Finite Group Databasebeta

← #13153883

Subgroups — #13153883

Order 1728.

1080 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#117C211yesmaximal, characteristic, complemented
2#11721819C211yesmaximal, characteristic
3#11721813C211yesmaximal, characteristic, complemented
4#11721754C212yesmaximal, complemented
5#11721752C212yesmaximal, complemented
6#11721731C212yesmaximal, complemented
7#11721729C212yesmaximal, complemented
8#11721712C211yesmaximal, characteristic, complemented
9C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
10#11721706C211yesmaximal, characteristic, complemented
11C4×#6226C211yesmaximal, characteristic, complemented
12#10596005—31nomaximal
13#10589636—33nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C4×C3⋊D9C2212yes—
18C2×#6226C2211yescharacteristic
19C2×#6226C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C22⋊C4C2211yescharacteristic
23C3×C9×C22⋊C4C2211yescharacteristic
24C3×C42×C9C2211yescharacteristic
25C3×C42×C9C2211yescharacteristic
26#88551C2212yes—
27#88536C2212yes—
28#88536C2212yes—
29#88536C2212yes—
30#88536C2212yes—
31C4⋊(C3⋊Dic9)C2212yes—
32C4⋊(C3⋊Dic9)C2211yescharacteristic
33C4⋊(C3⋊Dic9)C2211yescharacteristic
34#88534C2212yes—
35#88534C2212yes—
36C4×C3⋊Dic9C2212yes—
37C32×#117D311yescharacteristic
38#64139—31no—
39#64133—31no—
40#64107—32no—
41#64105—32no—
42#64094—32no—
43#64092—32no—
44#64084—31no—
45C4×C4⋊(C3⋊D3)—31no—
46#64078—31no—
47C4×#634—31no—
48C9×#117D313yescomplemented
49#63492—33no—
50#63486—33no—
51#63460—36no—
52#63458—36no—
53#63447—36no—
54#63445—36no—
55#63437—33no—
56C4×D36—33no—
57#63431—33no—
58C4×Dic18—33no—
59C23×C3×C9C2311yescharacteristic
60C22×C3⋊D9C2312yes—
61C2×C3×C4×C9—22no—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68#6233—42no—
69C2×C3⋊Dic9C2312yes—
70C2×C3⋊Dic9C2312yes—
71C2×C3⋊Dic9C2312yes—
72C4⋊(C3⋊D9)—22no—
73C4×C3⋊D9—42no—
74#6226—22no—
75#6226—42no—
76#5534—93no—
77#5534—91no—
78C22×C32×C4C2×D311yescharacteristic
79C2×#638—32no—
80C2×C4⋊(C3⋊D3)—31no—
81C2×C4×C3⋊D3—32no—
82C2×#634—31no—
83C2×#634—32no—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C22⋊C4C2×D311yescharacteristic
87C32×C22⋊C4C2×D311yescharacteristic
88C32×C42C2×D311yescharacteristic
89C32×C42C2×D311yescharacteristic
90#3757—32no—
91#3752—32no—
92#3752—32no—
93#3752—32no—
94#3752—32no—
95C4⋊(C3⋊Dic3)—31no—
96C4⋊(C3⋊Dic3)—31no—
97C4⋊(C3⋊Dic3)—32no—
98#3750—32no—
99#3750—32no—
100C4×C3⋊Dic3—32no—
101C22×C4×C9C2×D313yes—
102C2×#611—36no—
103C2×D36—33no—
104C2×C4×D9—36no—
105C2×Dic18—33no—
106C2×Dic18—36no—
107C9×C4⋊C4C2×D313yes—
108C9×C4⋊C4C2×D313yes—
109C9×C22⋊C4C2×D313yes—
110C9×C22⋊C4C2×D313yes—
111C42×C9C2×D313yes—
112C42×C9C2×D313yes—
113#3676—36no—
114(C2×D9)⋊C4—36no—
115(C2×D9)⋊C4—36no—
116(C2×D9)⋊C4—36no—
117(C2×D9)⋊C4—36no—
118C4⋊Dic9—33no—
119C4⋊Dic9—33no—
120C4⋊Dic9—36no—
121Dic9⋊C4—36no—
122Dic9⋊C4—36no—
123C4×Dic9—36no—
124C22×C3×C9—21no—
125C22×C3×C9—21no—
126C22×C3×C9—21no—
127C22×C3×C9C2411yescharacteristic
128C2×C3⋊D9—22no—
129C2×C3⋊D9—22no—
130C2×C3⋊D9—22no—
131C3×C4×C9—21no—
132C3×C4×C9—21no—
133C3×C4×C9—21no—
134C3×C4×C9—21no—
135C3×C4×C9—21no—
136C3×C4×C9#4112yes—
137C3⋊Dic9—22no—
138C3⋊Dic9—22no—
139C3⋊Dic9—22no—
140C3×#117D913yescomplemented
141C3×#117C3⋊D311yescharacteristic
142#930—93no—
143#930—91no—
144#924—93no—
145#924—91no—
146#898—96no—
147#898—92no—
148#896—96no—
149#896—92no—
150#885—96no—
151#885—92no—
152#883—96no—
153#883—92no—
154#875—93no—
155#875—91no—
156C4×D12—93no—
157C4×D12—91no—
158#869—93no—
159#869—91no—
160C4×Dic6—93no—
161C4×Dic6—91no—
162C23×C32C22×D311yescharacteristic
163C22×C3⋊D3—32no—
164C2×C32×C4—22no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171#638—122no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—32no—
175C4⋊(C3⋊D3)—62no—
176C4×C3⋊D3—122no—
177#634—62no—
178#634—122no—
179C23×C9C22×D313yes—
180C22×D9—36no—
181C2×C4×C9—26no—
182C2×C4×C9C22×D313yes—
183C2×C4×C9C22×D313yes—
184C2×C4×C9C22×D313yes—
185C2×C4×C9C22×D313yes—
186C2×C4×C9C22×D313yes—
187C2×C4×C9C22×D313yes—
188#611—126no—
189C2×Dic9—36no—
190C2×Dic9—36no—
191C2×Dic9—36no—
192D36—66no—
193C4×D9—126no—
194Dic18—66no—
195Dic18—126no—
196#542—271noSylow
197C2×C3×C9—41no—
198C2×C3×C9C2×#4111yescharacteristic
199C2×C3×C9H(5,𝔽2)11yescharacteristic
200C2×C3×C92−1+411yescharacteristic
201C3⋊D9—42no—
202C22×C3×C4C2×D913yes—
203C22×C3×C4C2×C3⋊D311yescharacteristic
204C2×#67—96no—
205C2×#67—92no—
206C2×D12—93no—
207C2×D12—91no—
208C2×C4×D3—96no—
209C2×C4×D3—92no—
210C2×Dic6—93no—
211C2×Dic6—91no—
212C2×Dic6—96no—
213C2×Dic6—92no—
214C3×C4⋊C4C2×D913yes—
215C3×C4⋊C4C2×D913yes—
216C3×C4⋊C4C2×C3⋊D311yescharacteristic
217C3×C4⋊C4C2×C3⋊D311yescharacteristic
218C3×C22⋊C4C2×D913yes—
219C3×C22⋊C4C2×D913yes—
220C3×C22⋊C4C2×C3⋊D311yescharacteristic
221C3×C22⋊C4C2×C3⋊D311yescharacteristic
222C3×C42C2×D913yes—
223C3×C42C2×D913yes—
224C3×C42C2×C3⋊D311yescharacteristic
225C3×C42C2×C3⋊D311yescharacteristic
226#217—96no—
227#217—92no—
228(C2×D3)⋊C4—96no—
229(C2×D3)⋊C4—96no—
230(C2×D3)⋊C4—96no—
231(C2×D3)⋊C4—96no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236C4⋊Dic3—93no—
237C4⋊Dic3—93no—
238C4⋊Dic3—96no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—92no—
242Dic3⋊C4—96no—
243Dic3⋊C4—96no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246C4×Dic3—96no—
247C4×Dic3—92no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32C23×D311yescharacteristic
252C2×C3⋊D3—62no—
253C2×C3⋊D3—62no—
254C2×C3⋊D3—62no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4#23512yes—
261C3⋊Dic3—62no—
262C3⋊Dic3—62no—
263C3⋊Dic3—62no—
264C22×C9—23no—
265C22×C9—23no—
266C22×C9—23no—
267C22×C9C23×D313yes—
268C2×D9—66no—
269C2×D9—66no—
270C2×D9—66no—
271C4×C9—23no—
272C4×C9—23no—
273C4×C9—23no—
274C4×C9—23no—
275C4×C9—23no—
276C4×C9#23516yes—
277Dic9—66no—
278Dic9—66no—
279Dic9—66no—
280C4⋊Q8—271no—
281#126—272no—
282#124—272no—
283#124—271no—
284#123—272no—
285C22⋊Q8—271no—
286C22⋊Q8—272no—
287#121—271no—
288C4×Q8—271no—
289C4×D4—271no—
290#117C3⋊D911yescharacteristic, complemented
291C3×C9#54211yescharacteristic, complemented, Sylow
292C23×C3C22×D913yes—
293C23×C3C22×C3⋊D311yescharacteristic
294C22×D3—92no—
295C22×D3—96no—
296C2×C3×C4—26no—
297C2×C3×C4—22no—
298C2×C3×C4C22×D913yes—
299C2×C3×C4C22×D913yes—
300C2×C3×C4C22×D913yes—
301C2×C3×C4C22×D913yes—
302C2×C3×C4C22×D913yes—
303C2×C3×C4C22×D913yes—
304C2×C3×C4C22×C3⋊D311yescharacteristic
305C2×C3×C4C22×C3⋊D311yescharacteristic
306C2×C3×C4C22×C3⋊D311yescharacteristic
307C2×C3×C4C22×C3⋊D311yescharacteristic
308C2×C3×C4C22×C3⋊D311yescharacteristic
309C2×C3×C4C22×C3⋊D311yescharacteristic
310#67—362no—
311#67—366no—
312C2×Dic3—92no—
313C2×Dic3—92no—
314C2×Dic3—92no—
315C2×Dic3—96no—
316C2×Dic3—96no—
317C2×Dic3—96no—
318D12—186no—
319D12—182no—
320C4×D3—362no—
321C4×D3—366no—
322Dic6—186no—
323Dic6—182no—
324Dic6—362no—
325Dic6—366no—
326C2×C32—41no—
327C2×C32C2×#23511yescharacteristic
328C2×C32#100911yescharacteristic
329C2×C32#101011yescharacteristic
330C3⋊D3—122no—
331C2×C9—43no—
332C2×C9C2×#23513yes—
333C2×C9#100913yes—
334C2×C9#101013yes—
335D9—126no—
336C2×Q8—271no—
337C2×Q8—272no—
338C2×D4—272no—
339C2×D4—271no—
340C22×C4—272no—
341C22×C4C2×C3⋊D911yescharacteristic
342C4⋊C4—271no—
343C4⋊C4—271no—
344C4⋊C4—272no—
345C4⋊C4—272no—
346C4⋊C4—272no—
347C4⋊C4C2×C3⋊D911yescharacteristic
348C4⋊C4C2×C3⋊D911yescharacteristic
349C22⋊C4—272no—
350C22⋊C4—272no—
351C22⋊C4—272no—
352C22⋊C4—272no—
353C22⋊C4—272no—
354C22⋊C4C2×C3⋊D911yescharacteristic
355C22⋊C4C2×C3⋊D911yescharacteristic
356C42—272no—
357C42C2×C3⋊D911yescharacteristic
358C42C2×C3⋊D911yescharacteristic
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—23no—
363C22×C3—23no—
364C22×C3—23no—
365C22×C3C23×D913yes—
366C22×C3C23×C3⋊D311yescharacteristic
367C2×D3—182no—
368C2×D3—182no—
369C2×D3—182no—
370C2×D3—186no—
371C2×D3—186no—
372C2×D3—186no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—23no—
379C3×C4—23no—
380C3×C4—23no—
381C3×C4—23no—
382C3×C4—23no—
383C3×C4#369716yes—
384C3×C4#382812yes—
385Dic3—182no—
386Dic3—182no—
387Dic3—182no—
388Dic3—186no—
389Dic3—186no—
390Dic3—186no—
391C32#553411yescharacteristic
392C9#553413yescomplemented
393C23—272no—
394C23C22×C3⋊D911yescharacteristic
395Q8—542no—
396Q8—1082no—
397D4—542no—
398D4—1082no—
399C2×C4—22no—
400C2×C4—272no—
401C2×C4—272no—
402C2×C4—272no—
403C2×C4—1082no—
404C2×C4C22×C3⋊D911yescharacteristic
405C2×C4C22×C3⋊D911yescharacteristic
406C2×C4C22×C3⋊D911yescharacteristic
407C2×C4C22×C3⋊D911yescharacteristic
408C2×C4C22×C3⋊D911yescharacteristic
409C2×C4C22×C3⋊D911yescharacteristic
410C2×C3—41no—
411C2×C3—43no—
412C2×C3C2×#369713yes—
413C2×C3#6371513yes—
414C2×C3#6371613yes—
415C2×C3C2×#382811yescharacteristic
416C2×C3#6436611yescharacteristic
417C2×C3#6436711yescharacteristic
418D3—362no—
419D3—366no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—542no—
424C22—542no—
425C22—542no—
426C22C23×C3⋊D911yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—542no—
433C4—542no—
434C4—542no—
435C4#8873612yes—
436C3#1058963613yescomplemented
437C3#1059600511yescharacteristic
438C2—41no—
439C2—1082no—
440C2C2×#8873611yescharacteristic
441C2#1172267211yescharacteristic
442C2#1172267311yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#117C2
socleC22×C32C23×D3
prime radical#11721819C2
hypercenter#117C3⋊D9
hypocenterC3×C9#542
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. #11721754C2semidirect
5. #11721752C2semidirect
6. #11721731C2semidirect
7. #11721729C2semidirect
8. #11721712C2semidirect
9. C4×C4⋊(C3⋊D9)C2semidirect
10. #11721706C2semidirect
11. C4×#6226C2semidirect
48. C9×#117D3semidirect
140. C3×#117D9semidirect
290. #117C3⋊D9semidirect
291. C3×C9#542semidirect
392. C9#5534semidirect
436. C3#10589636semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1172
#117218192
#117218132
#117217542
#117217522
#117217312
#117217292
#117217122
C4×C4⋊(C3⋊D9)2
#117217062
C4×#62262
#105960053
#105896363