Finite Group Databasebeta

← #13151153

Subgroups — #13151153

Order 1728.

1030 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic, complemented
2C2×#88639C211yesmaximal, characteristic, complemented
3#11721838C211yesmaximal, characteristic, complemented
4C9×#940C211yesmaximal, characteristic, complemented
5#11721289C212yesmaximal, complemented
6#11721288C212yesmaximal, complemented
7#67×Dic9C212yesmaximal, complemented
8#11721281C212yesmaximal, complemented
9#11721036C211yesmaximal, characteristic, complemented
10C4×#6192C211yesmaximal, characteristic, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10595158—31nomaximal
13#10589820—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×#67C2212yes—
18C2×C9×#67C2212yes—
19C2×C3×#611C2212yes—
20C22×C3×Dic9C2212yes—
21C3×#3699—24no—
22C2×C9×D12C2211yescharacteristic
23C2×C3×C4×D9C2211yescharacteristic
24C2×C3×Dic18C2211yescharacteristic
25C2×#6195C2212yes—
26C2×#6194C2212yes—
27C2×D3×Dic9C2212yes—
28C2×#6192C2211yescharacteristic
29#88639—24no—
30#88551C2212yes—
31C4⋊(C3⋊Dic9)C2211yescharacteristic
32C4×C9×Dic3C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34#88445C2211yescharacteristic
35(C2×D9)⋊Dic3C2211yescharacteristic
36Dic3⋊Dic9C2211yescharacteristic
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C2×C3×#237—31no—
40C2×#3796—31no—
41#64147—31no—
42C3×#940D311yescharacteristic
43#63979—32no—
44#63978—32no—
45Dic3×#67—32no—
46#63970—32no—
47#63908—31no—
48C4×#626—31no—
49Dic3×Dic6—31no—
50C2×#3699—31no—
51C2×#3699D311yescharacteristic, complemented
52C9×#127—31no—
53#63502—32no—
54#63501—32no—
55#63501—32no—
56#63500—31no—
57#63500—31no—
58#63500—61no—
59D4×Dic9—32no—
60C4×D36—31no—
61C4×Dic18—31no—
62C23×C3×C9C2312yes—
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2312yes—
65C22×C3×D9C2311yescharacteristic
66C3×D4×C9—22no—
67C2×C3×C4×C9C2311yescharacteristic
68#6233—42no—
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2312yes—
71C4×C3⋊D9—22no—
72C9×#67—24no—
73C9×#67—42no—
74C3×#611—24no—
75C2×C9×Dic3C2311yescharacteristic
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9—24no—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80C9×D12—22no—
81C3×C4×D9—22no—
82C3×Dic18D414yescomplemented
83#6195—24no—
84#6194—42no—
85D3×Dic9—24no—
86#6192—22no—
87#5834—91no—
88#5718—91no—
89C2×C32×D4C2×D311yescharacteristic
90C2×#638—32no—
91C2×C4×C3⋊D3—31no—
92C2×C3×#67—32no—
93C2×C3×#67C2×D312yes—
94C2×C3×#67C2×D312yes—
95C22×C3×Dic3—32no—
96C3×#237—64no—
97C2×C3×D12C2×D311yescharacteristic
98C2×C3×C4×D3—31no—
99C2×C3×Dic6—31no—
100C2×#626—32no—
101C2×#626—31no—
102C2×#625—32no—
103C2×D3×Dic3—32no—
104#3796—64no—
105#3757—32no—
106C4⋊(C3⋊Dic3)—31no—
107C3×C4×Dic3C2×D311yescharacteristic
108Dic3⋊Dic3—32no—
109Dic3⋊Dic3—31no—
110#3722—31no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—31no—
113Dic32—32no—
114C2×D4×C9—21no—
115C2×D4×C9—32no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C2×D4×C9C2×D311yescharacteristic
119C2×#611—32no—
120C2×#611—32no—
121C2×#611—32no—
122C2×#611—62no—
123C2×#611C2×D312yes—
124C22×Dic9—32no—
125C22×Dic9C2×D312yes—
126#3699—24no—
127#3699—64no—
128C2×D36—31no—
129C2×C4×D9—31no—
130C2×C4×D9—61no—
131C2×C4×D9C2×D311yescharacteristic
132C2×Dic18C2×D311yescharacteristic
133C42×C9—31no—
134#3676—32no—
135#3676—32no—
136#3676—62no—
137(C2×D9)⋊C4—31no—
138(C2×D9)⋊C4—31no—
139C4⋊Dic9—31no—
140C4⋊Dic9—31no—
141C4⋊Dic9—61no—
142Dic9⋊C4—32no—
143C4×Dic9—32no—
144C22×C3×C9—22no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—21no—
149C2×C3⋊D9—21no—
150C2×C3⋊D9—21no—
151C2×D3×C9—22no—
152C2×D3×C9—22no—
153C2×D3×C9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C3×C4×C9C2×D412yes—
158C3⋊Dic9—21no—
159C3⋊Dic9—22no—
160C9×Dic3—21no—
161C9×Dic3#4112yescomplemented
162C3×Dic9—21no—
163C3×Dic9C2×D414yes—
164C2×C3×#41—91no—
165C2×#239—91no—
166C2×#237—31no—
167C2×#237—91no—
168C3×#127—31no—
169Q8×Dic3—91no—
170#940—92no—
171#940—92no—
172#940D911yescharacteristic, complemented
173#939—92no—
174#939—92no—
175#938—91no—
176#938—91no—
177#938—91no—
178#938—92no—
179#938—181no—
180D4×Dic3—92no—
181#930—92no—
182#930—91no—
183C4×#67—92no—
184C4×#67—91no—
185C4×D12—91no—
186C4×Dic6—91no—
187C23×C32C22×D312yes—
188C22×C3⋊D3—31no—
189C22×C3×D3—31no—
190C22×C3×D3C22×D312yes—
191C32×D4—22no—
192C2×C32×C4C22×D311yescharacteristic
193#638—122no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—31no—
196C4×C3⋊D3—62no—
197C3×#67—24no—
198C3×#67—42no—
199C3×#67—64no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—64no—
203C2×C3×Dic3C22×D311yescharacteristic
204C2×C3×Dic3C22×D311yescharacteristic
205C3×D12—22no—
206C3×C4×D3—62no—
207C3×Dic6—34no—
208#626—64no—
209#626—62no—
210#625—122no—
211D3×Dic3—64no—
212C23×C9—22no—
213C23×C9—32no—
214C23×C9C22×D312yes—
215C22×D9—31no—
216C22×D9—61no—
217C22×D9C22×D311yescharacteristic
218D4×C9—22no—
219D4×C9—42no—
220D4×C9—64no—
221D4×C9—62no—
222D4×C9—122no—
223C2×C4×C9—21no—
224C2×C4×C9—31no—
225C2×C4×C9—31no—
226C2×C4×C9C22×D311yescharacteristic
227#611—24no—
228#611—64no—
229#611—122no—
230#611—122no—
231#611—124no—
232C2×Dic9—24no—
233C2×Dic9—32no—
234C2×Dic9—31no—
235C2×Dic9—62no—
236C2×Dic9—61no—
237C2×Dic9—64no—
238C2×Dic9C22×D311yescharacteristic
239C2×Dic9C22×D312yes—
240D36—62no—
241C4×D9—22no—
242C4×D9—62no—
243C4×D9—122no—
244Dic18#6714yescomplemented
245#550—271noSylow
246C2×C3×C9—42no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4111yescharacteristic
249C2×C3×C9H(5,𝔽2)11yescharacteristic
250C3⋊D9—41no—
251D3×C9—42no—
252C3×D9—41no—
253C3×#41—184no—
254C2×C3×Q8—91no—
255C2×C3×D4—21no—
256C2×C3×D4—32no—
257C2×C3×D4—32no—
258C2×C3×D4—31no—
259C2×C3×D4—92no—
260C2×C3×D4C2×D911yescharacteristic
261C2×C3×D4D3211yescharacteristic
262C22×C3×C4—92no—
263C22×C3×C4—91no—
264C2×#67—32no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—91no—
271C2×#67—182no—
272C2×#67C2×D912yes—
273C2×#67C2×D912yes—
274C22×Dic3—32no—
275C22×Dic3—92no—
276#239—184no—
277#237—64no—
278#237—184no—
279C2×D12—91no—
280C2×D12—92no—
281C2×D12C2×D911yescharacteristic
282C2×C4×D3—31no—
283C2×C4×D3—92no—
284C2×C4×D3—91no—
285C2×C4×D3—91no—
286C2×C4×D3—181no—
287C2×Dic6—31no—
288C3×C42—31no—
289#217—92no—
290#217—92no—
291#217—92no—
292#217—91no—
293#217—182no—
294(C2×D3)⋊C4—91no—
295(C2×D3)⋊C4—91no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—91no—
298C4⋊Dic3—91no—
299C4⋊Dic3—91no—
300C4⋊Dic3—91no—
301C4⋊Dic3—92no—
302C4⋊Dic3—181no—
303Dic3⋊C4—92no—
304Dic3⋊C4—91no—
305C4×Dic3—92no—
306C4×Dic3—92no—
307C4×Dic3C2×D911yescharacteristic
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32C23×D311yescharacteristic
312C2×C3⋊D3—61no—
313C2×C3⋊D3—61no—
314C2×C3⋊D3—61no—
315C2×C3×D3—22no—
316C2×C3×D3—22no—
317C2×C3×D3—22no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C32×C4D3×D412yes—
322C3⋊Dic3—61no—
323C3⋊Dic3—62no—
324C3×Dic3—21no—
325C3×Dic3—34no—
326C3×Dic3—61no—
327C3×Dic3#23712yes—
328C22×C9—22no—
329C22×C9—22no—
330C22×C9—22no—
331C22×C9—21no—
332C22×C9—42no—
333C22×C9—42no—
334C22×C9—42no—
335C22×C9—62no—
336C22×C9—62no—
337C22×C9—62no—
338C22×C9C23×D311yescharacteristic
339C2×D9—21no—
340C2×D9—21no—
341C2×D9—21no—
342C2×D9—62no—
343C2×D9—61no—
344C2×D9—61no—
345C2×D9—61no—
346C2×D9—121no—
347C2×D9—121no—
348C4×C9—22no—
349C4×C9—32no—
350C4×C9—61no—
351C4×C9C2×#6712yes—
352Dic9—21no—
353Dic9—62no—
354Dic9—61no—
355Dic9—121no—
356Dic9—122no—
357Dic9C2×#6714yes—
358C2×#41—91no—
359C2×#41—271no—
360#127—31no—
361#127—272no—
362#121—271no—
363#121—272no—
364#121—272no—
365#121—271no—
366C4×Q8—271no—
367C4×D4—271no—
368C4×D4—272no—
369C3×C9#55011yescharacteristic, complemented, Sylow
370C23×C3—22no—
371C23×C3—32no—
372C23×C3—91no—
373C23×C3C22×D912yes—
374C23×C3C2×D3212yes—
375C22×D3—31no—
376C22×D3—91no—
377C22×D3—91no—
378C22×D3—181no—
379C22×D3C22×D912yes—
380C3×Q8—94no—
381C3×D4—22no—
382C3×D4—22no—
383C3×D4—42no—
384C3×D4—64no—
385C3×D4—62no—
386C3×D4—122no—
387C3×D4—184no—
388C2×C3×C4—21no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—92no—
392C2×C3×C4—91no—
393C2×C3×C4—184no—
394C2×C3×C4—182no—
395C2×C3×C4C22×D911yescharacteristic
396C2×C3×C4C2×D3211yescharacteristic
397#67—24no—
398#67—42no—
399#67—64no—
400#67—184no—
401#67—182no—
402#67—362no—
403#67—362no—
404#67—362no—
405#67—362no—
406#67—364no—
407C2×Dic3—32no—
408C2×Dic3—31no—
409C2×Dic3—64no—
410C2×Dic3—92no—
411C2×Dic3—91no—
412C2×Dic3—92no—
413C2×Dic3—91no—
414C2×Dic3—182no—
415C2×Dic3—181no—
416C2×Dic3—184no—
417C2×Dic3C22×D911yescharacteristic
418C2×Dic3C22×D911yescharacteristic
419D12—22no—
420D12—182no—
421D12—184no—
422C4×D3—62no—
423C4×D3—184no—
424C4×D3—182no—
425C4×D3—182no—
426C4×D3—362no—
427Dic6—34no—
428C2×C32—42no—
429C2×C32C2×D3×D411yescharacteristic
430C2×C32C2×#23711yescharacteristic
431C2×C32#100411yescharacteristic
432C3⋊D3—121no—
433C3×D3—42no—
434C3×D3—121no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—42no—
439C2×C9—82no—
440C2×C9—122no—
441C2×C9D3×#4111yescharacteristic
442C2×C9#100911yescharacteristic
443C2×C9C22×#6711yescharacteristic
444D9—41no—
445D9—122no—
446D9—121no—
447#41—184no—
448#41—544no—
449C2×Q8—91no—
450C2×D4—32no—
451C2×D4—32no—
452C2×D4—31no—
453C2×D4—92no—
454C2×D4—271no—
455C2×D4—272no—
456C2×D4—272no—
457C2×D4—272no—
458C2×D4D3×D911yescharacteristic
459C22×C4—92no—
460C22×C4—91no—
461C22×C4—272no—
462C22×C4—271no—
463C4⋊C4—271no—
464C4⋊C4—272no—
465C4⋊C4—271no—
466C22⋊C4—271no—
467C22⋊C4—271no—
468C22⋊C4—272no—
469C22⋊C4—272no—
470C42—31no—
471C42—272no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—21no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—62no—
483C22×C3—62no—
484C22×C3—62no—
485C22×C3—181no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3C23×D911yescharacteristic
489C22×C3C22×D3211yescharacteristic
490C2×D3—22no—
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—61no—
494C2×D3—61no—
495C2×D3—61no—
496C2×D3—182no—
497C2×D3—181no—
498C2×D3—181no—
499C2×D3—181no—
500C2×D3—181no—
501C2×D3—181no—
502C2×D3—181no—
503C2×D3—361no—
504C2×D3—361no—
505C3×C4—22no—
506C3×C4—32no—
507C3×C4—61no—
508C3×C4—94no—
509C3×C4—181no—
510C3×C4D4×D912yes—
511C3×C4D3×#6712yes—
512Dic3—21no—
513Dic3—34no—
514Dic3—61no—
515Dic3—182no—
516Dic3—181no—
517Dic3—181no—
518Dic3—182no—
519Dic3—361no—
520Dic3—362no—
521Dic3#369912yescomplemented
522C32#571811yescharacteristic
523C9—21no—
524C9#583411yescharacteristic, complemented
525C23—32no—
526C23—91no—
527C23—271no—
528C23C2×D3×D912yes—
529Q8—94no—
530D4—22no—
531D4—64no—
532D4—62no—
533D4—122no—
534D4—184no—
535D4—542no—
536D4—544no—
537D4—1082no—
538D4—1082no—
539C2×C4—31no—
540C2×C4—31no—
541C2×C4—92no—
542C2×C4—91no—
543C2×C4—184no—
544C2×C4—182no—
545C2×C4—272no—
546C2×C4—271no—
547C2×C4—544no—
548C2×C4—542no—
549C2×C4C2×D3×D911yescharacteristic
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—42no—
554C2×C3—42no—
555C2×C3—82no—
556C2×C3—122no—
557C2×C3—361no—
558C2×C3C2×D4×D911yescharacteristic
559C2×C3C2×#369911yescharacteristic
560C2×C3#6371011yescharacteristic
561C2×C3D3×#23711yescharacteristic
562C2×C3#6431411yescharacteristic
563C2×C3C2×D3×#6711yescharacteristic
564D3—42no—
565D3—121no—
566D3—362no—
567D3—361no—
568D3—361no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—62no—
573C22—62no—
574C22—62no—
575C22—181no—
576C22—181no—
577C22—181no—
578C22—541no—
579C22—541no—
580C22—541no—
581C22C22×D3×D911yescharacteristic
582C4—32no—
583C4—61no—
584C4—94no—
585C4—181no—
586C4—542no—
587C4—541no—
588C4D9×#6712yes—
589C3—21no—
590C3#1058982011yescharacteristic, complemented
591C3#1059515811yescharacteristic
592C2—42no—
593C2—122no—
594C2—361no—
595C2—1081no—
596C2D3×#369911yescharacteristic
597C2#1172250011yescharacteristic
598C2C2×D9×#6711yescharacteristic
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×C3×#3699C2semidirect
2. C2×#88639C2semidirect
3. #11721838C2semidirect
4. C9×#940C2semidirect
5. #11721289C2semidirect
6. #11721288C2semidirect
7. #67×Dic9C2semidirect
8. #11721281C2semidirect
9. #11721036C2semidirect
10. C4×#6192C2semidirect
11. Dic3×Dic18C2semidirect
51. C2×#3699D3semidirect
82. C3×Dic18D4semidirect
161. C9×Dic3#41semidirect
172. #940D9semidirect
244. Dic18#67semidirect
369. C3×C9#550semidirect
521. Dic3#3699semidirect
524. C9#5834semidirect
590. C3#10589820semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C2×#886392
#117218382
C9×#9402
#117212892
#117212882
#67×Dic92
#117212812
#117210362
C4×#61922
Dic3×Dic182
#105951583
#105898203