Finite Group Databasebeta

← #13175362

Subgroups — #13175362

Order 1728.

1664 conjugacy classes of subgroups; only the 260 normal subgroups are shown.

normal subgroups only

This group has 1664 conjugacy classes of subgroups — too many to draw. Only its 260 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#11723911C211yesmaximal, characteristic, complemented
2C3×#64081C211yesmaximal, characteristic, complemented
3C32×#874C211yesmaximal, characteristic, complemented
4#11723522C212yesmaximal, complemented
5D12×C3⋊Dic3C212yesmaximal, complemented
6#11723474C214yesmaximal, complemented
7#11723473C214yesmaximal, complemented
8#10596190—31nomaximal
9#10593949—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12C2212yes—
12C2×C32×D12C2214yes—
13C2×#6290C2214yes—
14C2×D3×C3⋊Dic3C2214yes—
15C33×C42C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2212yes—
17#88857C2214yes—
18C3×#3752C2212yes—
19C3×#3750C2212yes—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)C2214yes—
22(C2×D3)⋊(C3⋊Dic3)C2214yes—
23C32×#127—31no—
24#64148—34no—
25#64147—32no—
26#64144—34no—
27D4×C3⋊Dic3—32no—
28#64081D311yescharacteristic, complemented
29#64079—31no—
30#64079—34no—
31#64079—64no—
32C3×#874D314yescomplemented
33C3×#872—34no—
34#63908—38no—
35Dic3×D12—38no—
36#63885—316no—
37#63884—316no—
38C22×C3×C3⋊D3C2311yescharacteristic
39C22×C32×D3C2314yes—
40C2×C33×C4C2312yes—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)C2314yes—
43C2×C3×C3⋊Dic3C2312yes—
44C2×C3×C3⋊Dic3C2311yescharacteristic
45C3×C4×C3⋊D3—22no—
46C32×D12—24no—
47C32×D12—44no—
48#6290—44no—
49D3×C3⋊Dic3—44no—
50#5719—94no—
51#5535—91no—
52C2×C32×D4—32no—
53C2×C32×D4—34no—
54C2×#638—34no—
55C22×C3⋊Dic3—34no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×D12C2×D318yes—
58C2×C3×D12C2×D3116yes—
59C2×C3×C4×D3—34no—
60C2×#625—316no—
61C2×D3×Dic3—316no—
62C32×C42—24no—
63C32×C42C2×D314yes—
64C32×C42C2×D311yescharacteristic
65#3757—34no—
66#3757—34no—
67#3752C2×D312yes—
68C4⋊(C3⋊Dic3)—32no—
69C4⋊(C3⋊Dic3)—38no—
70C4⋊(C3⋊Dic3)—68no—
71#3750—34no—
72#3750—316no—
73#3750—616no—
74#3750C2×D312yes—
75C4×C3⋊Dic3C2×D311yescharacteristic
76C3×(C2×D3)⋊C4—38no—
77C3×Dic3⋊C4—38no—
78C3×C4×Dic3—34no—
79(C2×D3)⋊Dic3—316no—
80(C2×D3)⋊Dic3—316no—
81C22×C33C2411yescharacteristic
82C2×C3×C3⋊D3—21no—
83C2×C3×C3⋊D3—22no—
84C2×C32×D3—24no—
85C2×C32×D3—24no—
86C2×C32×D3—24no—
87C33×C4—22no—
88C33×C4#4112yes—
89C3⋊(C3⋊Dic3)—24no—
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3—22no—
92C3×#127—34no—
93C3×#117—91no—
94#939—916no—
95#938—98no—
96#935—916no—
97D4×Dic3—98no—
98#930—92no—
99#894—94no—
100#884—94no—
101#874C3⋊D311yescharacteristic, complemented
102C4×D12—92no—
103#872—34no—
104#870—94no—
105#870—91no—
106#870—184no—
107C23×C32—34no—
108C22×C3⋊D3C22×D311yescharacteristic
109C22×C3×D3—34no—
110C22×C3×D3C22×D3116yes—
111C32×D4—64no—
112C32×D4—124no—
113C2×C32×C4—24no—
114C2×C32×C4—28no—
115C2×C32×C4C22×D318yes—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D312yes—
118C2×C32×C4C22×D311yescharacteristic
119#638—124no—
120C2×C3⋊Dic3—34no—
121C2×C3⋊Dic3—316no—
122C2×C3⋊Dic3—616no—
123C2×C3⋊Dic3—124no—
124C2×C3⋊Dic3C22×D312yes—
125C2×C3⋊Dic3C22×D311yescharacteristic
126C4×C3⋊D3—22no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—38no—
129C3×D12—216no—
130C3×D12—416no—
131C3×C4×D3—68no—
132#625—1216no—
133D3×Dic3—1216no—
134#540—271noSylow
135C2×C33C2×#4111yescharacteristic
136C2×C33H(5,𝔽2)12yes—
137C3×C3⋊D3—41no—
138C32×D3—44no—
139C2×C3×D4—38no—
140C2×C3×D4—316no—
141C22×C3×C4—91no—
142C2×#67—916no—
143C2×#67—94no—
144C22×Dic3—916no—
145C2×D12C2×C3⋊D312yes—
146C2×D12C2×C3⋊D314yes—
147C2×C4×D3—34no—
148C2×C4×D3—94no—
149C3×C4⋊C4—92no—
150C3×C22⋊C4—92no—
151C3×C42—24no—
152C3×C42—91no—
153C3×C42D3214yes—
154C3×C42C2×C3⋊D311yescharacteristic
155#217—916no—
156#217—916no—
157(C2×D3)⋊C4—38no—
158(C2×D3)⋊C4—94no—
159(C2×D3)⋊C4—94no—
160C4⋊Dic3—98no—
161C4⋊Dic3—92no—
162C4⋊Dic3—188no—
163Dic3⋊C4—38no—
164Dic3⋊C4—916no—
165Dic3⋊C4—94no—
166Dic3⋊C4—1816no—
167C4×Dic3—34no—
168C22×C32—24no—
169C22×C32—64no—
170C22×C32—64no—
171C22×C32—64no—
172C22×C32C23×D314yes—
173C22×C32C23×D311yescharacteristic
174C2×C3⋊D3—21no—
175C2×C3⋊D3—22no—
176C2×C3×D3—216no—
177C2×C3×D3—216no—
178C2×C3×D3—216no—
179C2×C3×D3—64no—
180C2×C3×D3—68no—
181C32×C4—22no—
182C32×C4—28no—
183C32×C4—28no—
184C32×C4—48no—
185C32×C4#23512yes—
186C32×C4#23718yes—
187C3⋊Dic3—21no—
188C3⋊Dic3—22no—
189C3⋊Dic3—64no—
190C3⋊Dic3—616no—
191C3⋊Dic3—1216no—
192C3×Dic3—64no—
193C3×Dic3—68no—
194#127—31no—
195#125—271no—
196#123—274no—
197#121—274no—
198#121—272no—
199C4×D4—272no—
200#117—91no—
201C33#54011yescharacteristic, complemented, Sylow
202C23×C3—316no—
203C23×C3—91no—
204C22×D3—34no—
205C22×D3C22×C3⋊D314yes—
206C3×D4—616no—
207C3×D4—1216no—
208C2×C3×C4—24no—
209C2×C3×C4—28no—
210C2×C3×C4—91no—
211C2×C3×C4—92no—
212C2×C3×C4—182no—
213C2×C3×C4C2×D3218yes—
214C2×C3×C4C2×D3214yes—
215C2×C3×C4C22×C3⋊D312yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217#67—364no—
218#67—3616no—
219C2×Dic3—38no—
220C2×Dic3—34no—
221C2×Dic3—916no—
222C2×Dic3—94no—
223C2×Dic3—1816no—
224C2×Dic3—3616no—
225D12—24no—
226D12—44no—
227C4×D3—68no—
228C4×D3—364no—
229C2×C32—24no—
230C2×C32—28no—
231C2×C32—124no—
232C2×C32C2×#23511yescharacteristic
233C2×C32C2×#23714yes—
234C2×C32#100418yes—
235C2×C32#100912yes—
236C3⋊D3—41no—
237C3×D3—416no—
238C3×D3—124no—
239C2×D4—32no—
240C2×D4—34no—
241C2×D4—274no—
242C22×C4—91no—
243C22×C4—274no—
244C4⋊C4—92no—
245C4⋊C4—272no—
246C4⋊C4—274no—
247C22⋊C4—92no—
248C22⋊C4—274no—
249C22⋊C4—274no—
250C42—91no—
251C42D3×C3⋊D311yescharacteristic
252C22×C3—24no—
253C22×C3—616no—
254C22×C3—616no—
255C22×C3—616no—
256C22×C3—181no—
257C22×C3—182no—
258C22×C3C22×D3214yes—
259C22×C3C23×C3⋊D311yescharacteristic
260C2×D3—24no—
261C2×D3—24no—
262C2×D3—24no—
263C2×D3—64no—
264C2×D3—68no—
265C3×C4—28no—
266C3×C4—22no—
267C3×C4—28no—
268C3×C4—48no—
269C3×C4—181no—
270C3×C4—182no—
271C3×C4#379518yes—
272C3×C4#383012yes—
273Dic3—64no—
274Dic3—68no—
275Dic3—1816no—
276Dic3—184no—
277Dic3—3616no—
278C32—24no—
279C32#553511yescharacteristic, complemented
280C32#571914yescomplemented
281C23—34no—
282C23—91no—
283D4—64no—
284D4—124no—
285D4—1084no—
286C2×C4—92no—
287C2×C4—91no—
288C2×C4—182no—
289C2×C4—274no—
290C2×C4—1084no—
291C2×C4C2×D3×C3⋊D312yes—
292C2×C4C2×D3×C3⋊D311yescharacteristic
293C2×C3—24no—
294C2×C3—28no—
295C2×C3—1216no—
296C2×C3—361no—
297C2×C3C2×#379514yes—
298C2×C3#6430718yes—
299C2×C3C2×#383011yescharacteristic
300C2×C3#6436112yes—
301D3—44no—
302D3—124no—
303C22—64no—
304C22—64no—
305C22—64no—
306C22—181no—
307C22—182no—
308C22C22×D3×C3⋊D311yescharacteristic
309C4—22no—
310C4—181no—
311C4—182no—
312C4—544no—
313C4#8901412yes—
314C3—24no—
315C3#1059394914yescomplemented
316C3#1059619011yescharacteristic, complemented
317C2—124no—
318C2—361no—
319C2C2×#8901411yescharacteristic
320C2#1172457912yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723911C2semidirect
2. C3×#64081C2semidirect
3. C32×#874C2semidirect
4. #11723522C2semidirect
5. D12×C3⋊Dic3C2semidirect
6. #11723474C2semidirect
7. #11723473C2semidirect
28. #64081D3semidirect
32. C3×#874D3semidirect
101. #874C3⋊D3semidirect
201. C33#540semidirect
279. C32#5535semidirect
280. C32#5719semidirect
315. C3#10593949semidirect
316. C3#10596190semidirect

Maximal subgroups

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

subgroupindex
#117239112
C3×#640812
C32×#8742
#117235222
D12×C3⋊Dic32
#117234742
#117234732
#105961903
#105939493