Finite Group Databasebeta

← #13147631

Subgroups — #13147631

Order 1728.

776 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 161 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#11721707C211yesmaximal, characteristic, complemented
2C9×#874C211yesmaximal, characteristic, complemented
3C3×#63434C211yesmaximal, characteristic, complemented
4#11721036C212yesmaximal, complemented
5D12×Dic9C212yesmaximal, complemented
6#11721016C214yesmaximal, complemented
7#11721015C214yesmaximal, complemented
8#10593949—31nomaximal
9#10589821—31nomaximal
10C2×C9×D12C2212yes—
11C2×C9×D12C2214yes—
12C2×C3×C4×D9C2211yescharacteristic
13C2×#6194C2214yes—
14C2×D3×Dic9C2214yes—
15C3×C42×C9C2211yescharacteristic
16C4⋊(C3⋊Dic9)C2212yes—
17#88534C2214yes—
18C3×(C2×D9)⋊C4C2212yes—
19C3×Dic9⋊C4C2212yes—
20C3×C4×Dic9C2211yescharacteristic
21(C2×D3)⋊Dic9C2214yes—
22(C2×D3)⋊Dic9C2214yes—
23#64079—31no—
24C3×#874D311yescharacteristic
25C3×#872—31no—
26#63908—32no—
27Dic3×D12—32no—
28#63885—34no—
29#63884—34no—
30C9×#127—31no—
31#63501—34no—
32#63500—32no—
33#63497—34no—
34D4×Dic9—32no—
35#63434D311yescharacteristic, complemented
36#63432—31no—
37#63432—61no—
38C22×D3×C9C2314yes—
39C22×C3×D9C2311yescharacteristic
40C2×C3×C4×C9C2312yes—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9C2314yes—
43C2×C3×Dic9C2312yes—
44C2×C3×Dic9C2311yescharacteristic
45C9×D12—24no—
46C9×D12—44no—
47C3×C4×D9—22no—
48#6194—44no—
49D3×Dic9—44no—
50#5719—91no—
51#5535—91no—
52C2×C3×D12C2×D312yes—
53C2×C3×D12C2×D314yes—
54C2×C3×C4×D3—31no—
55C2×#625—34no—
56C2×D3×Dic3—34no—
57C32×C42C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—32no—
59#3750—34no—
60C3×(C2×D3)⋊C4—32no—
61C3×Dic3⋊C4—32no—
62C3×C4×Dic3—31no—
63(C2×D3)⋊Dic3—34no—
64(C2×D3)⋊Dic3—34no—
65C2×D4×C9—32no—
66C2×D4×C9—34no—
67C2×#611—34no—
68C22×Dic9—34no—
69C2×C4×D9C2×D311yescharacteristic
70C42×C9—21no—
71C42×C9C2×D311yescharacteristic
72#3676—34no—
73#3676—34no—
74(C2×D9)⋊C4C2×D312yes—
75C4⋊Dic9—32no—
76C4⋊Dic9—62no—
77Dic9⋊C4—34no—
78Dic9⋊C4—64no—
79Dic9⋊C4C2×D312yes—
80C4×Dic9C2×D311yescharacteristic
81C22×C3×C9C2411yescharacteristic
82C2×D3×C9—24no—
83C2×D3×C9—24no—
84C2×D3×C9—24no—
85C2×C3×D9—21no—
86C2×C3×D9—22no—
87C3×C4×C9—22no—
88C3×C4×C9#4112yes—
89C3⋊Dic9—24no—
90C3×Dic9—21no—
91C3×Dic9—22no—
92C3×#127—31no—
93C3×#117—91no—
94#939—94no—
95#938—92no—
96#935—94no—
97D4×Dic3—92no—
98#930—92no—
99#894—94no—
100#884—94no—
101#874D911yescharacteristic, complemented
102C4×D12—92no—
103#872—31no—
104#870—91no—
105#870—91no—
106#870—181no—
107C22×C3×D3—31no—
108C22×C3×D3C22×D314yes—
109C2×C32×C4C22×D312yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—34no—
112C2×C3×Dic3—31no—
113C2×C3×Dic3—32no—
114C3×D12—24no—
115C3×D12—44no—
116C3×C4×D3—62no—
117#625—124no—
118D3×Dic3—124no—
119C23×C9—34no—
120C22×D9C22×D311yescharacteristic
121D4×C9—64no—
122D4×C9—124no—
123C2×C4×C9—21no—
124C2×C4×C9—22no—
125C2×C4×C9C22×D312yes—
126C2×C4×C9C22×D311yescharacteristic
127#611—124no—
128C2×Dic9—34no—
129C2×Dic9—64no—
130C2×Dic9—124no—
131C2×Dic9C22×D312yes—
132C2×Dic9C22×D311yescharacteristic
133C4×D9—22no—
134#540—271noSylow
135C2×C3×C9C2×#4111yescharacteristic
136C2×C3×C9H(5,𝔽2)12yes—
137D3×C9—44no—
138C3×D9—41no—
139C2×C3×D4—32no—
140C2×C3×D4—34no—
141C22×C3×C4—91no—
142C2×#67—94no—
143C2×#67—94no—
144C22×Dic3—94no—
145C2×D12C2×D912yes—
146C2×D12C2×D914yes—
147C2×C4×D3—31no—
148C2×C4×D3—94no—
149C3×C4⋊C4—92no—
150C3×C22⋊C4—92no—
151C3×C42—21no—
152C3×C42—91no—
153C3×C42C2×D911yescharacteristic
154C3×C42D3211yescharacteristic
155#217—94no—
156#217—94no—
157(C2×D3)⋊C4—32no—
158(C2×D3)⋊C4—94no—
159(C2×D3)⋊C4—94no—
160C4⋊Dic3—92no—
161C4⋊Dic3—92no—
162C4⋊Dic3—182no—
163Dic3⋊C4—32no—
164Dic3⋊C4—94no—
165Dic3⋊C4—94no—
166Dic3⋊C4—184no—
167C4×Dic3—31no—
168C22×C32C23×D311yescharacteristic
169C2×C3×D3—24no—
170C2×C3×D3—24no—
171C2×C3×D3—24no—
172C2×C3×D3—61no—
173C2×C3×D3—62no—
174C32×C4—22no—
175C32×C4#23712yes—
176C3⋊Dic3—64no—
177C3×Dic3—61no—
178C3×Dic3—62no—
179C22×C9—21no—
180C22×C9—64no—
181C22×C9—64no—
182C22×C9—64no—
183C22×C9C23×D311yescharacteristic
184C2×D9—21no—
185C2×D9—22no—
186C4×C9—22no—
187C4×C9—22no—
188C4×C9—42no—
189C4×C9#23512yes—
190Dic9—21no—
191Dic9—22no—
192Dic9—64no—
193Dic9—124no—
194#127—31no—
195#125—271no—
196#123—274no—
197#121—274no—
198#121—272no—
199C4×D4—272no—
200#117—91no—
201C3×C9#54011yescharacteristic, complemented, Sylow
202C23×C3—34no—
203C23×C3—91no—
204C22×D3—31no—
205C22×D3C22×D914yes—
206C3×D4—64no—
207C3×D4—124no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—91no—
211C2×C3×C4—92no—
212C2×C3×C4—182no—
213C2×C3×C4C22×D912yes—
214C2×C3×C4C22×D911yescharacteristic
215C2×C3×C4C2×D3212yes—
216C2×C3×C4C2×D3211yescharacteristic
217#67—364no—
218#67—364no—
219C2×Dic3—32no—
220C2×Dic3—31no—
221C2×Dic3—94no—
222C2×Dic3—94no—
223C2×Dic3—184no—
224C2×Dic3—364no—
225D12—24no—
226D12—44no—
227C4×D3—62no—
228C4×D3—364no—
229C2×C32C2×#23711yescharacteristic
230C2×C32#100412yes—
231C3×D3—44no—
232C3×D3—121no—
233C2×C9—21no—
234C2×C9—22no—
235C2×C9—124no—
236C2×C9C2×#23511yescharacteristic
237C2×C9#100912yes—
238D9—41no—
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×D911yescharacteristic
252C22×C3—21no—
253C22×C3—64no—
254C22×C3—64no—
255C22×C3—64no—
256C22×C3—181no—
257C22×C3—182no—
258C22×C3C23×D911yescharacteristic
259C22×C3C22×D3211yescharacteristic
260C2×D3—24no—
261C2×D3—24no—
262C2×D3—24no—
263C2×D3—61no—
264C2×D3—62no—
265C3×C4—22no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—42no—
269C3×C4—181no—
270C3×C4—182no—
271C3×C4#369912yes—
272C3×C4#379512yes—
273Dic3—61no—
274Dic3—62no—
275Dic3—184no—
276Dic3—184no—
277Dic3—364no—
278C32#571911yescharacteristic
279C9—21no—
280C9#553511yescharacteristic, complemented
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×D912yes—
292C2×C4C2×D3×D911yescharacteristic
293C2×C3—21no—
294C2×C3—22no—
295C2×C3—124no—
296C2×C3—361no—
297C2×C3C2×#369911yescharacteristic
298C2×C3#6371012yes—
299C2×C3C2×#379511yescharacteristic
300C2×C3#6430712yes—
301D3—44no—
302D3—121no—
303C22—64no—
304C22—64no—
305C22—64no—
306C22—181no—
307C22—182no—
308C22C22×D3×D911yescharacteristic
309C4—22no—
310C4—181no—
311C4—182no—
312C4—544no—
313C4#8863812yes—
314C3—21no—
315C3#1058982111yescharacteristic, complemented
316C3#1059394911yescharacteristic
317C2—124no—
318C2—361no—
319C2C2×#8863811yescharacteristic
320C2#1172247612yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#540
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721707C2semidirect
2. C9×#874C2semidirect
3. C3×#63434C2semidirect
4. #11721036C2semidirect
5. D12×Dic9C2semidirect
6. #11721016C2semidirect
7. #11721015C2semidirect
35. #63434D3semidirect
101. #874D9semidirect
201. C3×C9#540semidirect
280. C9#5535semidirect
315. C3#10589821semidirect

Maximal subgroups

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

subgroupindex
#117217072
C9×#8742
C3×#634342
#117210362
D12×Dic92
#117210162
#117210152
#105939493
#105898213