Finite Group Databasebeta

← #13144448

Subgroups — #13144448

Order 1728.

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

normal subgroups only

This group has 990 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#11722661C211yesmaximal, characteristic, complemented
2C3×C9×#120C211yesmaximal, characteristic, complemented
3#11721874C212yesmaximal, complemented
4#11720577C211yesmaximal, characteristic, complemented
5#11720523C212yesmaximal, complemented
6#10592731—31nomaximal
7#10588712—33nomaximal
8C24×C3×C9C2211yescharacteristic, complemented
9C2×C3×D4×C9—21no—
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×#6233—22no—
13C2×#6233C2212yes—
14#88742—22no—
15D4×C3⋊D9—22no—
16#88736—21no—
17C3×C9×C22⋊C4—21no—
18C3×C9×C22⋊C4C2211yescharacteristic, complemented
19#88551—22no—
20#88551C2212yescomplemented
21#64361—31no—
22C32×#120D311yescharacteristic
23#64162—32no—
24#63710—33no—
25#63661—31no—
26#63633—32no—
27C9×#120D313yescomplemented
28#63515—36no—
29#63393—33no—
30#63365—36no—
31C23×C3×C9—22no—
32C23×C3×C9—21no—
33C23×C3×C9—42no—
34C23×C3×C9D411yescharacteristic, complemented
35C23×C3×C9C2311yescharacteristic
36C22×C3⋊D9—21no—
37C22×C3⋊D9D412yescomplemented
38C3×D4×C9—22no—
39C3×D4×C9—42no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9D411yescharacteristic, complemented
42#6233—22no—
43#6233—22no—
44#6233—22no—
45#6233—44no—
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9—22no—
48C2×C3⋊Dic9D412yescomplemented
49C4⋊(C3⋊D9)—21no—
50C4×C3⋊D9—22no—
51#6226—21no—
52#5032—93no—
53#5032—91no—
54C24×C32C2×D311yescharacteristic
55C2×C32×D4—21no—
56C2×C32×D4C2×D311yescharacteristic
57C2×#638—32no—
58C2×#638—61no—
59C2×#638—62no—
60#3830—62no—
61D4×C3⋊D3—62no—
62#3828—61no—
63C24×C9C2×D313yescomplemented
64C32×C22⋊C4—21no—
65C32×C22⋊C4C2×D311yescharacteristic
66#3757—32no—
67#3757—62no—
68C2×D4×C9—23no—
69C2×D4×C9C2×D313yes—
70C2×#611—36no—
71C2×#611—63no—
72C2×#611—66no—
73#3699—66no—
74D4×D9—66no—
75#3697—63no—
76C9×C22⋊C4—23no—
77C9×C22⋊C4C2×D313yescomplemented
78#3676—36no—
79#3676—66no—
80C22×C3×C9—24no—
81C22×C3×C9—22no—
82C22×C3×C9—21no—
83C22×C3×C9—41no—
84C22×C3×C9—42no—
85C22×C3×C9—41no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9C2×D412yes—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3⋊D9—22no—
91C2×C3⋊D9—42no—
92C2×C3⋊D9—81no—
93C3×C4×C9—21no—
94C3×C4×C9—41no—
95C3⋊Dic9—22no—
96C3⋊Dic9—42no—
97#1004—93no—
98#1004—91no—
99C3×#120D913yescomplemented
100C3×#120C3⋊D311yescharacteristic
101#953—96no—
102#953—92no—
103#834—93no—
104#834—91no—
105#806—96no—
106#806—92no—
107C23×C32—22no—
108C23×C32—21no—
109C23×C32—42no—
110C23×C32#6711yescharacteristic
111C23×C32C22×D311yescharacteristic
112C22×C3⋊D3—32no—
113C22×C3⋊D3—61no—
114C32×D4—22no—
115C32×D4—42no—
116C2×C32×C4—21no—
117C2×C32×C4#6711yescharacteristic
118#638—62no—
119#638—62no—
120#638—62no—
121#638—124no—
122C2×C3⋊Dic3—32no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—62no—
125C4⋊(C3⋊D3)—61no—
126C4×C3⋊D3—62no—
127#634—61no—
128C23×C9—26no—
129C23×C9—23no—
130C23×C9—46no—
131C23×C9#6713yescomplemented
132C23×C9C22×D313yes—
133C22×D9—36no—
134C22×D9—63no—
135D4×C9—26no—
136D4×C9—46no—
137C2×C4×C9—23no—
138C2×C4×C9#6713yescomplemented
139#611—66no—
140#611—66no—
141#611—66no—
142#611—1212no—
143C2×Dic9—36no—
144C2×Dic9—63no—
145C2×Dic9—66no—
146D36—63no—
147C4×D9—66no—
148Dic18—63no—
149B(4,𝔽2)—271noSylow
150C2×C3×C9—22no—
151C2×C3×C9—21no—
152C2×C3×C9—41no—
153C2×C3×C9—42no—
154C2×C3×C9#12011yescharacteristic
155C3⋊D9—42no—
156C24×C3C2×D913yescomplemented
157C24×C3C2×C3⋊D311yescharacteristic
158C2×C3×D4—23no—
159C2×C3×D4—21no—
160C2×C3×D4C2×D913yes—
161C2×C3×D4C2×C3⋊D311yescharacteristic
162C2×#67—96no—
163C2×#67—92no—
164C2×#67—183no—
165C2×#67—181no—
166C2×#67—182no—
167C2×#67—186no—
168#237—182no—
169#237—186no—
170D3×D4—182no—
171D3×D4—186no—
172#235—183no—
173#235—181no—
174C3×C22⋊C4—23no—
175C3×C22⋊C4—21no—
176C3×C22⋊C4C2×D913yescomplemented
177C3×C22⋊C4C2×C3⋊D311yescharacteristic
178#217—96no—
179#217—92no—
180#217—182no—
181#217—186no—
182C22×C32—24no—
183C22×C32—22no—
184C22×C32—21no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32D3×D412yes—
191C22×C32C2×#6711yescharacteristic
192C2×C3⋊D3—62no—
193C2×C3⋊D3—122no—
194C2×C3⋊D3—241no—
195C32×C4—21no—
196C32×C4—41no—
197C3⋊Dic3—62no—
198C3⋊Dic3—122no—
199C22×C9—212no—
200C22×C9—26no—
201C22×C9—23no—
202C22×C9—43no—
203C22×C9—43no—
204C22×C9—46no—
205C22×C9—43no—
206C22×C9—43no—
207C22×C9D3×D416yes—
208C22×C9C2×#6713yes—
209C2×D9—66no—
210C2×D9—126no—
211C2×D9—243no—
212C4×C9—23no—
213C4×C9—43no—
214Dic9—66no—
215Dic9—126no—
216H(5,𝔽2)—271no—
217#120—272no—
218#120C3⋊D911yescharacteristic, complemented
219#99—271no—
220#99—272no—
221C3×C9B(4,𝔽2)11yescharacteristic, complemented, Sylow
222C23×C3—26no—
223C23×C3—23no—
224C23×C3—22no—
225C23×C3—21no—
226C23×C3—42no—
227C23×C3—46no—
228C23×C3#61113yescomplemented
229C23×C3C22×D913yes—
230C23×C3#63811yescharacteristic
231C23×C3C22×C3⋊D311yescharacteristic
232C22×D3—96no—
233C22×D3—92no—
234C22×D3—181no—
235C22×D3—183no—
236C3×D4—26no—
237C3×D4—22no—
238C3×D4—42no—
239C3×D4—46no—
240C2×C3×C4—23no—
241C2×C3×C4—21no—
242C2×C3×C4#61113yescomplemented
243C2×C3×C4#63811yescharacteristic
244#67—182no—
245#67—186no—
246#67—186no—
247#67—186no—
248#67—182no—
249#67—182no—
250#67—364no—
251#67—3612no—
252C2×Dic3—96no—
253C2×Dic3—92no—
254C2×Dic3—181no—
255C2×Dic3—183no—
256C2×Dic3—186no—
257C2×Dic3—182no—
258D12—181no—
259D12—183no—
260C4×D3—182no—
261C4×D3—186no—
262Dic6—181no—
263Dic6—183no—
264C2×C32—22no—
265C2×C32—21no—
266C2×C32—41no—
267C2×C32—42no—
268C2×C32#93711yescharacteristic
269C3⋊D3—122no—
270C2×C9—26no—
271C2×C9—23no—
272C2×C9—43no—
273C2×C9—46no—
274C2×C9#93713yes—
275D9—126no—
276C24C2×C3⋊D911yescharacteristic, complemented
277#41—542no—
278#41—541no—
279C2×D4—21no—
280C2×D4—272no—
281C2×D4—542no—
282C2×D4—541no—
283C2×D4—542no—
284C2×D4C2×C3⋊D911yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—272no—
287C22⋊C4—542no—
288C22⋊C4C2×C3⋊D911yescharacteristic, complemented
289C22×C3—24no—
290C22×C3—212no—
291C22×C3—22no—
292C22×C3—26no—
293C22×C3—23no—
294C22×C3—21no—
295C22×C3—43no—
296C22×C3—41no—
297C22×C3—43no—
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—41no—
301C22×C3—46no—
302C22×C3—43no—
303C22×C3—43no—
304C22×C3—41no—
305C22×C3D4×D916yes—
306C22×C3C2×#61113yes—
307C22×C3D4×C3⋊D312yes—
308C22×C3C2×#63811yescharacteristic
309C2×D3—186no—
310C2×D3—182no—
311C2×D3—362no—
312C2×D3—366no—
313C2×D3—721no—
314C2×D3—723no—
315C3×C4—23no—
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—43no—
319Dic3—186no—
320Dic3—182no—
321Dic3—362no—
322Dic3—366no—
323C32#503211yescharacteristic
324C9#503213yescomplemented
325C23—22no—
326C23—21no—
327C23—42no—
328C23—272no—
329C23—541no—
330C23#623311yescharacteristic, complemented
331C23C22×C3⋊D911yescharacteristic
332Q8—541no—
333D4—22no—
334D4—42no—
335D4—541no—
336D4—542no—
337D4—542no—
338D4—542no—
339D4—1084no—
340C2×C4—21no—
341C2×C4—272no—
342C2×C4—542no—
343C2×C4—541no—
344C2×C4—542no—
345C2×C4#623311yescharacteristic, complemented
346C2×C3—26no—
347C2×C3—23no—
348C2×C3—22no—
349C2×C3—21no—
350C2×C3—43no—
351C2×C3—46no—
352C2×C3—41no—
353C2×C3—42no—
354C2×C3#6349913yes—
355C2×C3#6414611yescharacteristic
356D3—366no—
357D3—362no—
358C22—22no—
359C22—24no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—42no—
365C22—41no—
366C22—542no—
367C22—1082no—
368C22—2161no—
369C22D4×C3⋊D912yes—
370C22C2×#623311yescharacteristic
371C4—21no—
372C4—41no—
373C4—542no—
374C4—1082no—
375C3#1058871213yescomplemented
376C3#1059273111yescharacteristic
377C2—21no—
378C2—22no—
379C2—42no—
380C2—41no—
381C2—1082no—
382C2#1172183711yescharacteristic
subgroupG/H
centerC2#11721837
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C22×C3⋊D3
FittingC3×C9×#120C2
socleC2×C32#937
hypercenter#120C3⋊D9
hypocenterC3×C9B(4,𝔽2)
LCSG ▷ C23×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #11722661C2semidirect
2. C3×C9×#120C2semidirect
3. #11721874C2semidirect
4. #11720577C2semidirect
5. #11720523C2semidirect
8. C24×C3×C9C22semidirect
18. C3×C9×C22⋊C4C22semidirect
20. #88551C22semidirect
27. C9×#120D3semidirect
34. C23×C3×C9D4semidirect
37. C22×C3⋊D9D4semidirect
41. C2×C3×C4×C9D4semidirect
48. C2×C3⋊Dic9D4semidirect
63. C24×C9C2×D3semidirect
77. C9×C22⋊C4C2×D3semidirect
99. C3×#120D9semidirect
131. C23×C9#67semidirect
138. C2×C4×C9#67semidirect
156. C24×C3C2×D9semidirect
176. C3×C22⋊C4C2×D9semidirect
218. #120C3⋊D9semidirect
221. C3×C9B(4,𝔽2)semidirect
228. C23×C3#611semidirect
242. C2×C3×C4#611semidirect
276. C24C2×C3⋊D9semidirect
288. C22⋊C4C2×C3⋊D9semidirect
324. C9#5032semidirect
330. C23#6233semidirect
345. C2×C4#6233semidirect
375. C3#10588712semidirect

Maximal subgroups

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

subgroupindex
#117226612
C3×C9×#1202
#117218742
#117205772
#117205232
#105927313
#105887123