Finite Group Databasebeta

← #13150230

Subgroups — #13150230

Order 1728.

970 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×#88658C212yesmaximal, complemented
2C4×#6233C211yesmaximal, characteristic, complemented
3C3×#63492C211yesmaximal, characteristic, complemented
4C2×C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
5#11721259C212yesmaximal, complemented
6#11721007C211yesmaximal, characteristic, complemented
7#11721000C211yesmaximal, characteristic, complemented
8#11720999C212yesmaximal, complemented
9#11720998C211yesmaximal, characteristic, complemented
10#11720971C211yesmaximal, characteristic, complemented
11#11720942C212yesmaximal, complemented
12#10594928—31nomaximal
13#10589609—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×#611C2212yes—
18C22×C9×Dic3C2212yes—
19C2×C3×D36C2211yescharacteristic
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22#88658—28no—
23C2×Dic3×D9C2212yes—
24C2×#6189C2212yes—
25#88551C2211yescharacteristic
26#88536C2211yescharacteristic
27#88534C2211yescharacteristic
28C4×C3⋊Dic9C2211yescharacteristic
29C3×(C2×D9)⋊C4C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic
31C9×C4⋊Dic3C2211yescharacteristic
32C9×C4⋊Dic3C2212yes—
33C3×C4⋊Dic9C2211yescharacteristic
34(C2×D9)⋊Dic3C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36#88442C2212yes—
37#88442C2212yes—
38C2×#3804—32no—
39C4×#638—31no—
40C3×#930—31no—
41C2×C3×C4⋊Dic3D311yescharacteristic
42#63961—32no—
43#63919—31no—
44#63916—31no—
45#63900—31no—
46#63899—32no—
47#63898—31no—
48#63854—32no—
49C2×#3697—32no—
50C2×C9×C4⋊C4—31no—
51#63492D311yescharacteristic, complemented
52#63491—32no—
53C4×#611—31no—
54C4×#611—61no—
55#63458—32no—
56#63457—31no—
57#63456—32no—
58#63455—31no—
59C4⋊C4×D9—31no—
60#63450—31no—
61C23×C3×C9C2311yescharacteristic
62C22×C3⋊D9C2311yescharacteristic
63C22×C3×D9C2312yes—
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67#6233D414yescomplemented
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C4×C3⋊D9—22no—
72C3×#611—24no—
73C2×C9×Dic3—24no—
74C2×C9×Dic3C2312yes—
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9C2312yes—
77C3×D36—22no—
78#6192—24no—
79(C3⋊D9)⋊C4—24no—
80Dic3×D9—24no—
81#6189—24no—
82#5611—91no—
83#5507—91no—
84C22×C32×C4C2×D311yescharacteristic
85C2×#638—31no—
86C2×C4×C3⋊D3—31no—
87C2×C3×#67—32no—
88C22×C3×Dic3C2×D312yes—
89C2×C3×D12—31no—
90C2×#627—32no—
91C2×#626—32no—
92C2×#624—32no—
93#3804—68no—
94C2×D3×Dic3—32no—
95#3757—31no—
96#3752—31no—
97#3750—31no—
98C4×C3⋊Dic3—31no—
99C3×(C2×D3)⋊C4—32no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3C2×D312yes—
102C3×C4⋊Dic3C2×D311yescharacteristic
103C3×C4⋊Dic3C2×D311yescharacteristic
104#3724—32no—
105#3724—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108C22×C4×C9—21no—
109C22×C4×C9—32no—
110C22×C4×C9C2×D311yescharacteristic
111C2×#611—31no—
112C2×#611—61no—
113C2×#611C2×D312yes—
114#3697—68no—
115C2×D36—32no—
116C2×D36C2×D311yescharacteristic
117C2×C4×D9—32no—
118C2×C4×D9—31no—
119C2×C4×D9—32no—
120C2×C4×D9—61no—
121C2×Dic18—32no—
122C9×C4⋊C4—31no—
123C9×C4⋊C4—31no—
124C9×C4⋊C4—32no—
125#3676—31no—
126#3676—61no—
127(C2×D9)⋊C4—32no—
128(C2×D9)⋊C4—32no—
129(C2×D9)⋊C4—31no—
130(C2×D9)⋊C4—61no—
131(C2×D9)⋊C4C2×D312yes—
132C4⋊Dic9C2×D311yescharacteristic
133Dic9⋊C4—31no—
134Dic9⋊C4—32no—
135Dic9⋊C4—32no—
136Dic9⋊C4—61no—
137C4×Dic9—31no—
138C4×Dic9—61no—
139C22×C3×C9—21no—
140C22×C3×C9—21no—
141C22×C3×C9C2×D412yes—
142C22×C3×C9C2411yescharacteristic
143C2×C3⋊D9—21no—
144C2×C3⋊D9—21no—
145C2×C3⋊D9C2×D412yes—
146C2×C3×D9—22no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C3×C4×C9—21no—
150C3×C4×C9#4112yescomplemented
151C3⋊Dic9—21no—
152C3⋊Dic9—21no—
153C3⋊Dic9C2×D412yes—
154C9×Dic3—22no—
155C9×Dic3—22no—
156C3×Dic9—22no—
157C2×#237—92no—
158C2×#235—92no—
159C3×#121—91no—
160C2×C3×C4⋊C4—31no—
161#938—91no—
162#935—92no—
163D4×Dic3—91no—
164#930—31no—
165#929—92no—
166C4×#67—91no—
167C4×#67—91no—
168C4×#67—181no—
169C2×C4⋊Dic3D911yescharacteristic, complemented
170#896—92no—
171#895—91no—
172#894—92no—
173#893—91no—
174D3×C4⋊C4—91no—
175D3×C4⋊C4—91no—
176#888—91no—
177#888—91no—
178#883—92no—
179#878—92no—
180C23×C32C22×D311yescharacteristic
181C22×C3⋊D3—31no—
182C22×C3×D3—32no—
183C2×C32×C4—22no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186#638—34no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—31no—
190C4×C3⋊D3—62no—
191C3×#67—64no—
192C2×C3×Dic3—24no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3C22×D312yes—
195C2×C3×Dic3C22×D312yes—
196C3×D12—62no—
197#627—64no—
198#626—64no—
199#624—64no—
200D3×Dic3—64no—
201C23×C9—21no—
202C23×C9C22×D311yescharacteristic
203C22×D9—31no—
204C22×D9—61no—
205C22×D9C22×D312yes—
206C2×C4×C9—22no—
207C2×C4×C9—21no—
208C2×C4×C9—21no—
209C2×C4×C9—32no—
210C2×C4×C9—32no—
211C2×C4×C9—42no—
212C2×C4×C9—64no—
213C2×C4×C9C22×D311yescharacteristic
214C2×C4×C9C22×D311yescharacteristic
215#611—24no—
216#611—34no—
217#611—64no—
218C2×Dic9—31no—
219C2×Dic9—31no—
220C2×Dic9—31no—
221C2×Dic9—61no—
222C2×Dic9—61no—
223C2×Dic9—61no—
224C2×Dic9C22×D312yes—
225D36—22no—
226D36—64no—
227C4×D9—64no—
228C4×D9—64no—
229C4×D9—62no—
230C4×D9—122no—
231Dic18—64no—
232#547—271noSylow
233C2×C3×C9—22no—
234C2×C3×C9C22×D411yescharacteristic
235C2×C3×C9C2×#4111yescharacteristic
236C2×C3×C92−1+411yescharacteristic
237C3⋊D9—22no—
238C3×D9—42no—
239C2×C3×D4—92no—
240C2×C3×D4—91no—
241C22×C3×C4—21no—
242C22×C3×C4—32no—
243C22×C3×C4C2×D911yescharacteristic
244C22×C3×C4D3211yescharacteristic
245C2×#67—32no—
246C2×#67—91no—
247C2×#67—91no—
248C2×#67—92no—
249C2×#67—181no—
250C22×Dic3—92no—
251C22×Dic3C2×D912yes—
252#237—188no—
253#235—188no—
254C2×D12—31no—
255C2×D12—92no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—91no—
259C2×C4×D3—91no—
260C2×C4×D3—92no—
261C2×C4×D3—181no—
262C2×Dic6—92no—
263C2×Dic6—92no—
264C3×C4⋊C4—31no—
265C3×C4⋊C4—31no—
266C3×C4⋊C4—32no—
267C3×C4⋊C4—91no—
268C3×C22⋊C4—92no—
269#217—91no—
270#217—91no—
271#217—92no—
272#217—92no—
273#217—181no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—91no—
278(C2×D3)⋊C4—91no—
279(C2×D3)⋊C4—181no—
280C4⋊Dic3—31no—
281C4⋊Dic3C2×D912yes—
282C4⋊Dic3C2×D911yescharacteristic
283C4⋊Dic3C2×D911yescharacteristic
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—91no—
287Dic3⋊C4—91no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—181no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—181no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32D3×D412yes—
297C22×C32C23×D311yescharacteristic
298C2×C3⋊D3—32no—
299C2×C3⋊D3—61no—
300C2×C3⋊D3—61no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C32×C4—21no—
305C32×C4#23512yes—
306C3⋊Dic3—32no—
307C3⋊Dic3—61no—
308C3⋊Dic3—61no—
309C3×Dic3—22no—
310C3×Dic3—22no—
311C3×Dic3—62no—
312C22×C9—21no—
313C22×C9—21no—
314C22×C9—21no—
315C22×C9—22no—
316C22×C9—41no—
317C22×C9—41no—
318C22×C9D3×D412yes—
319C22×C9C23×D311yescharacteristic
320C2×D9—22no—
321C2×D9—22no—
322C2×D9—22no—
323C2×D9—32no—
324C2×D9—62no—
325C2×D9—61no—
326C2×D9—61no—
327C2×D9—121no—
328C2×D9—121no—
329C4×C9—21no—
330C4×C9—22no—
331C4×C9—41no—
332C4×C9—62no—
333C4×C9—62no—
334C4×C9#23712yescomplemented
335Dic9—22no—
336Dic9—32no—
337Dic9—62no—
338Dic9—61no—
339Dic9—61no—
340Dic9—121no—
341Dic9—121no—
342C2×#41—272no—
343C4⋊Q8—271no—
344#123—272no—
345#123—272no—
346C22⋊Q8—272no—
347#121—91no—
348#121—271no—
349C4×D4—271no—
350C4×D4—271no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—271no—
353C3×C9#54711yescharacteristic, complemented, Sylow
354C23×C3—21no—
355C23×C3—92no—
356C23×C3C22×D911yescharacteristic
357C23×C3C2×D3211yescharacteristic
358C22×D3—32no—
359C22×D3—91no—
360C22×D3—91no—
361C22×D3—181no—
362C3×D4—184no—
363C3×D4—182no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—32no—
369C2×C3×C4—32no—
370C2×C3×C4—42no—
371C2×C3×C4—64no—
372C2×C3×C4—92no—
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C22×D911yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377#67—64no—
378#67—94no—
379#67—94no—
380#67—184no—
381#67—184no—
382C2×Dic3—24no—
383C2×Dic3—32no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—91no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—181no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—184no—
394C2×Dic3C22×D912yes—
395C2×Dic3C22×D912yes—
396D12—62no—
397D12—184no—
398C4×D3—184no—
399C4×D3—184no—
400C4×D3—184no—
401C4×D3—182no—
402C4×D3—182no—
403C4×D3—362no—
404Dic6—184no—
405Dic6—184no—
406C2×C32—22no—
407C2×C32C2×#23511yescharacteristic
408C2×C32C2×D3×D411yescharacteristic
409C2×C32#100711yescharacteristic
410C3⋊D3—62no—
411C3×D3—122no—
412C2×C9—22no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—42no—
417C2×C9C2×D3×D411yescharacteristic
418C2×C9C2×#23711yescharacteristic
419C2×C9#101011yescharacteristic
420D9—42no—
421D9—62no—
422D9—122no—
423#41—548no—
424C2×Q8—272no—
425C2×D4—92no—
426C2×D4—91no—
427C2×D4—272no—
428C2×D4—271no—
429C22×C4—32no—
430C22×C4—272no—
431C22×C4—272no—
432C22×C4—271no—
433C22×C4D3×D911yescharacteristic
434C4⋊C4—31no—
435C4⋊C4—31no—
436C4⋊C4—32no—
437C4⋊C4—91no—
438C4⋊C4—272no—
439C4⋊C4—272no—
440C4⋊C4—271no—
441C22⋊C4—92no—
442C22⋊C4—272no—
443C22⋊C4—272no—
444C22⋊C4—271no—
445C22⋊C4—271no—
446C42—271no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—22no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3—182no—
458C22×C3D4×D912yes—
459C22×C3C23×D911yescharacteristic
460C22×C3#380212yes—
461C22×C3C22×D3211yescharacteristic
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—92no—
466C2×D3—92no—
467C2×D3—182no—
468C2×D3—181no—
469C2×D3—181no—
470C2×D3—181no—
471C2×D3—181no—
472C2×D3—361no—
473C2×D3—361no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—22no—
477C3×C4—41no—
478C3×C4—62no—
479C3×C4—62no—
480C3×C4—182no—
481C3×C4#369712yescomplemented
482C3×C4#380412yes—
483Dic3—22no—
484Dic3—22no—
485Dic3—62no—
486Dic3—92no—
487Dic3—92no—
488Dic3—182no—
489Dic3—181no—
490Dic3—181no—
491Dic3—181no—
492Dic3—181no—
493Dic3—361no—
494Dic3—361no—
495C32#550711yescharacteristic
496C9—21no—
497C9#561111yescharacteristic, complemented
498C23—92no—
499C23—271no—
500C23C2×D3×D911yescharacteristic
501Q8—544no—
502D4—184no—
503D4—182no—
504D4—274no—
505D4—544no—
506C2×C4—22no—
507C2×C4—32no—
508C2×C4—32no—
509C2×C4—64no—
510C2×C4—92no—
511C2×C4—271no—
512C2×C4—271no—
513C2×C4—271no—
514C2×C4—544no—
515C2×C4—544no—
516C2×C4—542no—
517C2×C4C2×D3×D911yescharacteristic
518C2×C4C2×D3×D911yescharacteristic
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—362no—
526C2×C3C2×#369711yescharacteristic
527C2×C3C2×D4×D911yescharacteristic
528C2×C3#6371311yescharacteristic
529C2×C3#6429711yescharacteristic
530C2×C3C2×#380211yescharacteristic
531C2×C3C2×#380411yescharacteristic
532D3—122no—
533D3—182no—
534D3—182no—
535D3—362no—
536C22—21no—
537C22—21no—
538C22—182no—
539C22—182no—
540C22—182no—
541C22—272no—
542C22—541no—
543C22—541no—
544C22#8864612yes—
545C22C22×D3×D911yescharacteristic
546C4—21no—
547C4—62no—
548C4—62no—
549C4—182no—
550C4—272no—
551C4—541no—
552C4—541no—
553C4#8865812yescomplemented
554C3—21no—
555C3#1058960911yescharacteristic, complemented
556C3#1059492811yescharacteristic
557C2—22no—
558C2—362no—
559C2—542no—
560C2#1172245811yescharacteristic
561C2C2×#8864611yescharacteristic
562C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#547
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×#88658C2semidirect
2. C4×#6233C2semidirect
3. C3×#63492C2semidirect
4. C2×C9×C4⋊Dic3C2semidirect
5. #11721259C2semidirect
6. #11721007C2semidirect
7. #11721000C2semidirect
8. #11720999C2semidirect
9. #11720998C2semidirect
10. #11720971C2semidirect
11. #11720942C2semidirect
51. #63492D3semidirect
67. #6233D4semidirect
150. C3×C4×C9#41semidirect
169. C2×C4⋊Dic3D9semidirect
334. C4×C9#237semidirect
353. C3×C9#547semidirect
481. C3×C4#3697semidirect
497. C9#5611semidirect
553. C4#88658semidirect
555. C3#10589609semidirect

Maximal subgroups

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

subgroupindex
C2×#886582
C4×#62332
C3×#634922
C2×C9×C4⋊Dic32
#117212592
#117210072
#117210002
#117209992
#117209982
#117209712
#117209422
#105949283
#105896093