67—122no—
128#67—362no—
129D12—62no—
130D12—62no—
131D12—182no—
132D12—182no—
133D12—362no—
134D12—362no—
135C4×D3—62no—
136C3×C8—182no—
137C3⋊C8—62no—
138C3⋊C8—182no—
139C2×C32—22no—
140C2×C32—82no—
141C3⋊D3—82no—
142C3×D3—62no—
143C3×D3—62no—
144C3×D3—242no—
145C3×D3—242no—
146C2×D4—91no—
147C2×D4—272no—
148C2×D4—541no—
149C2×D4—541no—
150C22×C4—181no—
151D8—182no—
152D8—542no—
153C2×C8—272no—
154C4⋊C4—91no—
155C22⋊C4—181no—
156C42—91no—
157C22×C3—21no—
158C22×C3—91no—
159C22×C3—122no—
160C22×C3—181no—
161C22×C3—361no—
162C22×C3—361no—
163C22×C3—362no—
164C2×D3—62no—
165C2×D3—122no—
166C2×D3—122no—
167C2×D3—242no—
168C2×D3—242no—
169C2×D3—362no—
170C2×D3—362no—
171C2×D3—362no—
172C2×D3—722no—
173C2×D3—722no—
174C3×C4—62no—
175C3×C4—92no—
176C3×C4—91no—
177C3×C4—181no—
178C3×C4—361no—
179C3×C4C2×GO+(4,𝔽2)11yescharacteristic
180Dic3—62no—
181C32—22no—
182C23—181no—
183C23—542no—
184D4—91no—
185D4—181no—
186D4—182no—
187D4—542no—
188D4—1082no—
189D4—1082no—
190D4#1088⋊C211yescharacteristic, complemented
191C2×C4—91no—
192C2×C4—91no—
193C2×C4—91no—
194C2×C4—181no—
195C2×C4—362no—
196C8—182no—
197C2×C3—41no—
198C2×C3—62no—
199C2×C3—91no—
200C2×C3—91no—
201C2×C3—242no—
202C2×C3—361no—
203C2×C3—722no—
204C2×C3#6424211yescharacteristic
205D3—62no—
206D3—62no—
207D3—242no—
208D3—242no—
209D3—722no—
210C22—21no—
211C22—91no—
212C22—181no—
213C22—361no—
214C22—361no—
215C22—362no—
216C22—1082no—
217C22—1082no—
218C4—92no—
219C4—91no—
220C4—181no—
221C4—361no—
222C4C2×#1088⋊C211yescharacteristic
223C3—62no—
224C3#1059346411yescharacteristic
225C2—41no—
226C2—91no—
227C2—91no—
228C2—361no—
229C2—722no—
230C2#1172231611yescharacteristic
subgroupG/H
centerC2#11722316
commutator, cosocleC4×#272C23
FrattiniC3×C4C2×GO+(4,𝔽2)
FittingD4×H(3,𝔽3)D4
socleC2×C3#64242
hypercenterD4#1088⋊C2
hypocenterH(3,𝔽3)#459
LCSG ▷ C4×#272 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D4×#1088C2semidirect
2. #11722305C2semidirect
3. #11721190C2semidirect
4. #11720695C2semidirect
5. #11720693C2semidirect
26. D4×H(3,𝔽3)D4semidirect
28. C4⋊#272D4semidirect
51. #1088D8semidirect
112. H(3,𝔽3)#459semidirect
190. D4#1088⋊C2semidirect

Maximal subgroups

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

subgroupindex
D4×#10882
#117223052
#117211902
#117206952
#117206932
#50159