Finite Group Databasebeta

← #13150223

Subgroups — #13150223

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
2C2×#88634C211yesmaximal, characteristic, complemented
3C3×#63492C211yesmaximal, characteristic, complemented
4C9×#924C211yesmaximal, characteristic, complemented
5#11721278C212yesmaximal, complemented
6#611⋊Dic3C212yesmaximal, complemented
7#11721007C211yesmaximal, characteristic, complemented
8#11720978C212yesmaximal, complemented
9#11720941C212yesmaximal, complemented
10#11720929C211yesmaximal, characteristic, complemented
11#11720922C211yesmaximal, characteristic, complemented
12#10594922—31nomaximal
13#10589729—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C4×C3⋊D9—24no—
18C2×C4×C3⋊D9C2211yescharacteristic
19C2×C4×C3⋊D9C2211yescharacteristic
20C2×C4×C3⋊D9C2211yescharacteristic
21C2×C4×C3⋊D9C2211yescharacteristic
22C2×C3×#611C2212yes—
23C2×C3×D36C2211yescharacteristic, complemented
24C2×C9×Dic6C2211yescharacteristic, complemented
25C2×#6192C2212yes—
26C2×#6192C2212yes—
27C2×Dic3×D9C2212yescomplemented
28#88634—24no—
29C9×#217C2212yes—
30C3×(C2×D9)⋊C4C2212yescomplemented
31C9×C4⋊Dic3C2211yescharacteristic, complemented
32C9×Dic3⋊C4C2212yescomplemented
33C3×C4⋊Dic9C2211yescharacteristic, complemented
34#88445C2212yes—
35#88445C2212yes—
36(C2×D9)⋊Dic3C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C22×C4×C3⋊D3—31no—
40C2×#3796—31no—
41C3×#930—31no—
42C3×#924D311yescharacteristic
43#63981—32no—
44#67⋊Dic3—32no—
45#63916—31no—
46#63881—32no—
47#63871—31no—
48#63853—32no—
49#63849—31no—
50C2×#3701—31no—
51C22×C4×D9—31no—
52C22×C4×D9—61no—
53C9×C22⋊Q8—31no—
54#63508—31no—
55#63492D311yescharacteristic, complemented
56#63457—31no—
57#63456—32no—
58#63455—32no—
59#63454—31no—
60#63444—32no—
61#611⋊C4—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—22no—
64C22×C3⋊D9—22no—
65C22×C3⋊D9—22no—
66C22×C3⋊D9C2311yescharacteristic
67C22×C3⋊D9C2311yescharacteristic
68C22×C3×D9C2312yes—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9—22no—
73C2×C3⋊Dic9C2311yescharacteristic
74C2×C3⋊Dic9C2311yescharacteristic
75C4×C3⋊D9—22no—
76C4×C3⋊D9—22no—
77C4×C3⋊D9—22no—
78C4×C3⋊D9D414yescomplemented
79C3×#611—24no—
80C2×C9×Dic3C2312yes—
81C2×C9×Dic3C2312yes—
82C2×C3×Dic9C2312yes—
83C3×D36—22no—
84C9×Dic6—22no—
85#6192—24no—
86#6192—24no—
87Dic3×D9—24no—
88#5627—91no—
89#5606—91no—
90C23×C3⋊D3—31no—
91C22×C32×C4C2×D311yescharacteristic
92C22×C3⋊Dic3—31no—
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—31no—
95C2×C4×C3⋊D3—31no—
96C2×C4×C3⋊D3—31no—
97C2×C4×C3⋊D3—64no—
98C2×C3×#67—32no—
99C2×C3×D12—31no—
100C2×C3×Dic6C2×D311yescharacteristic
101C2×#626—32no—
102C2×#626—32no—
103C2×D3×Dic3—32no—
104#3796—64no—
105C23×D9—31no—
106C23×D9—61no—
107C3×#217C2×D312yes—
108C3×(C2×D3)⋊C4—32no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3C2×D311yescharacteristic
111C3×Dic3⋊C4C2×D312yes—
112Dic3⋊Dic3—32no—
113#3722—32no—
114#3722—32no—
115(C2×D3)⋊Dic3—32no—
116Dic32—32no—
117C2×Q8×C9—31no—
118C22×C4×C9—21no—
119C22×C4×C9C2×D311yescharacteristic
120C2×#611C2×D312yes—
121C22×Dic9—31no—
122C22×Dic9—61no—
123#3701—64no—
124C2×D36—32no—
125C2×D36—32no—
126C2×D36C2×D311yescharacteristic, complemented
127C2×C4×D9—31no—
128C2×C4×D9—32no—
129C2×C4×D9—31no—
130C2×C4×D9—31no—
131C2×C4×D9—31no—
132C2×C4×D9—64no—
133C2×C4×D9—61no—
134C2×C4×D9—61no—
135C2×C4×D9—61no—
136C2×C4×D9—61no—
137C2×C4×D9—68no—
138C9×C4⋊C4—32no—
139C9×C4⋊C4—31no—
140C9×C22⋊C4—32no—
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4—32no—
143(C2×D9)⋊C4—32no—
144(C2×D9)⋊C4C2×D312yescomplemented
145C4⋊Dic9C2×D311yescharacteristic, complemented
146Dic9⋊C4—32no—
147C4×Dic9—32no—
148C22×C3×C9—21no—
149C22×C3×C9—21no—
150C22×C3×C9#4112yescomplemented
151C22×C3×C9C2411yescharacteristic
152C2×C3⋊D9—28no—
153C2×C3⋊D9—21no—
154C2×C3⋊D9—21no—
155C2×C3⋊D9—21no—
156C2×C3⋊D9—21no—
157C2×C3⋊D9C2×D412yes—
158C2×C3⋊D9#4112yescomplemented
159C2×C3×D9—22no—
160C2×C3×D9—22no—
161C2×C3×D9—22no—
162C3×C4×C9—21no—
163C3×C4×C9C2×D412yes—
164C3⋊Dic9—21no—
165C3⋊Dic9C2×D412yes—
166C9×Dic3—22no—
167C9×Dic3—22no—
168C3×Dic9—22no—
169C2×#239—91no—
170C2×#237—91no—
171C22×C4×D3—91no—
172C22×C4×D3—91no—
173C22×C4×D3—181no—
174C3×C22⋊Q8—31no—
175C3×#121—91no—
176#946—91no—
177#938—91no—
178#937—92no—
179D4×Dic3—92no—
180#930—31no—
181#924D911yescharacteristic, complemented
182#897—91no—
183#895—91no—
184#894—92no—
185#893—92no—
186#892—91no—
187#892—91no—
188#883—92no—
189#882—92no—
190#67⋊C4—92no—
191#67⋊C4—92no—
192C23×C32C22×D311yescharacteristic
193C22×C3⋊D3—31no—
194C22×C3⋊D3—31no—
195C22×C3⋊D3—62no—
196C22×C3⋊D3—62no—
197C22×C3⋊D3—62no—
198C22×C3×D3—32no—
199C2×C32×C4—22no—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—62no—
205C4×C3⋊D3—34no—
206C4×C3⋊D3—62no—
207C4×C3⋊D3—62no—
208C4×C3⋊D3—62no—
209C3×#67—64no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3C22×D312yes—
212C2×C3×Dic3C22×D312yes—
213C3×D12—62no—
214C3×Dic6—22no—
215#626—64no—
216#626—64no—
217D3×Dic3—64no—
218C23×C9—21no—
219C23×C9C22×D311yescharacteristic
220C22×D9—31no—
221C22×D9—31no—
222C22×D9—61no—
223C22×D9—61no—
224C22×D9—62no—
225C22×D9—62no—
226C22×D9—64no—
227C22×D9—64no—
228C22×D9—64no—
229C22×D9—62no—
230C22×D9C22×D312yes—
231Q8×C9—62no—
232C2×C4×C9—24no—
233C2×C4×C9—22no—
234C2×C4×C9—21no—
235C2×C4×C9—21no—
236C2×C4×C9—32no—
237C2×C4×C9—32no—
238C2×C4×C9C22×D311yescharacteristic
239C2×C4×C9C22×D311yescharacteristic
240#611—24no—
241C2×Dic9—31no—
242C2×Dic9—31no—
243C2×Dic9—61no—
244C2×Dic9—61no—
245C2×Dic9—62no—
246C2×Dic9—64no—
247C2×Dic9C22×D312yes—
248D36—22no—
249D36—64no—
250D36—64no—
251C4×D9—34no—
252C4×D9—64no—
253C4×D9—62no—
254C4×D9—64no—
255C4×D9—62no—
256C4×D9—64no—
257C4×D9—64no—
258C4×D9—64no—
259C4×D9—62no—
260#525—271noSylow
261C2×C3×C9—22no—
262C2×C3×C9C22×D411yescharacteristic
263C2×C3×C9C2×#4111yescharacteristic
264C2×C3×C9C2×#4111yescharacteristic
265C3⋊D9—22no—
266C3⋊D9—22no—
267C3×D9—42no—
268C23×D3—91no—
269C23×D3—91no—
270C23×D3—181no—
271C2×C3×Q8—31no—
272C2×C3×D4—92no—
273C2×C3×D4—91no—
274C22×C3×C4—21no—
275C22×C3×C4C2×D911yescharacteristic
276C22×C3×C4D3211yescharacteristic
277C2×#67—32no—
278C2×#67—92no—
279C2×#67—92no—
280C22×Dic3—91no—
281C22×Dic3—91no—
282C22×Dic3—92no—
283C22×Dic3—181no—
284#239—184no—
285#237—184no—
286C2×D12—31no—
287C2×D12—92no—
288C2×D12—92no—
289C2×C4×D3—91no—
290C2×C4×D3—91no—
291C2×C4×D3—92no—
292C2×C4×D3—91no—
293C2×C4×D3—91no—
294C2×C4×D3—91no—
295C2×C4×D3—91no—
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×C4×D3—181no—
299C2×C4×D3—181no—
300C2×C4×D3—181no—
301C2×C4×D3—181no—
302C2×C4×D3—184no—
303C2×C4×D3—184no—
304C2×C4×D3—188no—
305C2×Dic6C2×D911yescharacteristic, complemented
306C3×C4⋊C4—31no—
307C3×C4⋊C4—32no—
308C3×C4⋊C4—91no—
309C3×C22⋊C4—32no—
310C3×C22⋊C4—92no—
311#217—92no—
312#217C2×D912yes—
313(C2×D3)⋊C4—32no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—92no—
317(C2×D3)⋊C4—92no—
318(C2×D3)⋊C4—92no—
319C4⋊Dic3—31no—
320C4⋊Dic3—92no—
321C4⋊Dic3C2×D911yescharacteristic, complemented
322Dic3⋊C4—92no—
323Dic3⋊C4C2×D912yescomplemented
324C4×Dic3—92no—
325C4×Dic3—92no—
326C22×C32—21no—
327C22×C32—21no—
328C22×C32#23912yes—
329C22×C32C23×D311yescharacteristic
330C2×C3⋊D3—32no—
331C2×C3⋊D3—32no—
332C2×C3⋊D3—68no—
333C2×C3⋊D3—61no—
334C2×C3⋊D3—61no—
335C2×C3⋊D3—61no—
336C2×C3⋊D3—61no—
337C2×C3×D3—62no—
338C2×C3×D3—62no—
339C2×C3×D3—62no—
340C32×C4—21no—
341C32×C4D3×D412yes—
342C3⋊Dic3—32no—
343C3⋊Dic3—61no—
344C3×Dic3—22no—
345C3×Dic3—22no—
346C3×Dic3—62no—
347C22×C9—22no—
348C22×C9—22no—
349C22×C9—21no—
350C22×C9—21no—
351C22×C9—21no—
352C22×C9—22no—
353C22×C9#23712yescomplemented
354C22×C9C23×D311yescharacteristic
355C2×D9—22no—
356C2×D9—22no—
357C2×D9—22no—
358C2×D9—32no—
359C2×D9—32no—
360C2×D9—62no—
361C2×D9—62no—
362C2×D9—62no—
363C2×D9—616no—
364C2×D9—68no—
365C2×D9—62no—
366C2×D9—62no—
367C2×D9—62no—
368C2×D9—61no—
369C2×D9—61no—
370C2×D9—61no—
371C2×D9—61no—
372C4×C9—22no—
373C4×C9—21no—
374C4×C9—22no—
375C4×C9—62no—
376C4×C9—62no—
377C4×C9D3×D412yes—
378Dic9—22no—
379Dic9—32no—
380Dic9—62no—
381Dic9—62no—
382Dic9—61no—
383C2×#41—271no—
384C23×C4—271no—
385#123—272no—
386C22⋊Q8—31no—
387C22⋊Q8—271no—
388#121—91no—
389#121—271no—
390#120—272no—
391C4×D4—272no—
392C4×D4—272no—
393#117—271no—
394C3×C9#52511yescharacteristic, complemented, Sylow
395C23×C3—21no—
396C23×C3—92no—
397C23×C3C22×D911yescharacteristic
398C23×C3C2×D3211yescharacteristic
399C22×D3—32no—
400C22×D3—91no—
401C22×D3—91no—
402C22×D3—91no—
403C22×D3—91no—
404C22×D3—181no—
405C22×D3—181no—
406C22×D3—182no—
407C22×D3—182no—
408C22×D3—182no—
409C22×D3—182no—
410C22×D3—184no—
411C22×D3—184no—
412C22×D3—184no—
413C22×D3—182no—
414C22×D3—182no—
415C3×Q8—62no—
416C3×D4—184no—
417C3×D4—182no—
418C2×C3×C4—24no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—32no—
424C2×C3×C4—32no—
425C2×C3×C4—92no—
426C2×C3×C4C22×D911yescharacteristic
427C2×C3×C4C22×D911yescharacteristic
428C2×C3×C4C2×D3211yescharacteristic
429C2×C3×C4C2×D3211yescharacteristic
430#67—64no—
431#67—184no—
432#67—184no—
433C2×Dic3—32no—
434C2×Dic3—91no—
435C2×Dic3—91no—
436C2×Dic3—91no—
437C2×Dic3—91no—
438C2×Dic3—181no—
439C2×Dic3—181no—
440C2×Dic3—184no—
441C2×Dic3—182no—
442C2×Dic3—182no—
443C2×Dic3—184no—
444C2×Dic3C22×D912yes—
445C2×Dic3C22×D912yes—
446D12—62no—
447D12—184no—
448D12—184no—
449C4×D3—94no—
450C4×D3—94no—
451C4×D3—184no—
452C4×D3—182no—
453C4×D3—182no—
454C4×D3—184no—
455C4×D3—182no—
456C4×D3—182no—
457C4×D3—184no—
458C4×D3—184no—
459C4×D3—184no—
460C4×D3—182no—
461C4×D3—182no—
462Dic6—22no—
463C2×C32—22no—
464C2×C32C2×D3×D411yescharacteristic
465C2×C32C2×#23911yescharacteristic
466C2×C32D3×#4111yescharacteristic
467C3⋊D3—62no—
468C3⋊D3—62no—
469C3×D3—122no—
470C2×C9—24no—
471C2×C9—22no—
472C2×C9—21no—
473C2×C9—21no—
474C2×C9—21no—
475C2×C9C2×D3×D411yescharacteristic
476C2×C9C2×#23711yescharacteristic
477C2×C9D3×#4111yescharacteristic
478D9—42no—
479D9—64no—
480D9—64no—
481D9—62no—
482D9—62no—
483C24—271no—
484#41—544no—
485C2×Q8—31no—
486C2×D4—92no—
487C2×D4—91no—
488C2×D4—272no—
489C2×D4—272no—
490C22×C4—272no—
491C22×C4—271no—
492C22×C4—271no—
493C22×C4—271no—
494C22×C4—271no—
495C22×C4—271no—
496C22×C4—544no—
497C22×C4D3×D911yescharacteristic
498C4⋊C4—31no—
499C4⋊C4—32no—
500C4⋊C4—91no—
501C4⋊C4—272no—
502C22⋊C4—32no—
503C22⋊C4—92no—
504C22⋊C4—272no—
505C22⋊C4—272no—
506C22⋊C4—272no—
507C42—272no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—21no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—22no—
515C22×C3—21no—
516C22×C3—182no—
517C22×C3—182no—
518C22×C3—182no—
519C22×C3#370112yescomplemented
520C22×C3C23×D911yescharacteristic
521C22×C3#379612yes—
522C22×C3C22×D3211yescharacteristic
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—92no—
527C2×D3—92no—
528C2×D3—92no—
529C2×D3—92no—
530C2×D3—182no—
531C2×D3—182no—
532C2×D3—182no—
533C2×D3—188no—
534C2×D3—1816no—
535C2×D3—188no—
536C2×D3—182no—
537C2×D3—182no—
538C2×D3—182no—
539C2×D3—181no—
540C2×D3—181no—
541C2×D3—181no—
542C2×D3—181no—
543C2×D3—181no—
544C2×D3—181no—
545C2×D3—181no—
546C2×D3—181no—
547C3×C4—22no—
548C3×C4—21no—
549C3×C4—21no—
550C3×C4—22no—
551C3×C4—62no—
552C3×C4—62no—
553C3×C4—182no—
554C3×C4D4×D912yes—
555C3×C4#381112yes—
556Dic3—22no—
557Dic3—22no—
558Dic3—62no—
559Dic3—92no—
560Dic3—92no—
561Dic3—182no—
562Dic3—182no—
563Dic3—181no—
564Dic3—181no—
565C32#562711yescharacteristic
566C9—21no—
567C9#560611yescharacteristic, complemented
568C23—92no—
569C23—271no—
570C23—271no—
571C23—542no—
572C23—542no—
573C23—542no—
574C23C2×D3×D911yescharacteristic
575Q8—62no—
576D4—184no—
577D4—182no—
578D4—544no—
579D4—544no—
580C2×C4—22no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—92no—
584C2×C4—271no—
585C2×C4—274no—
586C2×C4—271no—
587C2×C4—542no—
588C2×C4—542no—
589C2×C4—544no—
590C2×C4—542no—
591C2×C4—542no—
592C2×C4C2×D3×D911yescharacteristic
593C2×C4C2×D3×D911yescharacteristic
594C2×C3—24no—
595C2×C3—22no—
596C2×C3—22no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—362no—
601C2×C3C2×D4×D911yescharacteristic
602C2×C3C2×#370111yescharacteristic
603C2×C3#41×D911yescharacteristic
604C2×C3C2×#379611yescharacteristic
605C2×C3#6430611yescharacteristic
606C2×C3C2×#381111yescharacteristic
607D3—122no—
608D3—184no—
609D3—184no—
610D3—182no—
611D3—182no—
612D3—182no—
613D3—182no—
614C22—21no—
615C22—21no—
616C22—182no—
617C22—182no—
618C22—182no—
619C22—272no—
620C22—272no—
621C22—548no—
622C22—541no—
623C22—541no—
624C22—541no—
625C22—541no—
626C22#8863412yescomplemented
627C22C22×D3×D911yescharacteristic
628C4—21no—
629C4—62no—
630C4—62no—
631C4—182no—
632C4—272no—
633C4—541no—
634C4#8866812yes—
635C3—21no—
636C3#1058972911yescharacteristic, complemented
637C3#1059492211yescharacteristic
638C2—22no—
639C2—362no—
640C2—542no—
641C2—542no—
642C2C2×#8863411yescharacteristic
643C2#1172247511yescharacteristic
644C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#525
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×C4×C3⋊D9C2semidirect
2. C2×#88634C2semidirect
3. C3×#63492C2semidirect
4. C9×#924C2semidirect
5. #11721278C2semidirect
6. #611⋊Dic3C2semidirect
7. #11721007C2semidirect
8. #11720978C2semidirect
9. #11720941C2semidirect
10. #11720929C2semidirect
11. #11720922C2semidirect
23. C2×C3×D36C22semidirect
24. C2×C9×Dic6C22semidirect
27. C2×Dic3×D9C22semidirect
30. C3×(C2×D9)⋊C4C22semidirect
31. C9×C4⋊Dic3C22semidirect
32. C9×Dic3⋊C4C22semidirect
33. C3×C4⋊Dic9C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
55. #63492D3semidirect
78. C4×C3⋊D9D4semidirect
126. C2×D36C2×D3semidirect
144. (C2×D9)⋊C4C2×D3semidirect
145. C4⋊Dic9C2×D3semidirect
150. C22×C3×C9#41semidirect
158. C2×C3⋊D9#41semidirect
181. #924D9semidirect
305. C2×Dic6C2×D9semidirect
321. C4⋊Dic3C2×D9semidirect
323. Dic3⋊C4C2×D9semidirect
353. C22×C9#237semidirect
394. C3×C9#525semidirect
519. C22×C3#3701semidirect
567. C9#5606semidirect
626. C22#88634semidirect
636. C3#10589729semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C2×#886342
C3×#634922
C9×#9242
#117212782
#611⋊Dic32
#117210072
#117209782
#117209412
#117209292
#117209222
#105949223
#105897293