Finite Group Databasebeta

← #13143978

Subgroups — #13143978

Order 1728.

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

normal subgroups only

This group has 520 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#11721856C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic
3C3×C9×#103C211yesmaximal, characteristic, complemented
4#11720582C211yesmaximal, characteristic, complemented
5#11720542C211yesmaximal, characteristic
6#11720540C211yesmaximal, characteristic, complemented
7#11720526C211yesmaximal, characteristic, complemented
8#10592504—31nomaximal
9#10588485—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C3×C8×C9C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88536—21no—
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18#88534—21no—
19#88534—21no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64155—31no—
22C4⋊C4×C3⋊D3—31no—
23C32×#103D311yescharacteristic
24#63662—31no—
25#63647—31no—
26#63644—31no—
27#63635—31no—
28#63508—33no—
29C4⋊C4×D9—33no—
30C9×#103D313yescomplemented
31#63395—33no—
32(C2×D9)⋊C8—33no—
33#63377—33no—
34#63367—33no—
35C22×C3⋊D9D411yescharacteristic
36C3×Q8×C9—21no—
37C3×Q8×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C4×C3⋊D9—21no—
45C4×C3⋊D9—21no—
46C4×C3⋊D9—42no—
47C3×C8×C9—21no—
48C3⋊(C9⋊C8)—21no—
49#4805—93no—
50#4805—91no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—61no—
54C2×C32×C8C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56#3752—61no—
57C4⋊(C3⋊Dic3)—31no—
58C4⋊(C3⋊Dic3)—31no—
59#3750—61no—
60#3750—61no—
61C2×C3⋊(C3⋊C8)—31no—
62C2×Q8×C9C2×D313yes—
63C2×C4×D9—33no—
64C2×C4×D9—63no—
65C2×C8×C9C2×D313yes—
66C9×C4⋊C4C2×D313yes—
67(C2×D9)⋊C4—63no—
68C4⋊Dic9—33no—
69C4⋊Dic9—33no—
70Dic9⋊C4—63no—
71Dic9⋊C4—63no—
72C2×C9⋊C8—33no—
73C22×C3×C9C2×D411yescharacteristic
74C2×C3⋊D9—21no—
75C2×C3⋊D9—21no—
76C2×C3⋊D9SD1612yes—
77C3×C4×C9—41no—
78C3×C4×C9—41no—
79C3×C4×C9#4111yescharacteristic
80C3×C4×C9#4111yescharacteristic
81C3⋊Dic9—21no—
82C3⋊Dic9—41no—
83C3⋊Dic9—41no—
84#946—93no—
85#946—91no—
86D3×C4⋊C4—93no—
87D3×C4⋊C4—91no—
88C3×#103D913yescomplemented
89C3×#103C3⋊D311yescharacteristic
90#835—93no—
91#835—91no—
92(C2×D3)⋊C8—93no—
93(C2×D3)⋊C8—91no—
94#817—93no—
95#817—91no—
96#808—93no—
97#808—91no—
98C22×C3⋊D3—31no—
99C32×Q8—21no—
100C32×Q8—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—61no—
106C2×C3⋊Dic3—61no—
107C4×C3⋊D3—61no—
108C4×C3⋊D3—61no—
109C4×C3⋊D3—122no—
110C22×D9—33no—
111C32×C8—21no—
112C3⋊(C3⋊C8)—61no—
113Q8×C9—23no—
114Q8×C9—23no—
115C2×C4×C9—23no—
116C2×C4×C9—23no—
117C2×C4×C9C22×D313yes—
118C2×Dic9—33no—
119C2×Dic9—63no—
120C2×Dic9—63no—
121C4×D9—63no—
122C4×D9—63no—
123C4×D9—126no—
124C8×C9—23no—
125C9⋊C8—63no—
126#483—271noSylow
127C2×C3×C9#12311yescharacteristic
128C2×C3×C9C2×SD1611yescharacteristic
129C2×C3×C9#13711yescharacteristic
130C3⋊D9—22no—
131C2×C3×Q8C2×D913yes—
132C2×C3×Q8C2×C3⋊D311yescharacteristic
133C2×C4×D3—93no—
134C2×C4×D3—91no—
135C2×C4×D3—181no—
136C2×C4×D3—183no—
137C2×C3×C8C2×D913yes—
138C2×C3×C8C2×C3⋊D311yescharacteristic
139C3×C4⋊C4C2×D913yes—
140C3×C4⋊C4C2×C3⋊D311yescharacteristic
141(C2×D3)⋊C4—181no—
142(C2×D3)⋊C4—183no—
143C4⋊Dic3—93no—
144C4⋊Dic3—93no—
145C4⋊Dic3—91no—
146C4⋊Dic3—91no—
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149Dic3⋊C4—183no—
150Dic3⋊C4—183no—
151C2×C3⋊C8—93no—
152C2×C3⋊C8—91no—
153C22×C32D3×D411yescharacteristic
154C2×C3⋊D3—32no—
155C2×C3⋊D3—61no—
156C2×C3⋊D3—61no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4#23511yescharacteristic
160C32×C4#23711yescharacteristic
161C3⋊Dic3—61no—
162C3⋊Dic3—121no—
163C3⋊Dic3—121no—
164C22×C9D3×D413yes—
165C2×D9—36no—
166C2×D9—63no—
167C2×D9—63no—
168C4×C9—43no—
169C4×C9—43no—
170C4×C9#23513yes—
171C4×C9#23713yes—
172Dic9—63no—
173Dic9—123no—
174Dic9—123no—
175C22⋊Q8—271no—
176C2×C4⋊C4—271no—
177#106—271no—
178#106—271no—
179#103—271no—
180#103C3⋊D911yescharacteristic, complemented
181C22⋊C8—271no—
182C3×C9#48311yescharacteristic, complemented, Sylow
183C22×D3—93no—
184C22×D3—91no—
185C3×Q8—23no—
186C3×Q8—23no—
187C3×Q8—21no—
188C3×Q8—21no—
189C2×C3×C4—23no—
190C2×C3×C4—23no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4C22×D913yes—
194C2×C3×C4C22×C3⋊D311yescharacteristic
195C2×Dic3—93no—
196C2×Dic3—91no—
197C2×Dic3—183no—
198C2×Dic3—183no—
199C2×Dic3—181no—
200C2×Dic3—181no—
201C4×D3—183no—
202C4×D3—183no—
203C4×D3—181no—
204C4×D3—181no—
205C4×D3—362no—
206C4×D3—366no—
207C3×C8—23no—
208C3×C8—21no—
209C3⋊C8—183no—
210C3⋊C8—181no—
211C2×C32#88311yescharacteristic
212C2×C32D3×SD1611yescharacteristic
213C2×C32#91811yescharacteristic
214C3⋊D3—62no—
215C2×C9#88313yes—
216C2×C9D3×SD1613yes—
217C2×C9#91813yes—
218D9—66no—
219C2×Q8C2×C3⋊D911yescharacteristic
220C22×C4—271no—
221C22×C4—541no—
222C2×C8—271no—
223C2×C8C2×C3⋊D911yescharacteristic
224C4⋊C4—271no—
225C4⋊C4—271no—
226C4⋊C4—541no—
227C4⋊C4—541no—
228C4⋊C4C2×C3⋊D911yescharacteristic
229C22⋊C4—541no—
230C22×C3D4×D913yes—
231C22×C3D4×C3⋊D311yescharacteristic
232C2×D3—92no—
233C2×D3—96no—
234C2×D3—181no—
235C2×D3—181no—
236C2×D3—183no—
237C2×D3—183no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—43no—
241C3×C4—43no—
242C3×C4#369713yes—
243C3×C4#369913yes—
244C3×C4#382811yescharacteristic
245C3×C4#383011yescharacteristic
246Dic3—181no—
247Dic3—183no—
248Dic3—361no—
249Dic3—361no—
250Dic3—363no—
251Dic3—363no—
252C32#480511yescharacteristic
253C9#480513yescomplemented
254C23—271no—
255Q8—21no—
256Q8—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—271no—
260C2×C4—541no—
261C2×C4—541no—
262C2×C4—541no—
263C2×C4—541no—
264C2×C4—1082no—
265C2×C4C22×C3⋊D911yescharacteristic
266C8—21no—
267C8—541no—
268C2×C3#6344513yes—
269C2×C3SD16×D913yes—
270C2×C3#6348013yes—
271C2×C3#6409211yescharacteristic
272C2×C3SD16×C3⋊D311yescharacteristic
273C2×C3#6412711yescharacteristic
274D3—186no—
275D3—182no—
276C22—272no—
277C22—541no—
278C22—541no—
279C22D4×C3⋊D911yescharacteristic
280C4—41no—
281C4—41no—
282C4—541no—
283C4—1081no—
284C4—1081no—
285C4#8873611yescharacteristic
286C4#8874211yescharacteristic
287C3#1058848513yescomplemented
288C3#1059250411yescharacteristic
289C2—542no—
290C2#1172172911yescharacteristic
291C2SD16×C3⋊D911yescharacteristic
292C2#1172180411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenter#103C3⋊D9
hypocenterC3×C9#483
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
3. C3×C9×#103C2semidirect
4. #11720582C2semidirect
6. #11720540C2semidirect
7. #11720526C2semidirect
30. C9×#103D3semidirect
88. C3×#103D9semidirect
180. #103C3⋊D9semidirect
182. C3×C9#483semidirect
253. C9#4805semidirect
287. C3#10588485semidirect

Maximal subgroups

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

subgroupindex
#117218562
C4⋊C4×C3⋊D92
C3×C9×#1032
#117205822
#117205422
#117205402
#117205262
#105925043
#105884853