Finite Group Databasebeta

← #13144842

Subgroups — #13144842

Order 1728.

814 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88550C212yesmaximal, complemented
3#11721824C211yesmaximal, characteristic
4#11720583C212yesmaximal, complemented
5#11720573C211yesmaximal, characteristic
6C3×#5228—34nomaximal
7C2×Q8×H(3,𝔽3)—24no—
8C2×Q8×H(3,𝔽3)—21no—
9C2×Q8×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)—21no—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6229C2211yescharacteristic
13#88552—21no—
14#88550—24no—
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17C2×#6180C2212yes—
18C22×C32×Q8D314yes—
19C2×C3×#216—38no—
20C3×#924—34no—
21C3×#835—38no—
22C3×#830—34no—
23C23×H(3,𝔽3)D411yescharacteristic
24Q8×H(3,𝔽3)—24no—
25Q8×H(3,𝔽3)—22no—
26Q8×H(3,𝔽3)D414yes—
27C2×C4×H(3,𝔽3)—24no—
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34#6229—22no—
35#6180—22no—
36C3×#451—91no—
37#5228—34no—
38C2×C32×Q8—216no—
39C2×C32×Q8—24no—
40C2×C32×Q8C2×D318yes—
41C22×C32×C4—24no—
42C22×C32×C4C2×D314yes—
43C2×C3×Dic6—34no—
44C3×#217—64no—
45C3×#216—616no—
46C3×C4⋊Dic3—34no—
47C3×Dic3⋊C4—64no—
48C2×C3×C3⋊C8—38no—
49C22×H(3,𝔽3)—21no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)Q1612yescomplemented
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)—24no—
54C4×H(3,𝔽3)—21no—
55C4×H(3,𝔽3)C2×D412yes—
56#1084—41no—
57#1084—41no—
58C22×C3×Q8—34no—
59C22×C3×Q8C3⋊D311yescharacteristic
60C2×C3×Q16—92no—
61C3×C22⋊Q8—91no—
62C2×#216—38no—
63#924—34no—
64C3×#103—92no—
65C3×C22⋊C8—91no—
66#835—38no—
67#830—34no—
68C23×C32#6714yes—
69C32×Q8—216no—
70C32×Q8—28no—
71C32×Q8#67116yes—
72C2×C32×C4—216no—
73C2×C32×C4—28no—
74C2×C32×C4—28no—
75C2×C32×C4#6714yes—
76C2×C32×C4C22×D314yes—
77C2×C3×Dic3—64no—
78C2×C3×Dic3—64no—
79C3×Dic6—68no—
80C3×C3⋊C8—68no—
81#451—91noSylow
82C2×H(3,𝔽3)—22no—
83C2×H(3,𝔽3)#12011yescharacteristic
84C2×H(3,𝔽3)C2×Q1611yescharacteristic
85C2×H(3,𝔽3)#13711yescharacteristic
86C2×C3×Q8—24no—
87C2×C3×Q8—21no—
88C2×C3×Q8—38no—
89C2×C3×Q8—64no—
90C2×C3×Q8—616no—
91C2×C3×Q8—91no—
92C2×C3×Q8C2×C3⋊D312yes—
93C22×C3×C4—21no—
94C22×C3×C4—34no—
95C22×C3×C4—64no—
96C22×C3×C4C2×C3⋊D311yescharacteristic
97C2×Dic6—34no—
98C3×Q16—184no—
99C2×C3×C8—92no—
100C3×C4⋊C4—91no—
101C3×C4⋊C4—181no—
102C3×C22⋊C4—181no—
103#217—64no—
104#216—616no—
105C4⋊Dic3—34no—
106Dic3⋊C4—64no—
107C2×C3⋊C8—38no—
108C22×C32—24no—
109C22×C32—24no—
110C22×C32#21618yescomplemented
111C22×C32C2×#6714yes—
112C32×C4—216no—
113C32×C4—24no—
114C32×C4C2×#6718yes—
115C3×Dic3—124no—
116C3×Dic3—124no—
117C22×Q8#27211yescharacteristic
118C2×Q16—92no—
119C22⋊Q8—91no—
120#103—92no—
121C22⋊C8—91no—
122H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
123C23×C3—34no—
124C23×C3#63811yescharacteristic
125C3×Q8—24no—
126C3×Q8—22no—
127C3×Q8—316no—
128C3×Q8—68no—
129C3×Q8—616no—
130C3×Q8—182no—
131C3×Q8#63814yes—
132C2×C3×C4—24no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—34no—
136C2×C3×C4—34no—
137C2×C3×C4—68no—
138C2×C3×C4—68no—
139C2×C3×C4—616no—
140C2×C3×C4—181no—
141C2×C3×C4—181no—
142C2×C3×C4#63811yescharacteristic
143C2×C3×C4C22×C3⋊D311yescharacteristic
144C2×Dic3—64no—
145C2×Dic3—64no—
146Dic6—68no—
147C3×C8—182no—
148C3⋊C8—68no—
149C2×C32—28no—
150C2×C32#94214yes—
151C2×C32C2×#21614yes—
152C2×C32#95314yes—
153C2×Q8—24no—
154C2×Q8—21no—
155C2×Q8—91no—
156C2×Q8C2×#27212yes—
157C22×C4—21no—
158C22×C4C2×#27211yescharacteristic
159Q16—184no—
160C2×C8—92no—
161C4⋊C4—91no—
162C4⋊C4—181no—
163C22⋊C4—181no—
164C22×C3—21no—
165C22×C3—21no—
166C22×C3—34no—
167C22×C3—38no—
168C22×C3—64no—
169C22×C3—64no—
170C22×C3#375612yes—
171C22×C3C2×#63811yescharacteristic
172C3×C4—24no—
173C3×C4—21no—
174C3×C4—38no—
175C3×C4—64no—
176C3×C4—616no—
177C3×C4—361no—
178C3×C4—361no—
179C3×C4C2×#63812yes—
180Dic3—124no—
181Dic3—124no—
182C32#522814yescomplemented
183C23#623511yescharacteristic
184Q8—24no—
185Q8—22no—
186Q8—182no—
187Q8#623514yes—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—181no—
192C2×C4—181no—
193C2×C4#623511yescharacteristic
194C2×C4C22×#27211yescharacteristic
195C8—182no—
196C2×C3—22no—
197C2×C3—34no—
198C2×C3—34no—
199C2×C3—34no—
200C2×C3—68no—
201C2×C3#6415111yescharacteristic
202C2×C3C2×#375611yescharacteristic
203C2×C3#6416211yescharacteristic
204C22—21no—
205C22—21no—
206C22#8855012yescomplemented
207C22C2×#623511yescharacteristic
208C4—24no—
209C4—21no—
210C4—361no—
211C4—361no—
212C4C2×#623512yes—
213C3—34no—
214C3#1059292711yescharacteristic
215C2—22no—
216C2#1172185911yescharacteristic
217C2C2×#8855011yescharacteristic
218C2#1172187611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC22×Q8×H(3,𝔽3)C2
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C3×Q8C3⋊D3
hypocenterH(3,𝔽3)#451
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#88550C2semidirect
4. #11720583C2semidirect
51. C22×H(3,𝔽3)Q16semidirect
110. C22×C32#216semidirect
122. H(3,𝔽3)#451semidirect
182. C32#5228semidirect
206. C22#88550semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C2×#885502
#117218242
#117205832
#117205732
C3×#52283