Finite Group Databasebeta

← #13147668

Subgroups — #13147668

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721707C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic, complemented
3C3×C4×D36C211yesmaximal, characteristic, complemented
4#11721037C211yesmaximal, characteristic, complemented
5#11721037C211yesmaximal, characteristic, complemented
6#11721031C212yesmaximal, complemented
7D3×C4⋊Dic9C211yesmaximal, characteristic, complemented
8#11721016C212yesmaximal, complemented
9#11721012C212yesmaximal, complemented
10#11720946C212yesmaximal, complemented
11#11720906C211yesmaximal, characteristic, complemented
12#10593923—31nomaximal
13#10589667—31nomaximal
14C2×C9×D12C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×#6194C2212yes—
20C2×#6194C2212yes—
21C2×D3×Dic9C2212yes—
22C3×C42×C9C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25#88534C2212yes—
26#88534C2212yes—
27C9×(C2×D3)⋊C4C2211yescharacteristic
28C9×(C2×D3)⋊C4C2211yescharacteristic
29C3×(C2×D9)⋊C4C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32(C2×D3)⋊Dic9C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2212yes—
36Dic3×Dic9C2212yes—
37#64079—31no—
38C3×C4×D12—31no—
39C3×C4×D12D311yescharacteristic
40#63908—31no—
41#63908—31no—
42#63905—32no—
43D3×C4⋊Dic3—31no—
44#63885—32no—
45#63879—32no—
46#63858—32no—
47#63841—31no—
48C4×D4×C9—31no—
49#63501—32no—
50#63492—31no—
51#63492—31no—
52#63488—31no—
53C2×C4⋊Dic9—31no—
54#63460—32no—
55#63445—32no—
56#611⋊C4—32no—
57C4×D36D311yescharacteristic, complemented
58#63432—31no—
59#63432—61no—
60C22×D3×C9C2311yescharacteristic
61C22×D3×C9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C9×D12—22no—
72C4×D3×C9—22no—
73C4×D3×C9—22no—
74C3×D36—22no—
75C3×C4×D9—42no—
76#6194—24no—
77#6194—42no—
78D3×Dic9—42no—
79#5565—91no—
80#5558—91no—
81C2×C3×D12—31no—
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×C3×C4×D3C2×D311yescharacteristic
86C2×#625—32no—
87C2×#625—32no—
88C2×D3×Dic3—32no—
89C32×C42C2×D311yescharacteristic
90C4⋊(C3⋊Dic3)—31no—
91C4⋊(C3⋊Dic3)—31no—
92#3750—32no—
93#3750—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×(C2×D3)⋊C4C2×D311yescharacteristic
96C3×(C2×D3)⋊C4C2×D311yescharacteristic
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×D4×C9—31no—
105C22×C4×C9—31no—
106C22×C4×C9—31no—
107C2×#611—32no—
108C2×#611—32no—
109C22×Dic9—32no—
110C2×D36C2×D311yescharacteristic
111C2×C4×D9C2×D312yes—
112C9×C4⋊C4—31no—
113C9×C22⋊C4—31no—
114C9×C22⋊C4—31no—
115C42×C9—21no—
116C42×C9C2×D311yescharacteristic
117#3676—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4C2×D312yes—
121C4⋊Dic9—31no—
122C4⋊Dic9—31no—
123C4⋊Dic9—32no—
124C4⋊Dic9—61no—
125C4⋊Dic9—61no—
126C4⋊Dic9C2×D311yescharacteristic
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4—62no—
130Dic9⋊C4—62no—
131C4×Dic9—32no—
132C22×C3×C9C2411yescharacteristic
133C2×D3×C9—21no—
134C2×D3×C9—21no—
135C2×D3×C9—21no—
136C2×D3×C9—21no—
137C2×D3×C9—21no—
138C2×D3×C9#4112yes—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9—22no—
142C3×C4×C9—21no—
143C3×C4×C9—21no—
144C3×C4×C9#4112yes—
145C3⋊Dic9—22no—
146C3⋊Dic9—22no—
147C9×Dic3—21no—
148C9×Dic3—21no—
149C3×Dic9—22no—
150C3×C4×D4—31no—
151C3×C4×D4—91no—
152#939—92no—
153#938—91no—
154#938—91no—
155#930—91no—
156#930—91no—
157#929—92no—
158C4×#67—92no—
159#926—91no—
160C2×C4⋊Dic3—91no—
161#898—92no—
162#892—91no—
163D3×C4⋊C4—91no—
164#884—92no—
165#883—92no—
166#67⋊C4—92no—
167#879—92no—
168C4×D12—31no—
169C4×D12D911yescharacteristic, complemented
170#870—91no—
171#870—91no—
172#870—181no—
173C22×C3×D3—32no—
174C22×C3×D3C22×D311yescharacteristic
175C22×C3×D3C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D311yescharacteristic
184C3×D12—22no—
185C3×D12—62no—
186C3×C4×D3—22no—
187C3×C4×D3—22no—
188C3×C4×D3—122no—
189#625—64no—
190#625—122no—
191D3×Dic3—122no—
192C23×C9—31no—
193C23×C9—31no—
194C22×D9C22×D312yes—
195D4×C9—62no—
196C2×C4×C9—21no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—31no—
200C2×C4×C9—31no—
201C2×C4×C9—62no—
202C2×C4×C9—62no—
203C2×C4×C9C22×D311yescharacteristic
204C2×C4×C9C22×D311yescharacteristic
205C2×C4×C9C22×D311yescharacteristic
206#611—64no—
207#611—122no—
208C2×Dic9—32no—
209C2×Dic9—32no—
210C2×Dic9—62no—
211C2×Dic9—62no—
212C2×Dic9—122no—
213C2×Dic9C22×D312yes—
214D36—22no—
215C4×D9—42no—
216#553—271noSylow
217C2×C3×C9C2×#4111yescharacteristic
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9H(5,𝔽2)11yescharacteristic
220D3×C9—22no—
221D3×C9—41no—
222C3×D9—42no—
223C2×C3×D4—31no—
224C2×C3×D4—91no—
225C22×C3×C4—31no—
226C22×C3×C4—31no—
227C22×C3×C4—92no—
228C2×#67—92no—
229C2×#67—92no—
230C2×#67—92no—
231C2×#67—92no—
232C22×Dic3—92no—
233C2×D12—31no—
234C2×D12C2×D911yescharacteristic
235C2×C4×D3—32no—
236C2×C4×D3—92no—
237C2×C4×D3C2×D911yescharacteristic
238C2×C4×D3C2×D911yescharacteristic
239C3×C4⋊C4—31no—
240C3×C4⋊C4—91no—
241C3×C22⋊C4—31no—
242C3×C22⋊C4—31no—
243C3×C22⋊C4—92no—
244C3×C42—21no—
245C3×C42C2×D911yescharacteristic
246C3×C42D3211yescharacteristic
247#217—92no—
248#217—92no—
249#217—92no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4C2×D911yescharacteristic
255(C2×D3)⋊C4C2×D911yescharacteristic
256C4⋊Dic3—31no—
257C4⋊Dic3—92no—
258C4⋊Dic3—91no—
259C4⋊Dic3—91no—
260C4⋊Dic3—91no—
261C4⋊Dic3—91no—
262C4⋊Dic3—181no—
263C4⋊Dic3—181no—
264C4⋊Dic3C2×D911yescharacteristic
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—182no—
271Dic3⋊C4—182no—
272C4×Dic3—92no—
273C4×Dic3—92no—
274C22×C32C23×D311yescharacteristic
275C2×C3×D3—21no—
276C2×C3×D3—21no—
277C2×C3×D3—21no—
278C2×C3×D3—21no—
279C2×C3×D3—21no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3#23512yes—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4#23712yes—
287C3⋊Dic3—62no—
288C3⋊Dic3—62no—
289C3×Dic3—21no—
290C3×Dic3—21no—
291C3×Dic3—62no—
292C22×C9—21no—
293C22×C9—32no—
294C22×C9—61no—
295C22×C9—61no—
296C22×C9—61no—
297C22×C9—61no—
298C22×C9—61no—
299C22×C9C23×D311yescharacteristic
300C2×D9—22no—
301C2×D9—22no—
302C2×D9—22no—
303C4×C9—21no—
304C4×C9—21no—
305C4×C9—22no—
306C4×C9—41no—
307C4×C9—41no—
308C4×C9—61no—
309C4×C9—61no—
310C4×C9#23512yes—
311Dic9—22no—
312Dic9—62no—
313Dic9—62no—
314Dic9—122no—
315Dic9—122no—
316#126—272no—
317#125—271no—
318#123—272no—
319#121—272no—
320#121—271no—
321#121—271no—
322C4×D4—31no—
323C4×D4—91no—
324C4×D4—272no—
325#117—271no—
326C2×C4⋊C4—271no—
327C3×C9#55311yescharacteristic, complemented, Sylow
328C23×C3—31no—
329C23×C3—31no—
330C23×C3—92no—
331C22×D3—32no—
332C22×D3C22×D911yescharacteristic
333C22×D3C22×D911yescharacteristic
334C3×D4—62no—
335C3×D4—182no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—62no—
342C2×C3×C4—62no—
343C2×C3×C4—92no—
344C2×C3×C4—362no—
345C2×C3×C4C22×D911yescharacteristic
346C2×C3×C4C22×D911yescharacteristic
347C2×C3×C4C22×D911yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351#67—184no—
352#67—184no—
353#67—362no—
354#67—362no—
355C2×Dic3—32no—
356C2×Dic3—92no—
357C2×Dic3—92no—
358C2×Dic3—92no—
359C2×Dic3—92no—
360C2×Dic3—182no—
361C2×Dic3—182no—
362C2×Dic3—362no—
363C2×Dic3C22×D911yescharacteristic
364C2×Dic3C22×D911yescharacteristic
365D12—22no—
366D12—62no—
367C4×D3—22no—
368C4×D3—22no—
369C4×D3—122no—
370C4×D3—362no—
371C2×C32C2×#23511yescharacteristic
372C2×C32C2×#23711yescharacteristic
373C2×C32#100911yescharacteristic
374C3×D3—22no—
375C3×D3—41no—
376C3×D3—122no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—21no—
380C2×C9—62no—
381C2×C9—121no—
382C2×C9C2×#23511yescharacteristic
383C2×C9#100411yescharacteristic
384C2×C9D3×#4111yescharacteristic
385D9—42no—
386C2×D4—31no—
387C2×D4—91no—
388C2×D4—272no—
389C2×D4—272no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—92no—
393C22×C4—272no—
394C4⋊C4—31no—
395C4⋊C4—91no—
396C4⋊C4—272no—
397C4⋊C4—271no—
398C4⋊C4—271no—
399C4⋊C4—272no—
400C4⋊C4—272no—
401C22⋊C4—31no—
402C22⋊C4—31no—
403C22⋊C4—92no—
404C22⋊C4—272no—
405C22⋊C4—272no—
406C22⋊C4—272no—
407C42—272no—
408C42D3×D911yescharacteristic
409C22×C3—21no—
410C22×C3—32no—
411C22×C3—61no—
412C22×C3—61no—
413C22×C3—61no—
414C22×C3—61no—
415C22×C3—61no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—182no—
419C22×C3C23×D911yescharacteristic
420C22×C3C22×D3211yescharacteristic
421C2×D3—21no—
422C2×D3—21no—
423C2×D3—21no—
424C2×D3—21no—
425C2×D3—21no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3#369712yes—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—22no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—61no—
438C3×C4—61no—
439C3×C4—182no—
440C3×C4#369912yes—
441C3×C4#379512yes—
442Dic3—21no—
443Dic3—21no—
444Dic3—62no—
445Dic3—182no—
446Dic3—182no—
447Dic3—182no—
448Dic3—182no—
449Dic3—362no—
450Dic3—362no—
451C32#556511yescharacteristic
452C9—21no—
453C9#555811yescharacteristic, complemented
454C23—31no—
455C23—31no—
456C23—92no—
457D4—62no—
458D4—182no—
459D4—544no—
460D4—1082no—
461C2×C4—31no—
462C2×C4—31no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—92no—
466C2×C4—272no—
467C2×C4—272no—
468C2×C4—362no—
469C2×C4—1082no—
470C2×C4C2×D3×D911yescharacteristic
471C2×C4C2×D3×D911yescharacteristic
472C2×C4C2×D3×D911yescharacteristic
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—62no—
477C2×C3—121no—
478C2×C3—362no—
479C2×C3C2×#369711yescharacteristic
480C2×C3C2×#369911yescharacteristic
481C2×C3#6371511yescharacteristic
482C2×C3C2×#379511yescharacteristic
483C2×C3D3×#23511yescharacteristic
484C2×C3#6430711yescharacteristic
485D3—22no—
486D3—41no—
487D3—122no—
488C22—32no—
489C22—61no—
490C22—61no—
491C22—61no—
492C22—61no—
493C22—61no—
494C22—182no—
495C22—182no—
496C22—182no—
497C22C22×D3×D911yescharacteristic
498C4—21no—
499C4—21no—
500C4—61no—
501C4—61no—
502C4—182no—
503C4—542no—
504C4—542no—
505C4#8863812yes—
506C3—21no—
507C3#1058966711yescharacteristic, complemented
508C3#1059392311yescharacteristic
509C2—62no—
510C2—121no—
511C2—362no—
512C2C2×#8863811yescharacteristic
513C2D3×#369711yescharacteristic
514C2#1172247711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#553
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721707C2semidirect
2. C4×C9×D12C2semidirect
3. C3×C4×D36C2semidirect
4. #11721037C2semidirect
5. #11721037C2semidirect
6. #11721031C2semidirect
7. D3×C4⋊Dic9C2semidirect
8. #11721016C2semidirect
9. #11721012C2semidirect
10. #11720946C2semidirect
11. #11720906C2semidirect
57. C4×D36D3semidirect
169. C4×D12D9semidirect
327. C3×C9#553semidirect
453. C9#5558semidirect
507. C3#10589667semidirect

Maximal subgroups

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

subgroupindex
#117217072
C4×C9×D122
C3×C4×D362
#117210372
#117210312
D3×C4⋊Dic92
#117210162
#117210122
#117209462
#117209062
#105939233
#105896673