Finite Group Databasebeta

← #13150189

Subgroups — #13150189

Order 1728.

1030 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic, complemented
2C2×#88634C211yesmaximal, characteristic, complemented
3C4×#6233C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721289C212yesmaximal, complemented
6#11721270C212yesmaximal, complemented
7#11721045C211yesmaximal, characteristic, complemented
8#11721043C211yesmaximal, characteristic, complemented
9#11721034C212yesmaximal, complemented
10#11720976C212yesmaximal, complemented
11#11720911C211yesmaximal, characteristic, complemented
12#10595006—31nomaximal
13#10589936—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C2×C3×#611C2212yes—
19C9×#235—24no—
20C2×C9×D12C2211yescharacteristic
21C2×C4×D3×C9C2212yes—
22C2×C3×D36C2211yescharacteristic
23C2×C9×Dic6C2211yescharacteristic
24C2×#6195C2212yes—
25C2×#6194C2212yes—
26C2×#6194C2212yes—
27C2×#6192C2212yes—
28C2×Dic3×D9C2212yes—
29#88634—24no—
30#88551C2211yescharacteristic
31#88536C2211yescharacteristic
32#88534C2211yescharacteristic
33C4×C3⋊Dic9C2211yescharacteristic
34C3×(C2×D9)⋊C4C2212yes—
35C3×C4⋊Dic9C2211yescharacteristic
36(C2×D3)⋊Dic9C2212yes—
37Dic3⋊Dic9C2212yes—
38Dic3×Dic9C2212yes—
39C2×C3×#235D311yescharacteristic
40C2×#3796—31no—
41C4×#638—31no—
42C3×#930—31no—
43#63979—32no—
44#63970—32no—
45#63916—31no—
46#63914—31no—
47#63906—32no—
48#63879—32no—
49#63843—31no—
50C2×C9×#41—31no—
51C2×#3701—31no—
52Q8×Dic9—31no—
53#63502—32no—
54#63502—31no—
55#63500—32no—
56#63500—31no—
57#63492—32no—
58#63492D311yescharacteristic, complemented
59C4×#611—31no—
60C4×#611—32no—
61C4×#611—61no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2312yes—
65C22×C3×D9C2312yes—
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69#6233—22no—
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C4×C3⋊D9—22no—
74C9×#67—24no—
75C3×#611—42no—
76C2×C9×Dic3C2312yes—
77C2×C3×Dic9C2312yes—
78C9×D12—22no—
79C4×D3×C9—24no—
80C3×D36—22no—
81C9×Dic6D414yescomplemented
82#6195—42no—
83#6194—24no—
84#6194—42no—
85#6192—24no—
86Dic3×D9—24no—
87#5834—91no—
88#5599—91no—
89C22×C32×C4C2×D311yescharacteristic
90C2×#638—31no—
91C2×C4×C3⋊D3—31no—
92C2×C3×#67—32no—
93C2×C3×#67C2×D312yes—
94C3×#235—24no—
95C2×C3×D12—31no—
96C2×C3×D12C2×D311yescharacteristic
97C2×C3×C4×D3C2×D312yes—
98C2×C3×Dic6C2×D311yescharacteristic
99C2×#626—32no—
100C2×#626—32no—
101C2×#625—32no—
102C2×#625—32no—
103C2×D3×Dic3—32no—
104#3796—64no—
105#3757—31no—
106#3752—31no—
107#3750—31no—
108C4×C3⋊Dic3—31no—
109C3×(C2×D3)⋊C4—32no—
110C3×C4⋊Dic3—31no—
111Dic3⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113Dic32—32no—
114C9×#41—64no—
115C2×Q8×C9—31no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C22×C4×C9—21no—
119C22×C4×C9—32no—
120C22×C4×C9C2×D311yescharacteristic
121C2×#611—31no—
122C2×#611—32no—
123C2×#611—32no—
124C2×#611—32no—
125C2×#611—61no—
126C2×#611C2×D312yes—
127#3701—64no—
128C2×D36—32no—
129C2×D36C2×D311yescharacteristic
130C2×C4×D9—32no—
131C2×C4×D9—31no—
132C2×C4×D9—61no—
133#3676—31no—
134#3676—32no—
135#3676—61no—
136(C2×D9)⋊C4—31no—
137(C2×D9)⋊C4—61no—
138(C2×D9)⋊C4C2×D312yes—
139C4⋊Dic9—32no—
140C4⋊Dic9C2×D311yescharacteristic
141Dic9⋊C4—31no—
142Dic9⋊C4—61no—
143C4×Dic9—31no—
144C4×Dic9—32no—
145C4×Dic9—61no—
146C22×C3×C9—21no—
147C22×C3×C9—21no—
148C22×C3×C9—21no—
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—21no—
151C2×C3⋊D9—21no—
152C2×C3⋊D9—21no—
153C2×D3×C9—22no—
154C2×D3×C9—22no—
155C2×D3×C9—22no—
156C2×C3×D9—22no—
157C2×C3×D9—22no—
158C2×C3×D9—22no—
159C3×C4×C9—21no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—21no—
162C3⋊Dic9—21no—
163C3⋊Dic9#4112yescomplemented
164C9×Dic3C2×D414yes—
165C3×Dic9—22no—
166C2×C3×#41—31no—
167C2×#239—91no—
168C2×#237—91no—
169C2×#235D911yescharacteristic, complemented
170C3×#121—91no—
171Q8×Dic3—91no—
172#940—91no—
173#940—92no—
174#940—92no—
175#940—91no—
176#939—92no—
177#938—91no—
178#938—92no—
179#930—31no—
180#930—92no—
181C4×#67—92no—
182C4×#67—91no—
183C4×#67—91no—
184C4×#67—181no—
185#895—91no—
186#887—91no—
187#884—92no—
188#67⋊C4—92no—
189C23×C32C22×D311yescharacteristic
190C22×C3⋊D3—31no—
191C22×C3×D3—32no—
192C22×C3×D3C22×D312yes—
193C2×C32×C4—22no—
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196#638—62no—
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—31no—
200C4×C3⋊D3—62no—
201C3×#67—24no—
202C3×#67—122no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3C22×D312yes—
205C3×D12—22no—
206C3×D12—62no—
207C3×C4×D3—24no—
208C3×Dic6#6714yes—
209#626—64no—
210#626—122no—
211#625—64no—
212#625—122no—
213D3×Dic3—64no—
214C23×C9—21no—
215C23×C9—32no—
216C23×C9C22×D311yescharacteristic
217C22×D9—31no—
218C22×D9—61no—
219C22×D9C22×D312yes—
220Q8×C9—34no—
221D4×C9—64no—
222D4×C9—62no—
223C2×C4×C9—22no—
224C2×C4×C9—21no—
225C2×C4×C9—21no—
226C2×C4×C9—32no—
227C2×C4×C9—42no—
228C2×C4×C9—64no—
229C2×C4×C9C22×D311yescharacteristic
230C2×C4×C9C22×D311yescharacteristic
231#611—42no—
232#611—64no—
233#611—62no—
234#611—64no—
235#611—122no—
236#611—122no—
237C2×Dic9—31no—
238C2×Dic9—31no—
239C2×Dic9—31no—
240C2×Dic9—61no—
241C2×Dic9—61no—
242C2×Dic9—61no—
243C2×Dic9C22×D312yes—
244D36—22no—
245D36—64no—
246C4×D9—64no—
247C4×D9—62no—
248C4×D9—122no—
249#550—271noSylow
250C2×C3×C9—41no—
251C2×C3×C9C22×D411yescharacteristic
252C2×C3×C9C2×#4111yescharacteristic
253C2×C3×C9H(5,𝔽2)11yescharacteristic
254C3⋊D9—41no—
255D3×C9—42no—
256C3×D9—42no—
257C3×#41—64no—
258C2×C3×Q8—31no—
259C2×C3×D4—32no—
260C2×C3×D4—31no—
261C2×C3×D4—92no—
262C2×C3×D4—91no—
263C22×C3×C4—21no—
264C22×C3×C4—32no—
265C22×C3×C4C2×D911yescharacteristic
266C22×C3×C4D3211yescharacteristic
267C2×#67—32no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—91no—
271C2×#67—91no—
272C2×#67—92no—
273C2×#67—92no—
274C2×#67—92no—
275C2×#67—92no—
276C2×#67—181no—
277C2×#67C2×D912yes—
278C22×Dic3—92no—
279#239—184no—
280#237—184no—
281#235—24no—
282C2×D12—31no—
283C2×D12—92no—
284C2×D12—92no—
285C2×D12C2×D911yescharacteristic
286C2×C4×D3—92no—
287C2×C4×D3—91no—
288C2×C4×D3—91no—
289C2×C4×D3—181no—
290C2×C4×D3C2×D912yes—
291C2×Dic6C2×D911yescharacteristic
292C3×C4⋊C4—91no—
293C3×C22⋊C4—92no—
294#217—92no—
295#217—91no—
296#217—91no—
297#217—181no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—91no—
301(C2×D3)⋊C4—91no—
302(C2×D3)⋊C4—181no—
303C4⋊Dic3—31no—
304C4⋊Dic3—92no—
305Dic3⋊C4—91no—
306Dic3⋊C4—91no—
307Dic3⋊C4—92no—
308Dic3⋊C4—181no—
309C4×Dic3—92no—
310C4×Dic3—91no—
311C4×Dic3—91no—
312C4×Dic3—92no—
313C4×Dic3—181no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32C23×D311yescharacteristic
318C2×C3⋊D3—61no—
319C2×C3⋊D3—61no—
320C2×C3⋊D3—61no—
321C2×C3×D3—22no—
322C2×C3×D3—22no—
323C2×C3×D3—22no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C2×C3×D3—62no—
327C32×C4—21no—
328C32×C4C2×#6712yes—
329C3⋊Dic3—32no—
330C3⋊Dic3—61no—
331C3⋊Dic3—61no—
332C3×Dic3—62no—
333C3×Dic3C2×#6714yes—
334C22×C9—22no—
335C22×C9—21no—
336C22×C9—21no—
337C22×C9—21no—
338C22×C9—21no—
339C22×C9—41no—
340C22×C9—41no—
341C22×C9—62no—
342C22×C9—62no—
343C22×C9—62no—
344C22×C9C23×D311yescharacteristic
345C2×D9—22no—
346C2×D9—22no—
347C2×D9—22no—
348C2×D9—62no—
349C2×D9—61no—
350C2×D9—61no—
351C2×D9—61no—
352C2×D9—121no—
353C2×D9—121no—
354C4×C9—21no—
355C4×C9—22no—
356C4×C9—34no—
357C4×C9—41no—
358C4×C9D3×D412yes—
359Dic9—22no—
360Dic9—32no—
361Dic9—62no—
362Dic9—61no—
363Dic9—61no—
364Dic9—121no—
365Dic9—121no—
366C2×#41—31no—
367C2×#41—271no—
368#127—272no—
369#127—271no—
370#121—91no—
371#121—272no—
372#121—271no—
373#121—272no—
374C4×Q8—271no—
375C4×D4—272no—
376C4×D4—271no—
377C3×C9#55011yescharacteristic, complemented, Sylow
378C23×C3—21no—
379C23×C3—32no—
380C23×C3—92no—
381C23×C3C22×D911yescharacteristic
382C23×C3C2×D3211yescharacteristic
383C22×D3—32no—
384C22×D3—91no—
385C22×D3—91no—
386C22×D3—181no—
387C22×D3C22×D912yes—
388C3×Q8—34no—
389C3×D4—64no—
390C3×D4—62no—
391C3×D4—182no—
392C3×D4—362no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—32no—
398C2×C3×C4—42no—
399C2×C3×C4—64no—
400C2×C3×C4—92no—
401C2×C3×C4C22×D911yescharacteristic
402C2×C3×C4C22×D911yescharacteristic
403C2×C3×C4C2×D3211yescharacteristic
404C2×C3×C4C2×D3211yescharacteristic
405#67—24no—
406#67—122no—
407#67—184no—
408#67—184no—
409#67—182no—
410#67—182no—
411#67—184no—
412#67—184no—
413#67—362no—
414#67—362no—
415#67—362no—
416C2×Dic3—32no—
417C2×Dic3—91no—
418C2×Dic3—91no—
419C2×Dic3—91no—
420C2×Dic3—91no—
421C2×Dic3—91no—
422C2×Dic3—91no—
423C2×Dic3—181no—
424C2×Dic3—181no—
425C2×Dic3—181no—
426C2×Dic3—184no—
427C2×Dic3C22×D912yes—
428D12—22no—
429D12—62no—
430D12—184no—
431D12—362no—
432C4×D3—24no—
433C4×D3—184no—
434C4×D3—182no—
435C4×D3—182no—
436C4×D3—362no—
437Dic6#61114yescomplemented
438C2×C32—41no—
439C2×C32D3×#4111yescharacteristic
440C2×C32#100911yescharacteristic
441C2×C32C22×#6711yescharacteristic
442C3⋊D3—121no—
443C3×D3—42no—
444C3×D3—122no—
445C2×C9—21no—
446C2×C9—21no—
447C2×C9—21no—
448C2×C9—42no—
449C2×C9—41no—
450C2×C9—122no—
451C2×C9C2×D3×D411yescharacteristic
452C2×C9#100411yescharacteristic
453C2×C9D3×#4111yescharacteristic
454D9—42no—
455D9—122no—
456D9—121no—
457#41—64no—
458#41—544no—
459C2×Q8—31no—
460C2×D4—32no—
461C2×D4—31no—
462C2×D4—92no—
463C2×D4—91no—
464C2×D4—272no—
465C2×D4—272no—
466C2×D4—272no—
467C2×D4—271no—
468C2×D4—272no—
469C22×C4—32no—
470C22×C4—272no—
471C22×C4—271no—
472C22×C4D3×D911yescharacteristic
473C4⋊C4—91no—
474C4⋊C4—272no—
475C4⋊C4—271no—
476C22⋊C4—92no—
477C22⋊C4—272no—
478C22⋊C4—271no—
479C22⋊C4—271no—
480C42—272no—
481C42—271no—
482C22×C3—22no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—62no—
493C22×C3—62no—
494C22×C3—62no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3—182no—
498C22×C3C23×D911yescharacteristic
499C22×C3C22×D3211yescharacteristic
500C2×D3—22no—
501C2×D3—22no—
502C2×D3—22no—
503C2×D3—62no—
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—182no—
507C2×D3—181no—
508C2×D3—181no—
509C2×D3—181no—
510C2×D3—181no—
511C2×D3—181no—
512C2×D3—181no—
513C2×D3—361no—
514C2×D3—361no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—22no—
518C3×C4—34no—
519C3×C4—41no—
520C3×C4—182no—
521C3×C4C2×#61112yes—
522C3×C4D3×#6712yes—
523Dic3—62no—
524Dic3—92no—
525Dic3—92no—
526Dic3—182no—
527Dic3—181no—
528Dic3—181no—
529Dic3—181no—
530Dic3—181no—
531Dic3—361no—
532Dic3—361no—
533Dic3C2×#61114yes—
534C32#583411yescharacteristic
535C9—21no—
536C9#559911yescharacteristic, complemented
537C23—32no—
538C23—92no—
539C23—271no—
540C23C2×D3×D911yescharacteristic
541Q8—34no—
542D4—64no—
543D4—62no—
544D4—182no—
545D4—362no—
546D4—544no—
547D4—542no—
548D4—544no—
549D4—1082no—
550D4—1082no—
551C2×C4—22no—
552C2×C4—32no—
553C2×C4—64no—
554C2×C4—92no—
555C2×C4—271no—
556C2×C4—271no—
557C2×C4—271no—
558C2×C4—544no—
559C2×C4—542no—
560C2×C4C2×D3×D911yescharacteristic
561C2×C4C2×D3×D911yescharacteristic
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—42no—
566C2×C3—41no—
567C2×C3—41no—
568C2×C3—122no—
569C2×C3—362no—
570C2×C3#41×D911yescharacteristic
571C2×C3#6371511yescharacteristic
572C2×C3C22×#61111yescharacteristic
573C2×C3#6430611yescharacteristic
574C2×C3#6430711yescharacteristic
575C2×C3C2×D3×#6711yescharacteristic
576D3—42no—
577D3—122no—
578D3—362no—
579D3—361no—
580D3—361no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—62no—
585C22—62no—
586C22—62no—
587C22—182no—
588C22—182no—
589C22—182no—
590C22—541no—
591C22—541no—
592C22—541no—
593C22C22×D3×D911yescharacteristic
594C4—21no—
595C4—34no—
596C4—182no—
597C4—272no—
598C4—541no—
599C4—541no—
600C4D3×#61112yes—
601C3—21no—
602C3#1058993611yescharacteristic, complemented
603C3#1059500611yescharacteristic
604C2—41no—
605C2—122no—
606C2—362no—
607C2—1081no—
608C2#1172247511yescharacteristic
609C2#1172247711yescharacteristic
610C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#550
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×C9×#235C2semidirect
2. C2×#88634C2semidirect
3. C4×#6233C2semidirect
4. C3×#63492C2semidirect
5. #11721289C2semidirect
6. #11721270C2semidirect
7. #11721045C2semidirect
8. #11721043C2semidirect
9. #11721034C2semidirect
10. #11720976C2semidirect
11. #11720911C2semidirect
58. #63492D3semidirect
81. C9×Dic6D4semidirect
163. C3⋊Dic9#41semidirect
169. C2×#235D9semidirect
377. C3×C9#550semidirect
437. Dic6#611semidirect
536. C9#5599semidirect
602. C3#10589936semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C2×#886342
C4×#62332
C3×#634922
#117212892
#117212702
#117210452
#117210432
#117210342
#117209762
#117209112
#105950063
#105899363