Finite Group Databasebeta

← #13148212

Subgroups — #13148212

Order 1728.

1132 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6195C211yesmaximal, characteristic, complemented
2C2×#88662C211yesmaximal, characteristic, complemented
3#11721728C211yesmaximal, characteristic, complemented
4C9×#882C211yesmaximal, characteristic, complemented
5C3×#63439C211yesmaximal, characteristic, complemented
6#11721291C211yesmaximal, characteristic, complemented
7#11721287C211yesmaximal, characteristic, complemented
8#11721040C212yesmaximal, complemented
9#11721039C212yesmaximal, complemented
10#11721017C212yesmaximal, complemented
11#11720956C212yesmaximal, complemented
12#10594080—31nomaximal
13#10589694—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C23×D3×C9C2211yescharacteristic
16C2×#6233C2211yescharacteristic
17C2×C4⋊(C3⋊D9)C2212yescomplemented
18C2×C9×#67C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×C9×D12C2212yescomplemented
21C2×#6195—24no—
22C2×#6195C2212yes—
23C2×#6195C2212yes—
24C2×#6195C2211yescharacteristic, complemented
25C2×#6195C2212yescomplemented
26C2×#6195C2212yescomplemented
27#88662—24no—
28C2×D3×Dic9C2211yescharacteristic, complemented
29C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
30C2×#6189C2211yescharacteristic, complemented
31C3×C9×C22⋊C4C2211yescharacteristic
32#88536C2212yescomplemented
33C3×#3676C2211yescharacteristic
34C9×(C2×D3)⋊C4C2212yescomplemented
35C3×Dic9⋊C4C2212yescomplemented
36C3×C4×Dic9C2212yescomplemented
37(C2×D3)⋊Dic9C2212yescomplemented
38#88445C2212yescomplemented
39C22×#626—31no—
40C2×#3804—31no—
41#64091—31no—
42C3×#882D311yescharacteristic
43C3×#877—31no—
44#63981—31no—
45#63977—31no—
46#63911—32no—
47#63910—32no—
48#63886—32no—
49#63852—32no—
50C22×#611—31no—
51C2×#3699—31no—
52C9×#120—31no—
53#63515—31no—
54#63502—32no—
55#63501—32no—
56#63499—31no—
57#63447—32no—
58#63446—32no—
59#63444—31no—
60#63444—61no—
61#63439D311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—22no—
64C22×C3⋊D9—42no—
65C22×C3⋊D9C2312yes—
66C22×C3⋊D9C2311yescharacteristic
67C22×D3×C9—22no—
68C22×D3×C9—42no—
69C22×D3×C9C2312yes—
70C22×D3×C9C2311yescharacteristic
71C2×C3×C4×C9C2312yes—
72#6233—22no—
73C2×C3⋊Dic9C2311yescharacteristic
74C4⋊(C3⋊D9)—42no—
75C9×#67—22no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9D414yescomplemented
78C2×C3×Dic9C2312yes—
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2311yescharacteristic
81C9×D12—42no—
82#6195—22no—
83#6195—24no—
84#6195—24no—
85#6195—42no—
86#6195—42no—
87D3×Dic9—22no—
88(C3⋊D9)⋊C4—22no—
89#6189—22no—
90#5592—91no—
91#5525—91no—
92C23×C3⋊D3—31no—
93C23×C3×D3C2×D311yescharacteristic
94C2×#638—31no—
95C2×C4⋊(C3⋊D3)—32no—
96C2×C3×#67C2×D311yescharacteristic
97C22×C3×Dic3—31no—
98C2×C3×D12C2×D312yes—
99C2×#627—31no—
100C2×#626—31no—
101C2×#626—32no—
102C2×#626—32no—
103C2×#626—32no—
104C2×#626—32no—
105C2×#626—64no—
106C2×#624—31no—
107#3804—64no—
108C2×D3×Dic3—31no—
109C24×C9—31no—
110C23×D9—31no—
111C23×D9—61no—
112C32×C22⋊C4C2×D311yescharacteristic
113#3752—32no—
114C3×#217—31no—
115C3×(C2×D3)⋊C4C2×D312yes—
116C3×Dic3⋊C4—32no—
117C3×C4×Dic3—32no—
118#3722—32no—
119(C2×D3)⋊Dic3—32no—
120C2×D4×C9—31no—
121C2×D4×C9—32no—
122C2×#611—31no—
123C2×#611—32no—
124C2×#611—31no—
125C2×#611—32no—
126C2×#611—32no—
127C2×#611—32no—
128C2×#611—61no—
129C2×#611—64no—
130C22×Dic9—31no—
131C22×Dic9C2×D311yescharacteristic
132#3699—64no—
133C2×D36—32no—
134C2×D36—62no—
135C2×C4×D9—31no—
136C2×Dic18—31no—
137C9×C22⋊C4—21no—
138C9×C22⋊C4—32no—
139C9×C22⋊C4C2×D311yescharacteristic
140#3676—32no—
141#3676C2×D311yescharacteristic
142(C2×D9)⋊C4—32no—
143(C2×D9)⋊C4—32no—
144(C2×D9)⋊C4—62no—
145Dic9⋊C4C2×D312yescomplemented
146C4×Dic9C2×D312yescomplemented
147C22×C3×C9—22no—
148C22×C3×C9C2×D412yes—
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—21no—
151C2×C3⋊D9—22no—
152C2×C3⋊D9—22no—
153C2×C3⋊D9—22no—
154C2×C3⋊D9—22no—
155C2×C3⋊D9—44no—
156C2×D3×C9—21no—
157C2×D3×C9—22no—
158C2×D3×C9—22no—
159C2×D3×C9—22no—
160C2×D3×C9—22no—
161C2×D3×C9—44no—
162C3×C4×C9—22no—
163C3⋊Dic9—21no—
164C9×Dic3—21no—
165C3×Dic9—22no—
166C3×Dic9C2×D412yes—
167C3×Dic9C2×D412yes—
168C22×#67—91no—
169C2×#237—91no—
170C2×#235—91no—
171C22×D12—91no—
172C3×#120—31no—
173C3×#117—91no—
174#953—91no—
175#940—92no—
176#939—92no—
177#937—91no—
178#895—92no—
179#895—92no—
180#885—92no—
181#884—92no—
182#882—91no—
183#882—91no—
184#882—91no—
185#882—91no—
186#882—181no—
187#882D911yescharacteristic, complemented
188#877—31no—
189#875—92no—
190#874—92no—
191C23×C32C22×D311yescharacteristic
192C22×C3⋊D3—32no—
193C22×C3⋊D3—31no—
194C22×C3⋊D3—62no—
195C22×C3⋊D3—122no—
196C22×C3×D3—22no—
197C22×C3×D3—42no—
198C22×C3×D3C22×D312yes—
199C22×C3×D3C22×D311yescharacteristic
200C2×C32×C4C22×D312yes—
201#638—62no—
202C2×C3⋊Dic3—31no—
203C4⋊(C3⋊D3)—122no—
204C3×#67—22no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3C22×D311yescharacteristic
210C3×D12—42no—
211#627—62no—
212#626—62no—
213#626—64no—
214#626—64no—
215#626—122no—
216#626—122no—
217#624—62no—
218D3×Dic3—62no—
219C23×C9—21no—
220C23×C9—32no—
221C23×C9—31no—
222C23×C9—62no—
223C23×C9—122no—
224C23×C9C22×D311yescharacteristic
225C22×D9—32no—
226C22×D9—31no—
227C22×D9—62no—
228C22×D9—61no—
229C22×D9—62no—
230C22×D9—122no—
231C22×D9—122no—
232C22×D9—124no—
233D4×C9—62no—
234D4×C9—122no—
235C2×C4×C9—22no—
236C2×C4×C9—31no—
237C2×C4×C9C22×D312yes—
238#611—62no—
239#611—62no—
240#611—64no—
241#611—64no—
242#611—122no—
243#611—122no—
244#611—122no—
245C2×Dic9—31no—
246C2×Dic9—61no—
247C2×Dic9—62no—
248C2×Dic9D1214yescomplemented
249C2×Dic9C22×D312yes—
250C2×Dic9C22×D311yescharacteristic
251C2×Dic9C22×D311yescharacteristic
252D36—122no—
253D36—124no—
254C4×D9—62no—
255Dic18—62no—
256#535—271noSylow
257C2×C3×C9—22no—
258C2×C3×C9C22×D411yescharacteristic
259C2×C3×C9H(5,𝔽2)12yes—
260C3⋊D9—42no—
261C3⋊D9—41no—
262D3×C9—41no—
263D3×C9—42no—
264C24×C3—31no—
265C23×D3—91no—
266C23×D3—91no—
267C23×D3—181no—
268C23×D3C2×D911yescharacteristic
269C2×C3×D4—31no—
270C2×C3×D4—32no—
271C22×C3×C4—91no—
272C2×#67—92no—
273C2×#67—92no—
274C2×#67—91no—
275C2×#67—92no—
276C2×#67—92no—
277C2×#67—91no—
278C2×#67—91no—
279C2×#67—181no—
280C2×#67—184no—
281C2×#67C2×D911yescharacteristic
282C22×Dic3—31no—
283C22×Dic3—91no—
284#237—184no—
285#235—184no—
286C2×D12—91no—
287C2×D12—92no—
288C2×D12—92no—
289C2×D12—92no—
290C2×D12—92no—
291C2×D12—92no—
292C2×D12—92no—
293C2×D12—182no—
294C2×D12—184no—
295C2×D12C2×D912yescomplemented
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×C4×D3—91no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C3×C4⋊C4—92no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—32no—
304C3×C22⋊C4—91no—
305C3×C22⋊C4C2×D911yescharacteristic
306C3×C22⋊C4D3211yescharacteristic
307C3×C42—92no—
308#217—31no—
309#217—92no—
310(C2×D3)⋊C4—92no—
311(C2×D3)⋊C4—92no—
312(C2×D3)⋊C4—92no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—182no—
316(C2×D3)⋊C4C2×D912yescomplemented
317Dic3⋊C4—32no—
318C4×Dic3—32no—
319C22×C32—22no—
320C22×C32D3×D412yes—
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—62no—
323C2×C3⋊D3—61no—
324C2×C3⋊D3—62no—
325C2×C3⋊D3—62no—
326C2×C3⋊D3—62no—
327C2×C3⋊D3—124no—
328C2×C3×D3—21no—
329C2×C3×D3—22no—
330C2×C3×D3—22no—
331C2×C3×D3—22no—
332C2×C3×D3—22no—
333C2×C3×D3—44no—
334C32×C4—22no—
335C3⋊Dic3—61no—
336C3×Dic3—21no—
337C3×Dic3—32no—
338C3×Dic3—32no—
339C3×Dic3—62no—
340C22×C9—22no—
341C22×C9—21no—
342C22×C9—22no—
343C22×C9—42no—
344C22×C9—61no—
345C22×C9—62no—
346C22×C9—62no—
347C22×C9—62no—
348C22×C9—62no—
349C22×C9—124no—
350C22×C9C2×D1212yes—
351C22×C9C23×D311yescharacteristic
352C2×D9—64no—
353C2×D9—61no—
354C2×D9—62no—
355C2×D9—62no—
356C2×D9—62no—
357C2×D9—62no—
358C2×D9—122no—
359C2×D9—122no—
360C2×D9—122no—
361C2×D9—121no—
362C2×D9—124no—
363C2×D9—128no—
364C4×C9—22no—
365C4×C9—42no—
366C4×C9—61no—
367Dic9—22no—
368Dic9—61no—
369Dic9—121no—
370Dic9C2×D1212yes—
371Dic9C2×D1212yes—
372C2×#41—271no—
373C22×D4—271no—
374#127—272no—
375#124—272no—
376#121—272no—
377#121—272no—
378#120—31no—
379#120—271no—
380#120—271no—
381#120—271no—
382#117—91no—
383C3×C9#53511yescharacteristic, complemented, Sylow
384C23×C3—21no—
385C23×C3—32no—
386C23×C3—31no—
387C23×C3—62no—
388C23×C3—122no—
389C23×C3C22×D911yescharacteristic
390C23×C3C2×D3211yescharacteristic
391C22×D3—22no—
392C22×D3—42no—
393C22×D3—92no—
394C22×D3—91no—
395C22×D3—92no—
396C22×D3—91no—
397C22×D3—182no—
398C22×D3—181no—
399C22×D3—182no—
400C22×D3—182no—
401C22×D3—362no—
402C22×D3—362no—
403C22×D3—362no—
404C22×D3—364no—
405C22×D3C22×D912yes—
406C22×D3C22×D911yescharacteristic
407C3×D4—62no—
408C3×D4—122no—
409C2×C3×C4—22no—
410C2×C3×C4—31no—
411C2×C3×C4—94no—
412C2×C3×C4—91no—
413C2×C3×C4—91no—
414C2×C3×C4—92no—
415C2×C3×C4C22×D912yes—
416C2×C3×C4C2×D3212yes—
417#67—22no—
418#67—182no—
419#67—184no—
420#67—184no—
421#67—182no—
422#67—182no—
423#67—362no—
424#67—362no—
425#67—362no—
426C2×Dic3—31no—
427C2×Dic3—31no—
428C2×Dic3—34no—
429C2×Dic3—32no—
430C2×Dic3—91no—
431C2×Dic3—91no—
432C2×Dic3—181no—
433C2×Dic3—182no—
434C2×Dic3C22×D911yescharacteristic
435D12—42no—
436D12—184no—
437D12—184no—
438D12—182no—
439D12—362no—
440D12—362no—
441D12—364no—
442D12—362no—
443D12—362no—
444C4×D3—182no—
445C4×D3—182no—
446C4×D3—182no—
447Dic6—182no—
448Dic6—182no—
449C2×C32—22no—
450C2×C32C2×D3×D411yescharacteristic
451C2×C32#100412yes—
452C3⋊D3—122no—
453C3⋊D3—121no—
454C3×D3—41no—
455C3×D3—42no—
456C2×C9—22no—
457C2×C9—21no—
458C2×C9—22no—
459C2×C9—42no—
460C2×C9—121no—
461C2×C9—122no—
462C2×C9C22×D1211yescharacteristic
463C2×C9#100912yes—
464D9—124no—
465D9—121no—
466D9—122no—
467D9—241no—
468C24—31no—
469C24—271no—
470#41—544no—
471C2×Q8—271no—
472C2×D4—31no—
473C2×D4—32no—
474C2×D4—272no—
475C2×D4—272no—
476C2×D4—271no—
477C2×D4—272no—
478C2×D4—272no—
479C2×D4—271no—
480C2×D4—272no—
481C2×D4—544no—
482C22×C4—91no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—92no—
486C22⋊C4—32no—
487C22⋊C4—91no—
488C22⋊C4—272no—
489C22⋊C4—272no—
490C22⋊C4—272no—
491C22⋊C4D3×D911yescharacteristic
492C42—92no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—22no—
497C22×C3—42no—
498C22×C3—61no—
499C22×C3—62no—
500C22×C3—62no—
501C22×C3—62no—
502C22×C3—62no—
503C22×C3—124no—
504C22×C3D4×D912yes—
505C22×C3C23×D911yescharacteristic
506C22×C3D3×D1212yes—
507C22×C3C22×D3211yescharacteristic
508C2×D3—21no—
509C2×D3—22no—
510C2×D3—22no—
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—44no—
514C2×D3—184no—
515C2×D3—181no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—182no—
519C2×D3—182no—
520C2×D3—181no—
521C2×D3—182no—
522C2×D3—182no—
523C2×D3—182no—
524C2×D3—182no—
525C2×D3—362no—
526C2×D3—362no—
527C2×D3—362no—
528C2×D3—361no—
529C2×D3—364no—
530C2×D3—364no—
531C2×D3—368no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—61no—
536C3×C4—92no—
537C3×C4—92no—
538C3×C4—182no—
539Dic3—21no—
540Dic3—32no—
541Dic3—32no—
542Dic3—62no—
543Dic3—181no—
544Dic3—181no—
545Dic3—361no—
546C32#559211yescharacteristic
547C9—21no—
548C9#552511yescharacteristic, complemented
549C23—32no—
550C23—31no—
551C23—62no—
552C23—122no—
553C23—272no—
554C23—271no—
555C23—542no—
556C23—1082no—
557C23C2×D3×D911yescharacteristic
558Q8—542no—
559D4—62no—
560D4—122no—
561D4—544no—
562D4—544no—
563D4—542no—
564D4—542no—
565D4—1082no—
566D4—1082no—
567D4—1082no—
568C2×C4—31no—
569C2×C4—91no—
570C2×C4—91no—
571C2×C4—94no—
572C2×C4—92no—
573C2×C4—271no—
574C2×C4—542no—
575C2×C4—542no—
576C2×C4C2×D3×D912yes—
577C2×C3—22no—
578C2×C3—22no—
579C2×C3—22no—
580C2×C3—21no—
581C2×C3—42no—
582C2×C3—122no—
583C2×C3—121no—
584C2×C3C2×D4×D911yescharacteristic
585C2×C3#6371012yes—
586C2×C3C2×D3×D1211yescharacteristic
587C2×C3#6431412yes—
588D3—42no—
589D3—41no—
590D3—364no—
591D3—362no—
592D3—361no—
593D3—362no—
594D3—361no—
595D3—721no—
596C22—22no—
597C22—62no—
598C22—62no—
599C22—62no—
600C22—61no—
601C22—62no—
602C22—124no—
603C22—542no—
604C22—542no—
605C22—542no—
606C22—541no—
607C22—542no—
608C22—1084no—
609C22D9×D1212yes—
610C22C22×D3×D911yescharacteristic
611C4—22no—
612C4—61no—
613C4—92no—
614C4—92no—
615C4—182no—
616C4—541no—
617C3—21no—
618C3#1058969411yescharacteristic, complemented
619C3#1059408011yescharacteristic
620C2—22no—
621C2—122no—
622C2—121no—
623C2—1082no—
624C2—1081no—
625C2C2×D9×D1211yescharacteristic
626C2#1172250012yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#535
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. C22×#6195C2semidirect
2. C2×#88662C2semidirect
3. #11721728C2semidirect
4. C9×#882C2semidirect
5. C3×#63439C2semidirect
6. #11721291C2semidirect
7. #11721287C2semidirect
8. #11721040C2semidirect
9. #11721039C2semidirect
10. #11721017C2semidirect
11. #11720956C2semidirect
17. C2×C4⋊(C3⋊D9)C22semidirect
20. C2×C9×D12C22semidirect
24. C2×#6195C22semidirect
25. C2×#6195C22semidirect
26. C2×#6195C22semidirect
28. C2×D3×Dic9C22semidirect
29. C2×(C3⋊D9)⋊C4C22semidirect
30. C2×#6189C22semidirect
32. #88536C22semidirect
34. C9×(C2×D3)⋊C4C22semidirect
35. C3×Dic9⋊C4C22semidirect
36. C3×C4×Dic9C22semidirect
37. (C2×D3)⋊Dic9C22semidirect
38. #88445C22semidirect
61. #63439D3semidirect
77. C2×C3×Dic9D4semidirect
145. Dic9⋊C4C2×D3semidirect
146. C4×Dic9C2×D3semidirect
187. #882D9semidirect
248. C2×Dic9D12semidirect
295. C2×D12C2×D9semidirect
316. (C2×D3)⋊C4C2×D9semidirect
383. C3×C9#535semidirect
548. C9#5525semidirect
618. C3#10589694semidirect

Maximal subgroups

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

subgroupindex
C22×#61952
C2×#886622
#117217282
C9×#8822
C3×#634392
#117212912
#117212872
#117210402
#117210392
#117210172
#117209562
#105940803
#105896943