Finite Group Databasebeta

← #13148821

Subgroups — #13148821

Order 1728.

1024 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88635C212yesmaximal, complemented
2#11721751C211yesmaximal, characteristic, complemented
3C3×#63455C211yesmaximal, characteristic, complemented
4C9×#888C211yesmaximal, characteristic, complemented
5#11721007C211yesmaximal, characteristic, complemented
6#11721006C211yesmaximal, characteristic, complemented
7#11721005C212yesmaximal, complemented
8#11721003C212yesmaximal, complemented
9#11721002C212yesmaximal, complemented
10#11720992C211yesmaximal, characteristic, complemented
11C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
12#10594518—31nomaximal
13#10589827—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4⋊(C3⋊D9)C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×C9×Dic6C2212yes—
20C2×#6192C2212yes—
21C2×#6192C2212yes—
22C2×#6192C2212yes—
23C2×(C3⋊D9)⋊C4C2211yescharacteristic
24C2×(C3⋊D9)⋊C4C2212yes—
25C2×(C3⋊D9)⋊C4C2211yescharacteristic
26#88635—28no—
27C3×C9×C4⋊C4C2211yescharacteristic
28#88536C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic
31C9×Dic3⋊C4C2212yes—
32C4×C9×Dic3C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34#88445C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2211yescharacteristic
38C2×#3799—32no—
39#64104—31no—
40C3×#893—31no—
41C3×#888D311yescharacteristic
42#63916—31no—
43#63911—31no—
44#63910—32no—
45#63907—32no—
46#63904—32no—
47#63894—31no—
48C4×#624—31no—
49C2×#3701—32no—
50C9×C4⋊Q8—31no—
51#63509—32no—
52#63457—31no—
53#63457—31no—
54#63457—32no—
55#63457—61no—
56#63455—32no—
57#63455—31no—
58#63455D311yescharacteristic, complemented
59#63436—31no—
60C42×D9—31no—
61C22×C3⋊D9C2312yes—
62C22×C3⋊D9C2311yescharacteristic
63C22×C3×D9C2312yes—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C4⋊(C3⋊D9)—24no—
68C4⋊(C3⋊D9)—22no—
69C4×C3⋊D9—22no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C3×D36—22no—
76C3×C4×D9—24no—
77C9×Dic6—24no—
78#6192—24no—
79#6192—24no—
80#6192—24no—
81(C3⋊D9)⋊C4—22no—
82(C3⋊D9)⋊C4—24no—
83(C3⋊D9)⋊C4D414yescomplemented
84#5725—91no—
85#5548—91no—
86C2×C4⋊(C3⋊D3)—32no—
87C2×C4⋊(C3⋊D3)—31no—
88C2×C4×C3⋊D3—31no—
89C2×C3×D12—31no—
90C2×C3×C4×D3—32no—
91C2×C3×Dic6C2×D312yes—
92C2×#626—32no—
93C2×#626—32no—
94C2×#626—32no—
95C2×#624—31no—
96C2×#624—32no—
97C2×#624—31no—
98#3799—68no—
99C32×C4⋊C4C2×D311yescharacteristic
100#3752—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×Dic3⋊C4C2×D312yes—
104C3×C4×Dic3—31no—
105C3×C4×Dic3C2×D311yescharacteristic
106#3722—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—31no—
109Dic32—31no—
110C2×Q8×C9—32no—
111#3701—68no—
112C2×D36—32no—
113C2×D36—32no—
114C2×D36—32no—
115C2×D36—31no—
116C2×D36—32no—
117C2×D36—61no—
118C2×D36—62no—
119C2×D36C2×D311yescharacteristic
120C2×C4×D9—31no—
121C2×C4×D9—31no—
122C2×C4×D9—31no—
123C2×C4×D9—32no—
124C2×C4×D9—61no—
125C2×C4×D9C2×D312yes—
126C9×C4⋊C4—21no—
127C9×C4⋊C4—32no—
128C9×C4⋊C4—31no—
129C9×C4⋊C4C2×D311yescharacteristic
130C42×C9—31no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—62no—
135(C2×D9)⋊C4C2×D312yes—
136C4×Dic9—31no—
137C4×Dic9—31no—
138C4×Dic9C2×D311yescharacteristic
139C22×C3×C9C2411yescharacteristic
140C2×C3⋊D9—21no—
141C2×C3⋊D9—21no—
142C2×C3⋊D9—22no—
143C2×C3⋊D9—22no—
144C2×C3⋊D9—22no—
145C2×C3⋊D9C2×D412yes—
146C2×C3×D9—22no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C3×C4×C9—22no—
150C3×C4×C9#4112yescomplemented
151C3⋊Dic9—21no—
152C9×Dic3—22no—
153C9×Dic3C2×D412yes—
154C9×Dic3#4112yescomplemented
155C3×Dic9—21no—
156C3×Dic9C2×D412yes—
157C2×#239—92no—
158C2×#235—92no—
159C3×C4⋊Q8—31no—
160C3×C4×D4—91no—
161#947—92no—
162#940—91no—
163#938—91no—
164C4×#67—92no—
165#895—92no—
166#895—91no—
167#895—91no—
168#895—91no—
169#895—181no—
170#893—31no—
171#893—91no—
172#893—92no—
173#888D911yescharacteristic, complemented
174#885—92no—
175#884—92no—
176#874—91no—
177C4×D12—91no—
178C42×D3—91no—
179C42×D3—91no—
180C22×C3⋊D3—31no—
181C22×C3⋊D3—32no—
182C22×C3×D3—32no—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—31no—
186C4⋊(C3⋊D3)—64no—
187C4⋊(C3⋊D3)—62no—
188C4×C3⋊D3—62no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3C22×D312yes—
192C2×C3×Dic3C22×D311yescharacteristic
193C2×C3×Dic3C22×D311yescharacteristic
194C3×D12—62no—
195C3×C4×D3—64no—
196C3×Dic6—24no—
197#626—64no—
198#626—64no—
199#626—64no—
200#624—34no—
201#624—64no—
202#624—62no—
203C22×D9—31no—
204C22×D9—32no—
205C22×D9—61no—
206C22×D9—62no—
207C22×D9C22×D312yes—
208Q8×C9—64no—
209C2×C4×C9—21no—
210C2×C4×C9—22no—
211C2×C4×C9—31no—
212C2×C4×C9—31no—
213C2×C4×C9—32no—
214C2×C4×C9C22×D312yes—
215C2×C4×C9C22×D311yescharacteristic
216C2×Dic9—31no—
217C2×Dic9—61no—
218C2×Dic9C22×D311yescharacteristic
219C2×Dic9C22×D311yescharacteristic
220D36—22no—
221D36—64no—
222D36—64no—
223D36—64no—
224D36—64no—
225D36—62no—
226D36—122no—
227D36—124no—
228C4×D9—24no—
229C4×D9—34no—
230C4×D9—62no—
231C4×D9—62no—
232C4×D9—64no—
233C4×D9—122no—
234#532—271noSylow
235C2×C3×C9C22×D411yescharacteristic
236C2×C3×C9C2×#4111yescharacteristic
237C2×C3×C9C2×#4111yescharacteristic
238C3⋊D9—22no—
239C3⋊D9—42no—
240C3×D9—42no—
241C2×C3×Q8—32no—
242C2×C3×D4—91no—
243C22×C3×C4—92no—
244C2×#67—92no—
245C2×#67—92no—
246C2×#67—92no—
247#239—188no—
248#235—188no—
249C2×D12—31no—
250C2×D12—92no—
251C2×D12—92no—
252C2×D12—92no—
253C2×D12—91no—
254C2×D12—92no—
255C2×D12—91no—
256C2×D12—92no—
257C2×D12—181no—
258C2×D12—182no—
259C2×C4×D3—32no—
260C2×C4×D3—91no—
261C2×C4×D3—91no—
262C2×C4×D3—91no—
263C2×C4×D3—92no—
264C2×C4×D3—91no—
265C2×C4×D3—91no—
266C2×C4×D3—91no—
267C2×C4×D3—92no—
268C2×C4×D3—181no—
269C2×Dic6C2×D912yes—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—32no—
272C3×C4⋊C4—31no—
273C3×C4⋊C4C2×D911yescharacteristic
274C3×C4⋊C4D3211yescharacteristic
275C3×C22⋊C4—92no—
276C3×C42—31no—
277C3×C42—91no—
278#217—92no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—182no—
286C4⋊Dic3C2×D911yescharacteristic
287Dic3⋊C4C2×D912yes—
288C4×Dic3—31no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3C2×D911yescharacteristic
294C22×C32C23×D311yescharacteristic
295C2×C3⋊D3—32no—
296C2×C3⋊D3—61no—
297C2×C3⋊D3—61no—
298C2×C3⋊D3—62no—
299C2×C3⋊D3—62no—
300C2×C3⋊D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C32×C4—22no—
305C32×C4#23912yes—
306C3⋊Dic3—61no—
307C3×Dic3—22no—
308C3×Dic3—32no—
309C3×Dic3—61no—
310C3×Dic3D3×D412yes—
311C3×Dic3#23912yes—
312C22×C9—21no—
313C22×C9C23×D311yescharacteristic
314C2×D9—22no—
315C2×D9—22no—
316C2×D9—22no—
317C2×D9—32no—
318C2×D9—62no—
319C2×D9—61no—
320C2×D9—61no—
321C2×D9—62no—
322C2×D9—62no—
323C2×D9—62no—
324C2×D9—121no—
325C2×D9—121no—
326C2×D9—122no—
327C2×D9—122no—
328C2×D9—122no—
329C4×C9—22no—
330C4×C9—22no—
331C4×C9—32no—
332C4×C9—32no—
333C4×C9—42no—
334C4×C9—62no—
335C4×C9#23512yescomplemented
336Dic9—21no—
337Dic9—61no—
338Dic9—121no—
339Dic9D3×D412yes—
340C2×#41—272no—
341C4⋊Q8—31no—
342#127—271no—
343#124—272no—
344#121—272no—
345#121—271no—
346#121—271no—
347C4×D4—91no—
348C4×D4—271no—
349C4×D4—272no—
350C2×C42—271no—
351C3×C9#53211yescharacteristic, complemented, Sylow
352C23×C3—92no—
353C22×D3—32no—
354C22×D3—91no—
355C22×D3—91no—
356C22×D3—92no—
357C22×D3—92no—
358C22×D3—181no—
359C22×D3—182no—
360C3×Q8—64no—
361C3×D4—182no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—32no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4—184no—
370C2×C3×C4C22×D912yes—
371C2×C3×C4C22×D911yescharacteristic
372C2×C3×C4C2×D3212yes—
373C2×C3×C4C2×D3211yescharacteristic
374#67—184no—
375#67—184no—
376#67—184no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—181no—
382C2×Dic3C22×D912yes—
383C2×Dic3C22×D911yescharacteristic
384C2×Dic3C22×D911yescharacteristic
385D12—62no—
386D12—184no—
387D12—184no—
388D12—184no—
389D12—184no—
390D12—184no—
391D12—182no—
392D12—182no—
393D12—362no—
394D12—364no—
395C4×D3—64no—
396C4×D3—94no—
397C4×D3—94no—
398C4×D3—184no—
399C4×D3—184no—
400C4×D3—182no—
401C4×D3—182no—
402C4×D3—182no—
403C4×D3—182no—
404C4×D3—362no—
405Dic6—24no—
406C2×C32C2×D3×D411yescharacteristic
407C2×C32C2×#23911yescharacteristic
408C2×C32C2×#23911yescharacteristic
409C3⋊D3—62no—
410C3⋊D3—122no—
411C3×D3—122no—
412C2×C9—21no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9C2×#23511yescharacteristic
416C2×C9C2×D3×D411yescharacteristic
417C2×C9D3×#4111yescharacteristic
418D9—42no—
419D9—62no—
420D9—122no—
421D9—122no—
422D9—242no—
423#41—548no—
424C2×Q8—32no—
425C2×D4—91no—
426C2×D4—272no—
427C2×D4—272no—
428C2×D4—272no—
429C2×D4—271no—
430C2×D4—272no—
431C22×C4—92no—
432C22×C4—271no—
433C22×C4—271no—
434C22×C4—272no—
435C22×C4—271no—
436C4⋊C4—32no—
437C4⋊C4—31no—
438C4⋊C4D3×D911yescharacteristic
439C22⋊C4—92no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C42—31no—
444C42—91no—
445C42—271no—
446C42—271no—
447C22×C3—21no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—182no—
451C22×C3C23×D911yescharacteristic
452C22×C3C22×D3211yescharacteristic
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—92no—
457C2×D3—92no—
458C2×D3—182no—
459C2×D3—181no—
460C2×D3—181no—
461C2×D3—181no—
462C2×D3—181no—
463C2×D3—182no—
464C2×D3—182no—
465C2×D3—182no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—361no—
470C2×D3—361no—
471C2×D3—362no—
472C2×D3—362no—
473C2×D3—362no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—32no—
478C3×C4—32no—
479C3×C4—42no—
480C3×C4—62no—
481C3×C4—92no—
482C3×C4—181no—
483C3×C4#370112yescomplemented
484C3×C4#379912yes—
485Dic3—22no—
486Dic3—32no—
487Dic3—61no—
488Dic3—181no—
489Dic3—181no—
490Dic3—361no—
491Dic3D4×D912yes—
492Dic3#370112yescomplemented
493C32#572511yescharacteristic
494C9—21no—
495C9#554811yescharacteristic, complemented
496C23—92no—
497C23—271no—
498C23—272no—
499Q8—64no—
500D4—182no—
501D4—544no—
502D4—544no—
503D4—544no—
504D4—544no—
505D4—542no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—32no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—184no—
512C2×C4—274no—
513C2×C4—271no—
514C2×C4—542no—
515C2×C4—544no—
516C2×C4—542no—
517C2×C4C2×D3×D912yes—
518C2×C4C2×D3×D911yescharacteristic
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—362no—
523C2×C3C2×D4×D911yescharacteristic
524C2×C3C2×#370111yescharacteristic
525C2×C3C2×#370111yescharacteristic
526C2×C3C2×#379911yescharacteristic
527C2×C3D32×D411yescharacteristic
528C2×C3D3×#23911yescharacteristic
529D3—122no—
530D3—182no—
531D3—182no—
532D3—362no—
533D3—362no—
534D3—362no—
535D3—722no—
536C22—182no—
537C22—182no—
538C22—182no—
539C22—272no—
540C22—541no—
541C22—541no—
542C22—542no—
543C22—542no—
544C22—542no—
545C22C22×D3×D911yescharacteristic
546C4—22no—
547C4—32no—
548C4—32no—
549C4—62no—
550C4—92no—
551C4—181no—
552C4—541no—
553C4#8863512yescomplemented
554C3—21no—
555C3#1058982711yescharacteristic, complemented
556C3#1059451811yescharacteristic
557C2—362no—
558C2—542no—
559C2—1082no—
560C2C2×#8863511yescharacteristic
561C2D3×D4×D911yescharacteristic
562C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88635C2semidirect
2. #11721751C2semidirect
3. C3×#63455C2semidirect
4. C9×#888C2semidirect
5. #11721007C2semidirect
6. #11721006C2semidirect
7. #11721005C2semidirect
8. #11721003C2semidirect
9. #11721002C2semidirect
10. #11720992C2semidirect
11. C4×(C3⋊D9)⋊C4C2semidirect
58. #63455D3semidirect
83. (C3⋊D9)⋊C4D4semidirect
150. C3×C4×C9#41semidirect
154. C9×Dic3#41semidirect
173. #888D9semidirect
335. C4×C9#235semidirect
351. C3×C9#532semidirect
483. C3×C4#3701semidirect
492. Dic3#3701semidirect
495. C9#5548semidirect
553. C4#88635semidirect
555. C3#10589827semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
#117217512
C3×#634552
C9×#8882
#117210072
#117210062
#117210052
#117210032
#117210022
#117209922
C4×(C3⋊D9)⋊C42
#105945183
#105898273