Finite Group Databasebeta

← #13146651

Subgroups — #13146651

Order 1728.

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

normal subgroups only

This group has 328 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
1Q8×#1088C211yesmaximal, characteristic, complemented
2#11722305C211yesmaximal, characteristic
3#11721228C212yesmaximal, complemented
4#11720695C212yesmaximal, complemented
5#11720693C211yesmaximal, characteristic, complemented
6C2×#1088⋊C2—21no—
7C2×#1088⋊C2—21no—
8Q8×#272C2211yescharacteristic
9#88655C2212yes—
10C4⋊#1088—21no—
11C4⋊#1088C2211yescharacteristic
12C4×#1088—21no—
13C4×#1088C2211yescharacteristic
14#3755⋊C3—22no—
15#88438—22no—
16#88434C2212yes—
17D3×#215—62no—
18C2×((C3⋊D3)⋊C3)⋊C2—22no—
19C2×#1088—21no—
20C2×#1088D411yescharacteristic
21C2×#1088D411yescharacteristic
22#1088⋊C2—42no—
23#1088⋊C2—42no—
24Q8×H(3,𝔽3)D411yescharacteristic, complemented
25C4×#272—21no—
26C4×#272C2311yescharacteristic
27#6229—21no—
28#6229D411yescharacteristic, complemented
29C4⋊((C3⋊D3)⋊C3)—22no—
30#6198—22no—
31(C3⋊(C3⋊C8))⋊C3—22no—
32#5027—91nomaximal
33C3×D3×Q8—62no—
34D3×D12—62no—
35#3755—22no—
36C3×#215—62no—
37#3717—62no—
38#3715—62no—
39D3×C3⋊C8—62no—
40C2×#272C2×D411yescharacteristic
41C2×(C3⋊D3)⋊C3—42no—
42((C3⋊D3)⋊C3)⋊C2—42no—
43((C3⋊D3)⋊C3)⋊C2—42no—
44#1088—21no—
45#1088—41no—
46#1088SD1612yescomplemented
47C4×H(3,𝔽3)—21no—
48C4×H(3,𝔽3)C2×D411yescharacteristic
49#1084—21no—
50#1084#4111yescharacteristic
51C3×C4×Q8—91no—
52C2×#215—92no—
53D3×SD16—182no—
54#874—91no—
55#809—92no—
56C4⋊(C3⋊C8)—91no—
57C2×D32—122no—
58C32×Q8—22no—
59C4⋊(C3⋊D3)—22no—
60C3×D12—62no—
61C3×C4×D3—62no—
62C3×C4×D3—122no—
63C3×Dic6—62no—
64C3×Dic6—122no—
65#626—122no—
66C3⋊(C3⋊C8)—22no—
67C3×C3⋊C8—62no—
68#460—271noSylow
69C2×H(3,𝔽3)#12111yescharacteristic
70#272C2×SD1611yescharacteristic
71#272AGL(1,ℤ/8)11yescharacteristic
72(C3⋊D3)⋊C3—82no—
73C2×C3×Q8—91no—
74D3×Q8—62no—
75D3×D4—182no—
76C2×D12—92no—
77C2×D12—181no—
78C2×D12—181no—
79C3×SD16—182no—
80C3×C4⋊C4—91no—
81C3×C4⋊C4—181no—
82C3×C42—91no—
83C3×C42—181no—
84#215—62no—
85#215—182no—
86#215—182no—
87#214—182no—
88C2×C3⋊C8—92no—
89#204—182no—
90D3×C8—182no—
91C2×C3⋊D3—42no—
92C2×C3×D3—62no—
93C2×C3×D3—122no—
94D32—242no—
95D32—242no—
96C32×C4—22no—
97C32×C4—42no—
98C3×Dic3—62no—
99C3×Dic3—122no—
100C2×SD16—272no—
101#127—271no—
102C4×Q8—91no—
103C4⋊C8—271no—
104#102—272no—
105H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
106C22×D3—182no—
107C22×D3—362no—
108C3×Q8—62no—
109C3×Q8—91no—
110C3×Q8—181no—
111C3×Q8GO+(4,𝔽2)11yescharacteristic
112C3×D4—182no—
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—181no—
117C2×C3×C4—181no—
118#67—362no—
119D12—62no—
120D12—182no—
121D12—182no—
122D12—362no—
123D12—362no—
124C4×D3—62no—
125C4×D3—122no—
126Dic6—62no—
127Dic6—122no—
128C3×C8—182no—
129C3⋊C8—62no—
130C3⋊C8—182no—
131C2×C32—22no—
132C3⋊D3—82no—
133C3×D3—62no—
134C3×D3—62no—
135C3×D3—242no—
136C2×Q8—91no—
137C2×D4—272no—
138C2×D4—541no—
139C2×D4—541no—
140SD16—182no—
141SD16—542no—
142C2×C8—272no—
143C4⋊C4—91no—
144C4⋊C4—181no—
145C42—91no—
146C42—181no—
147C22×C3—91no—
148C22×C3—362no—
149C2×D3—62no—
150C2×D3—122no—
151C2×D3—362no—
152C2×D3—362no—
153C2×D3—362no—
154C2×D3—722no—
155C2×D3—722no—
156C3×C4—21no—
157C3×C4—62no—
158C3×C4—92no—
159C3×C4—91no—
160C3×C4—122no—
161C3×C4—181no—
162C3×C4—181no—
163C3×C4—361no—
164C3×C4C2×GO+(4,𝔽2)11yescharacteristic
165Dic3—62no—
166Dic3—122no—
167C32—22no—
168C23—542no—
169Q8—91no—
170Q8—181no—
171Q8#1088⋊C211yescharacteristic, complemented
172D4—182no—
173D4—542no—
174D4—1082no—
175D4—1082no—
176C2×C4—91no—
177C2×C4—91no—
178C2×C4—91no—
179C2×C4—181no—
180C2×C4—181no—
181C8—182no—
182C2×C3—62no—
183C2×C3—91no—
184C2×C3—91no—
185C2×C3—722no—
186C2×C3#6424211yescharacteristic
187D3—62no—
188D3—62no—
189D3—242no—
190D3—722no—
191C22—91no—
192C22—362no—
193C22—1082no—
194C22—1082no—
195C4—21no—
196C4—92no—
197C4—91no—
198C4—181no—
199C4—181no—
200C4—361no—
201C4C2×#1088⋊C211yescharacteristic
202C3—62no—
203C3#1059347611yescharacteristic
204C2—91no—
205C2—91no—
206C2—722no—
207C2#1172231611yescharacteristic
subgroupG/H
centerC2#11722316
commutator, cosocleC4×#272C23
FrattiniC3×C4C2×GO+(4,𝔽2)
FittingQ8×H(3,𝔽3)D4
socleC2×C3#64242
prime radical#11722305C2
hypercenterQ8#1088⋊C2
hypocenterH(3,𝔽3)#460
LCSG ▷ C4×#272 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#1088C2semidirect
3. #11721228C2semidirect
4. #11720695C2semidirect
5. #11720693C2semidirect
24. Q8×H(3,𝔽3)D4semidirect
28. #6229D4semidirect
46. #1088SD16semidirect
105. H(3,𝔽3)#460semidirect
171. Q8#1088⋊C2semidirect

Maximal subgroups

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

subgroupindex
Q8×#10882
#117223052
#117212282
#117206952
#117206932
#50279