Finite Group Databasebeta

← #13154014

Subgroups — #13154014

Order 1728.

1260 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#120C211yesmaximal, characteristic, complemented
2#11721874C212yesmaximal, complemented
3#11721840C211yesmaximal, characteristic, complemented
4#11721839C212yesmaximal, complemented
5#11721838C212yesmaximal, complemented
6#11721835C211yesmaximal, characteristic, complemented
7#11721730C212yesmaximal, complemented
8#11721729C212yesmaximal, complemented
9#11721725C212yesmaximal, complemented
10#10596066—31nomaximal
11#10589697—33nomaximal
12C24×C3×C9C2211yescharacteristic
13C2×C3×D4×C9C2211yescharacteristic
14C2×C3×D4×C9C2212yes—
15C2×#6233C2212yes—
16C2×#6233C2212yes—
17C2×#6233C2212yes—
18C2×#6233C2212yescomplemented
19C22×C3⋊Dic9C2211yescharacteristic, complemented
20C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
21C2×C4×C3⋊D9C2212yescomplemented
22C3×C9×C22⋊C4C2211yescharacteristic
23C3×C9×C22⋊C4C2212yes—
24#88551C2212yes—
25#88551C2212yes—
26#88551C2212yes—
27#88551C2211yescharacteristic, complemented
28#88536C2212yescomplemented
29C4⋊(C3⋊Dic9)C2212yescomplemented
30#88534C2212yescomplemented
31#88534C2212yescomplemented
32C4×C3⋊Dic9C2211yescharacteristic, complemented
33C32×#120D311yescharacteristic
34#64162—32no—
35#64149—31no—
36#64148—32no—
37#64147—32no—
38#64144—31no—
39#64093—32no—
40#64092—32no—
41#64088—32no—
42C9×#120D313yescomplemented
43#63515—36no—
44#63502—33no—
45#63501—36no—
46#63500—36no—
47#63497—33no—
48#63446—36no—
49#63445—36no—
50#63441—36no—
51C23×C3×C9—42no—
52C23×C3×C9—41no—
53C23×C3×C9C2311yescharacteristic
54C23×C3×C9C2312yes—
55C23×C3×C9C2311yescharacteristic
56C22×C3⋊D9C2312yes—
57C3×D4×C9—42no—
58C3×D4×C9—41no—
59C2×C3×C4×C9C2312yes—
60C2×C3×C4×C9C2311yescharacteristic
61#6233—42no—
62#6233—42no—
63#6233—42no—
64#6233—42no—
65C2×C3⋊Dic9—41no—
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2312yes—
69C4⋊(C3⋊D9)—41no—
70C4×C3⋊D9—42no—
71#5595—93no—
72#5595—91no—
73C24×C32C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C2×D312yes—
76C2×#638—32no—
77C2×#638—32no—
78C2×#638—32no—
79C2×#638—32no—
80C22×C3⋊Dic3—31no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4×C3⋊D3—32no—
83C24×C9C2×D313yes—
84C32×C22⋊C4C2×D312yes—
85C32×C22⋊C4C2×D311yescharacteristic
86#3757—32no—
87#3757—32no—
88#3757—32no—
89#3757—31no—
90#3752—32no—
91C4⋊(C3⋊Dic3)—32no—
92#3750—32no—
93#3750—32no—
94C4×C3⋊Dic3—31no—
95C2×D4×C9C2×D313yes—
96C2×D4×C9C2×D316yes—
97C2×#611—36no—
98C2×#611—36no—
99C2×#611—36no—
100C2×#611—36no—
101C22×Dic9—33no—
102C2×D36—33no—
103C2×C4×D9—36no—
104C9×C22⋊C4C2×D316yes—
105C9×C22⋊C4C2×D313yes—
106#3676—33no—
107#3676—36no—
108#3676—36no—
109#3676—36no—
110(C2×D9)⋊C4—36no—
111C4⋊Dic9—36no—
112Dic9⋊C4—36no—
113Dic9⋊C4—36no—
114C4×Dic9—33no—
115C22×C3×C9—21no—
116C22×C3×C9—22no—
117C22×C3×C9—22no—
118C22×C3×C9—22no—
119C22×C3×C9—22no—
120C22×C3×C9—22no—
121C22×C3×C9—21no—
122C22×C3×C9—44no—
123C22×C3×C9C2411yescharacteristic
124C2×C3⋊D9—22no—
125C2×C3⋊D9—22no—
126C2×C3⋊D9—22no—
127C3×C4×C9—22no—
128C3×C4×C9—21no—
129C3⋊Dic9—22no—
130C3⋊Dic9—22no—
131C3⋊Dic9—22no—
132C3×#120D913yescomplemented
133C3×#120C3⋊D311yescharacteristic
134#953—96no—
135#953—92no—
136#940—93no—
137#940—91no—
138#939—96no—
139#939—92no—
140#938—96no—
141#938—92no—
142#935—93no—
143#935—91no—
144#884—96no—
145#884—92no—
146#883—96no—
147#883—92no—
148#879—96no—
149#879—92no—
150C23×C32—42no—
151C23×C32—41no—
152C23×C32C22×D312yes—
153C23×C32C22×D311yescharacteristic
154C23×C32C22×D311yescharacteristic
155C22×C3⋊D3—32no—
156C32×D4—42no—
157C32×D4—41no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D312yes—
160#638—122no—
161#638—122no—
162#638—122no—
163#638—122no—
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—121no—
168C4⋊(C3⋊D3)—121no—
169C4×C3⋊D3—122no—
170C23×C9—46no—
171C23×C9—43no—
172C23×C9C22×D313yes—
173C23×C9C22×D316yes—
174C23×C9C22×D313yes—
175C22×D9—36no—
176D4×C9—46no—
177D4×C9—43no—
178C2×C4×C9C22×D313yes—
179C2×C4×C9C22×D316yes—
180#611—126no—
181#611—126no—
182#611—126no—
183#611—126no—
184C2×Dic9—36no—
185C2×Dic9—36no—
186C2×Dic9—36no—
187C2×Dic9—123no—
188D36—123no—
189C4×D9—126no—
190#560—271noSylow
191C2×C3×C9—41no—
192C2×C3×C9—42no—
193C2×C3×C9—41no—
194C2×C3×C9H(5,𝔽2)12yes—
195C2×C3×C9H(5,𝔽2)11yescharacteristic
196C3⋊D9—42no—
197C24×C3C2×D913yes—
198C24×C3C2×C3⋊D311yescharacteristic
199C2×C3×D4C2×D913yes—
200C2×C3×D4C2×D916yes—
201C2×C3×D4C2×C3⋊D311yescharacteristic
202C2×C3×D4C2×C3⋊D312yes—
203C2×#67—96no—
204C2×#67—96no—
205C2×#67—96no—
206C2×#67—96no—
207C2×#67—92no—
208C2×#67—92no—
209C2×#67—92no—
210C2×#67—92no—
211C22×Dic3—93no—
212C22×Dic3—91no—
213C2×D12—93no—
214C2×D12—91no—
215C2×C4×D3—96no—
216C2×C4×D3—92no—
217C3×C22⋊C4C2×D916yes—
218C3×C22⋊C4C2×D913yes—
219C3×C22⋊C4C2×C3⋊D312yes—
220C3×C22⋊C4C2×C3⋊D311yescharacteristic
221#217—96no—
222#217—96no—
223#217—96no—
224#217—93no—
225#217—92no—
226#217—92no—
227#217—92no—
228#217—91no—
229(C2×D3)⋊C4—96no—
230(C2×D3)⋊C4—92no—
231C4⋊Dic3—96no—
232C4⋊Dic3—92no—
233Dic3⋊C4—96no—
234Dic3⋊C4—96no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237C4×Dic3—93no—
238C4×Dic3—91no—
239C22×C32—21no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—21no—
246C22×C32—44no—
247C22×C32C23×D311yescharacteristic
248C2×C3⋊D3—62no—
249C2×C3⋊D3—62no—
250C2×C3⋊D3—62no—
251C32×C4—22no—
252C32×C4—21no—
253C3⋊Dic3—62no—
254C3⋊Dic3—62no—
255C3⋊Dic3—62no—
256C22×C9—23no—
257C22×C9—26no—
258C22×C9—26no—
259C22×C9—26no—
260C22×C9—26no—
261C22×C9—26no—
262C22×C9—23no—
263C22×C9—412no—
264C22×C9C23×D313yes—
265C2×D9—66no—
266C2×D9—66no—
267C2×D9—66no—
268C4×C9—26no—
269C4×C9—23no—
270Dic9—66no—
271Dic9—66no—
272Dic9—66no—
273#127—271no—
274#126—272no—
275#123—271no—
276#123—272no—
277#121—272no—
278#121—272no—
279#121—272no—
280#120—272no—
281#120C3⋊D911yescharacteristic, complemented
282C3×C9#56011yescharacteristic, complemented, Sylow
283C23×C3—42no—
284C23×C3—41no—
285C23×C3—46no—
286C23×C3—43no—
287C23×C3C22×D916yes—
288C23×C3C22×D913yes—
289C23×C3C22×D913yes—
290C23×C3C22×C3⋊D312yes—
291C23×C3C22×C3⋊D311yescharacteristic
292C23×C3C22×C3⋊D311yescharacteristic
293C22×D3—92no—
294C22×D3—96no—
295C3×D4—42no—
296C3×D4—41no—
297C3×D4—46no—
298C3×D4—43no—
299C2×C3×C4C22×D913yes—
300C2×C3×C4C22×D916yes—
301C2×C3×C4C22×C3⋊D311yescharacteristic
302C2×C3×C4C22×C3⋊D312yes—
303#67—362no—
304#67—362no—
305#67—362no—
306#67—362no—
307#67—366no—
308#67—366no—
309#67—366no—
310#67—366no—
311C2×Dic3—92no—
312C2×Dic3—92no—
313C2×Dic3—92no—
314C2×Dic3—96no—
315C2×Dic3—96no—
316C2×Dic3—96no—
317C2×Dic3—361no—
318C2×Dic3—363no—
319D12—361no—
320D12—363no—
321C4×D3—362no—
322C4×D3—366no—
323C2×C32—41no—
324C2×C32—42no—
325C2×C32—41no—
326C2×C32#100412yes—
327C2×C32#100411yescharacteristic
328C3⋊D3—122no—
329C2×C9—43no—
330C2×C9—46no—
331C2×C9—43no—
332C2×C9#100416yes—
333C2×C9#100413yes—
334D9—126no—
335C24C2×C3⋊D911yescharacteristic
336C2×D4—272no—
337C2×D4—272no—
338C2×D4—272no—
339C2×D4—272no—
340C2×D4—271no—
341C2×D4C2×C3⋊D912yes—
342C2×D4C2×C3⋊D911yescharacteristic
343C22×C4—271no—
344C22×C4—272no—
345C4⋊C4—272no—
346C4⋊C4—272no—
347C4⋊C4—272no—
348C22⋊C4—272no—
349C22⋊C4—272no—
350C22⋊C4—272no—
351C22⋊C4—271no—
352C22⋊C4—272no—
353C22⋊C4C2×C3⋊D912yes—
354C22⋊C4C2×C3⋊D911yescharacteristic
355C42—271no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—21no—
363C22×C3—23no—
364C22×C3—26no—
365C22×C3—26no—
366C22×C3—26no—
367C22×C3—26no—
368C22×C3—26no—
369C22×C3—23no—
370C22×C3—44no—
371C22×C3—412no—
372C22×C3C23×D913yes—
373C22×C3C23×C3⋊D311yescharacteristic
374C2×D3—182no—
375C2×D3—182no—
376C2×D3—182no—
377C2×D3—186no—
378C2×D3—186no—
379C2×D3—186no—
380C3×C4—22no—
381C3×C4—21no—
382C3×C4—26no—
383C3×C4—23no—
384Dic3—182no—
385Dic3—182no—
386Dic3—182no—
387Dic3—186no—
388Dic3—186no—
389Dic3—186no—
390C32#559511yescharacteristic
391C9#559513yescomplemented
392C23—42no—
393C23—41no—
394C23—272no—
395C23C22×C3⋊D912yes—
396C23C22×C3⋊D911yescharacteristic
397C23C22×C3⋊D911yescharacteristic
398D4—42no—
399D4—41no—
400D4—1081no—
401D4—1082no—
402D4—1082no—
403D4—1082no—
404D4—1082no—
405C2×C4—272no—
406C2×C4—272no—
407C2×C4—272no—
408C2×C4—1082no—
409C2×C4—1081no—
410C2×C4C22×C3⋊D911yescharacteristic
411C2×C4C22×C3⋊D912yes—
412C2×C3—46no—
413C2×C3—43no—
414C2×C3—43no—
415C2×C3—42no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3#6371016yes—
419C2×C3#6371013yes—
420C2×C3#6436112yes—
421C2×C3#6436111yescharacteristic
422D3—366no—
423D3—362no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—21no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—44no—
432C22—542no—
433C22—542no—
434C22—542no—
435C22C23×C3⋊D911yescharacteristic
436C4—22no—
437C4—21no—
438C4—542no—
439C4—542no—
440C4—542no—
441C3#1058969713yescomplemented
442C3#1059606611yescharacteristic
443C2—42no—
444C2—41no—
445C2—41no—
446C2—1082no—
447C2#1172266112yes—
448C2#1172266111yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#120C2
socleC22×C32C23×D3
hypercenter#120C3⋊D9
hypocenterC3×C9#560
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C3×C9×#120C2semidirect
2. #11721874C2semidirect
3. #11721840C2semidirect
4. #11721839C2semidirect
5. #11721838C2semidirect
6. #11721835C2semidirect
7. #11721730C2semidirect
8. #11721729C2semidirect
9. #11721725C2semidirect
18. C2×#6233C22semidirect
19. C22×C3⋊Dic9C22semidirect
20. C2×C4⋊(C3⋊D9)C22semidirect
21. C2×C4×C3⋊D9C22semidirect
27. #88551C22semidirect
28. #88536C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. #88534C22semidirect
31. #88534C22semidirect
32. C4×C3⋊Dic9C22semidirect
42. C9×#120D3semidirect
132. C3×#120D9semidirect
281. #120C3⋊D9semidirect
282. C3×C9#560semidirect
391. C9#5595semidirect
441. C3#10589697semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1202
#117218742
#117218402
#117218392
#117218382
#117218352
#117217302
#117217292
#117217252
#105960663
#105896973