Finite Group Databasebeta

← #13147747

Subgroups — #13147747

Order 1728.

816 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 149 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#11721713C211yesmaximal, characteristic, complemented
2C9×#876C211yesmaximal, characteristic, complemented
3C3×#63436C211yesmaximal, characteristic, complemented
4#11721049C213yesmaximal, complemented
5#11720983C213yesmaximal, complemented
6#11720982C213yesmaximal, complemented
7#11720981C213yesmaximal, complemented
8#10594022—31nomaximal
9#10589813—31nomaximal
10C22×D3×D9C2211yescharacteristic
11C2×C3×D36C2213yes—
12C2×C3×D36C2213yescomplemented
13C2×#6194C2213yes—
14C2×#6192C2213yes—
15C2×Dic3×D9C2213yescomplemented
16C3×C42×C9C2211yescharacteristic, complemented
17#88536C2213yes—
18#88534C2213yescomplemented
19C9×(C2×D3)⋊C4C2213yes—
20C9×Dic3⋊C4C2213yescomplemented
21(C2×D9)⋊Dic3C2213yes—
22(C2×D9)⋊Dic3C2213yescomplemented
23#64085—31no—
24C3×#876D311yescharacteristic
25C3×#874—31no—
26#63922—33no—
27#63886—33no—
28#63885—33no—
29#63884—33no—
30C9×#126—31no—
31#63457—33no—
32#63456—33no—
33#63446—33no—
34#63444—33no—
35#63438—31no—
36#63438—61no—
37#63436D311yescharacteristic, complemented
38C22×C3⋊D9C2311yescharacteristic
39C22×D3×C9C2311yescharacteristic
40C22×C3×D9C2311yescharacteristic
41C22×C3×D9C2313yes—
42C2×D3×D9—43no—
43C2×C3×C4×C9C2313yes—
44C2×C3⋊Dic9C2313yes—
45C2×C9×Dic3C2313yes—
46C3×D36—43no—
47C3×D36—43no—
48#6194—43no—
49#6192—43no—
50Dic3×D9—43no—
51#5720—91no—
52#5711—91no—
53C22×D32—31no—
54C2×C3×D12—33no—
55C2×C3×D12—33no—
56C2×#626—33no—
57C2×#625—33no—
58C2×D3×Dic3—33no—
59C23×D9—31no—
60C32×C42C2×D311yescharacteristic
61#3752—33no—
62#3750—33no—
63C3×(C2×D3)⋊C4C2×D313yes—
64C3×Dic3⋊C4C2×D313yes—
65(C2×D3)⋊Dic3—33no—
66(C2×D3)⋊Dic3—33no—
67C2×#611—33no—
68C2×D36—33no—
69C2×D36C2×D313yes—
70C2×D36C2×D313yescomplemented
71C2×C4×D9—33no—
72C9×C4⋊C4—33no—
73C9×C22⋊C4—33no—
74C42×C9—21no—
75C42×C9C2×D311yescharacteristic, complemented
76(C2×D9)⋊C4—33no—
77(C2×D9)⋊C4—33no—
78(C2×D9)⋊C4—33no—
79(C2×D9)⋊C4—63no—
80Dic9⋊C4—33no—
81Dic9⋊C4—63no—
82C22×C3×C9C2411yescharacteristic
83C2×C3⋊D9—23no—
84C2×D3×C9—23no—
85C2×C3×D9—23no—
86C2×C3×D9—23no—
87C2×C3×D9—23no—
88C2×C3×D9—23no—
89D3×D9—44no—
90C3×C4×C9—23no—
91C3⋊Dic9—23no—
92C9×Dic3—23no—
93C3×#127—91no—
94C3×#126—31no—
95#939—93no—
96#939—93no—
97#937—93no—
98#935—93no—
99#895—93no—
100#894—93no—
101#884—93no—
102#882—93no—
103#876—91no—
104#876—91no—
105#876—181no—
106#876D911yescharacteristic, complemented
107#874—31no—
108C22×C3⋊D3—31no—
109C22×C3×D3—31no—
110C22×C3×D3—33no—
111C22×C3×D3C22×D311yescharacteristic
112C2×D32—123no—
113C2×C32×C4C22×D313yes—
114C2×C3⋊Dic3—33no—
115C2×C3×Dic3C22×D313yes—
116C3×D12—123no—
117C3×D12—123no—
118#626—123no—
119#625—123no—
120D3×Dic3—123no—
121C23×C9—31no—
122C22×D9—31no—
123C22×D9—61no—
124C22×D9—123no—
125C22×D9C22×D311yescharacteristic
126C22×D9C22×D313yes—
127C2×C4×C9—23no—
128C2×C4×C9—33no—
129C2×C4×C9C22×D313yes—
130#611—123no—
131C2×Dic9—33no—
132C2×Dic9—63no—
133D36—43no—
134D36—43no—
135D36—123no—
136C4×D9—123no—
137#560—271noSylow
138C2×C3×C9H(5,𝔽2)13yes—
139C3⋊D9—41no—
140D3×C9—41no—
141C3×D9—41no—
142C3×D9—43no—
143C23×D3—91no—
144C23×D3—91no—
145C2×C3×D4—93no—
146C2×C3×D4—93no—
147C2×#67—93no—
148C2×#67—93no—
149C2×#67—93no—
150C22×Dic3—93no—
151C2×D12—33no—
152C2×D12—33no—
153C2×D12—93no—
154C2×C4×D3—93no—
155C3×C4⋊C4—33no—
156C3×C22⋊C4—33no—
157C3×C42—21no—
158C3×C42C2×D911yescharacteristic, complemented
159C3×C42D3211yescharacteristic
160#217—93no—
161#217—93no—
162(C2×D3)⋊C4—93no—
163(C2×D3)⋊C4—93no—
164(C2×D3)⋊C4—93no—
165(C2×D3)⋊C4—93no—
166(C2×D3)⋊C4—183no—
167(C2×D3)⋊C4C2×D913yes—
168Dic3⋊C4—93no—
169Dic3⋊C4—93no—
170Dic3⋊C4—183no—
171Dic3⋊C4C2×D913yescomplemented
172C22×C32C23×D311yescharacteristic
173C2×C3⋊D3—63no—
174C2×C3×D3—23no—
175C2×C3×D3—63no—
176C2×C3×D3—63no—
177C2×C3×D3—63no—
178C2×C3×D3—63no—
179D32—124no—
180C32×C4—23no—
181C3⋊Dic3—63no—
182C3×Dic3—23no—
183C22×C9—21no—
184C22×C9—63no—
185C22×C9C23×D311yescharacteristic
186C2×D9—23no—
187C2×D9—23no—
188C2×D9—23no—
189C2×D9—23no—
190C2×D9—63no—
191C2×D9—123no—
192C2×D9—124no—
193C4×C9—23no—
194C4×C9—43no—
195C4×C9—63no—
196Dic9—63no—
197Dic9—123no—
198#127—91no—
199#126—31no—
200#126—271no—
201#123—273no—
202#121—273no—
203#121—273no—
204#120—273no—
205C3×C9#56011yescharacteristic, complemented, Sylow
206C23×C3—31no—
207C23×C3—91no—
208C23×C3—93no—
209C22×D3—31no—
210C22×D3—33no—
211C22×D3—91no—
212C22×D3—91no—
213C22×D3—181no—
214C22×D3—363no—
215C22×D3—363no—
216C22×D3C22×D911yescharacteristic
217C3×D4—363no—
218C3×D4—363no—
219C2×C3×C4—23no—
220C2×C3×C4—33no—
221C2×C3×C4C22×D913yes—
222C2×C3×C4C2×D3213yes—
223#67—363no—
224#67—363no—
225#67—363no—
226C2×Dic3—93no—
227C2×Dic3—93no—
228C2×Dic3—183no—
229C2×Dic3—363no—
230C2×Dic3C22×D913yes—
231D12—123no—
232D12—123no—
233D12—363no—
234C4×D3—363no—
235C2×C32#100913yes—
236C3⋊D3—121no—
237C3×D3—41no—
238C3×D3—121no—
239C3×D3—123no—
240C2×C9—23no—
241C2×C9—121no—
242C2×C9#100413yes—
243D9—43no—
244D9—41no—
245D9—121no—
246D9—241no—
247C24—271no—
248C2×D4—93no—
249C2×D4—93no—
250C2×D4—273no—
251C2×D4—273no—
252C22×C4—273no—
253C4⋊C4—33no—
254C4⋊C4—273no—
255C22⋊C4—33no—
256C22⋊C4—273no—
257C22⋊C4—273no—
258C22⋊C4—273no—
259C42D3×D911yescharacteristic, complemented
260C22×C3—21no—
261C22×C3—63no—
262C22×C3—183no—
263C22×C3—183no—
264C22×C3—183no—
265C22×C3—183no—
266C22×C3C23×D911yescharacteristic
267C22×C3C22×D3211yescharacteristic
268C2×D3—23no—
269C2×D3—63no—
270C2×D3—63no—
271C2×D3—63no—
272C2×D3—63no—
273C2×D3—183no—
274C2×D3—183no—
275C2×D3—363no—
276C2×D3—364no—
277C2×D3—364no—
278C3×C4—23no—
279C3×C4—23no—
280C3×C4—43no—
281C3×C4—63no—
282Dic3—23no—
283Dic3—183no—
284Dic3—183no—
285Dic3—363no—
286C32#571111yescharacteristic
287C9—21no—
288C9#572011yescharacteristic, complemented
289C23—31no—
290C23—91no—
291C23—93no—
292C23—271no—
293C23—1083no—
294D4—363no—
295D4—363no—
296D4—1083no—
297D4—1083no—
298C2×C4—33no—
299C2×C4—273no—
300C2×C4—1083no—
301C2×C4C2×D3×D913yes—
302C2×C3—23no—
303C2×C3—121no—
304C2×C3—361no—
305C2×C3—363no—
306C2×C3#6371513yes—
307C2×C3#6430713yes—
308D3—41no—
309D3—121no—
310D3—123no—
311D3—361no—
312D3—361no—
313D3—721no—
314C22—63no—
315C22—183no—
316C22—183no—
317C22—183no—
318C22—183no—
319C22—543no—
320C22—1084no—
321C22C22×D3×D911yescharacteristic
322C4—23no—
323C4—63no—
324C4—543no—
325C3—21no—
326C3#1058981311yescharacteristic, complemented
327C3#1059402211yescharacteristic
328C2—121no—
329C2—361no—
330C2—363no—
331C2—1081no—
332C2#1172247713yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#560
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721713C2semidirect
2. C9×#876C2semidirect
3. C3×#63436C2semidirect
4. #11721049C2semidirect
5. #11720983C2semidirect
6. #11720982C2semidirect
7. #11720981C2semidirect
12. C2×C3×D36C22semidirect
15. C2×Dic3×D9C22semidirect
16. C3×C42×C9C22semidirect
18. #88534C22semidirect
20. C9×Dic3⋊C4C22semidirect
22. (C2×D9)⋊Dic3C22semidirect
37. #63436D3semidirect
70. C2×D36C2×D3semidirect
75. C42×C9C2×D3semidirect
106. #876D9semidirect
158. C3×C42C2×D9semidirect
171. Dic3⋊C4C2×D9semidirect
205. C3×C9#560semidirect
259. C42D3×D9semidirect
288. C9#5720semidirect
326. C3#10589813semidirect

Maximal subgroups

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

subgroupindex
#117217132
C9×#8762
C3×#634362
#117210492
#117209832
#117209822
#117209812
#105940223
#105898133