Finite Group Databasebeta

← #13144487

Subgroups — #13144487

Order 1728.

1034 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 127 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×C4⋊#272C211yesmaximal, characteristic
2H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#88549C212yesmaximal, complemented
4#11720573C211yesmaximal, characteristic, complemented
5#11720531C212yesmaximal, complemented
6C3×#5046—34nomaximal
7C23×#272—21no—
8C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272—21no—
11C2×C4⋊#272—24no—
12C2×C4⋊#272C2212yes—
13C4⋊C4×H(3,𝔽3)—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
15C22⋊C4×H(3,𝔽3)—21no—
16#88549—24no—
17C2×#6180C2212yescomplemented
18C22×C3×D12—34no—
19C32×C22⋊Q8D314yescomplemented
20C2×C3×#215—38no—
21C3×#830—34no—
22C3×#809—38no—
23C23×H(3,𝔽3)D411yescharacteristic
24C22×#272—24no—
25C22×#272—22no—
26C22×#272—42no—
27C22×#272—42no—
28Q8×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C4⋊#272—24no—
35C4⋊#272—22no—
36C4⋊#272D414yes—
37#6180—22no—
38C3×#450—91no—
39#5046—34no—
40C23×C3×D3—64no—
41C2×C32×Q8C2×D314yescomplemented
42C22×C32×C4C2×D314yes—
43C2×C3×D12—38no—
44C2×C3×D12—616no—
45C2×C3×D12—64no—
46C32×C4⋊C4—24no—
47C32×C4⋊C4C2×D314yescomplemented
48C32×C22⋊C4—24no—
49C3×#215—616no—
50C2×C3×C3⋊C8—38no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)—21no—
53C22×H(3,𝔽3)SD1612yescomplemented
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×#272—24no—
56C2×#272—48no—
57C2×#272—42no—
58C2×#272—42no—
59C4×H(3,𝔽3)—21no—
60C4×H(3,𝔽3)—41no—
61C4×H(3,𝔽3)—41no—
62C4×H(3,𝔽3)C2×D412yes—
63C22×C3×D4—91no—
64C22×D12—34no—
65C2×C3×SD16—92no—
66C3×C22⋊Q8—34no—
67C3×C22⋊Q8C3⋊D311yescharacteristic
68C2×#215—38no—
69C3×#102—92no—
70C3×C22⋊C8—91no—
71#830—34no—
72#809—38no—
73C23×C32#6714yes—
74C22×C3×D3—616no—
75C22×C3×D3—68no—
76C22×C3×D3—128no—
77C22×C3×D3—128no—
78C32×Q8—28no—
79C2×C32×C4—28no—
80C2×C32×C4—24no—
81C2×C32×C4—24no—
82C2×C32×C4#6714yes—
83C2×C32×C4C22×D314yes—
84C3×D12—316no—
85C3×D12—616no—
86C3×D12—68no—
87C3×C3⋊C8—68no—
88#450—91noSylow
89C2×H(3,𝔽3)—22no—
90C2×H(3,𝔽3)#12011yescharacteristic
91C2×H(3,𝔽3)C2×SD1611yescharacteristic
92C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
93#272—44no—
94C24×C3—181no—
95C23×D3—64no—
96C2×C3×Q8—34no—
97C2×C3×Q8C2×C3⋊D311yescharacteristic
98C2×C3×D4—92no—
99C2×C3×D4—184no—
100C2×C3×D4—181no—
101C22×C3×C4—34no—
102C22×C3×C4C2×C3⋊D311yescharacteristic
103C2×D12—38no—
104C2×D12—616no—
105C2×D12—64no—
106C3×SD16—184no—
107C2×C3×C8—92no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—34no—
110C3×C4⋊C4—64no—
111C3×C4⋊C4C2×C3⋊D311yescharacteristic
112C3×C22⋊C4—21no—
113C3×C22⋊C4—64no—
114#215—616no—
115C2×C3⋊C8—38no—
116C22×C32—24no—
117C22×C32—24no—
118C22×C32#21518yescomplemented
119C22×C32C2×#6714yes—
120C2×C3×D3—616no—
121C2×C3×D3—1232no—
122C2×C3×D3—128no—
123C2×C3×D3—128no—
124C32×C4—24no—
125C32×C4—44no—
126C32×C4—44no—
127C32×C4D3×D418yes—
128C22×D4—91no—
129C2×SD16—92no—
130C22⋊Q8#27211yescharacteristic, complemented
131#102—92no—
132C22⋊C8—91no—
133H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
134C23×C3—34no—
135C23×C3—184no—
136C23×C3—182no—
137C23×C3—362no—
138C23×C3—362no—
139C23×C3#63811yescharacteristic
140C22×D3—68no—
141C22×D3—616no—
142C22×D3—128no—
143C22×D3—128no—
144C3×Q8—22no—
145C3×Q8—68no—
146C3×D4—94no—
147C3×D4—184no—
148C3×D4—182no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—34no—
153C2×C3×C4—34no—
154C2×C3×C4—68no—
155C2×C3×C4—64no—
156C2×C3×C4—64no—
157C2×C3×C4#63811yescharacteristic
158C2×C3×C4C22×C3⋊D311yescharacteristic
159D12—316no—
160D12—68no—
161D12—616no—
162C3×C8—182no—
163C3⋊C8—68no—
164C2×C32—28no—
165C2×C32#93714yes—
166C2×C32C2×#21514yes—
167C2×C32#94914yes—
168C3×D3—1216no—
169C24—181no—
170C2×Q8C2×#27211yescharacteristic, complemented
171C2×D4—92no—
172C2×D4—184no—
173C2×D4—181no—
174C22×C4C2×#27211yescharacteristic
175SD16—184no—
176C2×C8—92no—
177C4⋊C4—21no—
178C4⋊C4C2×#27211yescharacteristic, complemented
179C22⋊C4—21no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—34no—
183C22×C3—38no—
184C22×C3—64no—
185C22×C3—64no—
186C22×C3—184no—
187C22×C3—368no—
188C22×C3—362no—
189C22×C3—362no—
190C22×C3#375512yes—
191C22×C3C2×#63811yescharacteristic
192C2×D3—616no—
193C2×D3—128no—
194C2×D3—128no—
195C2×D3—1232no—
196C3×C4—21no—
197C3×C4—38no—
198C3×C4—41no—
199C3×C4—41no—
200C3×C4—64no—
201C3×C4—124no—
202C3×C4—124no—
203C3×C4D4×C3⋊D312yes—
204C32#504614yescomplemented
205C23—184no—
206C23—182no—
207C23—362no—
208C23—362no—
209C23#623511yescharacteristic
210Q8—22no—
211D4—94no—
212D4—184no—
213D4—182no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4#623511yescharacteristic
218C2×C4C22×#27211yescharacteristic
219C8—182no—
220C2×C3—22no—
221C2×C3—34no—
222C2×C3—34no—
223C2×C3—34no—
224C2×C3—68no—
225C2×C3—364no—
226C2×C3#6414611yescharacteristic
227C2×C3C2×#375511yescharacteristic
228C2×C3#6415811yescharacteristic
229D3—1216no—
230C22—21no—
231C22—21no—
232C22—184no—
233C22—368no—
234C22—362no—
235C22—362no—
236C22#8854912yescomplemented
237C22C2×#623511yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4D4×#27212yes—
242C3—34no—
243C3#1059274511yescharacteristic
244C2—22no—
245C2—364no—
246C2#1172184711yescharacteristic
247C2C2×#8854911yescharacteristic
248C2#1172187011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC22×C4⋊#272C2
hypercenterC3×C22⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#450
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×C22⋊Q8C2semidirect
3. C2×#88549C2semidirect
4. #11720573C2semidirect
5. #11720531C2semidirect
8. C2×Q8×H(3,𝔽3)C22semidirect
14. C4⋊C4×H(3,𝔽3)C22semidirect
17. C2×#6180C22semidirect
19. C32×C22⋊Q8D3semidirect
41. C2×C32×Q8C2×D3semidirect
47. C32×C4⋊C4C2×D3semidirect
53. C22×H(3,𝔽3)SD16semidirect
118. C22×C32#215semidirect
130. C22⋊Q8#272semidirect
133. H(3,𝔽3)#450semidirect
170. C2×Q8C2×#272semidirect
178. C4⋊C4C2×#272semidirect
204. C32#5046semidirect
236. C22#88549semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
H(3,𝔽3)×C22⋊Q82
C2×#885492
#117205732
#117205312
C3×#50463