Finite Group Databasebeta

← #13144442

Subgroups — #13144442

Order 1728.

660 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic
3C4×C4⋊#272C211yesmaximal, characteristic
4#11720583C211yesmaximal, characteristic, complemented
5#11720531C211yesmaximal, characteristic
6#11720529C211yesmaximal, characteristic, complemented
7#11720520C211yesmaximal, characteristic, complemented
8C3×#5026—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272—21no—
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C42×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)C2211yescharacteristic
16#88549C414yescomplemented
17#88542—21no—
18C4⋊#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C32×C4×Q8D314yescomplemented
22C2×C3×#215—34no—
23C3×C4×D12—34no—
24C3×#835—34no—
25C3×#809—34no—
26C3×#807—34no—
27C3×#803—34no—
28C22×#272—21no—
29Q8×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)C2×C412yescomplemented
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×#1084—21no—
37C4⋊#272—21no—
38C4⋊#272C2×C412yes—
39C4×#272—42no—
40#6180—21no—
41#6180C2×C412yescomplemented
42C3×#440—91no—
43#5026—34no—
44C2×C32×Q8C2×D314yes—
45C2×C3×D12—34no—
46C2×C3×C4×D3—64no—
47C32×C4⋊C4—24no—
48C32×C4⋊C4C2×D314yes—
49C32×C42—24no—
50C32×C42C2×D314yes—
51C3×#215—316no—
52C3×(C2×D3)⋊C4—64no—
53C3×C4⋊Dic3—34no—
54C2×C3×C3⋊C8—34no—
55C2×C3×C3⋊C8—34no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C2×#272—22no—
58C2×#272—41no—
59C2×#272—41no—
60C4×H(3,𝔽3)—22no—
61C4×H(3,𝔽3)—21no—
62C4×H(3,𝔽3)—21no—
63C4×H(3,𝔽3)—41no—
64C4×H(3,𝔽3)C22×C411yescharacteristic
65C4×H(3,𝔽3)#4111yescharacteristic
66#1084—41no—
67C2×C3×SD16—91no—
68C3×C4×Q8—34no—
69C3×C4×Q8C3⋊D311yescharacteristic
70C3×C4×D4—91no—
71C2×#215—34no—
72C4×D12—34no—
73C3×#107—91no—
74C3×#103—91no—
75C3×#102—91no—
76C3×#97—91no—
77#835—34no—
78#809—34no—
79#807—34no—
80#803—34no—
81C22×C3×D3—64no—
82C32×Q8—24no—
83C32×Q8C4×D318yescomplemented
84C2×C32×C4—24no—
85C2×C32×C4—24no—
86C2×C32×C4#6714yes—
87C2×C32×C4#6714yes—
88C2×C32×C4C22×D314yes—
89C2×C3×Dic3—64no—
90C3×D12—38no—
91C3×D12—64no—
92C3×C4×D3—128no—
93C3×C3⋊C8—38no—
94C3×C3⋊C8—64no—
95#440—91noSylow
96C2×H(3,𝔽3)C4×D411yescharacteristic
97C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
98C2×H(3,𝔽3)#13711yescharacteristic
99#272—42no—
100C2×C3×Q8—34no—
101C2×C3×Q8C2×C3⋊D311yescharacteristic
102C2×C3×D4—91no—
103C22×C3×C4—181no—
104C2×D12—34no—
105C2×C4×D3—64no—
106C3×SD16—94no—
107C2×C3×C8—91no—
108C2×C3×C8—91no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—34no—
111C3×C4⋊C4—64no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4C2×C3⋊D311yescharacteristic
114C3×C22⋊C4—181no—
115C3×C42—21no—
116C3×C42—34no—
117C3×C42—64no—
118C3×C42C2×C3⋊D311yescharacteristic
119#215—316no—
120(C2×D3)⋊C4—64no—
121C4⋊Dic3—34no—
122C2×C3⋊C8—34no—
123C2×C3⋊C8—34no—
124C22×C32C2×#6714yes—
125C2×C3×D3—68no—
126C2×C3×D3—124no—
127C2×C3×D3—124no—
128C32×C4—28no—
129C32×C4—24no—
130C32×C4—24no—
131C32×C4—44no—
132C32×C4C2×C4×D314yes—
133C32×C4#23514yes—
134C3×Dic3—124no—
135C2×SD16—91no—
136C4×Q8#27211yescharacteristic, complemented
137C4×D4—91no—
138#107—91no—
139#103—91no—
140#102—91no—
141#97—91no—
142H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
143C23×C3—181no—
144C22×D3—64no—
145C3×Q8—21no—
146C3×Q8—38no—
147C3×Q8—64no—
148C3×Q8C4×C3⋊D312yes—
149C3×D4—92no—
150C3×D4—181no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—34no—
154C2×C3×C4—34no—
155C2×C3×C4—34no—
156C2×C3×C4—64no—
157C2×C3×C4—64no—
158C2×C3×C4—181no—
159C2×C3×C4—362no—
160C2×C3×C4#63811yescharacteristic
161C2×C3×C4#63811yescharacteristic
162C2×C3×C4C22×C3⋊D311yescharacteristic
163C2×Dic3—64no—
164D12—38no—
165D12—64no—
166C4×D3—128no—
167C3×C8—92no—
168C3×C8—181no—
169C3⋊C8—38no—
170C3⋊C8—64no—
171C2×C32C4×#6714yes—
172C2×C32#94214yes—
173C2×C32#94914yes—
174C3×D3—128no—
175C2×Q8C2×#27211yescharacteristic
176C2×D4—91no—
177C22×C4—181no—
178SD16—94no—
179C2×C8—91no—
180C2×C8—91no—
181C4⋊C4—21no—
182C4⋊C4—91no—
183C4⋊C4C2×#27211yescharacteristic
184C22⋊C4—181no—
185C42—21no—
186C42C2×#27211yescharacteristic
187C22×C3—34no—
188C22×C3—182no—
189C22×C3—361no—
190C22×C3—361no—
191C22×C3C2×#63811yescharacteristic
192C2×D3—68no—
193C2×D3—124no—
194C2×D3—124no—
195C3×C4—22no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—34no—
199C3×C4—34no—
200C3×C4—41no—
201C3×C4—64no—
202C3×C4—64no—
203C3×C4—68no—
204C3×C4—124no—
205C3×C4—361no—
206C3×C4C2×C4×C3⋊D311yescharacteristic
207C3×C4#382811yescharacteristic
208Dic3—124no—
209C32#502614yescomplemented
210C23—181no—
211Q8—21no—
212Q8C4×#27212yescomplemented
213D4—92no—
214D4—181no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—181no—
218C2×C4—362no—
219C2×C4#623511yescharacteristic
220C2×C4#623511yescharacteristic
221C2×C4C22×#27211yescharacteristic
222C8—92no—
223C8—181no—
224C2×C3—34no—
225C2×C3—34no—
226C2×C3—34no—
227C2×C3—362no—
228C2×C3C4×#63811yescharacteristic
229C2×C3#6415111yescharacteristic
230C2×C3#6415811yescharacteristic
231D3—128no—
232C22—182no—
233C22—361no—
234C22—361no—
235C22C2×#623511yescharacteristic
236C4—22no—
237C4—21no—
238C4—21no—
239C4—41no—
240C4—361no—
241C4C2×C4×#27211yescharacteristic
242C4#8874011yescharacteristic
243C3—34no—
244C3#1059272511yescharacteristic
245C2—362no—
246C2C4×#623511yescharacteristic
247C2#1172185911yescharacteristic
248C2#1172187011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC2×C4⋊#272C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C4×Q8C3⋊D3
hypocenterH(3,𝔽3)#440
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×H(3,𝔽3)C2semidirect
4. #11720583C2semidirect
6. #11720529C2semidirect
7. #11720520C2semidirect
16. #88549C4semidirect
21. C32×C4×Q8D3semidirect
30. Q8×H(3,𝔽3)C2×C4semidirect
41. #6180C2×C4semidirect
83. C32×Q8C4×D3semidirect
136. C4×Q8#272semidirect
142. H(3,𝔽3)#440semidirect
209. C32#5026semidirect
212. Q8C4×#272semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
C2×#885492
C4×C4⋊#2722
#117205832
#117205312
#117205292
#117205202
C3×#50263