Finite Group Databasebeta

← #13150372

Subgroups — #13150372

Order 1728.

964 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic18C211yesmaximal, characteristic, complemented
2C2×#88639C211yesmaximal, characteristic, complemented
3C4×#6233C211yesmaximal, characteristic, complemented
4C9×#930C211yesmaximal, characteristic, complemented
5#11721288C212yesmaximal, complemented
6#67×Dic9C212yesmaximal, complemented
7#11721035C212yesmaximal, complemented
8#11721018C212yesmaximal, complemented
9#11720967C211yesmaximal, characteristic, complemented
10#11720936C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10594900—31nomaximal
13#10589700—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C22×C3×Dic9C2212yes—
19C2×C9×D12C2211yescharacteristic
20C2×C3×Dic18—24no—
21C2×C3×Dic18C2211yescharacteristic
22C2×C3×Dic18C2212yes—
23C2×C3×Dic18C2211yescharacteristic
24C2×#6195C2212yes—
25C2×D3×Dic9C2212yes—
26#88639—24no—
27#88551C2211yescharacteristic
28#88536C2211yescharacteristic
29#88534C2211yescharacteristic
30C4×C3⋊Dic9C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yes—
32C9×C4⋊Dic3C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34(C2×D3)⋊Dic9C2212yes—
35#88445C2212yes—
36Dic3⋊Dic9C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C22×C3×Dic6—31no—
40C2×#3796—31no—
41C4×#638—31no—
42C3×#930D311yescharacteristic
43#63978—32no—
44Dic3×#67—32no—
45#63907—32no—
46#63887—32no—
47#63871—31no—
48#63863—31no—
49#63843—31no—
50C2×#3699—31no—
51C22×Dic18D311yescharacteristic, complemented
52C9×#121—31no—
53#63501—32no—
54#63498—31no—
55D4×Dic9—32no—
56C4×#611—31no—
57C4×#611—61no—
58#63459—31no—
59#63449—31no—
60#63447—32no—
61#63440—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2312yes—
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68#6233—22no—
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2311yescharacteristic
72C4×C3⋊D9—22no—
73C9×#67—24no—
74C2×C9×Dic3C2312yes—
75C2×C3×Dic9—24no—
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2312yes—
78C9×D12—22no—
79C3×Dic18—22no—
80C3×Dic18—24no—
81C3×Dic18D414yescomplemented
82#6195—24no—
83D3×Dic9—24no—
84#5817—91no—
85#5598—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#638—31no—
88C2×C4×C3⋊D3—31no—
89C2×C3×#67C2×D312yes—
90C22×C3×Dic3—32no—
91C2×C3×D12C2×D311yescharacteristic
92C2×C3×Dic6—31no—
93C2×C3×Dic6—31no—
94C2×C3×Dic6—32no—
95C2×C3×Dic6—64no—
96C2×#626—32no—
97C2×D3×Dic3—32no—
98#3796—64no—
99#3757—31no—
100#3752—31no—
101#3750—31no—
102C4×C3⋊Dic3—31no—
103C3×(C2×D3)⋊C4C2×D312yes—
104C3×C4⋊Dic3C2×D311yescharacteristic
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107#3722—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C2×D4×C9—32no—
112C2×D4×C9—31no—
113C22×C4×C9—21no—
114C22×C4×C9C2×D311yescharacteristic
115C2×#611—31no—
116C2×#611—32no—
117C2×#611—61no—
118C22×Dic9—32no—
119C22×Dic9C2×D312yes—
120#3699—64no—
121C2×C4×D9—31no—
122C2×C4×D9—61no—
123C2×Dic18—24no—
124C2×Dic18C2×D311yescharacteristic
125C2×Dic18C2×D312yes—
126C2×Dic18C2×D311yescharacteristic
127C9×C4⋊C4—31no—
128C9×C22⋊C4—32no—
129#3676—31no—
130#3676—32no—
131#3676—32no—
132#3676—61no—
133(C2×D9)⋊C4—31no—
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—61no—
136C4⋊Dic9—32no—
137Dic9⋊C4—31no—
138Dic9⋊C4—32no—
139Dic9⋊C4—61no—
140C4×Dic9—31no—
141C4×Dic9—32no—
142C4×Dic9—61no—
143C22×C3×C9—21no—
144C22×C3×C9—21no—
145C22×C3×C9#4112yescomplemented
146C22×C3×C9C2411yescharacteristic
147C2×C3⋊D9—21no—
148C2×C3⋊D9—21no—
149C2×C3⋊D9—21no—
150C2×D3×C9—22no—
151C2×D3×C9—22no—
152C2×D3×C9—22no—
153C3×C4×C9—21no—
154C3×C4×C9C2×D412yes—
155C3⋊Dic9—21no—
156C3⋊Dic9—21no—
157C3⋊Dic9—21no—
158C9×Dic3—22no—
159C3×Dic9—22no—
160C3×Dic9C2×D414yes—
161C22×C3×Q8—91no—
162C2×#239—91no—
163C2×#237—91no—
164C22×Dic6—31no—
165C3×#121—31no—
166#947—91no—
167#947—92no—
168#946—92no—
169#946—91no—
170Q8×Dic3—91no—
171#939—92no—
172#936—91no—
173D4×Dic3—92no—
174#930—92no—
175#930D911yescharacteristic, complemented
176C4×#67—91no—
177C4×#67—91no—
178C4×#67—92no—
179C4×#67—181no—
180#897—91no—
181#887—91no—
182#885—92no—
183#878—92no—
184C23×C32C22×D311yescharacteristic
185C22×C3⋊D3—31no—
186C22×C3×D3C22×D312yes—
187C2×C32×C4—22no—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190#638—62no—
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—31no—
194C4×C3⋊D3—62no—
195C3×#67—24no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—64no—
199C2×C3×Dic3C22×D312yes—
200C3×D12—22no—
201C3×Dic6—34no—
202C3×Dic6—64no—
203C3×Dic6—62no—
204#626—64no—
205D3×Dic3—64no—
206C23×C9—21no—
207C23×C9—32no—
208C23×C9C22×D311yescharacteristic
209C22×D9—31no—
210C22×D9—61no—
211D4×C9—64no—
212D4×C9—62no—
213C2×C4×C9—22no—
214C2×C4×C9—21no—
215C2×C4×C9—21no—
216C2×C4×C9—32no—
217C2×C4×C9—42no—
218C2×C4×C9C22×D311yescharacteristic
219C2×C4×C9C22×D311yescharacteristic
220#611—64no—
221#611—62no—
222#611—64no—
223C2×Dic9—24no—
224C2×Dic9—31no—
225C2×Dic9—31no—
226C2×Dic9—31no—
227C2×Dic9—61no—
228C2×Dic9—61no—
229C2×Dic9—61no—
230C2×Dic9—64no—
231C2×Dic9C22×D312yes—
232C2×Dic9C22×D312yes—
233C4×D9—62no—
234C4×D9—122no—
235Dic18—22no—
236Dic18—24no—
237Dic18#6714yescomplemented
238#548—271noSylow
239C2×C3×C9—22no—
240C2×C3×C9C22×D411yescharacteristic
241C2×C3×C9C2×#4111yescharacteristic
242C2×C3×C92−1+411yescharacteristic
243C3⋊D9—41no—
244D3×C9—42no—
245C2×C3×Q8—91no—
246C2×C3×Q8—92no—
247C2×C3×Q8—91no—
248C2×C3×Q8—184no—
249C2×C3×D4—32no—
250C2×C3×D4—31no—
251C22×C3×C4—21no—
252C22×C3×C4—92no—
253C22×C3×C4C2×D911yescharacteristic
254C22×C3×C4D3211yescharacteristic
255C2×#67—92no—
256C2×#67—91no—
257C2×#67—91no—
258C2×#67—181no—
259C2×#67C2×D912yes—
260C22×Dic3—32no—
261C22×Dic3—92no—
262#239—184no—
263#237—184no—
264C2×D12—92no—
265C2×D12C2×D911yescharacteristic
266C2×C4×D3—92no—
267C2×C4×D3—91no—
268C2×C4×D3—91no—
269C2×C4×D3—181no—
270C2×Dic6—31no—
271C2×Dic6—32no—
272C2×Dic6—31no—
273C2×Dic6—64no—
274C3×C4⋊C4—31no—
275C3×C22⋊C4—32no—
276#217—92no—
277#217—92no—
278#217—91no—
279#217—91no—
280#217—181no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—91no—
285(C2×D3)⋊C4—91no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—181no—
288(C2×D3)⋊C4C2×D912yes—
289C4⋊Dic3—92no—
290C4⋊Dic3—92no—
291C4⋊Dic3C2×D911yescharacteristic
292Dic3⋊C4—92no—
293Dic3⋊C4—91no—
294Dic3⋊C4—91no—
295Dic3⋊C4—92no—
296Dic3⋊C4—181no—
297C4×Dic3—92no—
298C4×Dic3—91no—
299C4×Dic3—91no—
300C4×Dic3—92no—
301C4×Dic3—181no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32#23712yes—
305C22×C32C23×D311yescharacteristic
306C2×C3⋊D3—61no—
307C2×C3⋊D3—61no—
308C2×C3⋊D3—61no—
309C2×C3×D3—22no—
310C2×C3×D3—22no—
311C2×C3×D3—22no—
312C32×C4—21no—
313C32×C4D3×D412yes—
314C3⋊Dic3—61no—
315C3⋊Dic3—61no—
316C3⋊Dic3—61no—
317C3×Dic3—22no—
318C3×Dic3—34no—
319C3×Dic3—62no—
320C22×C9—21no—
321C22×C9—21no—
322C22×C9—21no—
323C22×C9—22no—
324C22×C9—41no—
325C22×C9—41no—
326C22×C9—62no—
327C22×C9—62no—
328C22×C9—62no—
329C22×C9#23912yescomplemented
330C22×C9C23×D311yescharacteristic
331C2×D9—62no—
332C2×D9—61no—
333C2×D9—61no—
334C2×D9—61no—
335C2×D9—121no—
336C2×D9—121no—
337C4×C9—21no—
338C4×C9—22no—
339C4×C9—41no—
340C4×C9—62no—
341C4×C9C2×#6712yes—
342Dic9—22no—
343Dic9—62no—
344Dic9—61no—
345Dic9—61no—
346Dic9—61no—
347Dic9—121no—
348Dic9—121no—
349Dic9C2×#6714yes—
350C2×#41—271no—
351C22×Q8—91no—
352#124—272no—
353#124—271no—
354C22⋊Q8—272no—
355C22⋊Q8—271no—
356#121—31no—
357#121—272no—
358C4×Q8—271no—
359C4×D4—272no—
360C4×D4—271no—
361C3×C9#54811yescharacteristic, complemented, Sylow
362C23×C3—21no—
363C23×C3—32no—
364C23×C3C22×D911yescharacteristic
365C23×C3C2×D3211yescharacteristic
366C22×D3—91no—
367C22×D3—91no—
368C22×D3—181no—
369C22×D3C22×D912yes—
370C3×Q8—94no—
371C3×Q8—184no—
372C3×Q8—182no—
373C3×D4—64no—
374C3×D4—62no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—32no—
380C2×C3×C4—42no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—184no—
384C2×C3×C4C22×D911yescharacteristic
385C2×C3×C4C22×D911yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388#67—24no—
389#67—184no—
390#67—184no—
391#67—182no—
392#67—182no—
393C2×Dic3—32no—
394C2×Dic3—32no—
395C2×Dic3—64no—
396C2×Dic3—91no—
397C2×Dic3—91no—
398C2×Dic3—91no—
399C2×Dic3—91no—
400C2×Dic3—91no—
401C2×Dic3—91no—
402C2×Dic3—181no—
403C2×Dic3—181no—
404C2×Dic3—181no—
405C2×Dic3—184no—
406C2×Dic3C22×D912yes—
407D12—22no—
408D12—184no—
409C4×D3—184no—
410C4×D3—182no—
411C4×D3—182no—
412C4×D3—362no—
413Dic6—34no—
414Dic6—64no—
415Dic6—62no—
416C2×C32—22no—
417C2×C32C2×D3×D411yescharacteristic
418C2×C32C2×#23711yescharacteristic
419C2×C32#101011yescharacteristic
420C3⋊D3—121no—
421C3×D3—42no—
422C2×C9—22no—
423C2×C9—21no—
424C2×C9—21no—
425C2×C9—21no—
426C2×C9—42no—
427C2×C9—122no—
428C2×C9C2×#23911yescharacteristic
429C2×C9#100711yescharacteristic
430C2×C9C22×#6711yescharacteristic
431D9—122no—
432D9—121no—
433#41—544no—
434C2×Q8—91no—
435C2×Q8—92no—
436C2×Q8—91no—
437C2×Q8—184no—
438C2×D4—32no—
439C2×D4—31no—
440C2×D4—272no—
441C2×D4—271no—
442C22×C4—92no—
443C22×C4—272no—
444C22×C4—271no—
445C22×C4D3×D911yescharacteristic
446C4⋊C4—31no—
447C4⋊C4—272no—
448C4⋊C4—272no—
449C4⋊C4—271no—
450C22⋊C4—32no—
451C22⋊C4—272no—
452C22⋊C4—272no—
453C22⋊C4—272no—
454C22⋊C4—271no—
455C22⋊C4—271no—
456C42—272no—
457C42—271no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—22no—
463C22×C3—21no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3#369912yescomplemented
470C22×C3C23×D911yescharacteristic
471C22×C3#379612yes—
472C22×C3C22×D3211yescharacteristic
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—22no—
476C2×D3—182no—
477C2×D3—181no—
478C2×D3—181no—
479C2×D3—181no—
480C2×D3—181no—
481C2×D3—181no—
482C2×D3—181no—
483C2×D3—361no—
484C2×D3—361no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—22no—
488C3×C4—41no—
489C3×C4—62no—
490C3×C4—94no—
491C3×C4—182no—
492C3×C4D4×D912yes—
493C3×C4D3×#6712yes—
494Dic3—22no—
495Dic3—34no—
496Dic3—62no—
497Dic3—182no—
498Dic3—181no—
499Dic3—181no—
500Dic3—181no—
501Dic3—181no—
502Dic3—181no—
503Dic3—181no—
504Dic3—361no—
505Dic3—361no—
506C32#559811yescharacteristic
507C9—21no—
508C9#581711yescharacteristic, complemented
509C23—32no—
510C23—271no—
511C23C2×D3×D911yescharacteristic
512Q8—94no—
513Q8—184no—
514Q8—182no—
515D4—64no—
516D4—62no—
517D4—544no—
518D4—542no—
519C2×C4—22no—
520C2×C4—32no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—184no—
524C2×C4—271no—
525C2×C4—271no—
526C2×C4—271no—
527C2×C4—544no—
528C2×C4—542no—
529C2×C4C2×D3×D911yescharacteristic
530C2×C4C2×D3×D911yescharacteristic
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—42no—
537C2×C3—122no—
538C2×C3C2×D4×D911yescharacteristic
539C2×C3C2×#369911yescharacteristic
540C2×C3#6371611yescharacteristic
541C2×C3C2×#379611yescharacteristic
542C2×C3#6429711yescharacteristic
543C2×C3C2×D3×#6711yescharacteristic
544D3—42no—
545D3—362no—
546D3—361no—
547D3—361no—
548C22—21no—
549C22—21no—
550C22—62no—
551C22—62no—
552C22—62no—
553C22—541no—
554C22—541no—
555C22—541no—
556C22#8863912yescomplemented
557C22C22×D3×D911yescharacteristic
558C4—21no—
559C4—62no—
560C4—94no—
561C4—182no—
562C4—541no—
563C4—541no—
564C4—541no—
565C4D9×#6712yes—
566C3—21no—
567C3#1058970011yescharacteristic, complemented
568C3#1059490011yescharacteristic
569C2—22no—
570C2—122no—
571C2—1081no—
572C2C2×#8863911yescharacteristic
573C2#1172246411yescharacteristic
574C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#548
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. C22×C3×Dic18C2semidirect
2. C2×#88639C2semidirect
3. C4×#6233C2semidirect
4. C9×#930C2semidirect
5. #11721288C2semidirect
6. #67×Dic9C2semidirect
7. #11721035C2semidirect
8. #11721018C2semidirect
9. #11720967C2semidirect
10. #11720936C2semidirect
11. #11720910C2semidirect
51. C22×Dic18D3semidirect
81. C3×Dic18D4semidirect
145. C22×C3×C9#41semidirect
175. #930D9semidirect
237. Dic18#67semidirect
329. C22×C9#239semidirect
361. C3×C9#548semidirect
469. C22×C3#3699semidirect
508. C9#5817semidirect
556. C22#88639semidirect
567. C3#10589700semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×#886392
C4×#62332
C9×#9302
#117212882
#67×Dic92
#117210352
#117210182
#117209672
#117209362
#117209102
#105949003
#105897003