Finite Group Databasebeta

← #13148751

Subgroups — #13148751

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×#88640C212yesmaximal, complemented
2#11721749C211yesmaximal, characteristic, complemented
3C9×#893C211yesmaximal, characteristic, complemented
4C3×#63455C211yesmaximal, characteristic, complemented
5#11721043C211yesmaximal, characteristic, complemented
6#11721041C212yesmaximal, complemented
7#11721040C211yesmaximal, characteristic, complemented
8#11721034C212yesmaximal, complemented
9#11721033C212yesmaximal, complemented
10#11721006C211yesmaximal, characteristic, complemented
11C4×#6189C211yesmaximal, characteristic, complemented
12#10594520—31nomaximal
13#10589655—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C9×D12C2211yescharacteristic
17C2×C4×D3×C9C2212yes—
18C2×C3×D36C2211yescharacteristic
19C2×C3×C4×D9C2212yes—
20C2×#6195C2212yes—
21C2×#6195C2212yes—
22C2×#6194C2212yes—
23C2×#6194C2212yes—
24C2×#6192C2212yes—
25C2×#6192C2212yes—
26C2×#6189C2211yescharacteristic
27#88640—28no—
28C3×C9×C4⋊C4C2211yescharacteristic
29#88536C2212yes—
30C4×C3⋊Dic9C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C4×C9×Dic3C2211yescharacteristic
34C3×C4×Dic9C2211yescharacteristic
35Dic3⋊Dic9C2211yescharacteristic
36#88442C2211yescharacteristic
37Dic9⋊Dic3C2211yescharacteristic
38C2×#3798—32no—
39#64102—31no—
40C3×#893—31no—
41C3×#893D311yescharacteristic
42C4×#627—31no—
43#63914—31no—
44#63912—32no—
45#63911—31no—
46#63911—31no—
47#63906—32no—
48#63906—32no—
49C2×#3697—32no—
50C4×D4×C9—31no—
51#63502—31no—
52#63502—31no—
53#63492—32no—
54#63455—31no—
55#63455—61no—
56#63455D311yescharacteristic, complemented
57#63446—32no—
58#63446—32no—
59#63436—31no—
60C4×Dic18—31no—
61C22×C3⋊D9C2312yes—
62C22×D3×C9C2312yes—
63C22×C3×D9C2312yes—
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C4⋊(C3⋊D9)—22no—
69C4×C3⋊D9—24no—
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C9×D12—22no—
75C4×D3×C9—24no—
76C3×D36—22no—
77C3×C4×D9—24no—
78#6195—24no—
79#6195—42no—
80#6194—24no—
81#6194—42no—
82#6192—24no—
83#6192—42no—
84#6189D414yescomplemented
85#5553—91no—
86#5553—91no—
87C2×C4⋊(C3⋊D3)—31no—
88C2×C4×C3⋊D3—32no—
89C2×C3×D12—31no—
90C2×C3×D12C2×D311yescharacteristic
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3C2×D312yes—
93C2×#627—31no—
94C2×#626—32no—
95C2×#626—32no—
96C2×#626—32no—
97C2×#626—32no—
98C2×#625—32no—
99C2×#625—32no—
100#3798—68no—
101C32×C4⋊C4C2×D311yescharacteristic
102#3752—32no—
103C4×C3⋊Dic3—31no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4C2×D312yes—
106C3×C4×Dic3—31no—
107C3×C4×Dic3C2×D311yescharacteristic
108#3724—31no—
109Dic3⋊Dic3—31no—
110Dic3⋊Dic3—31no—
111C2×D4×C9—31no—
112C22×C4×C9—32no—
113C2×#611—32no—
114C2×#611—32no—
115C2×#611—32no—
116C2×#611—32no—
117#3697—68no—
118C2×D36—32no—
119C2×D36—31no—
120C2×D36—32no—
121C2×D36—61no—
122C2×D36C2×D311yescharacteristic
123C2×C4×D9—32no—
124C2×C4×D9—62no—
125C2×C4×D9C2×D312yes—
126C2×Dic18—31no—
127C9×C4⋊C4—21no—
128C9×C4⋊C4C2×D311yescharacteristic
129C9×C22⋊C4—32no—
130C42×C9—31no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—62no—
133(C2×D9)⋊C4C2×D312yes—
134C4⋊Dic9—31no—
135Dic9⋊C4—31no—
136Dic9⋊C4—31no—
137C4×Dic9—31no—
138C4×Dic9—61no—
139C4×Dic9C2×D311yescharacteristic
140C22×C3×C9C2411yescharacteristic
141C2×C3⋊D9—22no—
142C2×C3⋊D9—22no—
143C2×C3⋊D9—22no—
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—22no—
151C3×C4×C9#4112yescomplemented
152C3⋊Dic9—21no—
153C3⋊Dic9C2×D412yes—
154C9×Dic3—21no—
155C9×Dic3C2×D412yes—
156C3×Dic9—21no—
157C3×Dic9C2×D412yes—
158C2×#235—92no—
159C2×#235—92no—
160C3×C4×D4—31no—
161C3×C4×D4—91no—
162#940—91no—
163#940—91no—
164#940—91no—
165#940—91no—
166#930—92no—
167#930—92no—
168#893—31no—
169#893—91no—
170#893—91no—
171#893—181no—
172#893D911yescharacteristic, complemented
173#884—92no—
174#884—92no—
175#884—92no—
176#884—92no—
177#874—91no—
178#874—91no—
179C4×Dic6—91no—
180C4×Dic6—91no—
181C22×C3⋊D3—32no—
182C22×C3×D3—32no—
183C22×C3×D3C22×D312yes—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—31no—
188C4⋊(C3⋊D3)—62no—
189C4×C3⋊D3—64no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3C22×D311yescharacteristic
193C2×C3×Dic3C22×D311yescharacteristic
194C3×D12—22no—
195C3×D12—62no—
196C3×C4×D3—24no—
197C3×C4×D3—64no—
198#627—34no—
199#626—64no—
200#626—64no—
201#626—122no—
202#626—122no—
203#625—64no—
204#625—122no—
205C23×C9—32no—
206C22×D9—32no—
207C22×D9—62no—
208C22×D9C22×D312yes—
209D4×C9—62no—
210C2×C4×C9—21no—
211C2×C4×C9—22no—
212C2×C4×C9—31no—
213C2×C4×C9—31no—
214C2×C4×C9—64no—
215C2×C4×C9C22×D312yes—
216C2×C4×C9C22×D311yescharacteristic
217#611—64no—
218#611—64no—
219#611—122no—
220#611—122no—
221C2×Dic9—31no—
222C2×Dic9—31no—
223C2×Dic9—61no—
224C2×Dic9—61no—
225C2×Dic9C22×D311yescharacteristic
226C2×Dic9C22×D311yescharacteristic
227D36—22no—
228D36—64no—
229D36—64no—
230D36—62no—
231D36—122no—
232C4×D9—24no—
233C4×D9—64no—
234C4×D9—124no—
235Dic18—34no—
236#550—271noSylow
237C2×C3×C9C22×D411yescharacteristic
238C2×C3×C9C2×#4111yescharacteristic
239C2×C3×C9H(5,𝔽2)11yescharacteristic
240C3⋊D9—42no—
241D3×C9—42no—
242C3×D9—42no—
243C2×C3×D4—31no—
244C2×C3×D4—91no—
245C22×C3×C4—32no—
246C22×C3×C4—92no—
247C2×#67—92no—
248C2×#67—92no—
249C2×#67—92no—
250C2×#67—92no—
251C2×#67—92no—
252C2×#67—92no—
253C2×#67—92no—
254C2×#67—92no—
255#235—188no—
256#235—188no—
257C2×D12—31no—
258C2×D12—92no—
259C2×D12—92no—
260C2×D12—91no—
261C2×D12—91no—
262C2×D12—92no—
263C2×D12—92no—
264C2×D12—181no—
265C2×D12C2×D911yescharacteristic
266C2×C4×D3—32no—
267C2×C4×D3—92no—
268C2×C4×D3—92no—
269C2×C4×D3—182no—
270C2×C4×D3C2×D912yes—
271C2×Dic6—91no—
272C2×Dic6—91no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4C2×D911yescharacteristic
275C3×C4⋊C4D3211yescharacteristic
276C3×C22⋊C4—32no—
277C3×C22⋊C4—92no—
278C3×C42—31no—
279C3×C42—91no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—182no—
284(C2×D3)⋊C4C2×D912yes—
285C4⋊Dic3—91no—
286C4⋊Dic3—91no—
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×C32C23×D311yescharacteristic
297C2×C3⋊D3—62no—
298C2×C3⋊D3—62no—
299C2×C3⋊D3—62no—
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—22no—
307C32×C4#23512yes—
308C3⋊Dic3—32no—
309C3⋊Dic3—61no—
310C3×Dic3—21no—
311C3×Dic3—32no—
312C3×Dic3—61no—
313C3×Dic3D3×D412yes—
314C22×C9—21no—
315C22×C9—62no—
316C22×C9—62no—
317C22×C9—62no—
318C22×C9C23×D311yescharacteristic
319C2×D9—22no—
320C2×D9—22no—
321C2×D9—22no—
322C2×D9—64no—
323C2×D9—62no—
324C2×D9—62no—
325C2×D9—62no—
326C2×D9—122no—
327C2×D9—122no—
328C4×C9—22no—
329C4×C9—22no—
330C4×C9—32no—
331C4×C9—42no—
332C4×C9—61no—
333C4×C9#23512yescomplemented
334Dic9—21no—
335Dic9—32no—
336Dic9—62no—
337Dic9—61no—
338Dic9—121no—
339Dic9D3×D412yes—
340C2×#41—272no—
341#127—271no—
342#127—271no—
343#127—271no—
344#121—272no—
345#121—272no—
346#121—272no—
347C4×Q8—271no—
348C4×D4—31no—
349C4×D4—91no—
350C4×D4—271no—
351C3×C9#55011yescharacteristic, complemented, Sylow
352C23×C3—32no—
353C23×C3—92no—
354C22×D3—32no—
355C22×D3—92no—
356C22×D3—92no—
357C22×D3—182no—
358C22×D3C22×D912yes—
359C3×D4—62no—
360C3×D4—182no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—64no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—184no—
369C2×C3×C4C22×D912yes—
370C2×C3×C4C22×D911yescharacteristic
371C2×C3×C4C2×D3212yes—
372C2×C3×C4C2×D3211yescharacteristic
373#67—184no—
374#67—184no—
375#67—184no—
376#67—184no—
377#67—362no—
378#67—362no—
379#67—362no—
380#67—362no—
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—181no—
388C2×Dic3—181no—
389C2×Dic3C22×D911yescharacteristic
390C2×Dic3C22×D911yescharacteristic
391D12—22no—
392D12—62no—
393D12—184no—
394D12—184no—
395D12—184no—
396D12—182no—
397D12—182no—
398D12—362no—
399D12—362no—
400C4×D3—24no—
401C4×D3—64no—
402C4×D3—184no—
403C4×D3—184no—
404C4×D3—364no—
405Dic6—94no—
406Dic6—94no—
407C2×C32C2×#23511yescharacteristic
408C2×C32C2×D3×D411yescharacteristic
409C2×C32#100911yescharacteristic
410C3⋊D3—122no—
411C3×D3—42no—
412C3×D3—122no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—122no—
417C2×C9C2×#23511yescharacteristic
418C2×C9C2×D3×D411yescharacteristic
419C2×C9#100911yescharacteristic
420D9—42no—
421D9—124no—
422D9—122no—
423#41—548no—
424C2×Q8—271no—
425C2×D4—31no—
426C2×D4—91no—
427C2×D4—272no—
428C2×D4—272no—
429C2×D4—272no—
430C2×D4—272no—
431C2×D4—272no—
432C2×D4—272no—
433C2×D4—271no—
434C22×C4—32no—
435C22×C4—92no—
436C22×C4—272no—
437C4⋊C4—271no—
438C4⋊C4—271no—
439C4⋊C4—271no—
440C4⋊C4D3×D911yescharacteristic
441C22⋊C4—32no—
442C22⋊C4—92no—
443C22⋊C4—272no—
444C42—31no—
445C42—91no—
446C42—271no—
447C22×C3—21no—
448C22×C3—62no—
449C22×C3—62no—
450C22×C3—62no—
451C22×C3—182no—
452C22×C3—182no—
453C22×C3—182no—
454C22×C3C23×D911yescharacteristic
455C22×C3C22×D3211yescharacteristic
456C2×D3—22no—
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—184no—
463C2×D3—182no—
464C2×D3—182no—
465C2×D3—182no—
466C2×D3—182no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—362no—
470C2×D3—362no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—32no—
475C3×C4—42no—
476C3×C4—61no—
477C3×C4—92no—
478C3×C4—181no—
479C3×C4#369712yescomplemented
480C3×C4#379812yes—
481Dic3—21no—
482Dic3—32no—
483Dic3—61no—
484Dic3—92no—
485Dic3—92no—
486Dic3—182no—
487Dic3—181no—
488Dic3—181no—
489Dic3—361no—
490Dic3D4×D912yes—
491C32#555311yescharacteristic
492C9—21no—
493C9#555311yescharacteristic, complemented
494C23—32no—
495C23—92no—
496C23—272no—
497Q8—274no—
498D4—62no—
499D4—182no—
500D4—544no—
501D4—544no—
502D4—544no—
503D4—542no—
504D4—1082no—
505D4—1082no—
506D4—1082no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—64no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—184no—
513C2×C4—271no—
514C2×C4—271no—
515C2×C4—544no—
516C2×C4C2×D3×D912yes—
517C2×C4C2×D3×D911yescharacteristic
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—122no—
522C2×C3—362no—
523C2×C3C2×#369711yescharacteristic
524C2×C3C2×D4×D911yescharacteristic
525C2×C3#6371511yescharacteristic
526C2×C3C2×#379811yescharacteristic
527C2×C3D32×D411yescharacteristic
528C2×C3#6432011yescharacteristic
529D3—42no—
530D3—122no—
531D3—364no—
532D3—362no—
533D3—362no—
534C22—62no—
535C22—62no—
536C22—62no—
537C22—182no—
538C22—182no—
539C22—182no—
540C22—542no—
541C22—542no—
542C22—542no—
543C22C22×D3×D911yescharacteristic
544C4—22no—
545C4—32no—
546C4—61no—
547C4—92no—
548C4—181no—
549C4—272no—
550C4—541no—
551C4#8864012yescomplemented
552C3—21no—
553C3#1058965511yescharacteristic, complemented
554C3#1059452011yescharacteristic
555C2—122no—
556C2—362no—
557C2—1082no—
558C2C2×#8864011yescharacteristic
559C2D3×D4×D911yescharacteristic
560C2#1172251511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#550
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88640C2semidirect
2. #11721749C2semidirect
3. C9×#893C2semidirect
4. C3×#63455C2semidirect
5. #11721043C2semidirect
6. #11721041C2semidirect
7. #11721040C2semidirect
8. #11721034C2semidirect
9. #11721033C2semidirect
10. #11721006C2semidirect
11. C4×#6189C2semidirect
56. #63455D3semidirect
84. #6189D4semidirect
151. C3×C4×C9#41semidirect
172. #893D9semidirect
333. C4×C9#235semidirect
351. C3×C9#550semidirect
479. C3×C4#3697semidirect
493. C9#5553semidirect
551. C4#88640semidirect
553. C3#10589655semidirect

Maximal subgroups

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

subgroupindex
C2×#886402
#117217492
C9×#8932
C3×#634552
#117210432
#117210412
#117210402
#117210342
#117210332
#117210062
C4×#61892
#105945203
#105896553