Finite Group Databasebeta

← #13163409

Subgroups — #13163409

Order 1728.

2660 conjugacy classes of subgroups; only the 419 normal subgroups are shown.

normal subgroups only

This group has 2660 conjugacy classes of subgroups — too many to draw. Only its 419 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
1C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×#88850C211yesmaximal, characteristic, complemented
3#11723933C211yesmaximal, characteristic, complemented
4C33×#102C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722191C211yesmaximal, characteristic, complemented
7#11722179C211yesmaximal, characteristic, complemented
8#10592469—313nomaximal
9C23×C3⋊(C3⋊D3)—21no—
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12D4×C3⋊(C3⋊D3)—22no—
13D4×C3⋊(C3⋊D3)C414yescomplemented
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C33×C8C2211yescharacteristic, complemented
17C33×C4⋊C4C2211yescharacteristic, complemented
18#88859—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
20C4×C3⋊(C3⋊Dic3)—21no—
21C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
22#88850—22no—
23C2×D4×C3⋊D3—313no—
24C2×#3743—313no—
25#64101—313no—
26C32×#102D3113yescomplemented
27#63659—313no—
28#63648—313no—
29#63636—313no—
30C23×C33—21no—
31C22×C3⋊(C3⋊D3)—21no—
32C22×C3⋊(C3⋊D3)—24no—
33C22×C3⋊(C3⋊D3)—42no—
34C22×C3⋊(C3⋊D3)—42no—
35C22×C3⋊(C3⋊D3)D411yescharacteristic
36C33×D4—21no—
37C33×D4C2×C412yes—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40#6312—24no—
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yes—
45C4×C3⋊(C3⋊D3)C2×C412yes—
46C4×C3⋊(C3⋊D3)D412yes—
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49#4770—913no—
50C23×C3⋊D3—613no—
51C2×C32×D4C2×D3113yes—
52C2×#638—613no—
53D4×C3⋊D3—352no—
54D4×C3⋊D3—626no—
55C2×C4⋊(C3⋊D3)—313no—
56C2×C4×C3⋊D3—313no—
57C2×C32×C8C2×D3113yescomplemented
58C32×C4⋊C4C2×D3113yescomplemented
59#3752—613no—
60C4⋊(C3⋊Dic3)—313no—
61C4×C3⋊Dic3—613no—
62C2×C3⋊(C3⋊C8)—313no—
63#3743—626no—
64C22×C33—22no—
65C22×C33—41no—
66C22×C33—41no—
67C22×C33C2×D411yescharacteristic
68C2×C3⋊(C3⋊D3)—21no—
69C2×C3⋊(C3⋊D3)—21no—
70C2×C3⋊(C3⋊D3)—22no—
71C2×C3⋊(C3⋊D3)—48no—
72C2×C3⋊(C3⋊D3)—41no—
73C2×C3⋊(C3⋊D3)—41no—
74C2×C3⋊(C3⋊D3)C22⋊C412yes—
75C33×C4—41no—
76C33×C4C22×C411yescharacteristic
77C33×C4C2×D411yescharacteristic
78C3⋊(C3⋊Dic3)—41no—
79C3⋊(C3⋊Dic3)C22⋊C412yes—
80C2×D3×D4—913no—
81C2×ΓL(1,𝔽25)—913no—
82#892—913no—
83C3×#102C3⋊D3113yescomplemented
84#832—913no—
85#821—913no—
86#809—913no—
87C23×C32—213no—
88C22×C3⋊D3—313no—
89C22×C3⋊D3—652no—
90C22×C3⋊D3—613no—
91C22×C3⋊D3—1226no—
92C22×C3⋊D3—1226no—
93C32×D4—213no—
94C32×D4C4×D3126yes—
95C2×C32×C4—213no—
96C2×C32×C4C22×D3113yes—
97#638—652no—
98C2×C3⋊Dic3—313no—
99C2×C3⋊Dic3—613no—
100C4⋊(C3⋊D3)—326no—
101C4⋊(C3⋊D3)—613no—
102C4×C3⋊D3—326no—
103C4×C3⋊D3—326no—
104C32×C8—213no—
105C3⋊(C3⋊C8)—613no—
106#418—271noSylow
107C2×C33—42no—
108C2×C33C2×C22⋊C411yescharacteristic
109C2×C33AGL(1,ℤ/8)11yescharacteristic
110C2×C33AGL(1,ℤ/8)11yescharacteristic
111C3⋊(C3⋊D3)—22no—
112C3⋊(C3⋊D3)—42no—
113C23×D3—1813no—
114C2×C3×D4C2×C3⋊D3113yes—
115C2×#67—1813no—
116D3×D4—952no—
117D3×D4—1826no—
118C2×D12—913no—
119C2×C4×D3—913no—
120C2×C3×C8C2×C3⋊D3113yescomplemented
121C3×C4⋊C4C2×C3⋊D3113yescomplemented
122(C2×D3)⋊C4—1813no—
123C4⋊Dic3—913no—
124C4×Dic3—1813no—
125C2×C3⋊C8—913no—
126ΓL(1,𝔽25)—1826no—
127C22×C32—226no—
128C22×C32—413no—
129C22×C32—413no—
130C22×C32D3×D4113yes—
131C2×C3⋊D3—326no—
132C2×C3⋊D3—613no—
133C2×C3⋊D3—613no—
134C2×C3⋊D3—626no—
135C2×C3⋊D3—12104no—
136C2×C3⋊D3—1213no—
137C2×C3⋊D3—1213no—
138C32×C4—413no—
139C32×C4C2×C4×D3113yes—
140C32×C4D3×D4113yes—
141C3⋊Dic3—326no—
142C3⋊Dic3—1213no—
143C22×D4—271no—
144C2×M16—271no—
145#117—271no—
146#102—271no—
147#102—271no—
148#102—271no—
149#102C3⋊(C3⋊D3)11yescharacteristic, complemented
150C33#41811yescharacteristic, complemented, Sylow
151C23×C3—213no—
152C22×D3—913no—
153C22×D3—1852no—
154C22×D3—1813no—
155C22×D3—3626no—
156C22×D3—3626no—
157C3×D4—213no—
158C3×D4C4×C3⋊D3126yes—
159C2×C3×C4—213no—
160C2×C3×C4C22×C3⋊D3113yes—
161#67—1852no—
162C2×Dic3—913no—
163C2×Dic3—1813no—
164D12—926no—
165D12—1813no—
166C4×D3—926no—
167C4×D3—926no—
168C3×C8—213no—
169C3⋊C8—1813no—
170C2×C32—426no—
171C2×C32D3×C22⋊C4113yes—
172C2×C32#911113yes—
173C2×C32#914113yes—
174C3⋊D3—626no—
175C3⋊D3—1226no—
176C24—541no—
177C2×D4—274no—
178C2×D4—271no—
179C2×D4—542no—
180C2×D4—541no—
181C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
182C22×C4—271no—
183M16—542no—
184C2×C8—271no—
185C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
186C4⋊C4—271no—
187C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
188C22⋊C4—541no—
189C42—541no—
190C22×C3—226no—
191C22×C3—413no—
192C22×C3—413no—
193C22×C3D4×C3⋊D3113yes—
194C2×D3—926no—
195C2×D3—1813no—
196C2×D3—1813no—
197C2×D3—1826no—
198C2×D3—36104no—
199C2×D3—3613no—
200C2×D3—3613no—
201C3×C4—413no—
202C3×C4C2×C4×C3⋊D3113yes—
203C3×C4D4×C3⋊D3113yes—
204Dic3—926no—
205Dic3—3613no—
206C32#4770113yescomplemented
207C23—21no—
208C23—271no—
209C23—544no—
210C23—541no—
211C23—1082no—
212C23—1082no—
213D4—21no—
214D4—272no—
215D4—544no—
216D4—541no—
217D4C4×C3⋊(C3⋊D3)12yes—
218C2×C4—21no—
219C2×C4—271no—
220C2×C4—272no—
221C2×C4—272no—
222C2×C4—541no—
223C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
224C8—21no—
225C8—541no—
226C2×C3—426no—
227C2×C3C22⋊C4×C3⋊D3113yes—
228C2×C3#64120113yes—
229C2×C3#64123113yes—
230D3—1826no—
231D3—3626no—
232C22—22no—
233C22—41no—
234C22—41no—
235C22—272no—
236C22—541no—
237C22—541no—
238C22—542no—
239C22—1088no—
240C22—1081no—
241C22—1081no—
242C22D4×C3⋊(C3⋊D3)11yescharacteristic
243C4—41no—
244C4—272no—
245C4—1081no—
246C4C2×C4×C3⋊(C3⋊D3)11yescharacteristic
247C4D4×C3⋊(C3⋊D3)11yescharacteristic
248C3#10592469113yescomplemented
249C2—42no—
250C2—542no—
251C2—1082no—
252C2C22⋊C4×C3⋊(C3⋊D3)11yescharacteristic
253C2#1172395211yescharacteristic
254C2#1172395511yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×D4×C3⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenter#102C3⋊(C3⋊D3)
hypocenterC33#418
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#88850C2semidirect
3. #11723933C2semidirect
4. C33×#102C2semidirect
5. #11722202C2semidirect
6. #11722191C2semidirect
7. #11722179C2semidirect
13. D4×C3⋊(C3⋊D3)C4semidirect
16. C2×C33×C8C22semidirect
17. C33×C4⋊C4C22semidirect
19. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
21. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
26. C32×#102D3semidirect
57. C2×C32×C8C2×D3semidirect
58. C32×C4⋊C4C2×D3semidirect
83. C3×#102C3⋊D3semidirect
120. C2×C3×C8C2×C3⋊D3semidirect
121. C3×C4⋊C4C2×C3⋊D3semidirect
149. #102C3⋊(C3⋊D3)semidirect
150. C33#418semidirect
185. C2×C8C2×C3⋊(C3⋊D3)semidirect
187. C4⋊C4C2×C3⋊(C3⋊D3)semidirect
206. C32#4770semidirect
248. C3#10592469semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×#888502
#117239332
C33×#1022
#117222022
#117221912
#117221792
#105924693