Finite Group Databasebeta

← #13153869

Subgroups — #13153869

Order 1728.

1060 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#117C211yesmaximal, characteristic, complemented
2C4×#6233C211yesmaximal, characteristic
3#11721813C211yesmaximal, characteristic, complemented
4#11721753C211yesmaximal, characteristic, complemented
5#11721748C211yesmaximal, characteristic, complemented
6#11721744C211yesmaximal, characteristic, complemented
7#11721743C211yesmaximal, characteristic, complemented
8#11721731C212yesmaximal
9#11721725C212yesmaximal, complemented
10#11721713C212yesmaximal, complemented
11C4×#6226C212yesmaximal, complemented
12#10596004—31nomaximal
13#10589635—33nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×#6226C2212yes—
18C2×#6226C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C22⋊C4C2212yes—
22C3×C42×C9C2212yes—
23#88551C2212yes—
24#88551C2211yescharacteristic
25#88536C2212yes—
26#88536C2212yes—
27#88536C2211yescharacteristic
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C4⋊(C3⋊Dic9)C2212yes—
30#88534C2212yes—
31#88534C2212yes—
32#88534C2212yes—
33#88534C2211yescharacteristic
34C4×C3⋊Dic9C2212yes—
35C4×C3⋊Dic9C2211yescharacteristic
36C4×C3⋊Dic9C2212yes—
37C32×#117D311yescharacteristic
38C4×#638—31no—
39#64133—31no—
40#64106—31no—
41#64101—31no—
42#64097—31no—
43#64096—31no—
44#64094—32no—
45#64088—32no—
46#64085—32no—
47C4×#634—32no—
48C9×#117D313yescomplemented
49C4×#611—33no—
50#63486—33no—
51#63459—33no—
52#63454—33no—
53#63450—33no—
54#63449—33no—
55#63447—36no—
56#63441—36no—
57#63438—36no—
58C4×Dic18—36no—
59C23×C3×C9C2311yescharacteristic
60C22×C3⋊D9C2311yescharacteristic
61C2×C3×C4×C9—22no—
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66#6233—22no—
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C4×C3⋊D9—22no—
73#6226—22no—
74#6226—24no—
75#5533—93no—
76#5533—91no—
77C22×C32×C4C2×D311yescharacteristic
78C2×#638—31no—
79C2×C4×C3⋊D3—31no—
80C2×#634—32no—
81C2×#634—31no—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C22⋊C4C2×D312yes—
85C32×C42C2×D312yes—
86#3757—31no—
87#3757—32no—
88#3752—31no—
89#3752—32no—
90#3752—32no—
91C4⋊(C3⋊Dic3)—32no—
92C4⋊(C3⋊Dic3)—31no—
93#3750—31no—
94#3750—32no—
95#3750—32no—
96#3750—32no—
97C4×C3⋊Dic3—32no—
98C4×C3⋊Dic3—32no—
99C4×C3⋊Dic3—31no—
100C22×C4×C9C2×D313yes—
101C2×#611—33no—
102C2×C4×D9—33no—
103C2×Dic18—36no—
104C2×Dic18—33no—
105C9×C4⋊C4C2×D313yes—
106C9×C4⋊C4C2×D313yes—
107C9×C22⋊C4C2×D316yes—
108C42×C9C2×D316yes—
109#3676—33no—
110#3676—36no—
111(C2×D9)⋊C4—33no—
112(C2×D9)⋊C4—36no—
113(C2×D9)⋊C4—36no—
114C4⋊Dic9—36no—
115C4⋊Dic9—33no—
116Dic9⋊C4—33no—
117Dic9⋊C4—36no—
118Dic9⋊C4—36no—
119Dic9⋊C4—36no—
120C4×Dic9—33no—
121C4×Dic9—36no—
122C4×Dic9—36no—
123C22×C3×C9—21no—
124C22×C3×C9—21no—
125C22×C3×C9—21no—
126C22×C3×C9C2411yescharacteristic
127C2×C3⋊D9—21no—
128C2×C3⋊D9—21no—
129C2×C3⋊D9—21no—
130C3×C4×C9—22no—
131C3×C4×C9—21no—
132C3×C4×C9—22no—
133C3×C4×C9#4112yes—
134C3⋊Dic9—22no—
135C3⋊Dic9—22no—
136C3⋊Dic9—21no—
137C3⋊Dic9—21no—
138C3⋊Dic9#4112yes—
139C3×#117D913yescomplemented
140C3×#117C3⋊D311yescharacteristic
141C4×#67—93no—
142C4×#67—91no—
143#924—93no—
144#924—91no—
145#897—93no—
146#897—91no—
147#892—93no—
148#892—91no—
149#888—93no—
150#888—91no—
151#887—93no—
152#887—91no—
153#885—96no—
154#885—92no—
155#879—96no—
156#879—92no—
157#876—96no—
158#876—92no—
159C4×Dic6—96no—
160C4×Dic6—92no—
161C23×C32C22×D311yescharacteristic
162C22×C3⋊D3—31no—
163C2×C32×C4—22no—
164C2×C32×C4C22×D312yes—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D311yescharacteristic
168#638—62no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—32no—
174C4×C3⋊D3—62no—
175#634—64no—
176#634—62no—
177C23×C9C22×D313yes—
178C22×D9—33no—
179C2×C4×C9—26no—
180C2×C4×C9C22×D316yes—
181C2×C4×C9C22×D313yes—
182C2×C4×C9C22×D316yes—
183C2×C4×C9C22×D313yes—
184#611—66no—
185C2×Dic9—33no—
186C2×Dic9—33no—
187C2×Dic9—33no—
188C2×Dic9—36no—
189C2×Dic9—36no—
190C4×D9—66no—
191Dic18—612no—
192Dic18—66no—
193#556—271noSylow
194C2×C3×C9—41no—
195C2×C3×C9C2×#4111yescharacteristic
196C2×C3×C9C2×#4111yescharacteristic
197C2×C3×C92−1+411yescharacteristic
198C3⋊D9—41no—
199C22×C3×C4C2×D913yes—
200C22×C3×C4C2×C3⋊D311yescharacteristic
201C2×#67—93no—
202C2×#67—91no—
203C2×C4×D3—93no—
204C2×C4×D3—91no—
205C2×Dic6—96no—
206C2×Dic6—93no—
207C2×Dic6—92no—
208C2×Dic6—91no—
209C3×C4⋊C4C2×D913yes—
210C3×C4⋊C4C2×D913yes—
211C3×C4⋊C4C2×C3⋊D311yescharacteristic
212C3×C4⋊C4C2×C3⋊D311yescharacteristic
213C3×C22⋊C4C2×D916yes—
214C3×C22⋊C4C2×C3⋊D312yes—
215C3×C42C2×D916yes—
216C3×C42C2×C3⋊D312yes—
217#217—93no—
218#217—96no—
219#217—91no—
220#217—92no—
221(C2×D3)⋊C4—93no—
222(C2×D3)⋊C4—96no—
223(C2×D3)⋊C4—96no—
224(C2×D3)⋊C4—91no—
225(C2×D3)⋊C4—92no—
226(C2×D3)⋊C4—92no—
227C4⋊Dic3—96no—
228C4⋊Dic3—93no—
229C4⋊Dic3—92no—
230C4⋊Dic3—91no—
231Dic3⋊C4—93no—
232Dic3⋊C4—96no—
233Dic3⋊C4—96no—
234Dic3⋊C4—96no—
235Dic3⋊C4—91no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239C4×Dic3—96no—
240C4×Dic3—96no—
241C4×Dic3—93no—
242C4×Dic3—92no—
243C4×Dic3—92no—
244C4×Dic3—91no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32C23×D311yescharacteristic
249C2×C3⋊D3—61no—
250C2×C3⋊D3—61no—
251C2×C3⋊D3—61no—
252C32×C4—21no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4#23512yes—
256C3⋊Dic3—32no—
257C3⋊Dic3—62no—
258C3⋊Dic3—61no—
259C3⋊Dic3—62no—
260C3⋊Dic3—61no—
261C22×C9—23no—
262C22×C9—23no—
263C22×C9—23no—
264C22×C9C23×D313yes—
265C2×D9—63no—
266C2×D9—63no—
267C2×D9—63no—
268C4×C9—23no—
269C4×C9—26no—
270C4×C9—26no—
271C4×C9#23516yes—
272Dic9—36no—
273Dic9—66no—
274Dic9—63no—
275Dic9—66no—
276Dic9—63no—
277C4⋊Q8—271no—
278#126—272no—
279#126—272no—
280#124—272no—
281C22⋊Q8—271no—
282C22⋊Q8—271no—
283C4×Q8—271no—
284C4×Q8—272no—
285C4×D4—271no—
286#117—271no—
287#117C3⋊D911yescharacteristic, complemented
288C3×C9#55611yescharacteristic, complemented, Sylow
289C23×C3C22×D913yes—
290C23×C3C22×C3⋊D311yescharacteristic
291C22×D3—91no—
292C22×D3—93no—
293C2×C3×C4—26no—
294C2×C3×C4—22no—
295C2×C3×C4C22×D916yes—
296C2×C3×C4C22×D913yes—
297C2×C3×C4C22×D916yes—
298C2×C3×C4C22×D913yes—
299C2×C3×C4C22×C3⋊D312yes—
300C2×C3×C4C22×C3⋊D311yescharacteristic
301C2×C3×C4C22×C3⋊D312yes—
302C2×C3×C4C22×C3⋊D311yescharacteristic
303#67—182no—
304#67—186no—
305C2×Dic3—91no—
306C2×Dic3—93no—
307C2×Dic3—91no—
308C2×Dic3—91no—
309C2×Dic3—92no—
310C2×Dic3—92no—
311C2×Dic3—93no—
312C2×Dic3—93no—
313C2×Dic3—96no—
314C2×Dic3—96no—
315C4×D3—186no—
316C4×D3—182no—
317Dic6—184no—
318Dic6—1812no—
319Dic6—186no—
320Dic6—182no—
321C2×C32—41no—
322C2×C32C2×#23511yescharacteristic
323C2×C32D3×#4111yescharacteristic
324C2×C32#101011yescharacteristic
325C3⋊D3—121no—
326C2×C9—43no—
327C2×C9C2×#23513yes—
328C2×C9D3×#4113yes—
329C2×C9#101013yes—
330D9—123no—
331C2×Q8—271no—
332C2×Q8—272no—
333C2×D4—271no—
334C22×C4—271no—
335C22×C4C2×C3⋊D911yescharacteristic
336C4⋊C4—271no—
337C4⋊C4—271no—
338C4⋊C4—272no—
339C4⋊C4—272no—
340C4⋊C4—272no—
341C4⋊C4—272no—
342C4⋊C4C2×C3⋊D911yescharacteristic
343C4⋊C4C2×C3⋊D911yescharacteristic
344C22⋊C4—271no—
345C22⋊C4—272no—
346C22⋊C4—272no—
347C22⋊C4—271no—
348C22⋊C4—272no—
349C22⋊C4C2×C3⋊D912yes—
350C42—272no—
351C42—271no—
352C42—272no—
353C42C2×C3⋊D912yes—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—23no—
358C22×C3—23no—
359C22×C3—23no—
360C22×C3C23×D913yes—
361C22×C3C23×C3⋊D311yescharacteristic
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—181no—
365C2×D3—183no—
366C2×D3—183no—
367C2×D3—183no—
368C3×C4—21no—
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—23no—
372C3×C4—26no—
373C3×C4—26no—
374C3×C4#369716yes—
375C3×C4#382812yes—
376Dic3—92no—
377Dic3—96no—
378Dic3—182no—
379Dic3—181no—
380Dic3—182no—
381Dic3—181no—
382Dic3—186no—
383Dic3—183no—
384Dic3—186no—
385Dic3—183no—
386C32#553311yescharacteristic
387C9#553313yescomplemented
388C23—271no—
389C23C22×C3⋊D911yescharacteristic
390Q8—544no—
391Q8—542no—
392D4—542no—
393C2×C4—22no—
394C2×C4—271no—
395C2×C4—271no—
396C2×C4—271no—
397C2×C4—272no—
398C2×C4—272no—
399C2×C4—542no—
400C2×C4C22×C3⋊D912yes—
401C2×C4C22×C3⋊D911yescharacteristic
402C2×C4C22×C3⋊D912yes—
403C2×C4C22×C3⋊D911yescharacteristic
404C2×C3—41no—
405C2×C3—43no—
406C2×C3C2×#369713yes—
407C2×C3#41×D913yes—
408C2×C3#6371613yes—
409C2×C3C2×#382811yescharacteristic
410C2×C3#41×C3⋊D311yescharacteristic
411C2×C3#6436711yescharacteristic
412D3—361no—
413D3—363no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—541no—
418C22—541no—
419C22—541no—
420C22C23×C3⋊D911yescharacteristic
421C4—21no—
422C4—22no—
423C4—22no—
424C4—272no—
425C4—541no—
426C4—542no—
427C4—542no—
428C4—541no—
429C4#8873612yes—
430C3#1058963513yescomplemented
431C3#1059600411yescharacteristic
432C2—41no—
433C2—1081no—
434C2C2×#8873611yescharacteristic
435C2#41×C3⋊D911yescharacteristic
436C2#1172267311yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#117C2
socleC22×C32C23×D3
prime radicalC2×#6233C22
hypercenter#117C3⋊D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C3×C9×#117C2semidirect
3. #11721813C2semidirect
4. #11721753C2semidirect
5. #11721748C2semidirect
6. #11721744C2semidirect
7. #11721743C2semidirect
9. #11721725C2semidirect
10. #11721713C2semidirect
11. C4×#6226C2semidirect
48. C9×#117D3semidirect
139. C3×#117D9semidirect
287. #117C3⋊D9semidirect
288. C3×C9#556semidirect
387. C9#5533semidirect
430. C3#10589635semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1172
C4×#62332
#117218132
#117217532
#117217482
#117217442
#117217432
#117217312
#117217252
#117217132
C4×#62262
#105960043
#105896353