Finite Group Databasebeta

← #13163838

Subgroups — #13163838

Order 1728.

1512 conjugacy classes of subgroups; only the 349 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 349 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
1C33×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#11723990C211yesmaximal, characteristic, complemented
3#11723989C211yesmaximal, characteristic, complemented
4C2×#88860C211yesmaximal, characteristic, complemented
5#11722203C211yesmaximal, characteristic, complemented
6#11722193C211yesmaximal, characteristic, complemented
7#11722192C211yesmaximal, characteristic, complemented
8#10592898—313nomaximal
9C33×#41C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C33×SD16—21no—
13C33×D8—21no—
14C33×M16C2211yescharacteristic
15#88862—21no—
16#88860—21no—
17#88860—21no—
18#88860—21no—
19#88855—21no—
20#88855C2211yescharacteristic
21#88855C2211yescharacteristic
22C2×C3⋊(C3⋊(C3⋊C8))—21no—
23C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C32×AGL(1,ℤ/8)D3113yescomplemented
25#64158—313no—
26#64157—313no—
27C2×#3753—313no—
28#63660—313no—
29#63650—313no—
30#63649—313no—
31C23×C33—21no—
32C22×C3⋊(C3⋊D3)—21no—
33C33×Q8D411yescharacteristic
34C33×D4—21no—
35C33×D4—21no—
36C33×D4—21no—
37C33×D4D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C33×C8—21no—
43C3⋊(C3⋊(C3⋊C8))—21no—
44C3⋊(C3⋊(C3⋊C8))—21no—
45C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
46C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
47#5199—913no—
48C32×#41C2×D3113yes—
49C2×C32×D4C2×D3113yes—
50C2×C4⋊(C3⋊D3)—313no—
51C32×SD16—213no—
52C32×D8—213no—
53C32×M16C2×D3113yes—
54#3755—613no—
55#3753—613no—
56#3753—613no—
57#3753—613no—
58#3748—313no—
59#3748—313no—
60#3748—613no—
61C2×C3⋊(C3⋊C8)—313no—
62C2×C3⋊(C3⋊C8)—613no—
63C22×C33—21no—
64C22×C33—41no—
65C22×C33—81no—
66C22×C33#4111yescharacteristic
67C2×C3⋊(C3⋊D3)—41no—
68C2×C3⋊(C3⋊D3)—81no—
69C33×C4—21no—
70C33×C4C2×D411yescharacteristic
71C33×C4C2×D411yescharacteristic
72C3×AGL(1,ℤ/8)C3⋊D3113yescomplemented
73#949—913no—
74#948—913no—
75C2×#213—913no—
76#833—913no—
77#823—913no—
78#822—913no—
79C23×C32—213no—
80C22×C3⋊D3—613no—
81C32×Q8#67113yes—
82C32×D4—213no—
83C32×D4—213no—
84C32×D4—213no—
85C32×D4#67113yes—
86C2×C32×C4—213no—
87C2×C32×C4C22×D3113yes—
88C4⋊(C3⋊D3)—613no—
89C4⋊(C3⋊D3)—613no—
90C32×C8—213no—
91C3⋊(C3⋊C8)—313no—
92C3⋊(C3⋊C8)—313no—
93C3⋊(C3⋊C8)—613no—
94C3⋊(C3⋊C8)—613no—
95#472—271noSylow
96C2×C33—21no—
97C2×C33—41no—
98C2×C33—81no—
99C2×C33#12111yescharacteristic
100C3⋊(C3⋊D3)—81no—
101C3×#41C2×C3⋊D3113yes—
102C2×C3×D4C2×C3⋊D3113yes—
103C2×D12—913no—
104C3×SD16—213no—
105C3×D8—213no—
106C3×M16C2×C3⋊D3113yes—
107#215—1813no—
108#213—1813no—
109#213—1813no—
110#213—1813no—
111#208—913no—
112#208—913no—
113#208—1813no—
114C2×C3⋊C8—913no—
115C2×C3⋊C8—1813no—
116C22×C32—213no—
117C22×C32—413no—
118C22×C32—813no—
119C22×C32#237113yes—
120C2×C3⋊D3—1213no—
121C2×C3⋊D3—2413no—
122C32×C4—213no—
123C32×C4D3×D4113yes—
124C32×C4C2×#67113yes—
125AGL(1,ℤ/8)—271no—
126AGL(1,ℤ/8)C3⋊(C3⋊D3)11yescharacteristic, complemented
127C2×D8—271no—
128#131—271no—
129#108—271no—
130#100—271no—
131#100—271no—
132C33#47211yescharacteristic, complemented, Sylow
133C23×C3—213no—
134C22×D3—1813no—
135C3×Q8#638113yes—
136C3×D4—213no—
137C3×D4—213no—
138C3×D4—213no—
139C3×D4#638113yes—
140C2×C3×C4—213no—
141C2×C3×C4C22×C3⋊D3113yes—
142D12—1813no—
143D12—1813no—
144C3×C8—213no—
145C3⋊C8—913no—
146C3⋊C8—913no—
147C3⋊C8—1813no—
148C3⋊C8—1813no—
149C2×C32—213no—
150C2×C32—413no—
151C2×C32—813no—
152C2×C32#939113yes—
153C3⋊D3—2413no—
154#41C2×C3⋊(C3⋊D3)11yescharacteristic
155C2×D4—271no—
156C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
157SD16—21no—
158SD16—541no—
159D8—21no—
160D8—541no—
161D8—541no—
162D8—541no—
163M16—271no—
164M16—271no—
165M16—541no—
166M16C2×C3⋊(C3⋊D3)11yescharacteristic
167C2×C8—271no—
168C2×C8—541no—
169C22×C3—213no—
170C22×C3—413no—
171C22×C3—813no—
172C22×C3#3830113yes—
173C2×D3—3613no—
174C2×D3—7213no—
175C3×C4—213no—
176C3×C4D4×C3⋊D3113yes—
177C3×C4C2×#638113yes—
178C32#5199113yescomplemented
179C23—21no—
180C23—541no—
181Q8#631211yescharacteristic
182D4—21no—
183D4—21no—
184D4—21no—
185D4—541no—
186D4—541no—
187D4#631211yescharacteristic
188C2×C4—21no—
189C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
190C8—21no—
191C8—271no—
192C8—271no—
193C8—541no—
194C8—541no—
195C2×C3—213no—
196C2×C3—413no—
197C2×C3—813no—
198C2×C3#64148113yes—
199D3—7213no—
200C22—21no—
201C22—41no—
202C22—81no—
203C22—1081no—
204C22—2161no—
205C22#8907811yescharacteristic
206C4—21no—
207C4D4×C3⋊(C3⋊D3)11yescharacteristic
208C4C2×#631211yescharacteristic
209C3#10592898113yescomplemented
210C2—21no—
211C2—41no—
212C2—81no—
213C2—2161no—
214C2#1172398011yescharacteristic
subgroupG/H
centerC2#11723980
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×AGL(1,ℤ/8)C2
socleC2×C33#121
hypercenterAGL(1,ℤ/8)C3⋊(C3⋊D3)
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C33×AGL(1,ℤ/8)C2semidirect
2. #11723990C2semidirect
3. #11723989C2semidirect
4. C2×#88860C2semidirect
5. #11722203C2semidirect
6. #11722193C2semidirect
7. #11722192C2semidirect
24. C32×AGL(1,ℤ/8)D3semidirect
72. C3×AGL(1,ℤ/8)C3⋊D3semidirect
126. AGL(1,ℤ/8)C3⋊(C3⋊D3)semidirect
132. C33#472semidirect
178. C32#5199semidirect
209. C3#10592898semidirect

Maximal subgroups

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

subgroupindex
C33×AGL(1,ℤ/8)2
#117239902
#117239892
C2×#888602
#117222032
#117221932
#117221922
#105928983