Finite Group Databasebeta

← #13150203

Subgroups — #13150203

Order 1728.

970 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88658C212yesmaximal, complemented
2#11721819C211yesmaximal, characteristic, complemented
3C3×C4×#611C211yesmaximal, characteristic, complemented
4C2×C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
5#11721251C212yesmaximal, complemented
6#11721004C211yesmaximal, characteristic, complemented
7#11720994C212yesmaximal, complemented
8#11720992C211yesmaximal, characteristic, complemented
9#11720988C212yesmaximal, complemented
10D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#11720969C211yesmaximal, characteristic, complemented
12#10594906—31nomaximal
13#10589609—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C22×C9×Dic3C2212yes—
19C2×C3×C4×D9C2211yescharacteristic
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22#88658—28no—
23C2×Dic3×D9C2212yes—
24C2×#6189C2212yes—
25#88536C2212yes—
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27C3×#3676C2211yescharacteristic
28C3×(C2×D9)⋊C4C2211yescharacteristic
29C9×C4⋊Dic3C2211yescharacteristic
30C9×C4⋊Dic3C2211yescharacteristic
31C9×C4⋊Dic3C2212yes—
32C3×Dic9⋊C4C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34#88445C2212yes—
35#88445C2212yes—
36Dic9⋊Dic3C2212yes—
37Dic9⋊Dic3C2212yes—
38C2×#3804—32no—
39#64139—31no—
40C3×C4×#67—31no—
41C2×C3×C4⋊Dic3D311yescharacteristic
42#63966—32no—
43#63918—31no—
44#63909—31no—
45#63895—32no—
46#63894—31no—
47#63890—32no—
48D3×C4⋊Dic3—31no—
49C2×#3697—32no—
50C2×C9×C4⋊C4—31no—
51#63492—31no—
52#63492—61no—
53#63491—32no—
54C4×#611D311yescharacteristic, complemented
55#63458—32no—
56#63457—31no—
57#63456—32no—
58#63455—31no—
59C4⋊C4×D9—31no—
60#63450—31no—
61C23×C3×C9C2311yescharacteristic
62C22×C3⋊D9C2312yes—
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67#6233—24no—
68C2×C3⋊Dic9C2312yes—
69C4⋊(C3⋊D9)—22no—
70C3×#611D414yescomplemented
71C2×C9×Dic3—24no—
72C2×C9×Dic3C2312yes—
73C2×C9×Dic3C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C3×C4×D9—22no—
78#6192—24no—
79(C3⋊D9)⋊C4—24no—
80Dic3×D9—24no—
81#6189—24no—
82#5555—91no—
83#5507—91no—
84C22×C32×C4C2×D311yescharacteristic
85C2×#638—32no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C3×#67—31no—
88C22×C3×Dic3C2×D312yes—
89C2×C3×C4×D3—31no—
90C2×#627—32no—
91C2×#626—32no—
92C2×#624—32no—
93#3804—68no—
94C2×D3×Dic3—32no—
95#3752—32no—
96C4⋊(C3⋊Dic3)—31no—
97C3×#217—31no—
98C3×(C2×D3)⋊C4—31no—
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×C4⋊Dic3C2×D312yes—
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×Dic3⋊C4—31no—
103C3×C4×Dic3—31no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106#3722—32no—
107#3722—32no—
108C22×C4×C9—21no—
109C22×C4×C9—32no—
110C22×C4×C9C2×D311yescharacteristic
111C2×#611—32no—
112C2×#611—62no—
113C2×#611C2×D311yescharacteristic
114#3697—68no—
115C2×D36—32no—
116C2×D36—31no—
117C2×D36—61no—
118C2×C4×D9—32no—
119C2×C4×D9—32no—
120C2×C4×D9C2×D311yescharacteristic
121C2×Dic18—32no—
122C9×C4⋊C4—31no—
123C9×C4⋊C4—31no—
124C9×C4⋊C4—32no—
125#3676C2×D311yescharacteristic
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4—32no—
128(C2×D9)⋊C4—32no—
129(C2×D9)⋊C4—62no—
130(C2×D9)⋊C4C2×D311yescharacteristic
131C4⋊Dic9—31no—
132C4⋊Dic9—61no—
133Dic9⋊C4—32no—
134Dic9⋊C4—32no—
135Dic9⋊C4C2×D311yescharacteristic
136C4×Dic9C2×D311yescharacteristic
137C22×C3×C9—21no—
138C22×C3×C9—21no—
139C22×C3×C9C2×D412yes—
140C22×C3×C9C2411yescharacteristic
141C2×C3⋊D9—22no—
142C2×C3⋊D9—22no—
143C2×C3⋊D9—22no—
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×D412yes—
147C3×C4×C9—21no—
148C3×C4×C9#4112yescomplemented
149C3⋊Dic9—22no—
150C9×Dic3—22no—
151C9×Dic3—22no—
152C3×Dic9—21no—
153C3×Dic9—21no—
154C3×Dic9C2×D412yes—
155C2×#237—92no—
156C2×#235—92no—
157C3×C4×D4—91no—
158C2×C3×C4⋊C4—31no—
159#930—91no—
160#930—91no—
161#930—91no—
162#930—181no—
163#929—92no—
164C4×#67—31no—
165C2×C4⋊Dic3—91no—
166C2×C4⋊Dic3D911yescharacteristic, complemented
167#897—92no—
168#896—92no—
169#895—91no—
170#894—92no—
171#893—91no—
172D3×C4⋊C4—91no—
173#888—91no—
174#886—92no—
175#886—92no—
176C4×D12—91no—
177#869—91no—
178C23×C32C22×D311yescharacteristic
179C22×C3⋊D3—32no—
180C22×C3×D3—31no—
181C2×C32×C4—22no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184#638—64no—
185C2×C3⋊Dic3—32no—
186C4⋊(C3⋊D3)—62no—
187C3×#67—34no—
188C2×C3×Dic3—24no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3C22×D312yes—
193C2×C3×Dic3C22×D312yes—
194C3×C4×D3—62no—
195#627—64no—
196#626—64no—
197#624—64no—
198D3×Dic3—64no—
199C23×C9—21no—
200C23×C9C22×D311yescharacteristic
201C22×D9—32no—
202C22×D9—62no—
203C22×D9C22×D311yescharacteristic
204C2×C4×C9—22no—
205C2×C4×C9—21no—
206C2×C4×C9—21no—
207C2×C4×C9—32no—
208C2×C4×C9—32no—
209C2×C4×C9—42no—
210C2×C4×C9—64no—
211C2×C4×C9C22×D311yescharacteristic
212C2×C4×C9C22×D311yescharacteristic
213#611—64no—
214#611—124no—
215#611D1214yescomplemented
216C2×Dic9—32no—
217C2×Dic9—62no—
218C2×Dic9C22×D311yescharacteristic
219C2×Dic9C22×D311yescharacteristic
220C2×Dic9C22×D311yescharacteristic
221D36—64no—
222D36—62no—
223D36—122no—
224C4×D9—22no—
225C4×D9—64no—
226C4×D9—64no—
227Dic18—64no—
228#547—271noSylow
229C2×C3×C9—22no—
230C2×C3×C9C22×D411yescharacteristic
231C2×C3×C9C2×#4111yescharacteristic
232C2×C3×C92−1+411yescharacteristic
233C3⋊D9—42no—
234C3×D9—22no—
235C2×C3×D4—91no—
236C22×C3×C4—21no—
237C22×C3×C4—32no—
238C22×C3×C4—91no—
239C22×C3×C4C2×D911yescharacteristic
240C22×C3×C4D3211yescharacteristic
241C2×#67—31no—
242C2×#67—92no—
243C2×#67—92no—
244C2×#67—92no—
245C2×#67—182no—
246C22×Dic3—92no—
247C22×Dic3C2×D912yes—
248#237—188no—
249#235—188no—
250C2×D12—92no—
251C2×D12—91no—
252C2×D12—91no—
253C2×D12—181no—
254C2×C4×D3—31no—
255C2×C4×D3—92no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×Dic6—92no—
259C2×Dic6—92no—
260C3×C4⋊C4—31no—
261C3×C4⋊C4—31no—
262C3×C4⋊C4—32no—
263C3×C4⋊C4—91no—
264C3×C22⋊C4—91no—
265C3×C22⋊C4—91no—
266C3×C42—91no—
267#217—31no—
268(C2×D3)⋊C4—31no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—182no—
276C4⋊Dic3—91no—
277C4⋊Dic3—91no—
278C4⋊Dic3—92no—
279C4⋊Dic3—92no—
280C4⋊Dic3—181no—
281C4⋊Dic3C2×D911yescharacteristic
282C4⋊Dic3C2×D911yescharacteristic
283C4⋊Dic3C2×D912yes—
284Dic3⋊C4—31no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—31no—
288C22×C32—21no—
289C22×C32—21no—
290C22×C32D3×D412yes—
291C22×C32C23×D311yescharacteristic
292C2×C3⋊D3—62no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—62no—
295C2×C3×D3—32no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C32×C4—21no—
299C32×C4#23512yes—
300C3⋊Dic3—62no—
301C3×Dic3—22no—
302C3×Dic3—22no—
303C3×Dic3—32no—
304C3×Dic3—61no—
305C3×Dic3—61no—
306C22×C9—21no—
307C22×C9—21no—
308C22×C9—21no—
309C22×C9—22no—
310C22×C9—41no—
311C22×C9—41no—
312C22×C9C2×D1212yes—
313C22×C9C23×D311yescharacteristic
314C2×D9—21no—
315C2×D9—21no—
316C2×D9—62no—
317C2×D9—62no—
318C2×D9—62no—
319C2×D9—122no—
320C2×D9—122no—
321C2×D9—122no—
322C2×D9C2×D1212yes—
323C4×C9—21no—
324C4×C9—22no—
325C4×C9—41no—
326C4×C9—62no—
327C4×C9—62no—
328C4×C9#23712yescomplemented
329Dic9—21no—
330Dic9—21no—
331Dic9—62no—
332Dic9—122no—
333Dic9C2×D1212yes—
334C2×#41—272no—
335C4⋊Q8—271no—
336#123—272no—
337#123—272no—
338C22⋊Q8—272no—
339#121—271no—
340#121—271no—
341C4×D4—91no—
342C4×D4—271no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—271no—
345C3×C9#54711yescharacteristic, complemented, Sylow
346C23×C3—21no—
347C23×C3—91no—
348C23×C3C22×D911yescharacteristic
349C23×C3C2×D3211yescharacteristic
350C22×D3—31no—
351C22×D3—92no—
352C22×D3—92no—
353C22×D3—182no—
354C3×D4—94no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—32no—
360C2×C3×C4—32no—
361C2×C3×C4—42no—
362C2×C3×C4—64no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—182no—
367C2×C3×C4C22×D911yescharacteristic
368C2×C3×C4C22×D911yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371#67—34no—
372#67—184no—
373#67—184no—
374#67—184no—
375#67—364no—
376C2×Dic3—24no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—92no—
381C2×Dic3—92no—
382C2×Dic3—182no—
383C2×Dic3—184no—
384C2×Dic3C22×D912yes—
385C2×Dic3C22×D912yes—
386D12—184no—
387D12—182no—
388D12—182no—
389D12—362no—
390C4×D3—62no—
391C4×D3—184no—
392C4×D3—184no—
393C4×D3—184no—
394Dic6—184no—
395Dic6—184no—
396C2×C32—22no—
397C2×C32C2×#23511yescharacteristic
398C2×C32C2×D3×D411yescharacteristic
399C2×C32#100711yescharacteristic
400C3⋊D3—122no—
401C3×D3—62no—
402C2×C9—22no—
403C2×C9—21no—
404C2×C9—21no—
405C2×C9—21no—
406C2×C9—42no—
407C2×C9C22×D1211yescharacteristic
408C2×C9C2×#23711yescharacteristic
409C2×C9#101011yescharacteristic
410D9—22no—
411D9—122no—
412D9—242no—
413#41—548no—
414C2×Q8—272no—
415C2×D4—91no—
416C2×D4—272no—
417C2×D4—272no—
418C2×D4—271no—
419C22×C4—32no—
420C22×C4—91no—
421C22×C4—272no—
422C22×C4—272no—
423C22×C4D3×D911yescharacteristic
424C4⋊C4—31no—
425C4⋊C4—31no—
426C4⋊C4—32no—
427C4⋊C4—91no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C4⋊C4—271no—
431C22⋊C4—91no—
432C22⋊C4—91no—
433C22⋊C4—272no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C42—91no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—22no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—92no—
446C22×C3—181no—
447C22×C3—181no—
448C22×C3D4×D912yes—
449C22×C3C23×D911yescharacteristic
450C22×C3D3×D1212yes—
451C22×C3C22×D3211yescharacteristic
452C2×D3—32no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—182no—
456C2×D3—182no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—362no—
462C2×D3—362no—
463C2×D3—362no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—41no—
468C3×C4—62no—
469C3×C4—62no—
470C3×C4—92no—
471C3×C4—181no—
472C3×C4—181no—
473C3×C4#369712yescomplemented
474C3×C4#380412yes—
475Dic3—22no—
476Dic3—22no—
477Dic3—32no—
478Dic3—61no—
479Dic3—61no—
480Dic3—182no—
481Dic3—182no—
482Dic3—362no—
483C32#550711yescharacteristic
484C9—21no—
485C9#555511yescharacteristic, complemented
486C23—91no—
487C23—272no—
488C23C2×D3×D911yescharacteristic
489Q8—544no—
490D4—94no—
491D4—544no—
492D4—544no—
493D4—542no—
494C2×C4—22no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—64no—
498C2×C4—91no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—182no—
502C2×C4—272no—
503C2×C4—544no—
504C2×C4—544no—
505C2×C4C2×D3×D911yescharacteristic
506C2×C4C2×D3×D911yescharacteristic
507C2×C3—22no—
508C2×C3—22no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—42no—
513C2×C3—182no—
514C2×C3C2×#369711yescharacteristic
515C2×C3C2×D4×D911yescharacteristic
516C2×C3#6371311yescharacteristic
517C2×C3#6429711yescharacteristic
518C2×C3C2×D3×D1211yescharacteristic
519C2×C3C2×#380411yescharacteristic
520D3—62no—
521D3—362no—
522D3—362no—
523D3—722no—
524C22—21no—
525C22—21no—
526C22—92no—
527C22—181no—
528C22—181no—
529C22—542no—
530C22—542no—
531C22—542no—
532C22D9×D1212yes—
533C22C22×D3×D911yescharacteristic
534C4—21no—
535C4—62no—
536C4—62no—
537C4—92no—
538C4—181no—
539C4—181no—
540C4—542no—
541C4#8865812yescomplemented
542C3—21no—
543C3#1058960911yescharacteristic, complemented
544C3#1059490611yescharacteristic
545C2—22no—
546C2—182no—
547C2—1082no—
548C2#1172245811yescharacteristic
549C2C2×D9×D1211yescharacteristic
550C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#547
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88658C2semidirect
2. #11721819C2semidirect
3. C3×C4×#611C2semidirect
4. C2×C9×C4⋊Dic3C2semidirect
5. #11721251C2semidirect
6. #11721004C2semidirect
7. #11720994C2semidirect
8. #11720992C2semidirect
9. #11720988C2semidirect
10. D9×C4⋊Dic3C2semidirect
11. #11720969C2semidirect
54. C4×#611D3semidirect
70. C3×#611D4semidirect
148. C3×C4×C9#41semidirect
166. C2×C4⋊Dic3D9semidirect
215. #611D12semidirect
328. C4×C9#237semidirect
345. C3×C9#547semidirect
473. C3×C4#3697semidirect
485. C9#5555semidirect
541. C4#88658semidirect
543. C3#10589609semidirect

Maximal subgroups

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

subgroupindex
C2×#886582
#117218192
C3×C4×#6112
C2×C9×C4⋊Dic32
#117212512
#117210042
#117209942
#117209922
#117209882
D9×C4⋊Dic32
#117209692
#105949063
#105896093