Finite Group Databasebeta

← #13143864

Subgroups — #13143864

Order 1728.

790 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#88742C211yesmaximal, characteristic
2#11721723C212yesmaximal, complemented
3C3×C9×#99C211yesmaximal, characteristic, complemented
4#11720577C211yesmaximal, characteristic, complemented
5#11720523C212yesmaximal, complemented
6#10592439—31nomaximal
7#10588420—33nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×#6233C411yescharacteristic, complemented
10C2×#6233C2211yescharacteristic
11C22×C3⋊Dic9C411yescharacteristic, complemented
12C22×C3⋊Dic9C2211yescharacteristic
13#88742—22no—
14#88742—22no—
15C2×C4×C3⋊D9C411yescharacteristic, complemented
16C2×#6226C411yescharacteristic, complemented
17C3×C9×C22⋊C4C2212yes—
18#88551C2212yes—
19#88534—22no—
20C4×C3⋊Dic9—22no—
21C2×#3830—31no—
22#64086—32no—
23C2×#3699—33no—
24C32×#99D311yescharacteristic
25#63661—31no—
26#63633—32no—
27#63439—36no—
28C9×#99D313yescomplemented
29#63393—33no—
30#63365—36no—
31C23×C3×C9C2×C411yescharacteristic
32C23×C3×C9C2311yescharacteristic
33C22×C3⋊D9C2×C411yescharacteristic
34C3×D4×C9—22no—
35C2×C3×C4×C9—22no—
36C2×C3×C4×C9C2×C411yescharacteristic
37#6233—22no—
38#6233—22no—
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9—22no—
42C2×C3⋊Dic9C2×C411yescharacteristic
43C2×C3⋊Dic9C2×C411yescharacteristic
44C2×C3⋊Dic9C2×C411yescharacteristic
45C2×C3⋊Dic9D412yes—
46C2×C3⋊Dic9D412yes—
47C4×C3⋊D9—21no—
48C4×C3⋊D9—21no—
49#6226—22no—
50#4740—93no—
51#4740—91no—
52C2×C32×D4C2×D311yescharacteristic
53C2×#638—31no—
54C2×#638—31no—
55C22×C3⋊Dic3—31no—
56C22×C3⋊Dic3—31no—
57#3830—62no—
58#3830—62no—
59C2×C4×C3⋊D3—31no—
60C2×#634—31no—
61C32×C22⋊C4C2×D312yes—
62#3757—32no—
63#3750—62no—
64C4×C3⋊Dic3—62no—
65C2×D4×C9C2×D313yes—
66C2×#611—33no—
67C2×#611—33no—
68C22×Dic9—33no—
69C22×Dic9—33no—
70#3699—66no—
71#3699—66no—
72C2×C4×D9—33no—
73C2×Dic18—33no—
74C9×C22⋊C4C2×D316yes—
75#3676—36no—
76Dic9⋊C4—66no—
77C4×Dic9—66no—
78C22×C3×C9—21no—
79C22×C3×C9—41no—
80C22×C3×C9—41no—
81C22×C3×C9C22×C411yescharacteristic
82C22×C3×C9C2×D412yes—
83C2×C3⋊D9—21no—
84C2×C3⋊D9—41no—
85C3×C4×C9—21no—
86C3×C4×C9—42no—
87C3⋊Dic9—21no—
88C3⋊Dic9—42no—
89C3⋊Dic9C22⋊C412yes—
90C3⋊Dic9C22⋊C411yescharacteristic
91C3⋊Dic9C22⋊C411yescharacteristic
92C2×#237—93no—
93C2×#237—91no—
94#877—96no—
95#877—92no—
96C3×#99D913yescomplemented
97C3×#99C3⋊D311yescharacteristic
98#834—93no—
99#834—91no—
100#806—96no—
101#806—92no—
102C23×C32C4×D311yescharacteristic
103C23×C32C22×D311yescharacteristic
104C22×C3⋊D3—31no—
105C32×D4—22no—
106C2×C32×C4—22no—
107C2×C32×C4C4×D311yescharacteristic
108#638—62no—
109#638—62no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—31no—
112C2×C3⋊Dic3—32no—
113C2×C3⋊Dic3—32no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—62no—
118C4×C3⋊D3—61no—
119C4×C3⋊D3—61no—
120#634—62no—
121C23×C9C4×D313yes—
122C23×C9C22×D313yes—
123C22×D9—33no—
124D4×C9—26no—
125C2×C4×C9—26no—
126C2×C4×C9C4×D313yes—
127#611—66no—
128#611—66no—
129C2×Dic9—33no—
130C2×Dic9—33no—
131C2×Dic9—36no—
132C2×Dic9—36no—
133C2×Dic9—33no—
134C2×Dic9—63no—
135C2×Dic9—63no—
136C2×Dic9—66no—
137C4×D9—63no—
138C4×D9—63no—
139Dic18—66no—
140#410—271noSylow
141C2×C3×C9—22no—
142C2×C3×C9—21no—
143C2×C3×C9—41no—
144C2×C3×C9C2×C22⋊C411yescharacteristic
145C3⋊D9—41no—
146C2×C3×D4C2×D913yes—
147C2×C3×D4C2×C3⋊D311yescharacteristic
148C2×#67—93no—
149C2×#67—93no—
150C2×#67—91no—
151C2×#67—91no—
152C22×Dic3—93no—
153C22×Dic3—91no—
154C22×Dic3—93no—
155C22×Dic3—91no—
156#237—182no—
157#237—182no—
158#237—186no—
159#237—186no—
160C2×C4×D3—93no—
161C2×C4×D3—91no—
162C2×Dic6—93no—
163C2×Dic6—91no—
164C3×C22⋊C4C2×D916yes—
165C3×C22⋊C4C2×C3⋊D312yes—
166#217—96no—
167#217—92no—
168Dic3⋊C4—182no—
169Dic3⋊C4—186no—
170C4×Dic3—182no—
171C4×Dic3—186no—
172C22×C32—21no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32C2×C4×D311yescharacteristic
176C22×C32D3×D412yes—
177C2×C3⋊D3—61no—
178C2×C3⋊D3—121no—
179C32×C4—21no—
180C32×C4—42no—
181C3⋊Dic3—32no—
182C3⋊Dic3—31no—
183C3⋊Dic3—31no—
184C3⋊Dic3—61no—
185C3⋊Dic3—122no—
186C22×C9—23no—
187C22×C9—43no—
188C22×C9—43no—
189C22×C9C2×C4×D313yes—
190C22×C9D3×D416yes—
191C2×D9—63no—
192C2×D9—123no—
193C4×C9—23no—
194C4×C9—46no—
195Dic9—36no—
196Dic9—33no—
197Dic9—33no—
198Dic9—63no—
199Dic9—126no—
200C2×#41—271no—
201#117—272no—
202#99—271no—
203#99—272no—
204#99C3⋊D911yescharacteristic, complemented
205C3×C9#41011yescharacteristic, complemented, Sylow
206C23×C3C4×D913yes—
207C23×C3C22×D913yes—
208C23×C3C4×C3⋊D311yescharacteristic
209C23×C3C22×C3⋊D311yescharacteristic
210C22×D3—93no—
211C22×D3—91no—
212C3×D4—26no—
213C3×D4—22no—
214C2×C3×C4—26no—
215C2×C3×C4—22no—
216C2×C3×C4C4×D913yes—
217C2×C3×C4C4×C3⋊D311yescharacteristic
218#67—182no—
219#67—186no—
220#67—186no—
221#67—182no—
222C2×Dic3—93no—
223C2×Dic3—93no—
224C2×Dic3—96no—
225C2×Dic3—96no—
226C2×Dic3—91no—
227C2×Dic3—91no—
228C2×Dic3—92no—
229C2×Dic3—92no—
230C2×Dic3—93no—
231C2×Dic3—91no—
232C2×Dic3—181no—
233C2×Dic3—181no—
234C2×Dic3—183no—
235C2×Dic3—183no—
236C2×Dic3—186no—
237C2×Dic3—182no—
238C4×D3—181no—
239C4×D3—181no—
240C4×D3—183no—
241C4×D3—183no—
242Dic6—182no—
243Dic6—186no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—41no—
247C2×C32D3×C22⋊C411yescharacteristic
248C3⋊D3—121no—
249C2×C9—26no—
250C2×C9—23no—
251C2×C9—43no—
252C2×C9D3×C22⋊C413yes—
253D9—123no—
254#41—542no—
255#41—542no—
256C2×Q8—271no—
257C2×D4—271no—
258C2×D4—271no—
259C2×D4C2×C3⋊D911yescharacteristic
260C22×C4—271no—
261C22×C4—271no—
262C22×C4—271no—
263C4⋊C4—542no—
264C22⋊C4—272no—
265C22⋊C4C2×C3⋊D912yes—
266C42—542no—
267C22×C3—21no—
268C22×C3—23no—
269C22×C3—43no—
270C22×C3—41no—
271C22×C3—43no—
272C22×C3—41no—
273C22×C3C2×C4×D913yes—
274C22×C3D4×D916yes—
275C22×C3C2×C4×C3⋊D311yescharacteristic
276C22×C3D4×C3⋊D312yes—
277C2×D3—181no—
278C2×D3—183no—
279C2×D3—361no—
280C2×D3—363no—
281C3×C4—21no—
282C3×C4—23no—
283C3×C4—42no—
284C3×C4—46no—
285Dic3—92no—
286Dic3—91no—
287Dic3—91no—
288Dic3—96no—
289Dic3—93no—
290Dic3—93no—
291Dic3—181no—
292Dic3—183no—
293Dic3—362no—
294Dic3—366no—
295C32#474011yescharacteristic
296C9#474013yescomplemented
297C23—271no—
298C23C4×C3⋊D911yescharacteristic
299C23C22×C3⋊D911yescharacteristic
300Q8—542no—
301D4—22no—
302D4—542no—
303D4—542no—
304C2×C4—22no—
305C2×C4—271no—
306C2×C4—271no—
307C2×C4—272no—
308C2×C4—272no—
309C2×C4—271no—
310C2×C4—541no—
311C2×C4—541no—
312C2×C4—541no—
313C2×C4—541no—
314C2×C4—542no—
315C2×C4C4×C3⋊D911yescharacteristic
316C2×C3—26no—
317C2×C3—23no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—43no—
321C2×C3—41no—
322C2×C3C22⋊C4×D913yes—
323C2×C3C22⋊C4×C3⋊D311yescharacteristic
324D3—363no—
325D3—361no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—541no—
330C22—1081no—
331C22C2×C4×C3⋊D911yescharacteristic
332C22D4×C3⋊D912yes—
333C4—21no—
334C4—42no—
335C4—271no—
336C4—271no—
337C4—272no—
338C4—541no—
339C4—1082no—
340C3#1058842013yescomplemented
341C3#1059243911yescharacteristic
342C2—22no—
343C2—21no—
344C2—41no—
345C2—1081no—
346C2C22⋊C4×C3⋊D911yescharacteristic
subgroupG/H
centerC2C22⋊C4×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C22×C3⋊D3
FittingC3×C9×#99C2
socleC2×C32D3×C22⋊C4
prime radicalC2×#88742C2
cosocleC23×C3×C9C23
hypercenter#99C3⋊D9
hypocenterC3×C9#410
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #11721723C2semidirect
3. C3×C9×#99C2semidirect
4. #11720577C2semidirect
5. #11720523C2semidirect
9. C2×#6233C4semidirect
11. C22×C3⋊Dic9C4semidirect
15. C2×C4×C3⋊D9C4semidirect
16. C2×#6226C4semidirect
28. C9×#99D3semidirect
96. C3×#99D9semidirect
204. #99C3⋊D9semidirect
205. C3×C9#410semidirect
296. C9#4740semidirect
340. C3#10588420semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
#117217232
C3×C9×#992
#117205772
#117205232
#105924393
#105884203