Finite Group Databasebeta

← #13150914

Subgroups — #13150914

Order 1728.

1030 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88663C212yesmaximal, complemented
2D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C9×#940C211yesmaximal, characteristic, complemented
4C3×#63502C211yesmaximal, characteristic, complemented
5#11721290C212yesmaximal, complemented
6#11721281C212yesmaximal, complemented
7#11721270C212yesmaximal, complemented
8#11721043C211yesmaximal, characteristic, complemented
9D12×Dic9C211yesmaximal, characteristic, complemented
10Dic3×D36C211yesmaximal, characteristic, complemented
11C4×#6189C211yesmaximal, characteristic, complemented
12#10595144—31nomaximal
13#10589820—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×C9×#67C2212yes—
17C2×C9×#67C2212yes—
18C2×C3×#611C2212yes—
19C2×C3×#611C2212yes—
20C2×C9×D12C2211yescharacteristic
21C2×C3×D36C2211yescharacteristic
22C2×#6194C2212yes—
23C2×#6194C2212yes—
24#88663—28no—
25C2×D3×Dic9C2212yes—
26C2×Dic3×D9C2212yes—
27C2×#6189C2211yescharacteristic
28#88551C2212yes—
29C4⋊(C3⋊Dic9)C2211yescharacteristic
30C4×C3⋊Dic9C2211yescharacteristic
31C4×C9×Dic3C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2211yescharacteristic
36#88442C2211yescharacteristic
37Dic9⋊Dic3C2211yescharacteristic
38C2×#3805—32no—
39D4×C3⋊Dic3—31no—
40C3×#940—31no—
41C3×#940D311yescharacteristic
42#63980—32no—
43#63970—32no—
44#63970—32no—
45C4×#627—31no—
46#63914—31no—
47Dic3×D12—31no—
48Dic3×D12—31no—
49C2×#3699—32no—
50C9×#127—31no—
51#63502—31no—
52#63502D311yescharacteristic, complemented
53#63501—32no—
54#63501—32no—
55#63500—32no—
56D4×Dic9—31no—
57D4×Dic9—31no—
58D4×Dic9—61no—
59C4×D36—31no—
60C4×Dic18—31no—
61C23×C3×C9C2312yes—
62C22×D3×C9C2312yes—
63C22×C3×D9C2312yes—
64C3×D4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9—24no—
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C9×#67—24no—
71C9×#67—42no—
72C3×#611—24no—
73C3×#611—42no—
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C9×D12—22no—
79C3×D36—22no—
80#6194—24no—
81#6194—42no—
82D3×Dic9—24no—
83Dic3×D9—24no—
84#6189D414yescomplemented
85#5718—91no—
86#5718—91no—
87C2×C32×D4C2×D311yescharacteristic
88C22×C3⋊Dic3—32no—
89C2×C3×#67—32no—
90C2×C3×#67—32no—
91C2×C3×#67C2×D312yes—
92C2×C3×#67C2×D312yes—
93C2×C3×D12—31no—
94C2×C3×D12C2×D311yescharacteristic
95C2×#627—31no—
96C2×#625—32no—
97C2×#625—32no—
98#3805—68no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101#3757—32no—
102C4⋊(C3⋊Dic3)—31no—
103C4×C3⋊Dic3—31no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3C2×D311yescharacteristic
106#3724—31no—
107Dic3⋊Dic3—31no—
108Dic3⋊Dic3—31no—
109(C2×D3)⋊Dic3—32no—
110(C2×D3)⋊Dic3—32no—
111C2×D4×C9—21no—
112C2×D4×C9—32no—
113C2×D4×C9—32no—
114C2×D4×C9—31no—
115C2×D4×C9C2×D311yescharacteristic
116C2×#611—32no—
117C2×#611—32no—
118C2×#611C2×D312yes—
119C2×#611C2×D312yes—
120C22×Dic9—32no—
121C22×Dic9—32no—
122C22×Dic9—62no—
123#3699—68no—
124C2×D36C2×D311yescharacteristic
125C2×C4×D9—32no—
126C2×Dic18—31no—
127C42×C9—31no—
128#3676—32no—
129#3676—32no—
130#3676—62no—
131(C2×D9)⋊C4—32no—
132C4⋊Dic9—31no—
133C4⋊Dic9—31no—
134C4⋊Dic9—61no—
135Dic9⋊C4—31no—
136Dic9⋊C4—31no—
137C4×Dic9—31no—
138C4×Dic9—61no—
139C4×Dic9C2×D311yescharacteristic
140C22×C3×C9—22no—
141C22×C3×C9—22no—
142C22×C3×C9—22no—
143C22×C3×C9C2411yescharacteristic
144C2×D3×C9—22no—
145C2×D3×C9—22no—
146C2×D3×C9—22no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C2×C3×D9—22no—
150C3×C4×C9#4112yescomplemented
151C3⋊Dic9—22no—
152C3⋊Dic9—21no—
153C3⋊Dic9C2×D412yes—
154C9×Dic3—21no—
155C9×Dic3C2×D412yes—
156C3×Dic9—21no—
157C3×Dic9C2×D412yes—
158C2×#237—92no—
159C2×#237—92no—
160C3×#127—31no—
161C3×#127—91no—
162#940—31no—
163#940—91no—
164#940—91no—
165#940D911yescharacteristic, complemented
166#939—92no—
167#939—92no—
168#939—92no—
169#939—92no—
170#938—92no—
171#938—92no—
172D4×Dic3—91no—
173D4×Dic3—91no—
174D4×Dic3—91no—
175D4×Dic3—91no—
176D4×Dic3—181no—
177C4×D12—91no—
178C4×D12—91no—
179C4×Dic6—91no—
180C4×Dic6—91no—
181C23×C32C22×D312yes—
182C22×C3×D3—32no—
183C22×C3×D3C22×D312yes—
184C32×D4—22no—
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—64no—
190C3×#67—24no—
191C3×#67—42no—
192C3×#67—64no—
193C3×#67—122no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3C22×D311yescharacteristic
197C2×C3×Dic3C22×D311yescharacteristic
198C3×D12—22no—
199C3×D12—62no—
200#627—34no—
201#625—64no—
202#625—122no—
203D3×Dic3—64no—
204D3×Dic3—64no—
205C23×C9—22no—
206C23×C9—32no—
207C23×C9C22×D312yes—
208C22×D9C22×D312yes—
209D4×C9—24no—
210D4×C9—22no—
211D4×C9—64no—
212D4×C9—62no—
213D4×C9—122no—
214C2×C4×C9—21no—
215C2×C4×C9—31no—
216C2×C4×C9—31no—
217C2×C4×C9C22×D311yescharacteristic
218#611—24no—
219#611—42no—
220#611—64no—
221#611—122no—
222C2×Dic9—31no—
223C2×Dic9—31no—
224C2×Dic9—32no—
225C2×Dic9—61no—
226C2×Dic9—61no—
227C2×Dic9—62no—
228C2×Dic9—64no—
229C2×Dic9—64no—
230C2×Dic9—124no—
231C2×Dic9C22×D311yescharacteristic
232C2×Dic9C22×D311yescharacteristic
233D36—22no—
234C4×D9—64no—
235Dic18—34no—
236#550—271noSylow
237C2×C3×C9—42no—
238C2×C3×C9C22×D411yescharacteristic
239C2×C3×C9C2×#4111yescharacteristic
240C2×C3×C9H(5,𝔽2)11yescharacteristic
241D3×C9—42no—
242C3×D9—42no—
243C2×C3×D4—21no—
244C2×C3×D4—32no—
245C2×C3×D4—32no—
246C2×C3×D4—31no—
247C2×C3×D4—92no—
248C2×C3×D4—92no—
249C2×C3×D4—91no—
250C2×C3×D4C2×D911yescharacteristic
251C2×C3×D4D3211yescharacteristic
252C2×#67—32no—
253C2×#67—32no—
254C2×#67—92no—
255C2×#67—92no—
256C2×#67—92no—
257C2×#67—92no—
258C2×#67C2×D912yes—
259C2×#67C2×D912yes—
260C22×Dic3—92no—
261C22×Dic3—92no—
262C22×Dic3—92no—
263C22×Dic3—92no—
264C22×Dic3—182no—
265#237—188no—
266#237—188no—
267C2×D12—31no—
268C2×D12C2×D911yescharacteristic
269C2×C4×D3—92no—
270C2×C4×D3—92no—
271C2×Dic6—91no—
272C2×Dic6—91no—
273C3×C42—31no—
274C3×C42—91no—
275#217—92no—
276#217—92no—
277#217—92no—
278#217—92no—
279#217—182no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282C4⋊Dic3—91no—
283C4⋊Dic3—91no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—181no—
287Dic3⋊C4—91no—
288Dic3⋊C4—91no—
289Dic3⋊C4—91no—
290Dic3⋊C4—91no—
291C4×Dic3—31no—
292C4×Dic3—91no—
293C4×Dic3—91no—
294C4×Dic3—181no—
295C4×Dic3C2×D911yescharacteristic
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32C23×D311yescharacteristic
300C2×C3×D3—22no—
301C2×C3×D3—22no—
302C2×C3×D3—22no—
303C2×C3×D3—62no—
304C2×C3×D3—62no—
305C2×C3×D3—62no—
306C32×C4#23712yes—
307C3⋊Dic3—32no—
308C3⋊Dic3—62no—
309C3⋊Dic3—61no—
310C3×Dic3—21no—
311C3×Dic3—32no—
312C3×Dic3—61no—
313C3×Dic3D3×D412yes—
314C22×C9—24no—
315C22×C9—22no—
316C22×C9—22no—
317C22×C9—22no—
318C22×C9—21no—
319C22×C9—42no—
320C22×C9—42no—
321C22×C9—62no—
322C22×C9—62no—
323C22×C9—62no—
324C22×C9C23×D311yescharacteristic
325C2×D9—22no—
326C2×D9—22no—
327C2×D9—22no—
328C4×C9—22no—
329C4×C9—32no—
330C4×C9—61no—
331C4×C9#23712yescomplemented
332Dic9—21no—
333Dic9—32no—
334Dic9—62no—
335Dic9—62no—
336Dic9—61no—
337Dic9—121no—
338Dic9—122no—
339Dic9D3×D412yes—
340C2×#41—272no—
341#127—31no—
342#127—91no—
343#127—271no—
344#121—272no—
345#121—272no—
346#121—272no—
347C4×Q8—271no—
348C4×D4—271no—
349C4×D4—271no—
350C4×D4—271no—
351C3×C9#55011yescharacteristic, complemented, Sylow
352C23×C3—22no—
353C23×C3—32no—
354C23×C3—92no—
355C23×C3C22×D912yes—
356C23×C3C2×D3212yes—
357C22×D3—32no—
358C22×D3C22×D912yes—
359C3×D4—24no—
360C3×D4—22no—
361C3×D4—22no—
362C3×D4—64no—
363C3×D4—62no—
364C3×D4—122no—
365C3×D4—184no—
366C3×D4—182no—
367C3×D4—362no—
368C2×C3×C4—21no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4C22×D911yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375#67—24no—
376#67—42no—
377#67—64no—
378#67—122no—
379#67—184no—
380#67—184no—
381#67—362no—
382#67—362no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—91no—
388C2×Dic3—92no—
389C2×Dic3—91no—
390C2×Dic3—92no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—182no—
394C2×Dic3—184no—
395C2×Dic3—184no—
396C2×Dic3—184no—
397C2×Dic3—184no—
398C2×Dic3—364no—
399C2×Dic3C22×D911yescharacteristic
400C2×Dic3C22×D911yescharacteristic
401D12—22no—
402D12—62no—
403C4×D3—184no—
404C4×D3—184no—
405Dic6—94no—
406Dic6—94no—
407C2×C32—42no—
408C2×C32C2×D3×D411yescharacteristic
409C2×C32C2×#23711yescharacteristic
410C2×C32#100411yescharacteristic
411C3×D3—42no—
412C3×D3—122no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—44no—
417C2×C9—42no—
418C2×C9—122no—
419C2×C9C2×D3×D411yescharacteristic
420C2×C9C2×#23711yescharacteristic
421C2×C9#100411yescharacteristic
422D9—42no—
423#41—548no—
424C2×Q8—271no—
425C2×D4—32no—
426C2×D4—32no—
427C2×D4—31no—
428C2×D4—92no—
429C2×D4—92no—
430C2×D4—91no—
431C2×D4—272no—
432C2×D4—272no—
433C2×D4D3×D911yescharacteristic
434C22×C4—272no—
435C22×C4—272no—
436C22×C4—272no—
437C4⋊C4—271no—
438C4⋊C4—271no—
439C4⋊C4—271no—
440C4⋊C4—271no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C22⋊C4—272no—
444C42—31no—
445C42—91no—
446C42—271no—
447C22×C3—24no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—21no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—62no—
458C22×C3—62no—
459C22×C3—62no—
460C22×C3—182no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3C23×D911yescharacteristic
464C22×C3C22×D3211yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—62no—
471C3×C4—22no—
472C3×C4—32no—
473C3×C4—61no—
474C3×C4—92no—
475C3×C4—181no—
476C3×C4#369912yescomplemented
477C3×C4#380512yes—
478Dic3—21no—
479Dic3—32no—
480Dic3—61no—
481Dic3—92no—
482Dic3—92no—
483Dic3—182no—
484Dic3—182no—
485Dic3—181no—
486Dic3—182no—
487Dic3—181no—
488Dic3—362no—
489Dic3—361no—
490Dic3D4×D912yes—
491C32#571811yescharacteristic
492C9—21no—
493C9#571811yescharacteristic, complemented
494C23—32no—
495C23—92no—
496C23C2×D3×D912yes—
497Q8—274no—
498D4—22no—
499D4—64no—
500D4—62no—
501D4—122no—
502D4—184no—
503D4—182no—
504D4—362no—
505D4—544no—
506D4—1082no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—91no—
510C2×C4—91no—
511C2×C4—271no—
512C2×C4—271no—
513C2×C4—272no—
514C2×C4—544no—
515C2×C4—544no—
516C2×C4—544no—
517C2×C4C2×D3×D911yescharacteristic
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—44no—
522C2×C3—42no—
523C2×C3—42no—
524C2×C3—122no—
525C2×C3—362no—
526C2×C3C2×D4×D911yescharacteristic
527C2×C3C2×#369911yescharacteristic
528C2×C3#6371011yescharacteristic
529C2×C3D32×D411yescharacteristic
530C2×C3#6431311yescharacteristic
531C2×C3C2×#380511yescharacteristic
532D3—42no—
533D3—122no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—62no—
538C22—62no—
539C22—62no—
540C22—182no—
541C22—182no—
542C22—182no—
543C22C22×D3×D911yescharacteristic
544C4—32no—
545C4—61no—
546C4—92no—
547C4—181no—
548C4—272no—
549C4—541no—
550C4—542no—
551C4#8866312yescomplemented
552C3—21no—
553C3#1058982011yescharacteristic, complemented
554C3#1059514411yescharacteristic
555C2—42no—
556C2—122no—
557C2—362no—
558C2D3×D4×D911yescharacteristic
559C2#1172249911yescharacteristic
560C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#550
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88663C2semidirect
2. D4×C3⋊Dic9C2semidirect
3. C9×#940C2semidirect
4. C3×#63502C2semidirect
5. #11721290C2semidirect
6. #11721281C2semidirect
7. #11721270C2semidirect
8. #11721043C2semidirect
9. D12×Dic9C2semidirect
10. Dic3×D36C2semidirect
11. C4×#6189C2semidirect
52. #63502D3semidirect
84. #6189D4semidirect
150. C3×C4×C9#41semidirect
165. #940D9semidirect
331. C4×C9#237semidirect
351. C3×C9#550semidirect
476. C3×C4#3699semidirect
493. C9#5718semidirect
551. C4#88663semidirect
553. C3#10589820semidirect

Maximal subgroups

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

subgroupindex
C2×#886632
D4×C3⋊Dic92
C9×#9402
C3×#635022
#117212902
#117212812
#117212702
#117210432
D12×Dic92
Dic3×D362
C4×#61892
#105951443
#105898203