Finite Group Databasebeta

← #13150935

Subgroups — #13150935

Order 1728.

726 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721835C211yesmaximal, characteristic, complemented
2C9×#939C211yesmaximal, characteristic
3C3×#63498C211yesmaximal, characteristic, complemented
4#67⋊Dic9C211yesmaximal, characteristic, complemented
5#11721280C211yesmaximal, characteristic, complemented
6#11721267C211yesmaximal, characteristic, complemented
7#11721263C211yesmaximal, characteristic, complemented
8#11721263C211yesmaximal, characteristic, complemented
9#11721249C211yesmaximal, characteristic, complemented
10#11721247C211yesmaximal, characteristic, complemented
11#11721244C211yesmaximal, characteristic, complemented
12#11721018C211yesmaximal, characteristic, complemented
13#11720963C211yesmaximal, characteristic, complemented
14#11720904C211yesmaximal, characteristic, complemented
15#11720901C211yesmaximal, characteristic, complemented
16#10595201—31nomaximal
17#10589705—31nomaximal
18C2×C3×D4×C9C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C2×C9×#67C2211yescharacteristic
21C2×C9×#67C2211yescharacteristic
22C22×C9×Dic3C2211yescharacteristic
23C2×C3×Dic18C2211yescharacteristic
24C2×D3×Dic9C2211yescharacteristic
25C2×#6189C2211yescharacteristic
26C2×#6189C2211yescharacteristic
27#88551C2211yescharacteristic
28#88551C2211yescharacteristic
29#88551C2211yescharacteristic
30#88534C2211yescharacteristic
31#88534C2211yescharacteristic
32C9×#217C2211yescharacteristic
33C3×#3676C2211yescharacteristic
34C3×#3676C2211yescharacteristic
35C3×#3676C2211yescharacteristic
36C3×#3676C2211yescharacteristic
37C9×(C2×D3)⋊C4C2211yescharacteristic
38C9×Dic3⋊C4C2211yescharacteristic
39C3×C4×Dic9C2211yescharacteristic
40(C2×D3)⋊Dic9C2211yescharacteristic
41(C2×D3)⋊Dic9C2211yescharacteristic
42(C2×D3)⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46#88442C2211yescharacteristic
47#88442C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64144—31no—
54C3×#939D311yescharacteristic
55C3×#936—31no—
56#63982—31no—
57#67⋊Dic3—31no—
58#63969—31no—
59#63964—31no—
60#63959—31no—
61#63959—31no—
62#63956—31no—
63#63955—31no—
64#63887—31no—
65#63859—31no—
66#63839—31no—
67#63837—31no—
68C9×#121—31no—
69#63498—31no—
70#63498—31no—
71#63498D311yescharacteristic, complemented
72#63497—31no—
73#63497—31no—
74#63497—61no—
75D4×Dic9—31no—
76#63488—31no—
77#63486—31no—
78#63450—31no—
79#63449—31no—
80#63441—31no—
81#63441—31no—
82#63440—31no—
83#63440—31no—
84C23×C3×C9C2311yescharacteristic
85C23×C3×C9C2311yescharacteristic
86C22×D3×C9C2311yescharacteristic
87C3×D4×C9—41no—
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3⋊Dic9—41no—
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C3⋊Dic9C2311yescharacteristic
94C9×#67—22no—
95C9×#67—41no—
96C2×C9×Dic3—22no—
97C2×C9×Dic3C2311yescharacteristic
98C2×C9×Dic3C2311yescharacteristic
99C2×C9×Dic3C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C2×C3×Dic9C2311yescharacteristic
103C2×C3×Dic9C2311yescharacteristic
104C3×Dic18—41no—
105D3×Dic9—41no—
106#6189—22no—
107#6189—41no—
108#5669—91no—
109#5603—91no—
110C2×C32×D4C2×D311yescharacteristic
111C22×C3⋊Dic3—31no—
112C2×C3×#67C2×D311yescharacteristic
113C2×C3×#67C2×D311yescharacteristic
114C22×C3×Dic3C2×D311yescharacteristic
115C2×C3×Dic6—31no—
116C2×#627—31no—
117C2×#627—31no—
118C2×D3×Dic3—31no—
119#3757—31no—
120#3757—31no—
121#3757—31no—
122#3750—31no—
123#3750—31no—
124C3×#217—31no—
125C3×#217—31no—
126C3×#217—31no—
127C3×#217—31no—
128C3×#217C2×D311yescharacteristic
129C3×(C2×D3)⋊C4C2×D311yescharacteristic
130C3×Dic3⋊C4C2×D311yescharacteristic
131C3×C4×Dic3—31no—
132#3724—31no—
133#3724—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138Dic3⋊Dic3—31no—
139(C2×D3)⋊Dic3—31no—
140(C2×D3)⋊Dic3—31no—
141(C2×D3)⋊Dic3—31no—
142Dic32—31no—
143Dic32—31no—
144Dic32—31no—
145C2×D4×C9—21no—
146C2×D4×C9—31no—
147C2×D4×C9—31no—
148C2×D4×C9C2×D311yescharacteristic
149C22×C4×C9—31no—
150C22×Dic9—31no—
151C22×Dic9—31no—
152C22×Dic9—61no—
153C2×Dic18—31no—
154C2×Dic18—31no—
155C2×Dic18C2×D311yescharacteristic
156C9×C4⋊C4—31no—
157C9×C22⋊C4—31no—
158C9×C22⋊C4—31no—
159#3676—31no—
160#3676—31no—
161#3676—31no—
162#3676—31no—
163#3676—31no—
164#3676—31no—
165#3676—61no—
166#3676—61no—
167#3676—61no—
168#3676C2×D311yescharacteristic
169#3676C2×D311yescharacteristic
170#3676C2×D311yescharacteristic
171#3676C2×D311yescharacteristic
172C4⋊Dic9—31no—
173C4⋊Dic9—31no—
174C4⋊Dic9—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—31no—
179Dic9⋊C4—31no—
180Dic9⋊C4—31no—
181Dic9⋊C4—61no—
182Dic9⋊C4—61no—
183C4×Dic9—31no—
184C4×Dic9—31no—
185C4×Dic9—31no—
186C4×Dic9C2×D311yescharacteristic
187C22×C3×C9—21no—
188C22×C3×C9—21no—
189C22×C3×C9—21no—
190C22×C3×C9—21no—
191C22×C3×C9—21no—
192C22×C3×C9—21no—
193C22×C3×C9C2411yescharacteristic
194C2×D3×C9—21no—
195C2×D3×C9—21no—
196C2×D3×C9—21no—
197C3×C4×C9—21no—
198C3⋊Dic9—21no—
199C3⋊Dic9—21no—
200C3⋊Dic9—21no—
201C3⋊Dic9—21no—
202C9×Dic3—21no—
203C9×Dic3—21no—
204C9×Dic3#4112yes—
205C3×Dic9—21no—
206C3×Dic9—21no—
207C3×Dic9—21no—
208C3×Dic9—21no—
209C3×#124—91no—
210C3×#121—31no—
211#946—91no—
212#944—91no—
213#939D911yescharacteristic
214#936—31no—
215#936—91no—
216#936—91no—
217#935—91no—
218#935—91no—
219#935—91no—
220#935—181no—
221D4×Dic3—91no—
222#926—91no—
223#924—91no—
224#888—91no—
225#887—91no—
226#885—91no—
227#885—91no—
228#883—91no—
229#67⋊C4—91no—
230#879—91no—
231#879—91no—
232#879—91no—
233#878—91no—
234#878—91no—
235#878—91no—
236#878—91no—
237#877—91no—
238#876—91no—
239C4×Dic6—91no—
240C23×C32C22×D311yescharacteristic
241C23×C32C22×D311yescharacteristic
242C22×C3×D3C22×D311yescharacteristic
243C32×D4—41no—
244C2×C32×C4C22×D311yescharacteristic
245C2×C3⋊Dic3—31no—
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—31no—
249C2×C3⋊Dic3—121no—
250C3×#67—22no—
251C3×#67—41no—
252C2×C3×Dic3—22no—
253C2×C3×Dic3—31no—
254C2×C3×Dic3—31no—
255C2×C3×Dic3—31no—
256C2×C3×Dic3—31no—
257C2×C3×Dic3C22×D311yescharacteristic
258C2×C3×Dic3C22×D311yescharacteristic
259C2×C3×Dic3C22×D311yescharacteristic
260C3×Dic6—121no—
261#627—62no—
262#627—121no—
263D3×Dic3—121no—
264C23×C9—21no—
265C23×C9—21no—
266C23×C9—31no—
267C23×C9C22×D311yescharacteristic
268C23×C9C22×D311yescharacteristic
269D4×C9—41no—
270D4×C9—42no—
271D4×C9—62no—
272D4×C9—121no—
273C2×C4×C9—21no—
274C2×C4×C9—31no—
275C2×C4×C9—31no—
276C2×C4×C9—31no—
277C2×C4×C9—62no—
278C2×C4×C9C22×D311yescharacteristic
279C2×Dic9—31no—
280C2×Dic9—31no—
281C2×Dic9—31no—
282C2×Dic9—31no—
283C2×Dic9—61no—
284C2×Dic9—61no—
285C2×Dic9—61no—
286C2×Dic9—61no—
287C2×Dic9—121no—
288C2×Dic9—121no—
289C2×Dic9—122no—
290C2×Dic9C22×D311yescharacteristic
291C2×Dic9C22×D311yescharacteristic
292C2×Dic9C22×D311yescharacteristic
293C2×Dic9C22×D311yescharacteristic
294Dic18—41no—
295Dic18—62no—
296Dic18—121no—
297#542—271noSylow
298C2×C3×C9—41no—
299C2×C3×C9—41no—
300C2×C3×C9C2×#4111yescharacteristic
301C2×C3×C9H(5,𝔽2)11yescharacteristic
302C2×C3×C92−1+411yescharacteristic
303D3×C9—41no—
304C2×C3×Q8—91no—
305C2×C3×D4—21no—
306C2×C3×D4—31no—
307C2×C3×D4—31no—
308C2×C3×D4C2×D911yescharacteristic
309C2×C3×D4D3211yescharacteristic
310C22×C3×C4—31no—
311C2×#67C2×D911yescharacteristic
312C2×#67C2×D911yescharacteristic
313C22×Dic3—91no—
314C22×Dic3—91no—
315C22×Dic3—91no—
316C22×Dic3—181no—
317C22×Dic3C2×D911yescharacteristic
318C2×C4×D3—91no—
319C2×Dic6—31no—
320C2×Dic6—91no—
321C2×Dic6—91no—
322C2×Dic6—91no—
323C2×Dic6—91no—
324C3×C4⋊C4—31no—
325C3×C22⋊C4—31no—
326C3×C22⋊C4—31no—
327C3×C22⋊C4—91no—
328C3×C22⋊C4—91no—
329C3×C22⋊C4—91no—
330C3×C22⋊C4—91no—
331C3×C42—91no—
332#217—31no—
333#217—31no—
334#217—31no—
335#217—31no—
336#217—91no—
337#217—91no—
338#217—91no—
339#217—91no—
340#217—91no—
341#217—91no—
342#217—91no—
343#217—91no—
344#217—91no—
345#217—181no—
346#217—181no—
347#217—181no—
348#217C2×D911yescharacteristic
349(C2×D3)⋊C4—91no—
350(C2×D3)⋊C4—91no—
351(C2×D3)⋊C4—91no—
352(C2×D3)⋊C4C2×D911yescharacteristic
353C4⋊Dic3—91no—
354C4⋊Dic3—91no—
355C4⋊Dic3—91no—
356C4⋊Dic3—91no—
357C4⋊Dic3—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—91no—
361Dic3⋊C4—91no—
362Dic3⋊C4—91no—
363Dic3⋊C4—91no—
364Dic3⋊C4—91no—
365Dic3⋊C4—91no—
366Dic3⋊C4—91no—
367Dic3⋊C4—91no—
368Dic3⋊C4—91no—
369Dic3⋊C4—91no—
370Dic3⋊C4—91no—
371Dic3⋊C4—181no—
372Dic3⋊C4—181no—
373Dic3⋊C4C2×D911yescharacteristic
374C4×Dic3—31no—
375C4×Dic3—91no—
376C4×Dic3—91no—
377C4×Dic3—91no—
378C4×Dic3—91no—
379C4×Dic3—91no—
380C4×Dic3—91no—
381C22×C32—21no—
382C22×C32—21no—
383C22×C32—21no—
384C22×C32—21no—
385C22×C32—21no—
386C22×C32—21no—
387C22×C32C23×D311yescharacteristic
388C2×C3×D3—21no—
389C2×C3×D3—21no—
390C2×C3×D3—21no—
391C32×C4—21no—
392C3⋊Dic3—61no—
393C3⋊Dic3—61no—
394C3⋊Dic3—61no—
395C3⋊Dic3—61no—
396C3×Dic3—21no—
397C3×Dic3—21no—
398C3×Dic3—61no—
399C3×Dic3—61no—
400C3×Dic3—61no—
401C3×Dic3—61no—
402C3×Dic3#23712yes—
403C22×C9—22no—
404C22×C9—21no—
405C22×C9—21no—
406C22×C9—21no—
407C22×C9—21no—
408C22×C9—21no—
409C22×C9—21no—
410C22×C9—21no—
411C22×C9—41no—
412C22×C9—41no—
413C22×C9—41no—
414C22×C9—41no—
415C22×C9—41no—
416C22×C9—61no—
417C22×C9—61no—
418C22×C9—61no—
419C22×C9C23×D311yescharacteristic
420C4×C9—21no—
421C4×C9—32no—
422C4×C9—41no—
423C4×C9—61no—
424C4×C9—61no—
425Dic9—21no—
426Dic9—21no—
427Dic9—21no—
428Dic9—21no—
429Dic9—61no—
430Dic9—61no—
431Dic9—61no—
432Dic9—61no—
433Dic9—121no—
434Dic9—121no—
435Dic9—121no—
436Dic9—121no—
437C4⋊Q8—271no—
438#126—271no—
439#126—271no—
440#124—91no—
441#124—271no—
442#124—271no—
443#123—271no—
444#123—271no—
445C22⋊Q8—271no—
446C22⋊Q8—271no—
447C22⋊Q8—271no—
448#121—31no—
449C4×Q8—271no—
450C4×D4—271no—
451#117—271no—
452C3×C9#54211yescharacteristic, complemented, Sylow
453C23×C3—21no—
454C23×C3—21no—
455C23×C3—31no—
456C23×C3C22×D911yescharacteristic
457C23×C3C22×D911yescharacteristic
458C23×C3C2×D3211yescharacteristic
459C23×C3C2×D3211yescharacteristic
460C22×D3C22×D911yescharacteristic
461C3×Q8—361no—
462C3×D4—41no—
463C3×D4—41no—
464C3×D4—42no—
465C3×D4—62no—
466C3×D4—121no—
467C2×C3×C4—21no—
468C2×C3×C4—31no—
469C2×C3×C4—31no—
470C2×C3×C4—31no—
471C2×C3×C4—62no—
472C2×C3×C4—91no—
473C2×C3×C4—91no—
474C2×C3×C4—91no—
475C2×C3×C4—91no—
476C2×C3×C4C22×D911yescharacteristic
477C2×C3×C4C2×D3211yescharacteristic
478#67—22no—
479#67—41no—
480C2×Dic3—22no—
481C2×Dic3—31no—
482C2×Dic3—31no—
483C2×Dic3—31no—
484C2×Dic3—31no—
485C2×Dic3—91no—
486C2×Dic3—91no—
487C2×Dic3—91no—
488C2×Dic3—91no—
489C2×Dic3—91no—
490C2×Dic3—91no—
491C2×Dic3—91no—
492C2×Dic3—91no—
493C2×Dic3—181no—
494C2×Dic3—181no—
495C2×Dic3—181no—
496C2×Dic3—181no—
497C2×Dic3—361no—
498C2×Dic3—361no—
499C2×Dic3—361no—
500C2×Dic3—362no—
501C2×Dic3C22×D911yescharacteristic
502C2×Dic3C22×D911yescharacteristic
503C2×Dic3C22×D911yescharacteristic
504C4×D3—361no—
505Dic6—121no—
506Dic6—182no—
507Dic6—182no—
508Dic6—361no—
509Dic6—361no—
510C2×C32—41no—
511C2×C32—41no—
512C2×C32C2×#23711yescharacteristic
513C2×C32#100411yescharacteristic
514C2×C32#101011yescharacteristic
515C3×D3—41no—
516C2×C9—21no—
517C2×C9—21no—
518C2×C9—21no—
519C2×C9—42no—
520C2×C9—41no—
521C2×C9—41no—
522C2×C9—81no—
523C2×C9—121no—
524C2×C9#100411yescharacteristic
525C2×C9D3×#4111yescharacteristic
526C2×C9#101011yescharacteristic
527C2×Q8—91no—
528C2×Q8—271no—
529C2×Q8—271no—
530C2×D4—31no—
531C2×D4—31no—
532C2×D4D3×D911yescharacteristic
533C22×C4—31no—
534C22×C4—271no—
535C22×C4—271no—
536C4⋊C4—31no—
537C4⋊C4—271no—
538C4⋊C4—271no—
539C4⋊C4—271no—
540C4⋊C4—271no—
541C4⋊C4—271no—
542C4⋊C4—271no—
543C4⋊C4—271no—
544C4⋊C4—271no—
545C4⋊C4—271no—
546C22⋊C4—31no—
547C22⋊C4—31no—
548C22⋊C4—91no—
549C22⋊C4—91no—
550C22⋊C4—91no—
551C22⋊C4—91no—
552C22⋊C4—271no—
553C22⋊C4—271no—
554C22⋊C4—271no—
555C22⋊C4—271no—
556C22⋊C4—271no—
557C22⋊C4—271no—
558C42—91no—
559C42—271no—
560C42—271no—
561C42—271no—
562C22×C3—22no—
563C22×C3—21no—
564C22×C3—21no—
565C22×C3—21no—
566C22×C3—21no—
567C22×C3—21no—
568C22×C3—21no—
569C22×C3—21no—
570C22×C3—21no—
571C22×C3—21no—
572C22×C3—21no—
573C22×C3—21no—
574C22×C3—21no—
575C22×C3—21no—
576C22×C3—41no—
577C22×C3—41no—
578C22×C3—41no—
579C22×C3—41no—
580C22×C3—41no—
581C22×C3—61no—
582C22×C3—61no—
583C22×C3—61no—
584C22×C3C23×D911yescharacteristic
585C22×C3C22×D3211yescharacteristic
586C2×D3—21no—
587C2×D3—21no—
588C2×D3—21no—
589C3×C4—21no—
590C3×C4—21no—
591C3×C4—32no—
592C3×C4—41no—
593C3×C4—61no—
594C3×C4—61no—
595C3×C4—181no—
596C3×C4—181no—
597C3×C4—181no—
598C3×C4—181no—
599Dic3—21no—
600Dic3—21no—
601Dic3—61no—
602Dic3—61no—
603Dic3—61no—
604Dic3—61no—
605Dic3—181no—
606Dic3—181no—
607Dic3—181no—
608Dic3—181no—
609Dic3—181no—
610Dic3—181no—
611Dic3—181no—
612Dic3—181no—
613Dic3—361no—
614Dic3—361no—
615Dic3—361no—
616Dic3—361no—
617Dic3#369912yes—
618C32#560311yescharacteristic
619C9—21no—
620C9#566911yescharacteristic, complemented
621C23—31no—
622C23C2×D3×D911yescharacteristic
623C23C2×D3×D911yescharacteristic
624Q8—361no—
625Q8—542no—
626Q8—1081no—
627D4—41no—
628D4—62no—
629D4—121no—
630C2×C4—31no—
631C2×C4—31no—
632C2×C4—31no—
633C2×C4—62no—
634C2×C4—91no—
635C2×C4—91no—
636C2×C4—91no—
637C2×C4—91no—
638C2×C4—271no—
639C2×C4—271no—
640C2×C4—271no—
641C2×C4—271no—
642C2×C4—1081no—
643C2×C4—1081no—
644C2×C4C2×D3×D911yescharacteristic
645C2×C3—21no—
646C2×C3—21no—
647C2×C3—21no—
648C2×C3—42no—
649C2×C3—41no—
650C2×C3—41no—
651C2×C3—41no—
652C2×C3—41no—
653C2×C3—81no—
654C2×C3—121no—
655C2×C3C2×#369911yescharacteristic
656C2×C3#6371011yescharacteristic
657C2×C3#6371611yescharacteristic
658C2×C3#6430911yescharacteristic
659C2×C3D3×#23711yescharacteristic
660C2×C3#6433011yescharacteristic
661D3—41no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—61no—
669C22—61no—
670C22—61no—
671C22C22×D3×D911yescharacteristic
672C4—21no—
673C4—32no—
674C4—61no—
675C4—61no—
676C4—181no—
677C4—181no—
678C4—181no—
679C4—181no—
680C4—541no—
681C4—541no—
682C4—541no—
683C4—541no—
684C3—21no—
685C3#1058970511yescharacteristic, complemented
686C3#1059520111yescharacteristic
687C2—41no—
688C2—41no—
689C2—121no—
690C2#1172249411yescharacteristic
691C2D3×#369911yescharacteristic
692C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC3×#939D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#542
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. #11721835C2semidirect
3. C3×#63498C2semidirect
4. #67⋊Dic9C2semidirect
5. #11721280C2semidirect
6. #11721267C2semidirect
7. #11721263C2semidirect
8. #11721263C2semidirect
9. #11721249C2semidirect
10. #11721247C2semidirect
11. #11721244C2semidirect
12. #11721018C2semidirect
13. #11720963C2semidirect
14. #11720904C2semidirect
15. #11720901C2semidirect
71. #63498D3semidirect
452. C3×C9#542semidirect
620. C9#5669semidirect
685. C3#10589705semidirect

Maximal subgroups

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

subgroupindex
#117218352
C9×#9392
C3×#634982
#67⋊Dic92
#117212802
#117212672
#117212632
#117212492
#117212472
#117212442
#117210182
#117209632
#117209042
#117209012
#105952013
#105897053