Finite Group Databasebeta

← #13148831

Subgroups — #13148831

Order 1728.

1030 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88635C211yesmaximal, characteristic, complemented
2C2×#88634C211yesmaximal, characteristic, complemented
3#11721751C211yesmaximal, characteristic, complemented
4C3×#63457C211yesmaximal, characteristic, complemented
5C9×#887C211yesmaximal, characteristic, complemented
6#11721006C211yesmaximal, characteristic, complemented
7#11721006C212yesmaximal, complemented
8#11721005C212yesmaximal, complemented
9C4×#6192C211yesmaximal, characteristic, complemented
10#11720983C212yesmaximal, complemented
11#11720977C212yesmaximal, complemented
12#10594524—31nomaximal
13#10589679—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4⋊(C3⋊D9)C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×D36C2211yescharacteristic
18C2×C3×D36C2212yes—
19C2×C3×C4×D9C2211yescharacteristic
20C2×C9×Dic6C2211yescharacteristic
21C2×#6192C2212yes—
22C2×#6192C2212yes—
23C2×#6192C2211yescharacteristic
24C2×#6192C2212yes—
25C2×#6192C2212yes—
26C2×(C3⋊D9)⋊C4C2212yes—
27C2×Dic3×D9C2212yes—
28#88635—24no—
29#88634—24no—
30C3×C9×C4⋊C4C2211yescharacteristic
31#88536C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C9×Dic3⋊C4C2212yes—
34C4×C9×Dic3C2212yes—
35C4×C9×Dic3C2211yescharacteristic
36#88445C2211yescharacteristic
37(C2×D9)⋊Dic3C2211yescharacteristic
38Dic3⋊Dic9C2211yescharacteristic
39C2×#3799—31no—
40C2×#3796—31no—
41#64104—31no—
42C3×#895—31no—
43C3×#887D311yescharacteristic
44#63911—32no—
45#63911—31no—
46#63910—32no—
47C4×#626—31no—
48#63886—32no—
49#63880—32no—
50C2×#3701—31no—
51C2×#3701—31no—
52C4×Q8×C9—31no—
53#63457—32no—
54#63457—31no—
55#63457—32no—
56#63457—61no—
57#63457D311yescharacteristic, complemented
58#63436—32no—
59#63436—31no—
60C4×D36—31no—
61C4×D36—32no—
62C22×C3⋊D9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×C3×D9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C4⋊(C3⋊D9)—22no—
70C4⋊(C3⋊D9)—42no—
71C4×C3⋊D9—22no—
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2312yes—
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C3×D36—22no—
77C3×D36—42no—
78C3×C4×D9—22no—
79C9×Dic6D414yescomplemented
80#6192—24no—
81#6192—24no—
82#6192—24no—
83#6192—22no—
84#6192—42no—
85(C3⋊D9)⋊C4—24no—
86Dic3×D9—24no—
87#5599—91no—
88#5577—91no—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4⋊(C3⋊D3)—32no—
91C2×C4×C3⋊D3—31no—
92C2×C3×D12—31no—
93C2×C3×D12—32no—
94C2×C3×C4×D3—31no—
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#626—32no—
97C2×#626—32no—
98C2×#626—32no—
99C2×#626—31no—
100C2×#626—32no—
101C2×#624—32no—
102C2×D3×Dic3—32no—
103#3799—64no—
104#3796—64no—
105C32×C4⋊C4C2×D311yescharacteristic
106#3752—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×Dic3⋊C4C2×D312yes—
109C3×C4×Dic3C2×D312yes—
110C3×C4×Dic3C2×D311yescharacteristic
111Dic3⋊Dic3—31no—
112#3722—31no—
113(C2×D3)⋊Dic3—31no—
114C2×Q8×C9—31no—
115#3701—64no—
116#3701—64no—
117C2×D36—31no—
118C2×D36—32no—
119C2×D36—32no—
120C2×D36—32no—
121C2×D36—31no—
122C2×D36—32no—
123C2×D36—32no—
124C2×D36—61no—
125C2×D36—62no—
126C2×D36C2×D311yescharacteristic
127C2×D36C2×D312yes—
128C2×C4×D9—32no—
129C2×C4×D9—31no—
130C2×C4×D9—32no—
131C2×C4×D9—61no—
132C2×C4×D9C2×D311yescharacteristic
133C9×C4⋊C4—21no—
134C9×C4⋊C4—32no—
135C9×C4⋊C4C2×D311yescharacteristic
136C42×C9—32no—
137C42×C9—31no—
138(C2×D9)⋊C4—31no—
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—31no—
141(C2×D9)⋊C4—62no—
142(C2×D9)⋊C4C2×D312yes—
143C4⋊Dic9—31no—
144C22×C3×C9C2411yescharacteristic
145C2×C3⋊D9—22no—
146C2×C3⋊D9—21no—
147C2×C3⋊D9—21no—
148C2×C3⋊D9—22no—
149C2×C3⋊D9—21no—
150C2×C3⋊D9—22no—
151C2×C3×D9—22no—
152C2×C3×D9—21no—
153C2×C3×D9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—22no—
156C2×C3×D9—21no—
157C3×C4×C9—22no—
158C3×C4×C9C2×D412yes—
159C3⋊Dic9—21no—
160C9×Dic3—21no—
161C9×Dic3C2×D414yes—
162C9×Dic3#4112yescomplemented
163C3×Dic9—21no—
164C2×#239—91no—
165C2×#239—91no—
166C2×#237—91no—
167C2×#235—91no—
168C3×#121—91no—
169C3×C4×Q8—31no—
170#940—92no—
171#940—91no—
172#939—92no—
173C4×#67—91no—
174#895—31no—
175#895—92no—
176#895—92no—
177#895—91no—
178#895—91no—
179#895—181no—
180#887D911yescharacteristic, complemented
181#884—92no—
182#67⋊C4—92no—
183#874—91no—
184#874—92no—
185C4×D12—92no—
186C4×D12—91no—
187C22×C3⋊D3—31no—
188C22×C3⋊D3—32no—
189C22×C3×D3—31no—
190C22×C3×D3—32no—
191C2×C32×C4C22×D312yes—
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3—31no—
194C4⋊(C3⋊D3)—62no—
195C4⋊(C3⋊D3)—122no—
196C4×C3⋊D3—62no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3C22×D311yescharacteristic
199C2×C3×Dic3C22×D312yes—
200C2×C3×Dic3C22×D311yescharacteristic
201C3×D12—62no—
202C3×D12—122no—
203C3×C4×D3—62no—
204C3×Dic6D1214yes—
205#626—64no—
206#626—64no—
207#626—64no—
208#626—62no—
209#626—122no—
210#624—64no—
211D3×Dic3—64no—
212C22×D9—32no—
213C22×D9—31no—
214C22×D9—61no—
215C22×D9—62no—
216C22×D9C22×D311yescharacteristic
217C22×D9C22×D312yes—
218Q8×C9—34no—
219C2×C4×C9—21no—
220C2×C4×C9—22no—
221C2×C4×C9—31no—
222C2×C4×C9—32no—
223C2×C4×C9—31no—
224C2×C4×C9C22×D312yes—
225C2×C4×C9C22×D311yescharacteristic
226C2×Dic9—31no—
227C2×Dic9—61no—
228C2×Dic9C22×D311yescharacteristic
229D36—22no—
230D36—42no—
231D36—62no—
232D36—64no—
233D36—64no—
234D36—64no—
235D36—62no—
236D36—122no—
237D36—122no—
238D36—124no—
239D36—122no—
240C4×D9—22no—
241C4×D9—64no—
242C4×D9—64no—
243C4×D9—62no—
244C4×D9—122no—
245#550—271noSylow
246C2×C3×C9C22×D411yescharacteristic
247C2×C3×C9C2×#4111yescharacteristic
248C2×C3×C9H(5,𝔽2)11yescharacteristic
249C3⋊D9—41no—
250C3⋊D9—42no—
251C3×D9—42no—
252C3×D9—41no—
253C2×C3×Q8—31no—
254C2×C3×D4—91no—
255C2×C3×D4—92no—
256C22×C3×C4—91no—
257C2×#67—92no—
258C2×#67—92no—
259C2×#67—91no—
260C2×#67—92no—
261C2×#67—92no—
262C22×Dic3—92no—
263#239—184no—
264#239—184no—
265#237—184no—
266#235—184no—
267C2×D12—31no—
268C2×D12—32no—
269C2×D12—92no—
270C2×D12—91no—
271C2×D12—92no—
272C2×D12—92no—
273C2×D12—91no—
274C2×D12—91no—
275C2×D12—92no—
276C2×D12—92no—
277C2×D12—92no—
278C2×D12—181no—
279C2×D12—182no—
280C2×C4×D3—31no—
281C2×C4×D3—92no—
282C2×C4×D3—92no—
283C2×C4×D3—91no—
284C2×C4×D3—91no—
285C2×C4×D3—92no—
286C2×C4×D3—181no—
287C2×Dic6C2×D911yescharacteristic
288C3×C4⋊C4—21no—
289C3×C4⋊C4—32no—
290C3×C4⋊C4C2×D911yescharacteristic
291C3×C4⋊C4D3211yescharacteristic
292C3×C22⋊C4—92no—
293C3×C42—32no—
294C3×C42—31no—
295#217—91no—
296(C2×D3)⋊C4—32no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—91no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—91no—
302(C2×D3)⋊C4—182no—
303C4⋊Dic3—91no—
304Dic3⋊C4—91no—
305Dic3⋊C4C2×D912yes—
306C4×Dic3C2×D912yes—
307C4×Dic3C2×D911yescharacteristic
308C22×C32C23×D311yescharacteristic
309C2×C3⋊D3—61no—
310C2×C3⋊D3—62no—
311C2×C3⋊D3—61no—
312C2×C3⋊D3—61no—
313C2×C3⋊D3—62no—
314C2×C3⋊D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—61no—
321C32×C4—22no—
322C32×C4C2×D1212yes—
323C3⋊Dic3—61no—
324C3×Dic3—21no—
325C3×Dic3—61no—
326C3×Dic3C2×D1214yes—
327C3×Dic3#23912yes—
328C22×C9—21no—
329C22×C9C23×D311yescharacteristic
330C2×D9—21no—
331C2×D9—22no—
332C2×D9—21no—
333C2×D9—22no—
334C2×D9—21no—
335C2×D9—22no—
336C2×D9—62no—
337C2×D9—62no—
338C2×D9—61no—
339C2×D9—61no—
340C2×D9—62no—
341C2×D9—61no—
342C2×D9—62no—
343C2×D9—121no—
344C2×D9—121no—
345C2×D9—122no—
346C2×D9—122no—
347C2×D9—122no—
348C4×C9—22no—
349C4×C9—22no—
350C4×C9—34no—
351C4×C9—32no—
352C4×C9—42no—
353C4×C9—61no—
354C4×C9D3×D412yes—
355Dic9—21no—
356Dic9—61no—
357Dic9—121no—
358C2×#41—271no—
359C2×#41—271no—
360#127—271no—
361#127—272no—
362#121—91no—
363#121—272no—
364#121—272no—
365#121—271no—
366C4×Q8—31no—
367C4×D4—272no—
368C4×D4—271no—
369C3×C9#55011yescharacteristic, complemented, Sylow
370C23×C3—91no—
371C23×C3—92no—
372C22×D3—32no—
373C22×D3—31no—
374C22×D3—91no—
375C22×D3—92no—
376C22×D3—92no—
377C22×D3—91no—
378C22×D3—181no—
379C22×D3—182no—
380C3×Q8—34no—
381C3×D4—182no—
382C3×D4—362no—
383C2×C3×C4—21no—
384C2×C3×C4—22no—
385C2×C3×C4—31no—
386C2×C3×C4—32no—
387C2×C3×C4—31no—
388C2×C3×C4—91no—
389C2×C3×C4—182no—
390C2×C3×C4C22×D912yes—
391C2×C3×C4C22×D911yescharacteristic
392C2×C3×C4C2×D3212yes—
393C2×C3×C4C2×D3211yescharacteristic
394#67—184no—
395#67—184no—
396#67—184no—
397#67—182no—
398#67—362no—
399C2×Dic3—31no—
400C2×Dic3—91no—
401C2×Dic3—91no—
402C2×Dic3—181no—
403C2×Dic3—184no—
404C2×Dic3C22×D911yescharacteristic
405C2×Dic3C22×D912yes—
406C2×Dic3C22×D911yescharacteristic
407D12—62no—
408D12—122no—
409D12—184no—
410D12—182no—
411D12—184no—
412D12—184no—
413D12—182no—
414D12—182no—
415D12—362no—
416D12—362no—
417D12—362no—
418D12—364no—
419D12—362no—
420C4×D3—62no—
421C4×D3—184no—
422C4×D3—184no—
423C4×D3—182no—
424C4×D3—182no—
425C4×D3—184no—
426C4×D3—362no—
427Dic6D3614yescomplemented
428C2×C32C22×D1211yescharacteristic
429C2×C32C2×#23911yescharacteristic
430C2×C32#100911yescharacteristic
431C3⋊D3—121no—
432C3⋊D3—122no—
433C3×D3—122no—
434C3×D3—121no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9C2×D3×D411yescharacteristic
439C2×C9#100411yescharacteristic
440C2×C9D3×#4111yescharacteristic
441D9—41no—
442D9—42no—
443D9—122no—
444D9—121no—
445D9—122no—
446D9—242no—
447#41—544no—
448#41—544no—
449C2×Q8—31no—
450C2×D4—91no—
451C2×D4—92no—
452C2×D4—272no—
453C2×D4—271no—
454C2×D4—272no—
455C2×D4—272no—
456C2×D4—272no—
457C2×D4—271no—
458C2×D4—272no—
459C22×C4—91no—
460C22×C4—272no—
461C22×C4—272no—
462C22×C4—271no—
463C4⋊C4—32no—
464C4⋊C4—271no—
465C4⋊C4D3×D911yescharacteristic
466C22⋊C4—92no—
467C22⋊C4—271no—
468C22⋊C4—271no—
469C22⋊C4—272no—
470C42—32no—
471C42—31no—
472C22×C3—21no—
473C22×C3—182no—
474C22×C3—181no—
475C22×C3—181no—
476C22×C3—182no—
477C22×C3—182no—
478C22×C3—181no—
479C22×C3C23×D911yescharacteristic
480C22×C3C22×D3211yescharacteristic
481C2×D3—61no—
482C2×D3—62no—
483C2×D3—61no—
484C2×D3—62no—
485C2×D3—62no—
486C2×D3—61no—
487C2×D3—182no—
488C2×D3—181no—
489C2×D3—182no—
490C2×D3—181no—
491C2×D3—182no—
492C2×D3—181no—
493C2×D3—182no—
494C2×D3—182no—
495C2×D3—181no—
496C2×D3—181no—
497C2×D3—182no—
498C2×D3—181no—
499C2×D3—182no—
500C2×D3—361no—
501C2×D3—361no—
502C2×D3—362no—
503C2×D3—362no—
504C2×D3—362no—
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—34no—
509C3×C4—32no—
510C3×C4—42no—
511C3×C4—61no—
512C3×C4—181no—
513C3×C4C2×D3612yes—
514C3×C4D3×D1212yes—
515Dic3—21no—
516Dic3—61no—
517Dic3—181no—
518Dic3—181no—
519Dic3—361no—
520Dic3C2×D3614yes—
521Dic3#370112yescomplemented
522C32#557711yescharacteristic
523C9—21no—
524C9#559911yescharacteristic, complemented
525C23—92no—
526C23—91no—
527C23—271no—
528C23—272no—
529Q8—34no—
530D4—182no—
531D4—362no—
532D4—544no—
533D4—544no—
534D4—544no—
535D4—542no—
536D4—542no—
537D4—1082no—
538D4—1082no—
539C2×C4—31no—
540C2×C4—32no—
541C2×C4—31no—
542C2×C4—91no—
543C2×C4—182no—
544C2×C4—271no—
545C2×C4—544no—
546C2×C4—544no—
547C2×C4—542no—
548C2×C4C2×D3×D912yes—
549C2×C4C2×D3×D911yescharacteristic
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—362no—
554C2×C3—361no—
555C2×C3C22×D3611yescharacteristic
556C2×C3C2×#370111yescharacteristic
557C2×C3#6371511yescharacteristic
558C2×C3C2×D3×D1211yescharacteristic
559C2×C3#6431411yescharacteristic
560C2×C3D3×#23911yescharacteristic
561D3—121no—
562D3—122no—
563D3—362no—
564D3—361no—
565D3—362no—
566D3—361no—
567D3—362no—
568D3—722no—
569C22—181no—
570C22—182no—
571C22—182no—
572C22—181no—
573C22—181no—
574C22—182no—
575C22—541no—
576C22—542no—
577C22—541no—
578C22—542no—
579C22—541no—
580C22—542no—
581C22C22×D3×D911yescharacteristic
582C4—22no—
583C4—32no—
584C4—34no—
585C4—61no—
586C4—181no—
587C4—541no—
588C4D3×D3612yes—
589C3—21no—
590C3#1058967911yescharacteristic, complemented
591C3#1059452411yescharacteristic
592C2—361no—
593C2—362no—
594C2—1081no—
595C2—1082no—
596C2C2×D3×D3611yescharacteristic
597C2#1172250111yescharacteristic
598C2D3×#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#550
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. C2×#88634C2semidirect
3. #11721751C2semidirect
4. C3×#63457C2semidirect
5. C9×#887C2semidirect
6. #11721006C2semidirect
7. #11721006C2semidirect
8. #11721005C2semidirect
9. C4×#6192C2semidirect
10. #11720983C2semidirect
11. #11720977C2semidirect
57. #63457D3semidirect
79. C9×Dic6D4semidirect
162. C9×Dic3#41semidirect
180. #887D9semidirect
369. C3×C9#550semidirect
427. Dic6D36semidirect
521. Dic3#3701semidirect
524. C9#5599semidirect
590. C3#10589679semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
C2×#886342
#117217512
C3×#634572
C9×#8872
#117210062
#117210052
C4×#61922
#117209832
#117209772
#105945243
#105896793