Finite Group Databasebeta

← #13147649

Subgroups — #13147649

Order 1728.

1024 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88640C212yesmaximal, complemented
2C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C42×D3×C9C211yesmaximal, characteristic, complemented
4C3×C4×D36C211yesmaximal, characteristic, complemented
5#11721038C211yesmaximal, characteristic, complemented
6#11721037C211yesmaximal, characteristic, complemented
7#11721034C212yesmaximal, complemented
8#11721006C211yesmaximal, characteristic, complemented
9C4×#6192C212yesmaximal, complemented
10#11720970C211yesmaximal, characteristic, complemented
11#11720940C212yesmaximal, complemented
12#10593968—31nomaximal
13#10589579—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C4×D3×C9C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C4×D3×C9C2211yescharacteristic
19C2×C3×D36C2211yescharacteristic
20C2×C3×C4×D9C2212yes—
21C2×#6195C2212yes—
22C2×#6194C2212yes—
23C2×#6192C2212yes—
24C2×#6192C2212yes—
25C2×#6189C2212yes—
26#88640—28no—
27C3×C42×C9C2211yescharacteristic
28#88536C2212yes—
29C4⋊(C3⋊Dic9)C2211yescharacteristic
30C3×(C2×D9)⋊C4C2212yes—
31C4×C9×Dic3C2211yescharacteristic
32C4×C9×Dic3C2211yescharacteristic
33C4×C9×Dic3C2211yescharacteristic
34C3×C4⋊Dic9C2211yescharacteristic
35#88445C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37Dic3⋊Dic9C2212yes—
38C2×#3798—32no—
39C4×C4⋊(C3⋊D3)—31no—
40C3×C4×D12—31no—
41C3×C42×D3D311yescharacteristic
42#63918—31no—
43#63911—31no—
44#63909—31no—
45#63908—31no—
46#63906—32no—
47C4×#626—32no—
48#63852—32no—
49C2×#3697—32no—
50C2×C42×C9—31no—
51#63492—31no—
52#63492—32no—
53#63492—31no—
54#63437—32no—
55#63436—31no—
56C4×D36—32no—
57C4×D36—31no—
58C4×D36—61no—
59C4×D36D311yescharacteristic, complemented
60#63431—31no—
61C22×C3⋊D9C2312yes—
62C22×D3×C9C2311yescharacteristic
63C22×C3×D9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C4⋊(C3⋊D9)—22no—
69C4×C3⋊D9—24no—
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C4×D3×C9—22no—
75C4×D3×C9—22no—
76C4×D3×C9D414yescomplemented
77C3×D36—22no—
78C3×C4×D9—24no—
79#6195—24no—
80#6194—24no—
81#6192—24no—
82#6192—24no—
83#6189—24no—
84#5548—91no—
85#5477—91no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4×C3⋊D3—32no—
88C2×C3×D12—31no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3C2×D311yescharacteristic
91C2×C3×C4×D3C2×D311yescharacteristic
92C2×C3×C4×D3C2×D311yescharacteristic
93C2×#627—32no—
94C2×#626—32no—
95C2×#626—32no—
96C2×#626—32no—
97C2×#625—32no—
98#3798—68no—
99C32×C42C2×D311yescharacteristic
100#3752—32no—
101C4⋊(C3⋊Dic3)—31no—
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—31no—
104C3×C4×Dic3C2×D311yescharacteristic
105C3×C4×Dic3C2×D311yescharacteristic
106C3×C4×Dic3C2×D311yescharacteristic
107Dic3⋊Dic3—32no—
108#3722—32no—
109(C2×D3)⋊Dic3—32no—
110C22×C4×C9—31no—
111C22×C4×C9—31no—
112C22×C4×C9—31no—
113C2×#611—32no—
114C2×#611—32no—
115#3697—68no—
116C2×D36—32no—
117C2×D36—32no—
118C2×D36—31no—
119C2×D36—61no—
120C2×D36C2×D311yescharacteristic
121C2×C4×D9—32no—
122C2×C4×D9—62no—
123C2×C4×D9C2×D312yes—
124C2×Dic18—32no—
125C42×C9—21no—
126C42×C9—31no—
127C42×C9—31no—
128C42×C9—31no—
129C42×C9C2×D311yescharacteristic
130(C2×D9)⋊C4—32no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—62no—
134(C2×D9)⋊C4C2×D312yes—
135C4⋊Dic9—31no—
136C4⋊Dic9—32no—
137C4⋊Dic9—61no—
138C4⋊Dic9C2×D311yescharacteristic
139C22×C3×C9C2411yescharacteristic
140C2×C3⋊D9—22no—
141C2×C3⋊D9—22no—
142C2×C3⋊D9—22no—
143C2×D3×C9—21no—
144C2×D3×C9—21no—
145C2×D3×C9C2×D412yes—
146C2×C3×D9—22no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C3×C4×C9—21no—
150C3×C4×C9C2×D412yes—
151C3×C4×C9#4112yescomplemented
152C3⋊Dic9—22no—
153C9×Dic3—21no—
154C9×Dic3C2×D412yes—
155C9×Dic3#4112yescomplemented
156C3×Dic9—22no—
157C2×#235—92no—
158C2×#235—92no—
159C3×C4×D4—91no—
160C2×C3×C42—31no—
161#940—91no—
162#938—91no—
163#930—91no—
164#930—92no—
165#930—91no—
166C4×#67—92no—
167#895—91no—
168#888—91no—
169#885—92no—
170#884—92no—
171#875—92no—
172#874—91no—
173C4×D12—31no—
174C4×D12—92no—
175C4×D12—91no—
176C4×D12—91no—
177C4×D12—181no—
178C42×D3D911yescharacteristic, complemented
179#869—91no—
180C22×C3⋊D3—32no—
181C22×C3×D3—32no—
182C22×C3×D3C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3—32no—
187C4⋊(C3⋊D3)—62no—
188C4×C3⋊D3—64no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3C22×D311yescharacteristic
191C2×C3×Dic3C22×D311yescharacteristic
192C2×C3×Dic3C22×D311yescharacteristic
193C3×D12—62no—
194C3×C4×D3—22no—
195C3×C4×D3—22no—
196C3×C4×D3—64no—
197C3×C4×D3D1214yes—
198#627—64no—
199#626—64no—
200#626—64no—
201#626—64no—
202#625—64no—
203C23×C9—31no—
204C22×D9—32no—
205C22×D9—62no—
206C22×D9C22×D312yes—
207C2×C4×C9—21no—
208C2×C4×C9—21no—
209C2×C4×C9—21no—
210C2×C4×C9—31no—
211C2×C4×C9—31no—
212C2×C4×C9—34no—
213C2×C4×C9—31no—
214C2×C4×C9—62no—
215C2×C4×C9—62no—
216C2×C4×C9C22×D311yescharacteristic
217C2×C4×C9C22×D311yescharacteristic
218C2×C4×C9C22×D311yescharacteristic
219#611—64no—
220#611—64no—
221C2×Dic9—32no—
222C2×Dic9—62no—
223C2×Dic9C22×D312yes—
224D36—22no—
225D36—64no—
226D36—64no—
227D36—64no—
228D36—62no—
229C4×D9—24no—
230C4×D9—64no—
231C4×D9—124no—
232Dic18—64no—
233#532—271noSylow
234C2×C3×C9C22×D411yescharacteristic
235C2×C3×C9C2×#4111yescharacteristic
236C2×C3×C9C2×#4111yescharacteristic
237C3⋊D9—42no—
238D3×C9—22no—
239C3×D9—42no—
240C2×C3×D4—91no—
241C22×C3×C4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—31no—
244C22×C3×C4—92no—
245C2×#67—92no—
246C2×#67—92no—
247C2×#67—92no—
248C2×#67—92no—
249C2×#67—92no—
250#235—188no—
251#235—188no—
252C2×D12—31no—
253C2×D12—92no—
254C2×D12—92no—
255C2×D12—92no—
256C2×D12—91no—
257C2×D12—91no—
258C2×D12—181no—
259C2×C4×D3—32no—
260C2×C4×D3—92no—
261C2×C4×D3—92no—
262C2×C4×D3—182no—
263C2×C4×D3C2×D911yescharacteristic
264C2×C4×D3C2×D911yescharacteristic
265C2×C4×D3C2×D911yescharacteristic
266C2×Dic6—92no—
267C2×Dic6—92no—
268C3×C4⋊C4—91no—
269C3×C22⋊C4—92no—
270C3×C42—21no—
271C3×C42—31no—
272C3×C42—31no—
273C3×C42—31no—
274C3×C42C2×D911yescharacteristic
275C3×C42D3211yescharacteristic
276#217—92no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—182no—
284C4⋊Dic3—31no—
285C4⋊Dic3—92no—
286C4⋊Dic3—91no—
287C4⋊Dic3—91no—
288C4⋊Dic3—181no—
289Dic3⋊C4—92no—
290C4×Dic3C2×D911yescharacteristic
291C4×Dic3C2×D911yescharacteristic
292C4×Dic3C2×D911yescharacteristic
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—62no—
295C2×C3⋊D3—62no—
296C2×C3⋊D3—62no—
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3C2×D1212yes—
303C32×C4—21no—
304C32×C4C2×D1212yes—
305C32×C4#23512yes—
306C3⋊Dic3—62no—
307C3×Dic3—21no—
308C3×Dic3—62no—
309C3×Dic3C2×D1212yes—
310C3×Dic3#23512yes—
311C22×C9—21no—
312C22×C9—32no—
313C22×C9—61no—
314C22×C9—61no—
315C22×C9C23×D311yescharacteristic
316C2×D9—22no—
317C2×D9—22no—
318C2×D9—22no—
319C2×D9—64no—
320C2×D9—62no—
321C2×D9—62no—
322C2×D9—62no—
323C2×D9—122no—
324C2×D9—122no—
325C4×C9—21no—
326C4×C9—22no—
327C4×C9—22no—
328C4×C9—32no—
329C4×C9—32no—
330C4×C9—41no—
331C4×C9—61no—
332C4×C9#23512yescomplemented
333C4×C9D3×D412yes—
334Dic9—22no—
335Dic9—62no—
336Dic9—122no—
337C2×#41—272no—
338C4⋊Q8—271no—
339#127—271no—
340#124—272no—
341#121—272no—
342#121—271no—
343#121—271no—
344C4×D4—91no—
345C4×D4—272no—
346C4×D4—271no—
347C2×C42—31no—
348C3×C9#53211yescharacteristic, complemented, Sylow
349C23×C3—31no—
350C23×C3—92no—
351C22×D3—32no—
352C22×D3—92no—
353C22×D3—92no—
354C22×D3—182no—
355C22×D3C22×D911yescharacteristic
356C3×D4—182no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—34no—
363C2×C3×C4—31no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—92no—
367C2×C3×C4—184no—
368C2×C3×C4C22×D911yescharacteristic
369C2×C3×C4C22×D911yescharacteristic
370C2×C3×C4C22×D911yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3211yescharacteristic
373C2×C3×C4C2×D3211yescharacteristic
374#67—184no—
375#67—184no—
376#67—184no—
377#67—184no—
378#67—184no—
379C2×Dic3—32no—
380C2×Dic3—92no—
381C2×Dic3—92no—
382C2×Dic3—182no—
383C2×Dic3C22×D911yescharacteristic
384C2×Dic3C22×D911yescharacteristic
385C2×Dic3C22×D911yescharacteristic
386D12—62no—
387D12—184no—
388D12—184no—
389D12—184no—
390D12—184no—
391D12—182no—
392D12—182no—
393C4×D3—22no—
394C4×D3—22no—
395C4×D3—64no—
396C4×D3—184no—
397C4×D3—184no—
398C4×D3—364no—
399C4×D3D3614yescomplemented
400Dic6—184no—
401Dic6—184no—
402C2×C32C22×D1211yescharacteristic
403C2×C32C2×#23511yescharacteristic
404C2×C32C2×#23511yescharacteristic
405C3⋊D3—122no—
406C3×D3—22no—
407C3×D3—122no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—62no—
412C2×C9C2×#23511yescharacteristic
413C2×C9C2×D3×D411yescharacteristic
414C2×C9D3×#4111yescharacteristic
415D9—42no—
416D9—124no—
417D9—122no—
418#41—548no—
419C2×Q8—272no—
420C2×D4—91no—
421C2×D4—272no—
422C2×D4—272no—
423C2×D4—272no—
424C2×D4—272no—
425C2×D4—271no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—92no—
430C22×C4—272no—
431C4⋊C4—91no—
432C4⋊C4—272no—
433C4⋊C4—271no—
434C22⋊C4—92no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C22⋊C4—272no—
438C42—31no—
439C42—31no—
440C42—31no—
441C42D3×D911yescharacteristic
442C22×C3—21no—
443C22×C3—32no—
444C22×C3—61no—
445C22×C3—61no—
446C22×C3—182no—
447C22×C3—182no—
448C22×C3—182no—
449C22×C3C23×D911yescharacteristic
450C22×C3C22×D3211yescharacteristic
451C2×D3—21no—
452C2×D3—21no—
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—184no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—362no—
464C2×D3—362no—
465C2×D3C2×D3612yes—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—32no—
471C3×C4—32no—
472C3×C4—41no—
473C3×C4—61no—
474C3×C4—182no—
475C3×C4C2×D3612yes—
476C3×C4#369712yescomplemented
477C3×C4#379812yes—
478C3×C4D3×D1212yes—
479Dic3—21no—
480Dic3—62no—
481Dic3—182no—
482Dic3—182no—
483Dic3—362no—
484Dic3C2×D3612yes—
485Dic3#369712yescomplemented
486C32#547711yescharacteristic
487C9—21no—
488C9#554811yescharacteristic, complemented
489C23—31no—
490C23—92no—
491C23—272no—
492Q8—544no—
493D4—182no—
494D4—544no—
495D4—544no—
496D4—544no—
497D4—544no—
498D4—542no—
499C2×C4—31no—
500C2×C4—31no—
501C2×C4—34no—
502C2×C4—31no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—92no—
506C2×C4—184no—
507C2×C4—272no—
508C2×C4—544no—
509C2×C4C2×D3×D911yescharacteristic
510C2×C4C2×D3×D911yescharacteristic
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—62no—
516C2×C3—362no—
517C2×C3C22×D3611yescharacteristic
518C2×C3C2×#369711yescharacteristic
519C2×C3C2×#369711yescharacteristic
520C2×C3C2×#379811yescharacteristic
521C2×C3C2×D3×D1211yescharacteristic
522C2×C3D3×#23511yescharacteristic
523D3—22no—
524D3—122no—
525D3—364no—
526D3—362no—
527D3—362no—
528C22—32no—
529C22—61no—
530C22—61no—
531C22—182no—
532C22—182no—
533C22—182no—
534C22—542no—
535C22—542no—
536C22—542no—
537C22C22×D3×D911yescharacteristic
538C4—21no—
539C4—32no—
540C4—32no—
541C4—61no—
542C4—182no—
543C4—542no—
544C4#8864012yescomplemented
545C4D3×D3612yes—
546C3—21no—
547C3#1058957911yescharacteristic, complemented
548C3#1059396811yescharacteristic
549C2—62no—
550C2—362no—
551C2—1082no—
552C2C2×#8864011yescharacteristic
553C2C2×D3×D3611yescharacteristic
554C2D3×#369711yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88640C2semidirect
2. C4×C4⋊(C3⋊D9)C2semidirect
3. C42×D3×C9C2semidirect
4. C3×C4×D36C2semidirect
5. #11721038C2semidirect
6. #11721037C2semidirect
7. #11721034C2semidirect
8. #11721006C2semidirect
9. C4×#6192C2semidirect
10. #11720970C2semidirect
11. #11720940C2semidirect
59. C4×D36D3semidirect
76. C4×D3×C9D4semidirect
151. C3×C4×C9#41semidirect
155. C9×Dic3#41semidirect
178. C42×D3D9semidirect
332. C4×C9#235semidirect
348. C3×C9#532semidirect
399. C4×D3D36semidirect
476. C3×C4#3697semidirect
485. Dic3#3697semidirect
488. C9#5548semidirect
544. C4#88640semidirect
547. C3#10589579semidirect

Maximal subgroups

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

subgroupindex
C2×#886402
C4×C4⋊(C3⋊D9)2
C42×D3×C92
C3×C4×D362
#117210382
#117210372
#117210342
#117210062
C4×#61922
#117209702
#117209402
#105939683
#105895793