Finite Group Databasebeta

← #13143960

Subgroups — #13143960

Order 1728.

600 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 137 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#11721854C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic
3C2×#88524C211yesmaximal, characteristic
4#11721749C211yesmaximal, characteristic
5C3×C9×#103C211yesmaximal, characteristic, complemented
6#11720537C211yesmaximal, characteristic, complemented
7#11720527C211yesmaximal, characteristic, complemented
8#10592517—31nomaximal
9#10588498—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9—21no—
13C2×#6226C2211yescharacteristic
14C2×C3×C8×C9C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16#88548—22no—
17#88536—21no—
18#88534—21no—
19#88534—21no—
20C4×C3⋊Dic9C2211yescharacteristic
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22#88524—22no—
23#64153—31no—
24C2×#3755—31no—
25C2×#3744—31no—
26#64102—31no—
27C32×#103D311yescharacteristic
28#63641—31no—
29#63637—31no—
30#63506—33no—
31C2×#3675—33no—
32C2×#3664—33no—
33#63455—33no—
34C9×#103D313yescomplemented
35Dic9⋊C8—33no—
36#63368—33no—
37C22×C3⋊D9—21no—
38C3×Q8×C9—21no—
39C3×Q8×C9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)D412yes—
48C4×C3⋊D9—42no—
49#6226—21no—
50#6226—21no—
51C3×C8×C9—21no—
52C3⋊(C9⋊C8)—21no—
53#4818—93no—
54#4818—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4×C3⋊D3—61no—
58C2×#634—31no—
59C2×C32×C8C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61#3755—62no—
62#3752—61no—
63#3750—61no—
64#3750—61no—
65C4×C3⋊Dic3—31no—
66C2×C3⋊(C3⋊C8)—31no—
67#3744—62no—
68C2×Q8×C9C2×D313yes—
69C2×D36—33no—
70C2×C4×D9—63no—
71C2×Dic18—33no—
72C2×C8×C9C2×D313yes—
73C9×C4⋊C4C2×D313yes—
74#3675—66no—
75(C2×D9)⋊C4—63no—
76Dic9⋊C4—63no—
77Dic9⋊C4—63no—
78C4×Dic9—33no—
79C2×C9⋊C8—33no—
80#3664—66no—
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—22no—
83C2×C3⋊D9—41no—
84C2×C3⋊D9—41no—
85C3×C4×C9—41no—
86C3×C4×C9—41no—
87C3×C4×C9C2×D411yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—21no—
90C3⋊Dic9—41no—
91C3⋊Dic9SD1612yescomplemented
92#944—93no—
93#944—91no—
94C2×#215—93no—
95C2×#215—91no—
96C2×#204—93no—
97C2×#204—91no—
98#893—93no—
99#893—91no—
100C3×#103D913yescomplemented
101C3×#103C3⋊D311yescharacteristic
102Dic3⋊C8—93no—
103Dic3⋊C8—91no—
104#810—93no—
105#810—91no—
106C22×C3⋊D3—61no—
107C32×Q8—21no—
108C32×Q8—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—61no—
115C4⋊(C3⋊D3)—32no—
116C4⋊(C3⋊D3)—61no—
117C4×C3⋊D3—122no—
118#634—61no—
119#634—61no—
120C22×D9—63no—
121C32×C8—21no—
122C3⋊(C3⋊C8)—61no—
123Q8×C9—23no—
124Q8×C9—23no—
125C2×C4×C9—23no—
126C2×C4×C9—23no—
127C2×C4×C9C22×D313yes—
128C2×Dic9—33no—
129C2×Dic9—33no—
130C2×Dic9—63no—
131D36—36no—
132D36—63no—
133C4×D9—126no—
134Dic18—63no—
135Dic18—63no—
136C8×C9—23no—
137C9⋊C8—63no—
138#461—271noSylow
139C2×C3×C9#12111yescharacteristic
140C2×C3×C9C2×SD1611yescharacteristic
141C2×C3×C9#13711yescharacteristic
142C3⋊D9—42no—
143C2×C3×Q8C2×D913yes—
144C2×C3×Q8C2×C3⋊D311yescharacteristic
145C2×D12—93no—
146C2×D12—91no—
147C2×C4×D3—181no—
148C2×C4×D3—183no—
149C2×Dic6—93no—
150C2×Dic6—91no—
151C2×C3×C8C2×D913yes—
152C2×C3×C8C2×C3⋊D311yescharacteristic
153C3×C4⋊C4C2×D913yes—
154C3×C4⋊C4C2×C3⋊D311yescharacteristic
155#215—182no—
156#215—186no—
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—183no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—183no—
162Dic3⋊C4—183no—
163C4×Dic3—93no—
164C4×Dic3—91no—
165C2×C3⋊C8—93no—
166C2×C3⋊C8—91no—
167#204—182no—
168#204—186no—
169C22×C32D3×D411yescharacteristic
170C2×C3⋊D3—62no—
171C2×C3⋊D3—121no—
172C2×C3⋊D3—121no—
173C32×C4—41no—
174C32×C4—41no—
175C32×C4#23511yescharacteristic
176C32×C4D3×D411yescharacteristic
177C3⋊Dic3—32no—
178C3⋊Dic3—61no—
179C3⋊Dic3—121no—
180C22×C9D3×D413yes—
181C2×D9—66no—
182C2×D9—123no—
183C2×D9—123no—
184C4×C9—43no—
185C4×C9—43no—
186C4×C9#23513yes—
187C4×C9D3×D413yes—
188Dic9—36no—
189Dic9—63no—
190Dic9—123no—
191C2×SD16—271no—
192C2×SD16—271no—
193C4⋊Q8—271no—
194C4×D4—271no—
195C4⋊C8—271no—
196#103—271no—
197#103C3⋊D911yescharacteristic, complemented
198C3×C9#46111yescharacteristic, complemented, Sylow
199C22×D3—183no—
200C22×D3—181no—
201C3×Q8—23no—
202C3×Q8—23no—
203C3×Q8—21no—
204C3×Q8—21no—
205C2×C3×C4—23no—
206C2×C3×C4—23no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4C22×D913yes—
210C2×C3×C4C22×C3⋊D311yescharacteristic
211C2×Dic3—93no—
212C2×Dic3—91no—
213C2×Dic3—93no—
214C2×Dic3—91no—
215C2×Dic3—183no—
216C2×Dic3—181no—
217D12—96no—
218D12—92no—
219D12—183no—
220D12—181no—
221C4×D3—362no—
222C4×D3—366no—
223Dic6—183no—
224Dic6—183no—
225Dic6—181no—
226Dic6—181no—
227C3×C8—23no—
228C3×C8—21no—
229C3⋊C8—183no—
230C3⋊C8—181no—
231C2×C32#88411yescharacteristic
232C2×C32D3×SD1611yescharacteristic
233C2×C32#91811yescharacteristic
234C3⋊D3—122no—
235C2×C9#88413yes—
236C2×C9D3×SD1613yes—
237C2×C9#91813yes—
238D9—126no—
239C2×Q8—271no—
240C2×Q8C2×C3⋊D911yescharacteristic
241C2×D4—271no—
242C22×C4—541no—
243SD16—542no—
244SD16—542no—
245C2×C8—271no—
246C2×C8C2×C3⋊D911yescharacteristic
247C4⋊C4—541no—
248C4⋊C4—541no—
249C4⋊C4C2×C3⋊D911yescharacteristic
250C22⋊C4—541no—
251C42—271no—
252C22×C3D4×D913yes—
253C22×C3D4×C3⋊D311yescharacteristic
254C2×D3—182no—
255C2×D3—186no—
256C2×D3—361no—
257C2×D3—361no—
258C2×D3—363no—
259C2×D3—363no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—43no—
263C3×C4—43no—
264C3×C4#369713yes—
265C3×C4D4×D913yes—
266C3×C4#382811yescharacteristic
267C3×C4D4×C3⋊D311yescharacteristic
268Dic3—92no—
269Dic3—96no—
270Dic3—181no—
271Dic3—183no—
272Dic3—361no—
273Dic3—363no—
274C32#481811yescharacteristic
275C9#481813yescomplemented
276C23—541no—
277Q8—21no—
278Q8—21no—
279Q8—541no—
280Q8—541no—
281D4—272no—
282D4—541no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—271no—
286C2×C4—271no—
287C2×C4—541no—
288C2×C4—1082no—
289C2×C4C22×C3⋊D911yescharacteristic
290C8—21no—
291C8—541no—
292C2×C3#6344613yes—
293C2×C3SD16×D913yes—
294C2×C3#6348013yes—
295C2×C3#6409311yescharacteristic
296C2×C3SD16×C3⋊D311yescharacteristic
297C2×C3#6412711yescharacteristic
298D3—362no—
299D3—366no—
300C22—542no—
301C22—1081no—
302C22—1081no—
303C22D4×C3⋊D911yescharacteristic
304C4—41no—
305C4—41no—
306C4—272no—
307C4—541no—
308C4—1081no—
309C4#8873611yescharacteristic
310C4D4×C3⋊D911yescharacteristic
311C3#1058849813yescomplemented
312C3#1059251711yescharacteristic
313C2—1082no—
314C2#1172173011yescharacteristic
315C2SD16×C3⋊D911yescharacteristic
316C2#1172180411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenter#103C3⋊D9
hypocenterC3×C9#461
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
1. #11721854C2semidirect
5. C3×C9×#103C2semidirect
6. #11720537C2semidirect
7. #11720527C2semidirect
34. C9×#103D3semidirect
91. C3⋊Dic9SD16semidirect
100. C3×#103D9semidirect
197. #103C3⋊D9semidirect
198. C3×C9#461semidirect
275. C9#4818semidirect
311. C3#10588498semidirect

Maximal subgroups

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

subgroupindex
#117218542
C2×#885482
C2×#885242
#117217492
C3×C9×#1032
#117205372
#117205272
#105925173
#105884983