Finite Group Databasebeta

← #13143915

Subgroups — #13143915

Order 1728.

570 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 135 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#11721838C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×C9×#102C211yesmaximal, characteristic, complemented
4#11720576C211yesmaximal, characteristic, complemented
5#11720542C211yesmaximal, characteristic, complemented
6#11720541C211yesmaximal, characteristic, complemented
7#11720525C211yesmaximal, characteristic, complemented
8#10592473—31nomaximal
9#10588454—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4×C3⋊D9—21no—
13C2×C4×C3⋊D9C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16#88551—21no—
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19#88534—21no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64147—31no—
22C4⋊C4×C3⋊D3—31no—
23C32×#102D311yescharacteristic
24#63659—31no—
25#63647—31no—
26#63645—31no—
27#63634—31no—
28#63500—33no—
29C4⋊C4×D9—33no—
30C9×#102D313yescomplemented
31#63392—33no—
32(C2×D9)⋊C8—33no—
33#63378—33no—
34#63366—33no—
35C23×C3×C9—21no—
36C22×C3⋊D9D411yescharacteristic
37C3×D4×C9—21no—
38C3×D4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41#6233—42no—
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C4×C3⋊D9—21no—
46C4×C3⋊D9—21no—
47C4×C3⋊D9—42no—
48C3×C8×C9—21no—
49C3⋊(C9⋊C8)—21no—
50#4774—93no—
51#4774—91no—
52C2×C32×D4C2×D311yescharacteristic
53C2×#638—61no—
54C2×C4×C3⋊D3—31no—
55C2×C4×C3⋊D3—61no—
56C2×C32×C8C2×D311yescharacteristic
57C32×C4⋊C4C2×D311yescharacteristic
58#3757—61no—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—31no—
61#3750—61no—
62C2×C3⋊(C3⋊C8)—31no—
63C2×D4×C9C2×D313yes—
64C2×#611—63no—
65C2×C4×D9—33no—
66C2×C4×D9—63no—
67C2×C8×C9C2×D313yes—
68C9×C4⋊C4C2×D313yes—
69#3676—63no—
70C4⋊Dic9—33no—
71C4⋊Dic9—33no—
72Dic9⋊C4—63no—
73C2×C9⋊C8—33no—
74C22×C3×C9—41no—
75C22×C3×C9—41no—
76C22×C3×C9—41no—
77C22×C3×C9C2×D411yescharacteristic
78C2×C3⋊D9—21no—
79C2×C3⋊D9—21no—
80C2×C3⋊D9D812yes—
81C3×C4×C9—41no—
82C3×C4×C9#4111yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—21no—
85C3⋊Dic9—41no—
86C3⋊Dic9—41no—
87#938—93no—
88#938—91no—
89D3×C4⋊C4—93no—
90D3×C4⋊C4—91no—
91C3×#102D913yescomplemented
92C3×#102C3⋊D311yescharacteristic
93#832—93no—
94#832—91no—
95(C2×D3)⋊C8—93no—
96(C2×D3)⋊C8—91no—
97#818—93no—
98#818—91no—
99#807—93no—
100#807—91no—
101C23×C32—21no—
102C22×C3⋊D3—31no—
103C32×D4—21no—
104C32×D4—21no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107#638—122no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—61no—
111C4×C3⋊D3—61no—
112C4×C3⋊D3—61no—
113C4×C3⋊D3—122no—
114C23×C9—23no—
115C22×D9—33no—
116C32×C8—21no—
117C3⋊(C3⋊C8)—61no—
118D4×C9—23no—
119D4×C9—23no—
120C2×C4×C9—23no—
121C2×C4×C9C22×D313yes—
122#611—126no—
123C2×Dic9—33no—
124C2×Dic9—63no—
125C2×Dic9—63no—
126C4×D9—63no—
127C4×D9—63no—
128C4×D9—126no—
129C8×C9—23no—
130C9⋊C8—63no—
131#480—271noSylow
132C2×C3×C9—81no—
133C2×C3×C9#12311yescharacteristic
134C2×C3×C9C2×D811yescharacteristic
135C2×C3×C9#13711yescharacteristic
136C3⋊D9—22no—
137C2×C3×D4C2×D913yes—
138C2×C3×D4C2×C3⋊D311yescharacteristic
139C2×#67—181no—
140C2×#67—183no—
141C2×C4×D3—93no—
142C2×C4×D3—91no—
143C2×C4×D3—181no—
144C2×C4×D3—183no—
145C2×C3×C8C2×D913yes—
146C2×C3×C8C2×C3⋊D311yescharacteristic
147C3×C4⋊C4C2×D913yes—
148C3×C4⋊C4C2×C3⋊D311yescharacteristic
149#217—181no—
150#217—183no—
151C4⋊Dic3—93no—
152C4⋊Dic3—93no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155Dic3⋊C4—181no—
156Dic3⋊C4—183no—
157C2×C3⋊C8—93no—
158C2×C3⋊C8—91no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32D3×D411yescharacteristic
163C2×C3⋊D3—32no—
164C2×C3⋊D3—61no—
165C2×C3⋊D3—61no—
166C32×C4—41no—
167C32×C4#23511yescharacteristic
168C32×C4#23711yescharacteristic
169C3⋊Dic3—61no—
170C3⋊Dic3—121no—
171C3⋊Dic3—121no—
172C22×C9—43no—
173C22×C9—43no—
174C22×C9—43no—
175C22×C9D3×D413yes—
176C2×D9—36no—
177C2×D9—63no—
178C2×D9—63no—
179C4×C9—43no—
180C4×C9#23513yes—
181C4×C9#23713yes—
182Dic9—63no—
183Dic9—123no—
184Dic9—123no—
185#121—271no—
186C2×C4⋊C4—271no—
187#107—271no—
188#107—271no—
189#102—271no—
190#102C3⋊D911yescharacteristic, complemented
191C22⋊C8—271no—
192C3×C9#48011yescharacteristic, complemented, Sylow
193C23×C3—23no—
194C23×C3—21no—
195C22×D3—93no—
196C22×D3—91no—
197C3×D4—23no—
198C3×D4—23no—
199C3×D4—21no—
200C3×D4—21no—
201C2×C3×C4—23no—
202C2×C3×C4—21no—
203C2×C3×C4C22×D913yes—
204C2×C3×C4C22×C3⋊D311yescharacteristic
205#67—362no—
206#67—366no—
207C2×Dic3—93no—
208C2×Dic3—91no—
209C2×Dic3—183no—
210C2×Dic3—183no—
211C2×Dic3—181no—
212C2×Dic3—181no—
213C4×D3—183no—
214C4×D3—183no—
215C4×D3—181no—
216C4×D3—181no—
217C4×D3—362no—
218C4×D3—366no—
219C3×C8—23no—
220C3×C8—21no—
221C3⋊C8—183no—
222C3⋊C8—181no—
223C2×C32—81no—
224C2×C32#88311yescharacteristic
225C2×C32D3×D811yescharacteristic
226C2×C32#91511yescharacteristic
227C3⋊D3—62no—
228C2×C9—83no—
229C2×C9#88313yes—
230C2×C9D3×D813yes—
231C2×C9#91513yes—
232D9—66no—
233C2×D4—541no—
234C2×D4C2×C3⋊D911yescharacteristic
235C22×C4—271no—
236C22×C4—541no—
237C2×C8—271no—
238C2×C8C2×C3⋊D911yescharacteristic
239C4⋊C4—271no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C4⋊C4C2×C3⋊D911yescharacteristic
243C22⋊C4—541no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—43no—
248C22×C3—43no—
249C22×C3—43no—
250C22×C3D4×D913yes—
251C22×C3D4×C3⋊D311yescharacteristic
252C2×D3—92no—
253C2×D3—96no—
254C2×D3—181no—
255C2×D3—181no—
256C2×D3—183no—
257C2×D3—183no—
258C3×C4—41no—
259C3×C4—43no—
260C3×C4#369713yes—
261C3×C4#369913yes—
262C3×C4#382811yescharacteristic
263C3×C4#383011yescharacteristic
264Dic3—181no—
265Dic3—183no—
266Dic3—361no—
267Dic3—361no—
268Dic3—363no—
269Dic3—363no—
270C32#477411yescharacteristic
271C9#477413yescomplemented
272C23—21no—
273C23—271no—
274D4—21no—
275D4—21no—
276D4—1082no—
277C2×C4—21no—
278C2×C4—271no—
279C2×C4—541no—
280C2×C4—541no—
281C2×C4—541no—
282C2×C4—541no—
283C2×C4—1082no—
284C2×C4C22×C3⋊D911yescharacteristic
285C8—21no—
286C8—541no—
287C2×C3—83no—
288C2×C3—81no—
289C2×C3#6344513yes—
290C2×C3D8×D913yes—
291C2×C3#6347713yes—
292C2×C3#6409211yescharacteristic
293C2×C3D8×C3⋊D311yescharacteristic
294C2×C3#6412411yescharacteristic
295D3—186no—
296D3—182no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—272no—
301C22—541no—
302C22—541no—
303C22D4×C3⋊D911yescharacteristic
304C4—41no—
305C4—541no—
306C4—1081no—
307C4—1081no—
308C4#8873611yescharacteristic
309C4#8874211yescharacteristic
310C3#1058845413yescomplemented
311C3#1059247311yescharacteristic
312C2—81no—
313C2—542no—
314C2#1172172911yescharacteristic
315C2D8×C3⋊D911yescharacteristic
316C2#1172179711yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
prime radical#11721838C2
hypercenter#102C3⋊D9
hypocenterC3×C9#480
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C3⋊D9C2semidirect
3. C3×C9×#102C2semidirect
4. #11720576C2semidirect
5. #11720542C2semidirect
6. #11720541C2semidirect
7. #11720525C2semidirect
30. C9×#102D3semidirect
91. C3×#102D9semidirect
190. #102C3⋊D9semidirect
192. C3×C9#480semidirect
271. C9#4774semidirect
310. C3#10588454semidirect

Maximal subgroups

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

subgroupindex
#117218382
C4⋊C4×C3⋊D92
C3×C9×#1022
#117205762
#117205422
#117205412
#117205252
#105924733
#105884543