Finite Group Databasebeta

← #13146635

Subgroups — #13146635

Order 1728.

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

normal subgroups only

This group has 316 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
2#11722300C211yesmaximal, characteristic, complemented
3#11721188C212yesmaximal
4#11720696C212yesmaximal, complemented
5#11720693C211yesmaximal, characteristic, complemented
6C22×#1088—21no—
7D4×#272C2211yescharacteristic
8#88651C2212yes—
9C22⋊#1088—21no—
10C4⋊#1088C2211yescharacteristic
11C4×#1088C2211yescharacteristic
12#88590—21no—
13#88590—21no—
14#3754⋊C3—22no—
15#88437—22no—
16#88434C2212yes—
17D3×#214—62no—
18C22×#272—21no—
19C2×#1088—21no—
20C2×#1088—42no—
21C2×#1088D411yescharacteristic
22C2×#1088D411yescharacteristic
23D4×H(3,𝔽3)D411yescharacteristic
24#6235—21no—
25C4⋊#272D411yescharacteristic
26C4×#272C2311yescharacteristic
27#634⋊C3—22no—
28#6199—22no—
29#6196—22no—
30(C3⋊(C3⋊C8))⋊C3—22no—
31#5019—91nomaximal
32C3×D3×D4—62no—
33D3×Dic6—62no—
34#3754—22no—
35C3×#214—62no—
36#3716—62no—
37#3715—62no—
38D3×C3⋊C8—62no—
39C22×H(3,𝔽3)—21no—
40C2×#272—21no—
41C2×#272—41no—
42C2×#272—41no—
43C2×#272C2×D411yescharacteristic
44#1088—21no—
45#1088—41no—
46#1088SD1612yescomplemented
47C4×H(3,𝔽3)C2×D411yescharacteristic
48#1084#4111yescharacteristic
49(C3⋊Dic3)⋊C3—42no—
50C3×C4×D4—91no—
51C2×#214—92no—
52D3×SD16—182no—
53#869—91no—
54#810—92no—
55C4⋊(C3⋊C8)—91no—
56C22×C3×D3—122no—
57C32×D4—22no—
58#634—22no—
59C3×#67—122no—
60C3×D12—62no—
61C3×C4×D3—62no—
62C3×Dic6—62no—
63#627—122no—
64D3×Dic3—122no—
65C3⋊(C3⋊C8)—22no—
66C3×C3⋊C8—62no—
67#461—271noSylow
68C2×H(3,𝔽3)—41no—
69C2×H(3,𝔽3)#12111yescharacteristic
70#272—41no—
71#272C2×SD1611yescharacteristic
72#272#13711yescharacteristic
73C2×C3×D4—91no—
74C22×C3×C4—181no—
75D3×Q8—182no—
76D3×D4—62no—
77C2×Dic6—92no—
78C3×SD16—182no—
79C3×C4⋊C4—91no—
80C3×C22⋊C4—181no—
81C3×C42—91no—
82#215—182no—
83#214—62no—
84#214—182no—
85#214—182no—
86C4⋊Dic3—181no—
87C4⋊Dic3—181no—
88C2×C3⋊C8—92no—
89#204—182no—
90D3×C8—182no—
91C22×C32—42no—
92C2×C3×D3—62no—
93C2×C3×D3—122no—
94C2×C3×D3—242no—
95C2×C3×D3—242no—
96C32×C4—22no—
97C3⋊Dic3—42no—
98C3×Dic3—62no—
99C3×Dic3—122no—
100C2×SD16—272no—
101C4⋊Q8—271no—
102C4×D4—91no—
103C4⋊C8—271no—
104#103—272no—
105H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
106C23×C3—181no—
107C22×D3—122no—
108C3×Q8—182no—
109C3×D4—62no—
110C3×D4—91no—
111C3×D4—181no—
112C3×D4GO+(4,𝔽2)11yescharacteristic
113C2×C3×C4—91no—
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—181no—
117C2×C3×C4—362no—
118#67—122no—
119C2×Dic3—182no—
120D12—62no—
121C4×D3—62no—
122C4×D3—362no—
123Dic6—62no—
124Dic6—182no—
125Dic6—182no—
126Dic6—362no—
127C3×C8—182no—
128C3⋊C8—62no—
129C3⋊C8—182no—
130C2×C32—22no—
131C2×C32—82no—
132C3×D3—62no—
133C3×D3—62no—
134C3×D3—242no—
135C2×Q8—272no—
136C2×D4—91no—
137C22×C4—181no—
138SD16—182no—
139SD16—542no—
140C2×C8—272no—
141C4⋊C4—91no—
142C4⋊C4—541no—
143C4⋊C4—541no—
144C22⋊C4—181no—
145C42—91no—
146C22×C3—21no—
147C22×C3—91no—
148C22×C3—122no—
149C22×C3—181no—
150C22×C3—361no—
151C22×C3—361no—
152C2×D3—62no—
153C2×D3—122no—
154C2×D3—242no—
155C2×D3—242no—
156C3×C4—62no—
157C3×C4—92no—
158C3×C4—91no—
159C3×C4—181no—
160C3×C4—362no—
161C3×C4—361no—
162C3×C4C2×GO+(4,𝔽2)11yescharacteristic
163Dic3—62no—
164Dic3—122no—
165Dic3—362no—
166C32—22no—
167C23—181no—
168Q8—182no—
169Q8—542no—
170D4—91no—
171D4—181no—
172D4#1088⋊C211yescharacteristic
173C2×C4—91no—
174C2×C4—91no—
175C2×C4—91no—
176C2×C4—181no—
177C2×C4—362no—
178C2×C4—542no—
179C8—182no—
180C2×C3—41no—
181C2×C3—62no—
182C2×C3—91no—
183C2×C3—91no—
184C2×C3—242no—
185C2×C3—361no—
186C2×C3#6424211yescharacteristic
187D3—62no—
188D3—62no—
189D3—242no—
190C22—21no—
191C22—91no—
192C22—181no—
193C22—361no—
194C22—361no—
195C4—92no—
196C4—91no—
197C4—181no—
198C4—361no—
199C4—362no—
200C4C2×#1088⋊C211yescharacteristic
201C3—62no—
202C3#1059346511yescharacteristic
203C2—41no—
204C2—91no—
205C2—91no—
206C2—361no—
207C2#1172231611yescharacteristic
subgroupG/H
centerC2#11722316
commutator, cosocleC4×#272C23
FrattiniC3×C4C2×GO+(4,𝔽2)
FittingD4×H(3,𝔽3)D4
socleC2×C3#64242
prime radicalD4×#272C22
hypercenterD4#1088⋊C2
hypocenterH(3,𝔽3)#461
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
2. #11722300C2semidirect
4. #11720696C2semidirect
5. #11720693C2semidirect
46. #1088SD16semidirect
105. H(3,𝔽3)#461semidirect

Maximal subgroups

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

subgroupindex
D4×#10882
#117223002
#117211882
#117206962
#117206932
#50199