Finite Group Databasebeta

← #13179763

Subgroups — #13179763

Order 1728.

3192 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C22⋊Q8C211yesmaximal, characteristic, complemented
2#11723987C211yesmaximal, characteristic, complemented
3Q8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C4×#6312C211yesmaximal, characteristic
5C2×C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
6#11723938C212yesmaximal, complemented
7C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
8#11723930C212yesmaximal, complemented
9#11723929C211yesmaximal, characteristic, complemented
10#11723922C212yesmaximal
11#11723919C212yesmaximal, complemented
12#10596106—313nomaximal
13C2×C33×Q8C2211yescharacteristic
14C22×C33×C4C2211yescharacteristic
15C2×#6312C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2212yes—
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×C4×C3⋊(C3⋊D3)C2212yes—
19C2×#6308C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21C33×C4⋊C4C2212yes—
22C33×C22⋊C4C2212yes—
23#88864C2211yescharacteristic
24#88859C2212yes—
25#88859C2211yescharacteristic
26C4⋊(C3⋊(C3⋊Dic3))C2212yes—
27C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
28C4⋊(C3⋊(C3⋊Dic3))C2212yes—
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
30#88857C2212yes—
31#88857C2212yes—
32#88857C2212yes—
33#88857C2211yescharacteristic
34C4×C3⋊(C3⋊Dic3)C2212yes—
35C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
36C32×C22⋊Q8D3113yescomplemented
37#64155—313no—
38Q8×C3⋊Dic3—313no—
39C4×#638—313no—
40C2×C4⋊(C3⋊Dic3)—313no—
41#64106—326no—
42C4⋊C4×C3⋊D3—313no—
43#64098—326no—
44#64097—313no—
45#64090—326no—
46#64087—326no—
47C23×C33C2311yescharacteristic
48C22×C3⋊(C3⋊D3)C2311yescharacteristic
49C33×Q8—22no—
50C2×C33×C4—22no—
51C2×C33×C4C2312yes—
52C2×C33×C4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2311yescharacteristic
55#6312Q814yescomplemented
56C2×C3⋊(C3⋊Dic3)—42no—
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C2×C3⋊(C3⋊Dic3)C2312yes—
61C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
62C4×C3⋊(C3⋊D3)—22no—
63C4×C3⋊(C3⋊D3)—42no—
64#6308—42no—
65#5635—913no—
66C2×C32×Q8C2×D3113yes—
67C22×C32×C4C2×D3113yes—
68C2×#638—313no—
69C22×C3⋊Dic3—326no—
70C2×C4×C3⋊D3—313no—
71C2×C4×C3⋊D3—326no—
72C2×#634—326no—
73C32×C4⋊C4C2×D3113yes—
74C32×C4⋊C4C2×D3126yes—
75C32×C22⋊C4C2×D3126yes—
76#3757—313no—
77#3752—326no—
78#3752—313no—
79C4⋊(C3⋊Dic3)—326no—
80C4⋊(C3⋊Dic3)—313no—
81C4⋊(C3⋊Dic3)—326no—
82C4⋊(C3⋊Dic3)—313no—
83#3750—326no—
84#3750—326no—
85#3750—326no—
86#3750—313no—
87C4×C3⋊Dic3—326no—
88C4×C3⋊Dic3—313no—
89C22×C33—21no—
90C22×C33—21no—
91C22×C33C2×Q812yes—
92C22×C33C2411yescharacteristic
93C2×C3⋊(C3⋊D3)—21no—
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3⋊(C3⋊D3)C2×Q812yes—
96C33×C4—21no—
97C33×C4—22no—
98C33×C4—22no—
99C33×C4#4112yes—
100C3⋊(C3⋊Dic3)—21no—
101C3⋊(C3⋊Dic3)—21no—
102C3⋊(C3⋊Dic3)—22no—
103C3⋊(C3⋊Dic3)—22no—
104C3⋊(C3⋊Dic3)C2×Q812yes—
105C3×C22⋊Q8C3⋊D3113yescomplemented
106#946—913no—
107Q8×Dic3—913no—
108C4×#67—913no—
109C2×C4⋊Dic3—913no—
110#897—926no—
111D3×C4⋊C4—913no—
112#889—926no—
113#888—913no—
114#67⋊C4—926no—
115#878—926no—
116C23×C32C22×D3113yes—
117C22×C3⋊D3—313no—
118C32×Q8—226no—
119C2×C32×C4—226no—
120C2×C32×C4C22×D3126yes—
121C2×C32×C4C22×D3126yes—
122C2×C32×C4C22×D3113yes—
123C2×C32×C4C22×D3113yes—
124#638—352no—
125C2×C3⋊Dic3—313no—
126C2×C3⋊Dic3—326no—
127C2×C3⋊Dic3—313no—
128C2×C3⋊Dic3—313no—
129C2×C3⋊Dic3—326no—
130C2×C3⋊Dic3—1226no—
131C4×C3⋊D3—626no—
132C4×C3⋊D3—1226no—
133#634—1226no—
134#554—271noSylow
135C2×C33—22no—
136C2×C33C22×Q811yescharacteristic
137C2×C33C2×#4111yescharacteristic
138C2×C33H(5,𝔽2)11yescharacteristic
139C3⋊(C3⋊D3)—22no—
140C2×C3×Q8C2×C3⋊D3113yes—
141C22×C3×C4C2×C3⋊D3113yes—
142C2×#67—913no—
143C22×Dic3—926no—
144C2×C4×D3—913no—
145C2×C4×D3—926no—
146C2×Dic6—926no—
147C3×C4⋊C4C2×C3⋊D3113yes—
148C3×C4⋊C4C2×C3⋊D3126yes—
149C3×C22⋊C4C2×C3⋊D3126yes—
150#217—913no—
151(C2×D3)⋊C4—926no—
152(C2×D3)⋊C4—913no—
153C4⋊Dic3—926no—
154C4⋊Dic3—913no—
155C4⋊Dic3—926no—
156C4⋊Dic3—913no—
157Dic3⋊C4—926no—
158Dic3⋊C4—926no—
159Dic3⋊C4—926no—
160Dic3⋊C4—913no—
161C4×Dic3—926no—
162C4×Dic3—913no—
163C22×C32—213no—
164C22×C32—213no—
165C22×C32D3×Q8126yes—
166C22×C32C23×D3113yes—
167C2×C3⋊D3—326no—
168C2×C3⋊D3—613no—
169C2×C3⋊D3—613no—
170C32×C4—213no—
171C32×C4—226no—
172C32×C4—226no—
173C32×C4#237126yes—
174C3⋊Dic3—326no—
175C3⋊Dic3—613no—
176C3⋊Dic3—613no—
177C3⋊Dic3—626no—
178C3⋊Dic3—626no—
179C4⋊Q8—271no—
180#125—272no—
181C22⋊Q8—271no—
182C22⋊Q8—272no—
183C22⋊Q8—272no—
184C22⋊Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
185C4×Q8—271no—
186C4×D4—271no—
187C4×D4—272no—
188C2×C4⋊C4—271no—
189C2×C4⋊C4—271no—
190C33#55411yescharacteristic, complemented, Sylow
191C23×C3C22×C3⋊D3113yes—
192C22×D3—913no—
193C3×Q8—226no—
194C2×C3×C4—226no—
195C2×C3×C4C22×C3⋊D3126yes—
196C2×C3×C4C22×C3⋊D3126yes—
197C2×C3×C4C22×C3⋊D3113yes—
198C2×C3×C4C22×C3⋊D3113yes—
199#67—952no—
200C2×Dic3—913no—
201C2×Dic3—926no—
202C2×Dic3—913no—
203C2×Dic3—913no—
204C2×Dic3—926no—
205C2×Dic3—3626no—
206C4×D3—1826no—
207C4×D3—3626no—
208Dic6—3626no—
209C2×C32—226no—
210C2×C32C2×#237113yes—
211C2×C32C2×D3×Q8113yes—
212C2×C32#1009113yes—
213C3⋊D3—626no—
214C2×Q8—272no—
215C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
216C2×D4—271no—
217C22×C4—272no—
218C22×C4—271no—
219C22×C4—272no—
220C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
221C4⋊C4—271no—
222C4⋊C4—272no—
223C4⋊C4—272no—
224C4⋊C4—271no—
225C4⋊C4—271no—
226C4⋊C4—272no—
227C4⋊C4—272no—
228C4⋊C4—272no—
229C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
230C4⋊C4C2×C3⋊(C3⋊D3)12yes—
231C22⋊C4—271no—
232C22⋊C4—271no—
233C22⋊C4—272no—
234C22⋊C4C2×C3⋊(C3⋊D3)12yes—
235C42—272no—
236C42—271no—
237C22×C3—213no—
238C22×C3—213no—
239C22×C3Q8×C3⋊D3126yes—
240C22×C3C23×C3⋊D3113yes—
241C2×D3—926no—
242C2×D3—1813no—
243C2×D3—1813no—
244C3×C4—213no—
245C3×C4—226no—
246C3×C4—226no—
247C3×C4#3830126yes—
248Dic3—926no—
249Dic3—1813no—
250Dic3—1813no—
251Dic3—1826no—
252Dic3—1826no—
253C32#5635113yescomplemented
254C23—271no—
255C23C22×C3⋊(C3⋊D3)11yescharacteristic
256Q8—22no—
257Q8—1082no—
258D4—274no—
259C2×C4—22no—
260C2×C4—271no—
261C2×C4—271no—
262C2×C4—272no—
263C2×C4—271no—
264C2×C4—272no—
265C2×C4—542no—
266C2×C4—1082no—
267C2×C4—1082no—
268C2×C4C22×C3⋊(C3⋊D3)12yes—
269C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
270C2×C4C22×C3⋊(C3⋊D3)12yes—
271C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
272C2×C3—226no—
273C2×C3C2×#3830113yes—
274C2×C3C2×Q8×C3⋊D3113yes—
275C2×C3#64366113yes—
276D3—1826no—
277C22—21no—
278C22—21no—
279C22—272no—
280C22—541no—
281C22—541no—
282C22Q8×C3⋊(C3⋊D3)12yes—
283C22C23×C3⋊(C3⋊D3)11yescharacteristic
284C4—21no—
285C4—22no—
286C4—22no—
287C4—272no—
288C4—541no—
289C4—542no—
290C4—542no—
291C4—541no—
292C4#8907812yes—
293C3#10596106113yescomplemented
294C2—22no—
295C2—542no—
296C2C2×#8907811yescharacteristic
297C2C2×Q8×C3⋊(C3⋊D3)11yescharacteristic
298C2#1172468411yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊Q8C2
prime radicalC2×#6312C22
hypercenterC22⋊Q8C3⋊(C3⋊D3)
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C22⋊Q8C2semidirect
2. #11723987C2semidirect
3. Q8×C3⋊(C3⋊Dic3)C2semidirect
5. C2×C4⋊(C3⋊(C3⋊Dic3))C2semidirect
6. #11723938C2semidirect
7. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
8. #11723930C2semidirect
9. #11723929C2semidirect
11. #11723919C2semidirect
36. C32×C22⋊Q8D3semidirect
55. #6312Q8semidirect
105. C3×C22⋊Q8C3⋊D3semidirect
184. C22⋊Q8C3⋊(C3⋊D3)semidirect
190. C33#554semidirect
253. C32#5635semidirect
293. C3#10596106semidirect

Maximal subgroups

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

subgroupindex
C33×C22⋊Q82
#117239872
Q8×C3⋊(C3⋊Dic3)2
C4×#63122
C2×C4⋊(C3⋊(C3⋊Dic3))2
#117239382
C4⋊C4×C3⋊(C3⋊D3)2
#117239302
#117239292
#117239222
#117239192
#105961063