Finite Group Databasebeta

← #13150030

Subgroups — #13150030

Order 1728.

950 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88736C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3C2×C9×#208C211yesmaximal, characteristic, complemented
4#11720873C212yesmaximal, complemented
5#11720871C212yesmaximal, complemented
6#11720869C212yesmaximal, complemented
7#11720868C212yesmaximal, complemented
8#11720867C214yesmaximal, complemented
9#10594795—31nomaximal
10#10589850—31nomaximal
11C22×C3×C4×C9C2211yescharacteristic
12C2×#6233—21no—
13#88736—21no—
14#88736—21no—
15#88736C2212yes—
16#88736C2212yescomplemented
17C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
18C2×C4×C3⋊D9—21no—
19C2×#6226C2211yescharacteristic, complemented
20C2×C3×#611—21no—
21C3×#3697—21no—
22C3×#3697—21no—
23C3×#3697C2212yes—
24C3×#3697C2212yescomplemented
25C2×C3×D36C2211yescharacteristic, complemented
26C2×C3×C4×D9—21no—
27C2×C3×Dic18C2211yescharacteristic, complemented
28C9×#208C2212yescomplemented
29C9×#208C2212yescomplemented
30C2×C9×C3⋊C8C2212yescomplemented
31#88418C2214yescomplemented
32#88417C2214yescomplemented
33#88416C2214yescomplemented
34#88415C2214yescomplemented
35C2×#3828—31no—
36C2×C3×#235—31no—
37C2×C3×#208D311yescharacteristic
38#63833—32no—
39#63831—32no—
40#63830—32no—
41#63829—34no—
42#63825—32no—
43C2×#3697—31no—
44C2×#3697—61no—
45C2×#3697D311yescharacteristic, complemented
46C2×C9×M16—31no—
47#63471—32no—
48#63471—32no—
49#63470—32no—
50#63470—32no—
51#63467—34no—
52C23×C3×C9D411yescharacteristic
53C22×C3⋊D9—21no—
54C22×C3×D9—21no—
55C2×C3×C4×C9D411yescharacteristic
56C2×C3×C4×C9D411yescharacteristic
57C2×C3×C4×C9D411yescharacteristic
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×C4×C9C2311yescharacteristic
61#6233—22no—
62#6233—22no—
63C2×C3⋊Dic9—21no—
64C4⋊(C3⋊D9)—21no—
65C4⋊(C3⋊D9)C2312yes—
66C4×C3⋊D9—22no—
67C4×C3⋊D9—22no—
68#6226—21no—
69#6226C2312yes—
70C3×#611—22no—
71C3×#611—22no—
72C2×C3×Dic9—21no—
73C3×D36—21no—
74C3×D36C2312yes—
75C3×C4×D9—22no—
76C3×C4×D9—22no—
77C3×Dic18—21no—
78C3×Dic18C2312yes—
79C9×C3⋊C8C2314yes—
80#5822—91no—
81#5748—91no—
82C22×C32×C4C2×D311yescharacteristic
83C2×#638—61no—
84#3828—32no—
85#3828—32no—
86#3828—61no—
87#3828—61no—
88C2×C4⋊(C3⋊D3)—31no—
89C2×C4×C3⋊D3—61no—
90C2×#634—31no—
91C2×C3×#67—61no—
92C3×#235—32no—
93C3×#235—32no—
94C3×#235—61no—
95C3×#235—61no—
96C2×C3×D12—31no—
97C2×C3×C4×D3—61no—
98C2×C3×Dic6—31no—
99C3×#208C2×D312yes—
100C3×#208C2×D312yes—
101C2×C3×C3⋊C8C2×D312yes—
102#3719—34no—
103#3717—34no—
104#3716—34no—
105#3714—34no—
106C22×C4×C9—21no—
107C22×C4×C9C2×D311yescharacteristic
108C2×#611—21no—
109C2×#611—62no—
110C2×#611—61no—
111#3697—21no—
112#3697—21no—
113#3697—32no—
114#3697—32no—
115#3697—62no—
116#3697—62no—
117#3697—62no—
118#3697—62no—
119#3697—61no—
120#3697—61no—
121#3697C2×D312yes—
122#3697C2×D312yescomplemented
123C2×D36—31no—
124C2×D36—61no—
125C2×D36C2×D311yescharacteristic, complemented
126C2×C4×D9—21no—
127C2×C4×D9—62no—
128C2×C4×D9—61no—
129C2×Dic18—31no—
130C2×Dic18—61no—
131C2×Dic18C2×D311yescharacteristic, complemented
132C9×M16—32no—
133C9×M16—32no—
134C2×C8×C9—32no—
135D72—34no—
136#3664—34no—
137#3664—34no—
138Dic36—34no—
139C22×C3×C9—41no—
140C22×C3×C9C2×D411yescharacteristic
141C22×C3×C9C2×D411yescharacteristic
142C22×C3×C9C2×D411yescharacteristic
143C2×C3⋊D9—22no—
144C2×C3⋊D9—42no—
145C2×C3×D9—22no—
146C2×C3×D9—42no—
147C3×C4×C9C2×D411yescharacteristic
148C3×C4×C9C2×D411yescharacteristic
149C3×C4×C9C2×D411yescharacteristic
150C3×C4×C9C2411yescharacteristic
151C3⋊Dic9—22no—
152C3×Dic9—22no—
153C2×C3×#41—91no—
154C2×#235—31no—
155C2×#235—91no—
156C2×#235—91no—
157C2×#235—181no—
158C2×C3×M16—31no—
159#951—92no—
160#950—94no—
161#949—92no—
162#942—92no—
163#932—92no—
164C2×#208D911yescharacteristic, complemented
165#909—92no—
166#909—92no—
167#908—92no—
168#908—92no—
169#904—94no—
170C23×C32D1211yescharacteristic
171C22×C3⋊D3—61no—
172C22×C3×D3—61no—
173C2×C32×C4D1211yescharacteristic
174C2×C32×C4D1211yescharacteristic
175C2×C32×C4D1211yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179#638—62no—
180#638—62no—
181C2×C3⋊Dic3—61no—
182C4⋊(C3⋊D3)—32no—
183C4⋊(C3⋊D3)—61no—
184C4×C3⋊D3—62no—
185C4×C3⋊D3—62no—
186#634—32no—
187#634—61no—
188C3×#67—62no—
189C3×#67—62no—
190C2×C3×Dic3—61no—
191C3×D12—32no—
192C3×D12—61no—
193C3×C4×D3—62no—
194C3×C4×D3—62no—
195C3×Dic6—32no—
196C3×Dic6—61no—
197C23×C9—21no—
198C23×C9#6711yescharacteristic
199C22×D9—21no—
200C22×D9—62no—
201C22×D9—61no—
202C3×C3⋊C8C22×D314yes—
203C2×C4×C9—21no—
204C2×C4×C9—21no—
205C2×C4×C9—21no—
206C2×C4×C9—21no—
207C2×C4×C9—21no—
208C2×C4×C9—21no—
209C2×C4×C9#6711yescharacteristic
210C2×C4×C9#6711yescharacteristic
211C2×C4×C9#6711yescharacteristic
212C2×C4×C9C22×D311yescharacteristic
213C2×C4×C9C22×D311yescharacteristic
214C2×C4×C9C22×D311yescharacteristic
215#611—22no—
216#611—22no—
217#611—64no—
218#611—64no—
219#611—62no—
220#611—62no—
221C2×Dic9—21no—
222C2×Dic9—62no—
223C2×Dic9—61no—
224D36—21no—
225D36—32no—
226D36—62no—
227D36—62no—
228D36—61no—
229D36C22×D312yes—
230C4×D9—22no—
231C4×D9—22no—
232C4×D9—64no—
233C4×D9—64no—
234C4×D9—62no—
235C4×D9—62no—
236Dic18—21no—
237Dic18—32no—
238Dic18—62no—
239Dic18—62no—
240Dic18—61no—
241Dic18C22×D312yes—
242C8×C9—34no—
243#575—271noSylow
244C2×C3×C9—21no—
245C2×C3×C9—21no—
246C2×C3×C9—21no—
247C2×C3×C9C22×D411yescharacteristic
248C3⋊D9—42no—
249C3×D9—42no—
250C3×#41—92no—
251C3×#41—92no—
252C3×#41—181no—
253C3×#41—181no—
254C2×C3×Q8—91no—
255C2×C3×D4—91no—
256C2×C3×D4—181no—
257C22×C3×C4—21no—
258C22×C3×C4—181no—
259C22×C3×C4C2×D911yescharacteristic
260C22×C3×C4D3211yescharacteristic
261C2×#67—61no—
262C2×#67—182no—
263C2×#67—181no—
264C2×#67—181no—
265#235—32no—
266#235—32no—
267#235—61no—
268#235—61no—
269#235—92no—
270#235—92no—
271#235—92no—
272#235—92no—
273#235—182no—
274#235—182no—
275#235—182no—
276#235—182no—
277#235—181no—
278#235—181no—
279#235—181no—
280#235—181no—
281C2×D12—31no—
282C2×D12—91no—
283C2×D12—91no—
284C2×D12—181no—
285C2×C4×D3—61no—
286C2×C4×D3—182no—
287C2×C4×D3—181no—
288C2×C4×D3—181no—
289C2×Dic6—31no—
290C2×Dic6—91no—
291C2×Dic6—91no—
292C2×Dic6—181no—
293C3×M16—32no—
294C3×M16—32no—
295C2×C3×C8—32no—
296#216—94no—
297#215—94no—
298#214—94no—
299#213—94no—
300#208C2×D912yescomplemented
301#208C2×D912yescomplemented
302C2×C3⋊C8C2×D912yescomplemented
303Dic12—94no—
304D24—94no—
305#204—94no—
306#204—94no—
307C22×C32—41no—
308C22×C32C2×D1211yescharacteristic
309C22×C32C2×D1211yescharacteristic
310C22×C32C2×D1211yescharacteristic
311C2×C3⋊D3—62no—
312C2×C3⋊D3—122no—
313C2×C3×D3—62no—
314C2×C3×D3—122no—
315C32×C4C2×D1211yescharacteristic
316C32×C4C2×D1211yescharacteristic
317C32×C4C2×D1211yescharacteristic
318C32×C4C23×D311yescharacteristic
319C3⋊Dic3—62no—
320C3×Dic3—62no—
321C22×C9—21no—
322C22×C9—21no—
323C22×C9—21no—
324C22×C9—41no—
325C22×C9—42no—
326C22×C9C2×#6711yescharacteristic
327C22×C9C2×#6711yescharacteristic
328C22×C9C2×#6711yescharacteristic
329C2×D9—22no—
330C2×D9—42no—
331C2×D9—64no—
332C2×D9—62no—
333C2×D9—124no—
334C2×D9—122no—
335C4×C9—21no—
336C4×C9—21no—
337C4×C9—21no—
338C4×C9—21no—
339C4×C9C2×#6711yescharacteristic
340C4×C9C2×#6711yescharacteristic
341C4×C9C2×#6711yescharacteristic
342C4×C9C23×D311yescharacteristic
343Dic9—22no—
344Dic9—64no—
345Dic9—62no—
346C2×#41—91no—
347C2×#41—271no—
348#137—272no—
349#137—272no—
350AGL(1,ℤ/8)—272no—
351AGL(1,ℤ/8)—272no—
352#135—274no—
353C2×M16—31no—
354C3×C9#57511yescharacteristic, complemented, Sylow
355C23×C3—21no—
356C23×C3—181no—
357C23×C3D3611yescharacteristic
358C23×C3#62611yescharacteristic
359C22×D3—61no—
360C22×D3—182no—
361C22×D3—181no—
362C22×D3—181no—
363C3×Q8—92no—
364C3×Q8—181no—
365C3×D4—92no—
366C3×D4—182no—
367C3×D4—182no—
368C3×D4—181no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—181no—
376C2×C3×C4—182no—
377C2×C3×C4—182no—
378C2×C3×C4D3611yescharacteristic
379C2×C3×C4D3611yescharacteristic
380C2×C3×C4D3611yescharacteristic
381C2×C3×C4C22×D911yescharacteristic
382C2×C3×C4C22×D911yescharacteristic
383C2×C3×C4C22×D911yescharacteristic
384C2×C3×C4#62611yescharacteristic
385C2×C3×C4#62611yescharacteristic
386C2×C3×C4#62611yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390#67—62no—
391#67—62no—
392#67—184no—
393#67—184no—
394#67—182no—
395#67—182no—
396#67—182no—
397#67—182no—
398C2×Dic3—61no—
399C2×Dic3—182no—
400C2×Dic3—181no—
401C2×Dic3—181no—
402D12—32no—
403D12—61no—
404D12—92no—
405D12—92no—
406D12—182no—
407D12—182no—
408D12—181no—
409D12—181no—
410C4×D3—62no—
411C4×D3—62no—
412C4×D3—184no—
413C4×D3—184no—
414C4×D3—182no—
415C4×D3—182no—
416C4×D3—182no—
417C4×D3—182no—
418Dic6—32no—
419Dic6—61no—
420Dic6—92no—
421Dic6—92no—
422Dic6—182no—
423Dic6—182no—
424Dic6—181no—
425Dic6—181no—
426C3×C8—34no—
427C3⋊C8C22×D914yes—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32C22×D1211yescharacteristic
432C3⋊D3—122no—
433C3×D3—122no—
434C2×C9—22no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—21no—
439C2×C9—41no—
440C2×C9—41no—
441C2×C9C22×#6711yescharacteristic
442D9—42no—
443D9—124no—
444D9—122no—
445#41—92no—
446#41—92no—
447#41—181no—
448#41—181no—
449#41—272no—
450#41—272no—
451#41—541no—
452#41—541no—
453C2×Q8—91no—
454C2×Q8—271no—
455C2×D4—91no—
456C2×D4—181no—
457C2×D4—271no—
458C2×D4—541no—
459C22×C4—181no—
460C22×C4—541no—
461C22×C4D3×D911yescharacteristic
462Q16—274no—
463SD16—274no—
464SD16—274no—
465D8—274no—
466M16—32no—
467M16—32no—
468C2×C8—32no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—42no—
475C22×C3—182no—
476C22×C3—362no—
477C22×C3C2×D3611yescharacteristic
478C22×C3C2×D3611yescharacteristic
479C22×C3C2×D3611yescharacteristic
480C22×C3C2×#62611yescharacteristic
481C22×C3C2×#62611yescharacteristic
482C22×C3C2×#62611yescharacteristic
483C2×D3—62no—
484C2×D3—122no—
485C2×D3—184no—
486C2×D3—182no—
487C2×D3—182no—
488C2×D3—362no—
489C2×D3—364no—
490C2×D3—362no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—182no—
496C3×C4C2×D3611yescharacteristic
497C3×C4C2×D3611yescharacteristic
498C3×C4C2×D3611yescharacteristic
499C3×C4C23×D911yescharacteristic
500C3×C4C2×#62611yescharacteristic
501C3×C4C2×#62611yescharacteristic
502C3×C4C2×#62611yescharacteristic
503C3×C4C22×D3211yescharacteristic
504Dic3—62no—
505Dic3—184no—
506Dic3—182no—
507Dic3—182no—
508C32#574811yescharacteristic
509C9—21no—
510C9#582211yescharacteristic, complemented
511C23—181no—
512C23—541no—
513C23#619211yescharacteristic
514Q8—92no—
515Q8—181no—
516Q8—272no—
517Q8—541no—
518D4—92no—
519D4—182no—
520D4—182no—
521D4—181no—
522D4—272no—
523D4—542no—
524D4—542no—
525D4—541no—
526C2×C4—181no—
527C2×C4—182no—
528C2×C4—182no—
529C2×C4—541no—
530C2×C4—542no—
531C2×C4—542no—
532C2×C4#619211yescharacteristic
533C2×C4#619211yescharacteristic
534C2×C4#619211yescharacteristic
535C2×C4C2×D3×D911yescharacteristic
536C2×C4C2×D3×D911yescharacteristic
537C2×C4C2×D3×D911yescharacteristic
538C8—34no—
539C2×C3—22no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—362no—
550C2×C3C22×D3611yescharacteristic
551C2×C3C22×#62611yescharacteristic
552D3—122no—
553D3—364no—
554D3—362no—
555D3—362no—
556C22—41no—
557C22—182no—
558C22—362no—
559C22—542no—
560C22—1082no—
561C22C2×#619211yescharacteristic
562C22C2×#619211yescharacteristic
563C22C2×#619211yescharacteristic
564C4—182no—
565C4—542no—
566C4C2×#619211yescharacteristic
567C4C2×#619211yescharacteristic
568C4C2×#619211yescharacteristic
569C4C22×D3×D911yescharacteristic
570C3—21no—
571C3#1058985011yescharacteristic, complemented
572C3#1059479511yescharacteristic
573C2—21no—
574C2—21no—
575C2—21no—
576C2—362no—
577C2—1082no—
578C2C22×#619211yescharacteristic
subgroupG/H
centerC4C2×#6192
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC22×C3×C4×C9C22
socleC2×C32C22×D12
hypercenterC22×C4D3×D9
hypocenterC3×C9#575
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88736C2semidirect
2. C2×C3×#3697C2semidirect
3. C2×C9×#208C2semidirect
4. #11720873C2semidirect
5. #11720871C2semidirect
6. #11720869C2semidirect
7. #11720868C2semidirect
8. #11720867C2semidirect
16. #88736C22semidirect
17. C2×C4⋊(C3⋊D9)C22semidirect
19. C2×#6226C22semidirect
24. C3×#3697C22semidirect
25. C2×C3×D36C22semidirect
27. C2×C3×Dic18C22semidirect
28. C9×#208C22semidirect
29. C9×#208C22semidirect
30. C2×C9×C3⋊C8C22semidirect
31. #88418C22semidirect
32. #88417C22semidirect
33. #88416C22semidirect
34. #88415C22semidirect
45. C2×#3697D3semidirect
122. #3697C2×D3semidirect
125. C2×D36C2×D3semidirect
131. C2×Dic18C2×D3semidirect
164. C2×#208D9semidirect
300. #208C2×D9semidirect
301. #208C2×D9semidirect
302. C2×C3⋊C8C2×D9semidirect
354. C3×C9#575semidirect
510. C9#5822semidirect
571. C3#10589850semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×C3×#36972
C2×C9×#2082
#117208732
#117208712
#117208692
#117208682
#117208672
#105947953
#105898503