Finite Group Databasebeta

← #13138872

Subgroups — #13138872

Order 1728.

513 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#11721136C212yesmaximal, complemented
2C2×#88432C211yesmaximal, characteristic
3C8×#1084C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#11720315C211yesmaximal, characteristic, complemented
6#10590745—32nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
8C2×((C3⋊D3)⋊C3)⋊C4—22no—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#88450—22no—
14#88432C414yescomplemented
15#63894—32no—
16Dic3×D12—32no—
17C2×#3714—32no—
18#63648D312yescomplemented
19C3×#821—32no—
20C3×C8×Dic3—32no—
21C3×C8×Dic3—61no—
22#63575—32no—
23C22×(C3⋊D3)⋊C3—22no—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084D411yescharacteristic
26C2×#1084D411yescharacteristic
27C2×(C3⋊Dic3)⋊C3—22no—
28C4⋊((C3⋊D3)⋊C3)—22no—
29C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
30((C3⋊D3)⋊C3)⋊C4—44no—
31C8×H(3,𝔽3)—21no—
32#6180C2×C412yescomplemented
33#4756—92no—
34C4×D24—91no—
35C2×C4⋊(C3⋊D3)C2×D312yes—
36C2×C3×D12—32no—
37C2×#624—62no—
38C2×D3×Dic3—62no—
39C2×C32×C8—21no—
40C2×C32×C8C2×D312yes—
41C4⋊(C3⋊Dic3)C2×D312yes—
42C3×C4⋊Dic3—32no—
43C3×C4×Dic3—32no—
44C3×C4×Dic3—61no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47#3722—62no—
48(C2×D3)⋊Dic3—62no—
49#3714—38no—
50C22×H(3,𝔽3)C2×D411yescharacteristic
51C2×(C3⋊D3)⋊C3—24no—
52C2×(C3⋊D3)⋊C3—42no—
53C2×(C3⋊D3)⋊C3—42no—
54C4×H(3,𝔽3)C22×C411yescharacteristic
55C4×H(3,𝔽3)#4111yescharacteristic
56#1084—21no—
57#1084D812yescomplemented
58(C3⋊Dic3)⋊C3—42no—
59D4×Dic3—92no—
60C2×#213—92no—
61C2×D24—91no—
62#893—92no—
63C4×D12—92no—
64C3×#102—92no—
65C3×C4×C8—91no—
66#821—32no—
67#821—92no—
68#818—91no—
69C8×Dic3—32no—
70C8×Dic3—61no—
71#807—92no—
72C22×C3⋊D3—22no—
73C22×C3×D3—62no—
74C2×C32×C4—21no—
75C2×C32×C4C22×D312yes—
76C2×C3⋊Dic3—22no—
77C4⋊(C3⋊D3)—22no—
78C4⋊(C3⋊D3)C4×D314yescomplemented
79C2×C3×Dic3—32no—
80C2×C3×Dic3—32no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—62no—
84C3×D12—34no—
85C3×D12—62no—
86#624—124no—
87D3×Dic3—124no—
88C32×C8—22no—
89C32×C8—22no—
90C3×C3⋊C8—34no—
91C3×C3⋊C8—62no—
92C4×D8—271noSylow
93C2×H(3,𝔽3)C4×D411yescharacteristic
94C2×H(3,𝔽3)C2×D811yescharacteristic
95C2×H(3,𝔽3)#13511yescharacteristic
96(C3⋊D3)⋊C3—44no—
97C2×C3×D4—92no—
98C22×Dic3—182no—
99C2×D12—32no—
100C2×D12—92no—
101C2×C4×D3—182no—
102C2×C4×D3—182no—
103C2×C3×C8—32no—
104C2×C3×C8—61no—
105C2×C3×C8—91no—
106C2×C3×C8D3211yescharacteristic
107C3×C4⋊C4—92no—
108C3×C42—91no—
109#217—182no—
110#213—98no—
111(C2×D3)⋊C4—182no—
112(C2×D3)⋊C4—182no—
113C4⋊Dic3—32no—
114C4⋊Dic3—92no—
115C4×Dic3—32no—
116C4×Dic3—61no—
117C2×C3⋊C8—32no—
118C2×C3⋊C8—61no—
119D24—94no—
120C22×C32—21no—
121C22×C32D3×D412yes—
122C2×C3⋊D3—24no—
123C2×C3⋊D3—42no—
124C2×C3⋊D3—42no—
125C2×C3×D3—64no—
126C2×C3×D3—122no—
127C2×C3×D3—122no—
128C32×C4—21no—
129C32×C4—21no—
130C32×C4C2×C4×D312yes—
131C32×C4#23712yes—
132C3⋊Dic3—42no—
133C3×Dic3—34no—
134C3×Dic3—62no—
135C3×Dic3—62no—
136C3×Dic3—62no—
137C3×Dic3—122no—
138C2×D8—271no—
139C4×D4—272no—
140#107—271no—
141#102—92no—
142C4×C8—91no—
143H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
144C23×C3—182no—
145C22×D3—62no—
146C22×D3—182no—
147C3×D4—94no—
148C3×D4—182no—
149C2×C3×C4—32no—
150C2×C3×C4—61no—
151C2×C3×C4—91no—
152C2×C3×C4—91no—
153C2×C3×C4—182no—
154C2×C3×C4C2×D3211yescharacteristic
155C2×Dic3—32no—
156C2×Dic3—32no—
157C2×Dic3—61no—
158C2×Dic3—61no—
159C2×Dic3—62no—
160C2×Dic3—182no—
161C2×Dic3—364no—
162D12—34no—
163D12—62no—
164D12—94no—
165D12—182no—
166C4×D3—364no—
167C4×D3—364no—
168C3×C8—21no—
169C3×C8—62no—
170C3×C8—62no—
171C3×C8—92no—
172C3⋊C8—34no—
173C3⋊C8—62no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—21no—
177C2×C32#67⋊C412yes—
178C2×C32D3×D812yes—
179C2×C32#91612yes—
180C3⋊D3—44no—
181C3×D3—124no—
182C2×D4—92no—
183C22×C4—542no—
184D8—274no—
185C2×C8—91no—
186C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
187C4⋊C4—92no—
188C22⋊C4—542no—
189C42—91no—
190C22×C3—32no—
191C22×C3—61no—
192C22×C3—184no—
193C22×C3—362no—
194C22×C3—362no—
195C22×C3#380211yescharacteristic
196C2×D3—64no—
197C2×D3—122no—
198C2×D3—122no—
199C2×D3—184no—
200C2×D3—362no—
201C2×D3—362no—
202C3×C4—32no—
203C3×C4—32no—
204C3×C4—61no—
205C3×C4—61no—
206C3×C4—92no—
207C3×C4—181no—
208C3×C4—362no—
209C3×C4C4×D3211yescharacteristic
210C3×C4#380511yescharacteristic
211Dic3—34no—
212Dic3—62no—
213Dic3—62no—
214Dic3—62no—
215Dic3—122no—
216Dic3—362no—
217C32—21no—
218C32#475612yescomplemented
219C23—182no—
220D4—94no—
221D4—182no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—182no—
225C2×C4—1084no—
226C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
227C8—21no—
228C8—92no—
229C2×C3—32no—
230C2×C3—32no—
231C2×C3—32no—
232C2×C3—61no—
233C2×C3—61no—
234C2×C3—61no—
235C2×C3—364no—
236C2×C3#6379711yescharacteristic
237C2×C3#6380911yescharacteristic
238C2×C3#6390211yescharacteristic
239D3—124no—
240D3—364no—
241C22—184no—
242C22—362no—
243C22—362no—
244C22#8865511yescharacteristic
245C4—92no—
246C4—181no—
247C4—362no—
248C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
249C4#8867211yescharacteristic
250C3—32no—
251C3—61no—
252C3#1059074011yescharacteristic
253C2—364no—
254C2#1172083811yescharacteristic
255C2#1172084311yescharacteristic
256C2#1172113811yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88432C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C4×D8
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
1. #11721136C2semidirect
3. C8×#1084C2semidirect
4. #63648⋊C3C2semidirect
5. #11720315C2semidirect
14. #88432C4semidirect
18. #63648D3semidirect
29. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
32. #6180C2×C4semidirect
57. #1084D8semidirect
78. C4⋊(C3⋊D3)C4×D3semidirect
143. H(3,𝔽3)C4×D8semidirect
218. C32#4756semidirect

Maximal subgroups

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

subgroupindex
#117211362
C2×#884322
C8×#10842
#63648⋊C32
#117203152
#105907453