Finite Group Databasebeta

← #13150231

Subgroups — #13150231

Order 1728.

1038 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88663C212yesmaximal, complemented
2C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721290C212yesmaximal, complemented
6#11721044C212yesmaximal, complemented
7#11721043C211yesmaximal, characteristic, complemented
8#11721028C212yesmaximal, complemented
9#11721025C211yesmaximal, characteristic, complemented
10#11720998C211yesmaximal, characteristic, complemented
11#11720971C211yesmaximal, characteristic, complemented
12#10594929—31nomaximal
13#10589698—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C2×C3×#611C2212yes—
19C2×C9×D12C2211yescharacteristic
20C2×C3×D36C2211yescharacteristic
21C2×#6194C2212yes—
22C2×#6194C2212yes—
23#88663—28no—
24C2×D3×Dic9C2212yes—
25C2×Dic3×D9C2212yes—
26C2×#6189C2212yes—
27C4×C3⋊Dic9C2211yescharacteristic
28C4×C3⋊Dic9C2211yescharacteristic
29C4×C3⋊Dic9C2211yescharacteristic
30C4×C3⋊Dic9C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C9×C4⋊Dic3C2211yescharacteristic
34C3×C4⋊Dic9C2211yescharacteristic
35(C2×D3)⋊Dic9C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37#88442C2212yes—
38C2×#3805—32no—
39C2×C4×C3⋊Dic3—31no—
40C3×#930—31no—
41C3×#930D311yescharacteristic
42#63980—32no—
43#63919—31no—
44#63915—32no—
45#63914—31no—
46#63902—32no—
47#63898—31no—
48#63898—31no—
49C2×#3699—32no—
50C9×#121—31no—
51#63502—31no—
52#63501—32no—
53D4×Dic9—31no—
54#63492D311yescharacteristic, complemented
55C2×C4×Dic9—31no—
56C2×C4×Dic9—61no—
57#63455—31no—
58#63450—31no—
59#63447—32no—
60#611⋊C4—32no—
61C23×C3×C9C2311yescharacteristic
62C22×D3×C9C2312yes—
63C22×C3×D9C2312yes—
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9—22no—
68C2×C3⋊Dic9D414yescomplemented
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C9×#67—24no—
74C3×#611—24no—
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9C2312yes—
77C9×D12—22no—
78C3×D36—22no—
79#6194—24no—
80#6194—24no—
81D3×Dic9—24no—
82Dic3×D9—24no—
83#6189—24no—
84#5596—91no—
85#5596—91no—
86C22×C32×C4C2×D311yescharacteristic
87C22×C3⋊Dic3—31no—
88C22×C3⋊Dic3—31no—
89C2×C3×#67—32no—
90C2×C3×#67C2×D312yes—
91C2×C3×D12—31no—
92C2×C3×D12C2×D311yescharacteristic
93C2×#627—32no—
94C2×#625—32no—
95C2×#625—32no—
96#3805—68no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99C4×C3⋊Dic3—31no—
100C4×C3⋊Dic3—31no—
101C4×C3⋊Dic3—31no—
102C4×C3⋊Dic3—31no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4C2×D312yes—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3C2×D311yescharacteristic
107#3724—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110C2×D4×C9—32no—
111C2×D4×C9—31no—
112C22×C4×C9—21no—
113C22×C4×C9C2×D311yescharacteristic
114C2×#611—32no—
115C2×#611—32no—
116C2×#611C2×D312yes—
117C22×Dic9—31no—
118C22×Dic9—31no—
119C22×Dic9—32no—
120C22×Dic9—61no—
121C22×Dic9—61no—
122#3699—68no—
123C2×D36C2×D311yescharacteristic
124C2×C4×D9—32no—
125C2×Dic18—32no—
126C9×C4⋊C4—31no—
127C9×C22⋊C4—32no—
128#3676—32no—
129(C2×D9)⋊C4—32no—
130(C2×D9)⋊C4C2×D312yes—
131C4⋊Dic9C2×D311yescharacteristic
132Dic9⋊C4—32no—
133C4×Dic9—31no—
134C4×Dic9—31no—
135C4×Dic9—31no—
136C4×Dic9—31no—
137C4×Dic9—61no—
138C4×Dic9—61no—
139C4×Dic9—61no—
140C4×Dic9—61no—
141C22×C3×C9—21no—
142C22×C3×C9—21no—
143C22×C3×C9C2×D412yes—
144C22×C3×C9C2411yescharacteristic
145C2×D3×C9—22no—
146C2×D3×C9—22no—
147C2×D3×C9—22no—
148C2×C3×D9—22no—
149C2×C3×D9—22no—
150C2×C3×D9—22no—
151C3×C4×C9—21no—
152C3×C4×C9#4112yescomplemented
153C3⋊Dic9—21no—
154C3⋊Dic9C2×D412yes—
155C3⋊Dic9C2×D412yes—
156C3⋊Dic9#4112yescomplemented
157C9×Dic3—22no—
158C3×Dic9—22no—
159C2×#237—92no—
160C2×#237—92no—
161C3×#121—31no—
162C3×#121—91no—
163#940—91no—
164#940—91no—
165#939—92no—
166#939—92no—
167D4×Dic3—91no—
168D4×Dic3—91no—
169#930—31no—
170#930D911yescharacteristic, complemented
171C2×C4×Dic3—91no—
172C2×C4×Dic3—91no—
173C2×C4×Dic3—181no—
174#893—91no—
175#893—91no—
176#888—91no—
177#888—91no—
178#885—92no—
179#885—92no—
180#67⋊C4—92no—
181#67⋊C4—92no—
182C23×C32C22×D311yescharacteristic
183C22×C3×D3—32no—
184C22×C3×D3C22×D312yes—
185C2×C32×C4—22no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—62no—
194C3×#67—24no—
195C3×#67—64no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3C22×D312yes—
198C3×D12—22no—
199C3×D12—62no—
200#627—64no—
201#625—64no—
202#625—64no—
203D3×Dic3—64no—
204D3×Dic3—64no—
205C23×C9—21no—
206C23×C9—32no—
207C23×C9C22×D311yescharacteristic
208C22×D9C22×D312yes—
209D4×C9—64no—
210D4×C9—62no—
211C2×C4×C9—24no—
212C2×C4×C9—22no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—32no—
216C2×C4×C9C22×D311yescharacteristic
217C2×C4×C9C22×D311yescharacteristic
218#611—24no—
219#611—64no—
220#611—64no—
221C2×Dic9—34no—
222C2×Dic9—31no—
223C2×Dic9—31no—
224C2×Dic9—31no—
225C2×Dic9—31no—
226C2×Dic9—61no—
227C2×Dic9—61no—
228C2×Dic9—61no—
229C2×Dic9—61no—
230C2×Dic9—64no—
231C2×Dic9—64no—
232C2×Dic9—62no—
233C2×Dic9—64no—
234C2×Dic9C22×D312yes—
235D36—22no—
236C4×D9—64no—
237Dic18—64no—
238#532—271noSylow
239C2×C3×C9—22no—
240C2×C3×C9C22×D411yescharacteristic
241C2×C3×C9C2×#4111yescharacteristic
242C2×C3×C9C2×#4111yescharacteristic
243D3×C9—42no—
244C3×D9—42no—
245C2×C3×D4—32no—
246C2×C3×D4—31no—
247C2×C3×D4—92no—
248C2×C3×D4—91no—
249C22×C3×C4—21no—
250C22×C3×C4C2×D911yescharacteristic
251C22×C3×C4D3211yescharacteristic
252C2×#67—32no—
253C2×#67—92no—
254C2×#67—92no—
255C2×#67—92no—
256C2×#67—92no—
257C2×#67C2×D912yes—
258C22×Dic3—92no—
259C22×Dic3—91no—
260C22×Dic3—91no—
261C22×Dic3—91no—
262C22×Dic3—91no—
263C22×Dic3—92no—
264C22×Dic3—181no—
265C22×Dic3—181no—
266#237—188no—
267#237—188no—
268C2×D12—31no—
269C2×D12C2×D911yescharacteristic
270C2×C4×D3—92no—
271C2×C4×D3—92no—
272C2×Dic6—92no—
273C2×Dic6—92no—
274C3×C4⋊C4—31no—
275C3×C4⋊C4—91no—
276C3×C22⋊C4—32no—
277C3×C22⋊C4—92no—
278#217—92no—
279#217—92no—
280(C2×D3)⋊C4—32no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4C2×D912yes—
284C4⋊Dic3—31no—
285C4⋊Dic3C2×D911yescharacteristic
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—91no—
294C4×Dic3—91no—
295C4×Dic3—91no—
296C4×Dic3—181no—
297C4×Dic3—181no—
298C4×Dic3—181no—
299C4×Dic3—181no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32D3×D412yes—
303C22×C32C23×D311yescharacteristic
304C2×C3×D3—22no—
305C2×C3×D3—22no—
306C2×C3×D3—22no—
307C2×C3×D3—62no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C32×C4—21no—
311C32×C4#23712yes—
312C3⋊Dic3—32no—
313C3⋊Dic3—32no—
314C3⋊Dic3—32no—
315C3⋊Dic3—61no—
316C3×Dic3—22no—
317C3×Dic3—62no—
318C22×C9—22no—
319C22×C9—22no—
320C22×C9—21no—
321C22×C9—21no—
322C22×C9—21no—
323C22×C9—22no—
324C22×C9—62no—
325C22×C9—62no—
326C22×C9—62no—
327C22×C9D3×D412yes—
328C22×C9C23×D311yescharacteristic
329C2×D9—22no—
330C2×D9—22no—
331C2×D9—22no—
332C4×C9—22no—
333C4×C9—21no—
334C4×C9—22no—
335C4×C9—62no—
336C4×C9#23712yescomplemented
337Dic9—22no—
338Dic9—32no—
339Dic9—32no—
340Dic9—32no—
341Dic9—62no—
342Dic9—62no—
343Dic9—62no—
344Dic9—62no—
345Dic9—61no—
346C2×#41—272no—
347C4⋊Q8—271no—
348#127—271no—
349#124—272no—
350#121—31no—
351#121—91no—
352#121—272no—
353C4×D4—272no—
354C4×D4—271no—
355C4×D4—271no—
356C2×C42—271no—
357C3×C9#53211yescharacteristic, complemented, Sylow
358C23×C3—21no—
359C23×C3—32no—
360C23×C3—92no—
361C23×C3C22×D911yescharacteristic
362C23×C3C2×D3211yescharacteristic
363C22×D3—32no—
364C22×D3C22×D912yes—
365C3×D4—64no—
366C3×D4—62no—
367C3×D4—184no—
368C3×D4—182no—
369C2×C3×C4—24no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—32no—
375C2×C3×C4—92no—
376C2×C3×C4C22×D911yescharacteristic
377C2×C3×C4C22×D911yescharacteristic
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C2×D3211yescharacteristic
380#67—24no—
381#67—64no—
382#67—184no—
383#67—184no—
384#67—184no—
385#67—184no—
386C2×Dic3—32no—
387C2×Dic3—91no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—94no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—94no—
395C2×Dic3—91no—
396C2×Dic3—91no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—181no—
401C2×Dic3—184no—
402C2×Dic3—184no—
403C2×Dic3—184no—
404C2×Dic3—182no—
405C2×Dic3—182no—
406C2×Dic3—184no—
407C2×Dic3C22×D912yes—
408D12—22no—
409D12—62no—
410C4×D3—184no—
411C4×D3—184no—
412Dic6—184no—
413Dic6—184no—
414C2×C32—22no—
415C2×C32C2×D3×D411yescharacteristic
416C2×C32C2×#23711yescharacteristic
417C2×C32D3×#4111yescharacteristic
418C3×D3—42no—
419C3×D3—122no—
420C2×C9—24no—
421C2×C9—22no—
422C2×C9—21no—
423C2×C9—21no—
424C2×C9—21no—
425C2×C9—122no—
426C2×C9C2×D3×D411yescharacteristic
427C2×C9C2×#23711yescharacteristic
428C2×C9D3×#4111yescharacteristic
429D9—42no—
430#41—548no—
431C2×Q8—272no—
432C2×D4—32no—
433C2×D4—31no—
434C2×D4—92no—
435C2×D4—91no—
436C2×D4—272no—
437C2×D4—272no—
438C22×C4—272no—
439C22×C4—272no—
440C22×C4—271no—
441C22×C4—271no—
442C22×C4D3×D911yescharacteristic
443C4⋊C4—31no—
444C4⋊C4—91no—
445C4⋊C4—272no—
446C22⋊C4—32no—
447C22⋊C4—92no—
448C22⋊C4—272no—
449C22⋊C4—272no—
450C42—271no—
451C42—271no—
452C42—271no—
453C42—271no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—22no—
462C22×C3—62no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—182no—
468C22×C3D4×D912yes—
469C22×C3C23×D911yescharacteristic
470C22×C3#380212yes—
471C22×C3C22×D3211yescharacteristic
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—62no—
478C3×C4—22no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—22no—
482C3×C4—62no—
483C3×C4—182no—
484C3×C4#369912yescomplemented
485C3×C4#380512yes—
486Dic3—22no—
487Dic3—62no—
488Dic3—92no—
489Dic3—92no—
490Dic3—92no—
491Dic3—92no—
492Dic3—92no—
493Dic3—92no—
494Dic3—182no—
495Dic3—182no—
496Dic3—182no—
497Dic3—182no—
498Dic3—181no—
499Dic3—181no—
500C32#559611yescharacteristic
501C9—21no—
502C9#559611yescharacteristic, complemented
503C23—32no—
504C23—92no—
505C23C2×D3×D911yescharacteristic
506Q8—544no—
507D4—64no—
508D4—62no—
509D4—184no—
510D4—182no—
511D4—544no—
512D4—544no—
513C2×C4—22no—
514C2×C4—32no—
515C2×C4—92no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—271no—
519C2×C4—274no—
520C2×C4—271no—
521C2×C4—544no—
522C2×C4—544no—
523C2×C4—542no—
524C2×C4C2×D3×D911yescharacteristic
525C2×C4C2×D3×D911yescharacteristic
526C2×C3—24no—
527C2×C3—22no—
528C2×C3—22no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—122no—
533C2×C3—362no—
534C2×C3C2×D4×D911yescharacteristic
535C2×C3C2×#369911yescharacteristic
536C2×C3#41×D911yescharacteristic
537C2×C3C2×#380211yescharacteristic
538C2×C3#6430611yescharacteristic
539C2×C3C2×#380511yescharacteristic
540D3—42no—
541D3—122no—
542C22—21no—
543C22—21no—
544C22—62no—
545C22—62no—
546C22—62no—
547C22—182no—
548C22—182no—
549C22—182no—
550C22#8864612yes—
551C22C22×D3×D911yescharacteristic
552C4—21no—
553C4—62no—
554C4—182no—
555C4—272no—
556C4—272no—
557C4—272no—
558C4—541no—
559C4#8866312yescomplemented
560C3—21no—
561C3#1058969811yescharacteristic, complemented
562C3#1059492911yescharacteristic
563C2—22no—
564C2—122no—
565C2—362no—
566C2C2×#8864611yescharacteristic
567C2#1172247511yescharacteristic
568C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#532
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×#88663C2semidirect
2. C2×C4×C3⋊Dic9C2semidirect
3. C9×#930C2semidirect
4. C3×#63492C2semidirect
5. #11721290C2semidirect
6. #11721044C2semidirect
7. #11721043C2semidirect
8. #11721028C2semidirect
9. #11721025C2semidirect
10. #11720998C2semidirect
11. #11720971C2semidirect
54. #63492D3semidirect
68. C2×C3⋊Dic9D4semidirect
152. C3×C4×C9#41semidirect
156. C3⋊Dic9#41semidirect
170. #930D9semidirect
336. C4×C9#237semidirect
357. C3×C9#532semidirect
484. C3×C4#3699semidirect
502. C9#5596semidirect
559. C4#88663semidirect
561. C3#10589698semidirect

Maximal subgroups

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

subgroupindex
C2×#886632
C2×C4×C3⋊Dic92
C9×#9302
C3×#634922
#117212902
#117210442
#117210432
#117210282
#117210252
#117209982
#117209712
#105949293
#105896983