Finite Group Databasebeta

← #13147557

Subgroups — #13147557

Order 1728.

724 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721712C211yesmaximal, characteristic, complemented
2C3×#63434C211yesmaximal, characteristic, complemented
3C9×#869C211yesmaximal, characteristic, complemented
4#11721004C211yesmaximal, characteristic
5#11721003C212yesmaximal, complemented
6#11720992C211yesmaximal, characteristic, complemented
7#11720986C212yesmaximal, complemented
8#11720945C211yesmaximal, characteristic, complemented
9#11720942C212yesmaximal, complemented
10#11720920C212yesmaximal, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10593877—31nomaximal
13#10589835—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2212yes—
18C2×#6192C2212yes—
19C2×(C3⋊D9)⋊C4C2212yes—
20C3×C42×C9C2211yescharacteristic
21#88536C2212yes—
22#88536C2212yes—
23C3×(C2×D9)⋊C4C2211yescharacteristic
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C9×C4⋊Dic3C2211yescharacteristic
26C9×C4⋊Dic3C2211yescharacteristic
27C9×C4⋊Dic3C2212yes—
28C3×Dic9⋊C4C2211yescharacteristic
29C3×Dic9⋊C4C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31#88445C2212yes—
32#88445C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64084—31no—
38C3×#872—31no—
39C3×#869D311yescharacteristic
40#63909—31no—
41#63907—32no—
42#63894—31no—
43#63888—32no—
44#63857—31no—
45#63854—32no—
46#63848—32no—
47#63844—31no—
48C9×C4⋊Q8—31no—
49#63509—32no—
50#63508—32no—
51#63460—32no—
52#63459—31no—
53#63457—31no—
54#63456—32no—
55#63455—31no—
56#63449—31no—
57#63437—31no—
58#63437—61no—
59#63434D311yescharacteristic, complemented
60C22×C3⋊D9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C4⋊(C3⋊D9)—22no—
67#6226—22no—
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C3×C4×D9—22no—
74C9×Dic6—42no—
75#6192—42no—
76(C3⋊D9)⋊C4—42no—
77#5733—91no—
78#5534—91no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×#634—31no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6C2×D312yes—
83C2×#626—32no—
84C2×#624—32no—
85C32×C42C2×D311yescharacteristic
86#3752—32no—
87#3752—32no—
88C3×(C2×D3)⋊C4—31no—
89C3×(C2×D3)⋊C4—31no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D312yes—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4—31no—
94C3×Dic3⋊C4—31no—
95C3×C4×Dic3—31no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98#3722—32no—
99#3722—32no—
100(C2×D3)⋊Dic3—32no—
101Dic32—32no—
102C2×Q8×C9—32no—
103C2×D36—31no—
104C2×D36—32no—
105C2×D36—61no—
106C2×C4×D9—32no—
107C2×C4×D9C2×D311yescharacteristic
108C2×Dic18—31no—
109C2×Dic18—61no—
110C9×C4⋊C4—31no—
111C9×C4⋊C4—31no—
112C9×C4⋊C4—32no—
113C42×C9—21no—
114C42×C9C2×D311yescharacteristic
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—62no—
121(C2×D9)⋊C4—62no—
122(C2×D9)⋊C4C2×D311yescharacteristic
123(C2×D9)⋊C4C2×D311yescharacteristic
124C4⋊Dic9—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4C2×D311yescharacteristic
127Dic9⋊C4C2×D311yescharacteristic
128C4×Dic9—32no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×C3⋊D9—22no—
132C2×C3⋊D9—22no—
133C2×C3⋊D9—22no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C3×C4×C9—21no—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C9×Dic3—22no—
142C9×Dic3—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×C4⋊Q8—31no—
147C3×#117—91no—
148#947—92no—
149#946—92no—
150#930—91no—
151#924—91no—
152#898—92no—
153#898—92no—
154#897—91no—
155#896—92no—
156#895—91no—
157#894—92no—
158#893—91no—
159#887—91no—
160#885—92no—
161#883—92no—
162#875—91no—
163#875—91no—
164#875—181no—
165C4×D12—91no—
166#872—31no—
167#869D911yescharacteristic, complemented
168C4×Dic6—91no—
169C22×C3⋊D3—32no—
170C22×C3×D3—31no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—32no—
175C4⋊(C3⋊D3)—62no—
176#634—62no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D312yes—
181C2×C3×Dic3C22×D312yes—
182C3×C4×D3—62no—
183C3×Dic6—42no—
184#626—122no—
185#624—122no—
186C22×D9—32no—
187C22×D9—62no—
188C22×D9C22×D311yescharacteristic
189Q8×C9—122no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—32no—
194C2×C4×C9—32no—
195C2×C4×C9C22×D311yescharacteristic
196C2×C4×C9C22×D311yescharacteristic
197C2×C4×C9C22×D311yescharacteristic
198C2×Dic9—32no—
199C2×Dic9—62no—
200C2×Dic9C22×D311yescharacteristic
201C2×Dic9C22×D311yescharacteristic
202C2×Dic9C22×D311yescharacteristic
203D36—62no—
204D36—122no—
205D36—122no—
206C4×D9—22no—
207C4×D9—122no—
208Dic18—62no—
209Dic18—122no—
210#542—271noSylow
211C2×C3×C9C2×#4111yescharacteristic
212C2×C3×C9H(5,𝔽2)11yescharacteristic
213C2×C3×C92−1+411yescharacteristic
214C3⋊D9—42no—
215C3×D9—41no—
216C2×C3×Q8—32no—
217C22×C3×C4—91no—
218C2×#67—92no—
219C2×D12—91no—
220C2×D12—91no—
221C2×D12—92no—
222C2×D12—181no—
223C2×C4×D3—31no—
224C2×C4×D3—92no—
225C2×C4×D3—92no—
226C2×Dic6—91no—
227C2×Dic6—91no—
228C2×Dic6—181no—
229C2×Dic6C2×D912yes—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—91no—
235C3×C22⋊C4—91no—
236C3×C22⋊C4—91no—
237C3×C42—21no—
238C3×C42—91no—
239C3×C42C2×D911yescharacteristic
240C3×C42D3211yescharacteristic
241#217—92no—
242(C2×D3)⋊C4—31no—
243(C2×D3)⋊C4—31no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—182no—
254(C2×D3)⋊C4—182no—
255C4⋊Dic3—92no—
256C4⋊Dic3C2×D911yescharacteristic
257C4⋊Dic3C2×D911yescharacteristic
258C4⋊Dic3C2×D912yes—
259Dic3⋊C4—31no—
260Dic3⋊C4—31no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264C4×Dic3—31no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C22×C32C23×D311yescharacteristic
268C2×C3⋊D3—62no—
269C2×C3⋊D3—62no—
270C2×C3⋊D3—62no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—61no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4#23912yes—
277C3⋊Dic3—62no—
278C3×Dic3—22no—
279C3×Dic3—22no—
280C3×Dic3—61no—
281C3×Dic3—61no—
282C3×Dic3—61no—
283C22×C9—21no—
284C22×C9C23×D311yescharacteristic
285C2×D9—21no—
286C2×D9—21no—
287C2×D9—21no—
288C2×D9—62no—
289C2×D9—62no—
290C2×D9—62no—
291C2×D9—122no—
292C2×D9—122no—
293C2×D9—122no—
294C4×C9—21no—
295C4×C9—21no—
296C4×C9—22no—
297C4×C9—41no—
298C4×C9—41no—
299C4×C9—62no—
300C4×C9—62no—
301C4×C9#23512yes—
302Dic9—21no—
303Dic9—21no—
304Dic9—21no—
305Dic9—62no—
306Dic9—122no—
307C4⋊Q8—31no—
308#126—272no—
309#124—272no—
310#124—271no—
311#123—272no—
312C22⋊Q8—271no—
313C22⋊Q8—272no—
314#121—271no—
315C4×Q8—271no—
316C4×D4—271no—
317#117—91no—
318C3×C9#54211yescharacteristic, complemented, Sylow
319C23×C3—91no—
320C22×D3—31no—
321C22×D3—92no—
322C22×D3—92no—
323C22×D3—182no—
324C3×Q8—122no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—91no—
333C2×C3×C4—182no—
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340#67—362no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—182no—
347C2×Dic3C22×D912yes—
348C2×Dic3C22×D912yes—
349D12—182no—
350D12—182no—
351D12—362no—
352D12—362no—
353C4×D3—62no—
354C4×D3—362no—
355C4×D3—362no—
356Dic6—42no—
357Dic6—182no—
358Dic6—182no—
359Dic6—362no—
360C2×C32#100411yescharacteristic
361C2×C32C2×#23911yescharacteristic
362C2×C32#100711yescharacteristic
363C3⋊D3—122no—
364C3×D3—121no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9C2×#23511yescharacteristic
369C2×C9#100911yescharacteristic
370C2×C9#101011yescharacteristic
371D9—41no—
372D9—122no—
373D9—242no—
374C2×Q8—32no—
375C2×Q8—271no—
376C2×D4—272no—
377C2×D4—271no—
378C22×C4—91no—
379C22×C4—272no—
380C4⋊C4—31no—
381C4⋊C4—31no—
382C4⋊C4—32no—
383C4⋊C4—91no—
384C4⋊C4—91no—
385C4⋊C4—272no—
386C4⋊C4—272no—
387C22⋊C4—91no—
388C22⋊C4—91no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—91no—
395C42—272no—
396C42D3×D911yescharacteristic
397C22×C3—21no—
398C22×C3—181no—
399C22×C3—181no—
400C22×C3—181no—
401C22×C3C23×D911yescharacteristic
402C22×C3C22×D3211yescharacteristic
403C2×D3—61no—
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—182no—
407C2×D3—182no—
408C2×D3—182no—
409C2×D3—182no—
410C2×D3—182no—
411C2×D3—182no—
412C2×D3—362no—
413C2×D3—362no—
414C2×D3—362no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—22no—
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—62no—
423C3×C4—62no—
424C3×C4—181no—
425C3×C4—181no—
426C3×C4—181no—
427C3×C4#370112yes—
428C3×C4#379912yes—
429Dic3—22no—
430Dic3—22no—
431Dic3—61no—
432Dic3—61no—
433Dic3—61no—
434Dic3—182no—
435Dic3—182no—
436Dic3—362no—
437C32#573311yescharacteristic
438C9—21no—
439C9#553411yescharacteristic, complemented
440C23—91no—
441C23—272no—
442Q8—122no—
443Q8—542no—
444D4—542no—
445D4—1082no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—182no—
452C2×C4—272no—
453C2×C4—1082no—
454C2×C4C2×D3×D911yescharacteristic
455C2×C4C2×D3×D911yescharacteristic
456C2×C4C2×D3×D911yescharacteristic
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—361no—
461C2×C3#6371011yescharacteristic
462C2×C3C2×#370111yescharacteristic
463C2×C3#6371311yescharacteristic
464C2×C3#6429711yescharacteristic
465C2×C3C2×#379911yescharacteristic
466C2×C3#6430711yescharacteristic
467D3—121no—
468D3—362no—
469D3—362no—
470D3—722no—
471C22—181no—
472C22—181no—
473C22—181no—
474C22—542no—
475C22—542no—
476C22—542no—
477C22C22×D3×D911yescharacteristic
478C4—21no—
479C4—21no—
480C4—62no—
481C4—62no—
482C4—181no—
483C4—181no—
484C4—181no—
485C4—542no—
486C4#8863512yes—
487C3—21no—
488C3#1058983511yescharacteristic, complemented
489C3#1059387711yescharacteristic
490C2—361no—
491C2—1082no—
492C2#1172245811yescharacteristic
493C2C2×#8863511yescharacteristic
494C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721004C2
hypercenterC42D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721712C2semidirect
2. C3×#63434C2semidirect
3. C9×#869C2semidirect
5. #11721003C2semidirect
6. #11720992C2semidirect
7. #11720986C2semidirect
8. #11720945C2semidirect
9. #11720942C2semidirect
10. #11720920C2semidirect
11. #11720913C2semidirect
59. #63434D3semidirect
167. #869D9semidirect
318. C3×C9#542semidirect
439. C9#5534semidirect
488. C3#10589835semidirect

Maximal subgroups

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

subgroupindex
#117217122
C3×#634342
C9×#8692
#117210042
#117210032
#117209922
#117209862
#117209452
#117209422
#117209202
#117209132
#105938773
#105898353