Finite Group Databasebeta

← #13138908

Subgroups — #13138908

Order 1728.

531 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×C8×#272C211yesmaximal, characteristic
2#11721152C212yesmaximal
3#63645⋊C3C212yesmaximal, complemented
4#11720315C212yesmaximal, complemented
5D3×#818—32nomaximal
6C2×C4×#272C2211yescharacteristic
7C2×#6199—22no—
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9C2×#6180C2211yescharacteristic
10C8×#272C414yescomplemented
11C4⋊((C3⋊Dic3)⋊C3)C2214yes—
12#88449—22no—
13C2×C3×D3×C8—32no—
14C2×C3×D3×C8—61no—
15D3×C4⋊Dic3—34no—
16#63645D312yescomplemented
17C3×#818—32no—
18#63575—34no—
19C22×#272D411yescharacteristic
20C2×C4×H(3,𝔽3)C2311yescharacteristic
21C2×#1084D411yescharacteristic
22C4×#272C2×C412yes—
23C4×#272Q812yes—
24C2×(C3⋊Dic3)⋊C3—24no—
25#6199—44no—
26C8×H(3,𝔽3)C2×C412yescomplemented
27#6180C2×C412yescomplemented
28C2×#818—91no—
29D3×#107—92no—
30C2×C3×C4×D3—32no—
31C2×C3×C4×D3—61no—
32C2×D3×Dic3—64no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D312yes—
35C4⋊(C3⋊Dic3)C2×D314yes—
36C3×C4⋊Dic3—34no—
37C2×C3×C3⋊C8—32no—
38C2×C3×C3⋊C8—61no—
39C3×D3×C8—38no—
40C3×D3×C8—64no—
41Dic3⋊Dic3—64no—
42C22×H(3,𝔽3)C2×D411yescharacteristic
43C2×#272C4⋊C412yes—
44C2×#272D812yes—
45C2×#272Q1612yes—
46C4×H(3,𝔽3)C22×C411yescharacteristic
47C4×H(3,𝔽3)C2×Q811yescharacteristic
48#1084C4⋊C412yes—
49(C3⋊Dic3)⋊C3—44no—
50C22×C3×C8—91no—
51C2×C4⋊Dic3—92no—
52C2×D3×C8—32no—
53C2×D3×C8—61no—
54D3×C4⋊C4—94no—
55C3×#107—92no—
56#818—32no—
57#818—92no—
58#818—92no—
59#807—94no—
60C22×C3×D3—32no—
61C22×C3×D3—61no—
62C2×C32×C4—21no—
63C2×C32×C4C22×D312yes—
64C2×C3⋊Dic3—24no—
65C2×C3×Dic3—32no—
66C2×C3×Dic3—61no—
67C2×C3×Dic3—64no—
68C3×C4×D3—34no—
69C3×C4×D3—34no—
70C3×C4×D3—62no—
71C3×C4×D3—62no—
72D3×Dic3—128no—
73C32×C8—22no—
74C32×C8C4×D314yescomplemented
75C3×C3⋊C8—34no—
76C3×C3⋊C8—62no—
77C2×#107—271noSylow
78C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
79C2×H(3,𝔽3)C2×D811yescharacteristic
80C2×H(3,𝔽3)C2×Q1611yescharacteristic
81#272#10714yescomplemented
82C22×C3×C4—91no—
83C22×Dic3—182no—
84C2×C4×D3—32no—
85C2×C4×D3—61no—
86C2×C4×D3—184no—
87C2×C3×C8—32no—
88C2×C3×C8—61no—
89C2×C3×C8—91no—
90C2×C3×C8—94no—
91C2×C3×C8D3211yescharacteristic
92C3×C4⋊C4—94no—
93C4⋊Dic3—34no—
94C4⋊Dic3—94no—
95C4⋊Dic3—182no—
96Dic3⋊C4—184no—
97C2×C3⋊C8—32no—
98C2×C3⋊C8—61no—
99D3×C8—38no—
100D3×C8—64no—
101C22×C32—21no—
102C22×C32D3×D412yes—
103C2×C3×D3—34no—
104C2×C3×D3—34no—
105C2×C3×D3—34no—
106C2×C3×D3—62no—
107C2×C3×D3—62no—
108C2×C3×D3—62no—
109C32×C4—21no—
110C32×C4—21no—
111C32×C4C2×C4×D312yes—
112C32×C4D3×Q812yes—
113C3⋊Dic3—44no—
114C3×Dic3—34no—
115C3×Dic3—62no—
116C3×Dic3—124no—
117C22×C8—91no—
118C2×C4⋊C4—272no—
119#107—92no—
120#107—272no—
121H(3,𝔽3)C2×#10711yescharacteristic, complemented, Sylow
122C23×C3—91no—
123C22×D3—32no—
124C22×D3—61no—
125C2×C3×C4—32no—
126C2×C3×C4—61no—
127C2×C3×C4—91no—
128C2×C3×C4—92no—
129C2×C3×C4—92no—
130C2×C3×C4—184no—
131C2×C3×C4C2×D3211yescharacteristic
132C2×Dic3—32no—
133C2×Dic3—61no—
134C2×Dic3—64no—
135C2×Dic3—184no—
136C2×Dic3—364no—
137C4×D3—34no—
138C4×D3—34no—
139C4×D3—62no—
140C4×D3—62no—
141C4×D3—368no—
142C3×C8—34no—
143C3×C8—62no—
144C3×C8—92no—
145C3×C8#62412yes—
146C3⋊C8—34no—
147C3⋊C8—62no—
148C2×C32—21no—
149C2×C32—21no—
150C2×C32—21no—
151C2×C32D3×C4⋊C412yes—
152C2×C32D3×D812yes—
153C2×C32D3×Q1612yes—
154C3×D3—38no—
155C3×D3—64no—
156C22×C4—91no—
157C22×C4—542no—
158C2×C8—91no—
159C2×C8—94no—
160C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
161C4⋊C4—94no—
162C4⋊C4—542no—
163C22×C3—32no—
164C22×C3—61no—
165C22×C3—92no—
166C22×C3—92no—
167C22×C3—92no—
168C22×C3#380211yescharacteristic
169C2×D3—34no—
170C2×D3—34no—
171C2×D3—34no—
172C2×D3—62no—
173C2×D3—62no—
174C2×D3—62no—
175C3×C4—32no—
176C3×C4—32no—
177C3×C4—61no—
178C3×C4—61no—
179C3×C4—92no—
180C3×C4—364no—
181C3×C4#379711yescharacteristic
182C3×C4C2×#62411yescharacteristic
183Dic3—34no—
184Dic3—62no—
185Dic3—124no—
186Dic3—364no—
187C32—21no—
188C32D3×#10712yescomplemented
189C23—91no—
190C2×C4—91no—
191C2×C4—92no—
192C2×C4—92no—
193C2×C4—184no—
194C2×C4—1084no—
195C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
196C8—92no—
197C8#619912yescomplemented
198C2×C3—32no—
199C2×C3—32no—
200C2×C3—32no—
201C2×C3—61no—
202C2×C3—61no—
203C2×C3—61no—
204C2×C3—94no—
205C2×C3#6379711yescharacteristic
206C2×C3#6380211yescharacteristic
207C2×C3#6390011yescharacteristic
208D3—38no—
209D3—64no—
210C22—92no—
211C22—92no—
212C22—92no—
213C22#8865511yescharacteristic
214C4—92no—
215C4—364no—
216C4#8865111yescharacteristic
217C4C2×#619911yescharacteristic
218C3—32no—
219C3—61no—
220C3#1059068711yescharacteristic
221C2—94no—
222C2#1172083311yescharacteristic
223C2#1172083811yescharacteristic
224C2#1172115211yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#107
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #63645⋊C3C2semidirect
4. #11720315C2semidirect
10. C8×#272C4semidirect
16. #63645D3semidirect
26. C8×H(3,𝔽3)C2×C4semidirect
27. #6180C2×C4semidirect
74. C32×C8C4×D3semidirect
81. #272#107semidirect
121. H(3,𝔽3)C2×#107semidirect
188. C32D3×#107semidirect
197. C8#6199semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
#117211522
#63645⋊C32
#117203152
D3×#8183