Finite Group Databasebeta

← #13163449

Subgroups — #13163449

Order 1728.

1680 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 1680 conjugacy classes of subgroups — too many to draw. Only its 351 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#11723987C211yesmaximal, characteristic, complemented
2C2×#88862C211yesmaximal, characteristic, complemented
3C2×#88850C211yesmaximal, characteristic, complemented
4#11723936C211yesmaximal, characteristic
5C33×#103C211yesmaximal, characteristic, complemented
6#11722191C211yesmaximal, characteristic, complemented
7#11722177C211yesmaximal, characteristic, complemented
8#10592511—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C4⋊(C3⋊(C3⋊D3))—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15#88862—22no—
16#88859—21no—
17#88859—21no—
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19#88857—21no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21#88850—22no—
22#64155—313no—
23C2×#3755—313no—
24C2×#3743—313no—
25#64104—313no—
26C32×#103D3113yescomplemented
27#63648—313no—
28#63634—313no—
29C22×C3⋊(C3⋊D3)—21no—
30C22×C3⋊(C3⋊D3)D411yescharacteristic
31C33×Q8—21no—
32C33×Q8—21no—
33C2×C33×C4—21no—
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3⋊(C3⋊Dic3)—21no—
37C2×C3⋊(C3⋊Dic3)D411yescharacteristic
38C4⋊(C3⋊(C3⋊D3))—21no—
39C4⋊(C3⋊(C3⋊D3))—21no—
40C4⋊(C3⋊(C3⋊D3))—42no—
41C4×C3⋊(C3⋊D3)—21no—
42C4×C3⋊(C3⋊D3)—21no—
43C33×C8—21no—
44C3⋊(C3⋊(C3⋊C8))D412yescomplemented
45#4812—913no—
46C2×C32×Q8C2×D3113yes—
47C2×C4⋊(C3⋊D3)—313no—
48C2×C4⋊(C3⋊D3)—613no—
49C2×C4×C3⋊D3—313no—
50C2×C32×C8C2×D3113yes—
51C32×C4⋊C4C2×D3113yes—
52#3755—626no—
53#3752—613no—
54#3752—613no—
55C4⋊(C3⋊Dic3)—313no—
56#3750—613no—
57C2×C3⋊(C3⋊C8)—313no—
58#3743—626no—
59C22×C33C2×D411yescharacteristic
60C2×C3⋊(C3⋊D3)—21no—
61C2×C3⋊(C3⋊D3)—21no—
62C2×C3⋊(C3⋊D3)—21no—
63C2×C3⋊(C3⋊D3)—41no—
64C2×C3⋊(C3⋊D3)—41no—
65C2×C3⋊(C3⋊D3)—41no—
66C33×C4—41no—
67C33×C4—41no—
68C33×C4C2×D411yescharacteristic
69C33×C4#4111yescharacteristic
70C3⋊(C3⋊Dic3)—21no—
71C3⋊(C3⋊Dic3)—41no—
72#946—913no—
73C2×#215—913no—
74C2×ΓL(1,𝔽25)—913no—
75#895—913no—
76C3×#103C3⋊D3113yescomplemented
77#821—913no—
78#807—913no—
79C22×C3⋊D3—313no—
80C22×C3⋊D3—613no—
81C32×Q8—213no—
82C32×Q8—213no—
83C2×C32×C4—213no—
84C2×C32×C4—213no—
85C2×C32×C4C22×D3113yes—
86C2×C3⋊Dic3—313no—
87C2×C3⋊Dic3—613no—
88C4⋊(C3⋊D3)—613no—
89C4⋊(C3⋊D3)—613no—
90C4⋊(C3⋊D3)—1226no—
91C4×C3⋊D3—613no—
92C4×C3⋊D3—613no—
93C32×C8—213no—
94C3⋊(C3⋊C8)—326no—
95#468—271noSylow
96C2×C33#12111yescharacteristic
97C2×C33AGL(1,ℤ/8)11yescharacteristic
98C2×C33#13711yescharacteristic
99C3⋊(C3⋊D3)—41no—
100C3⋊(C3⋊D3)—81no—
101C2×C3×Q8C2×C3⋊D3113yes—
102C2×D12—913no—
103C2×D12—1813no—
104C2×C4×D3—913no—
105C2×C3×C8C2×C3⋊D3113yes—
106C3×C4⋊C4C2×C3⋊D3113yes—
107#215—1826no—
108(C2×D3)⋊C4—1813no—
109(C2×D3)⋊C4—1813no—
110C4⋊Dic3—913no—
111Dic3⋊C4—1813no—
112C2×C3⋊C8—913no—
113ΓL(1,𝔽25)—1826no—
114C22×C32D3×D4113yes—
115C2×C3⋊D3—613no—
116C2×C3⋊D3—613no—
117C2×C3⋊D3—613no—
118C2×C3⋊D3—1213no—
119C2×C3⋊D3—1213no—
120C2×C3⋊D3—1213no—
121C32×C4—413no—
122C32×C4—413no—
123C32×C4#235113yes—
124C32×C4D3×D4113yes—
125C3⋊Dic3—613no—
126C3⋊Dic3—1213no—
127C2×SD16—271no—
128C2×M16—271no—
129C22⋊Q8—271no—
130#121—271no—
131#107—271no—
132#103C3⋊(C3⋊D3)11yescharacteristic, complemented
133#102—271no—
134C33#46811yescharacteristic, complemented, Sylow
135C22×D3—913no—
136C22×D3—1813no—
137C3×Q8—213no—
138C3×Q8—213no—
139C2×C3×C4—213no—
140C2×C3×C4—213no—
141C2×C3×C4C22×C3⋊D3113yes—
142C2×Dic3—913no—
143C2×Dic3—1813no—
144D12—1813no—
145D12—1813no—
146D12—3626no—
147C4×D3—1813no—
148C4×D3—1813no—
149C3×C8—213no—
150C3⋊C8—926no—
151C2×C32#884113yes—
152C2×C32#914113yes—
153C2×C32#918113yes—
154C3⋊D3—1213no—
155C3⋊D3—2413no—
156C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
157C2×D4—271no—
158C2×D4—541no—
159C22×C4—271no—
160SD16—542no—
161M16—542no—
162C2×C8—271no—
163C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
164C4⋊C4—271no—
165C4⋊C4—541no—
166C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
167C22⋊C4—541no—
168C22⋊C4—541no—
169C22×C3D4×C3⋊D3113yes—
170C2×D3—1813no—
171C2×D3—1813no—
172C2×D3—1813no—
173C2×D3—3613no—
174C2×D3—3613no—
175C2×D3—3613no—
176C3×C4—413no—
177C3×C4—413no—
178C3×C4#3828113yes—
179C3×C4D4×C3⋊D3113yes—
180Dic3—1813no—
181Dic3—3613no—
182C32#4812113yescomplemented
183C23—271no—
184C23—541no—
185Q8—21no—
186Q8—21no—
187D4—541no—
188D4—541no—
189D4—1082no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—271no—
193C2×C4—541no—
194C2×C4—541no—
195C2×C4—541no—
196C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
197C8—21no—
198C8—272no—
199C2×C3#64093113yes—
200C2×C3#64123113yes—
201C2×C3#64127113yes—
202D3—3613no—
203D3—7213no—
204C22—541no—
205C22—541no—
206C22—541no—
207C22—1081no—
208C22—1081no—
209C22—1081no—
210C22D4×C3⋊(C3⋊D3)11yescharacteristic
211C4—41no—
212C4—41no—
213C4—541no—
214C4—1081no—
215C4#8907611yescharacteristic
216C4D4×C3⋊(C3⋊D3)11yescharacteristic
217C3#10592511113yescomplemented
218C2—1081no—
219C2—2161no—
220C2#1172392511yescharacteristic
221C2#1172395511yescharacteristic
222C2#1172395911yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#103C2
socleC22×C33C2×D4
prime radical#11723936C2
hypercenter#103C3⋊(C3⋊D3)
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #11723987C2semidirect
2. C2×#88862C2semidirect
3. C2×#88850C2semidirect
5. C33×#103C2semidirect
6. #11722191C2semidirect
7. #11722177C2semidirect
26. C32×#103D3semidirect
44. C3⋊(C3⋊(C3⋊C8))D4semidirect
76. C3×#103C3⋊D3semidirect
132. #103C3⋊(C3⋊D3)semidirect
134. C33#468semidirect
182. C32#4812semidirect
217. C3#10592511semidirect

Maximal subgroups

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

subgroupindex
#117239872
C2×#888622
C2×#888502
#117239362
C33×#1032
#117221912
#117221772
#105925113