Finite Group Databasebeta

← #13148361

Subgroups — #13148361

Order 1728.

722 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721724C211yesmaximal, characteristic, complemented
2C9×#885C211yesmaximal, characteristic, complemented
3C3×#63445C211yesmaximal, characteristic, complemented
4#11721290C211yesmaximal, characteristic
5#11721280C211yesmaximal, characteristic, complemented
6#11721257C211yesmaximal, characteristic, complemented
7#11721248C211yesmaximal, characteristic, complemented
8#11721042C211yesmaximal, characteristic, complemented
9#11721026C211yesmaximal, characteristic, complemented
10#11721012C211yesmaximal, characteristic, complemented
11#11720965C211yesmaximal, characteristic, complemented
12#11720947C211yesmaximal, characteristic, complemented
13#11720944C211yesmaximal, characteristic, complemented
14#11720912C211yesmaximal, characteristic, complemented
15#11720903C211yesmaximal, characteristic, complemented
16#10594160—31nomaximal
17#10589771—31nomaximal
18C22×C3⋊Dic9C2211yescharacteristic
19C2×#6226C2211yescharacteristic
20C2×C9×#67C2211yescharacteristic
21C2×C3×#611C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×C9×Dic6C2211yescharacteristic
24C2×#6194C2211yescharacteristic
25C2×D3×Dic9C2211yescharacteristic
26C2×#6189C2211yescharacteristic
27C3×C9×C22⋊C4C2211yescharacteristic
28#88551C2211yescharacteristic
29C4⋊(C3⋊Dic9)C2211yescharacteristic
30#88534C2211yescharacteristic
31#88534C2211yescharacteristic
32C9×#217C2211yescharacteristic
33C3×#3676C2211yescharacteristic
34C9×(C2×D3)⋊C4C2211yescharacteristic
35C9×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D9)⋊C4C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C3×C4⋊Dic9C2211yescharacteristic
39C3×Dic9⋊C4C2211yescharacteristic
40(C2×D3)⋊Dic9C2211yescharacteristic
41(C2×D3)⋊Dic9C2211yescharacteristic
42(C2×D9)⋊Dic3C2211yescharacteristic
43(C2×D9)⋊Dic3C2211yescharacteristic
44(C2×D9)⋊Dic3C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64087—31no—
54C3×#885D311yescharacteristic
55C3×#883—31no—
56#63980—31no—
57#63969—31no—
58#63959—31no—
59#63957—31no—
60#63913—31no—
61#63899—31no—
62#63879—31no—
63#63861—31no—
64#63859—31no—
65#63856—31no—
66#63844—31no—
67#63839—31no—
68C9×#124—31no—
69#63508—31no—
70#63506—31no—
71#63501—31no—
72#63497—31no—
73#63447—31no—
74#63447—31no—
75#63445D311yescharacteristic, complemented
76#611⋊C4—31no—
77#63441—31no—
78#63440—31no—
79#63440—31no—
80#63440—61no—
81#63439—31no—
82#63438—31no—
83C4×Dic18—31no—
84C23×C3×C9C2311yescharacteristic
85C22×D3×C9C2311yescharacteristic
86C22×C3×D9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3⋊Dic9—22no—
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C3⋊Dic9C2311yescharacteristic
94#6226—22no—
95C9×#67—41no—
96C3×#611—41no—
97C2×C9×Dic3C2311yescharacteristic
98C2×C9×Dic3C2311yescharacteristic
99C2×C9×Dic3C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C2×C3×Dic9C2311yescharacteristic
103C3×C4×D9—41no—
104C9×Dic6—41no—
105#6194—22no—
106D3×Dic9—41no—
107#6189—41no—
108#5669—91no—
109#5662—91no—
110C22×C3⋊Dic3—31no—
111C2×#634—31no—
112C2×C3×#67—31no—
113C2×C3×#67C2×D311yescharacteristic
114C2×C3×C4×D3—31no—
115C2×C3×Dic6C2×D311yescharacteristic
116C2×#627—31no—
117C2×#625—31no—
118C2×D3×Dic3—31no—
119C32×C22⋊C4C2×D311yescharacteristic
120#3757—31no—
121C4⋊(C3⋊Dic3)—31no—
122#3750—31no—
123#3750—31no—
124C3×#217—31no—
125C3×#217C2×D311yescharacteristic
126C3×(C2×D3)⋊C4—31no—
127C3×(C2×D3)⋊C4C2×D311yescharacteristic
128C3×(C2×D3)⋊C4C2×D311yescharacteristic
129C3×C4⋊Dic3—31no—
130C3×Dic3⋊C4—31no—
131C3×C4×Dic3C2×D311yescharacteristic
132#3724—31no—
133#3724—31no—
134#3724—31no—
135#3724—31no—
136Dic3⋊Dic3—31no—
137(C2×D3)⋊Dic3—31no—
138(C2×D3)⋊Dic3—31no—
139(C2×D3)⋊Dic3—31no—
140(C2×D3)⋊Dic3—31no—
141(C2×D3)⋊Dic3—31no—
142Dic32—31no—
143Dic32—31no—
144Dic32—31no—
145C2×Q8×C9—31no—
146C2×D4×C9—31no—
147C2×#611—31no—
148C2×#611C2×D311yescharacteristic
149C22×Dic9—31no—
150C22×Dic9—31no—
151C22×Dic9—61no—
152C2×C4×D9C2×D311yescharacteristic
153C2×Dic18—31no—
154C2×Dic18—31no—
155C2×Dic18—61no—
156C9×C22⋊C4—21no—
157C9×C22⋊C4—31no—
158C9×C22⋊C4—31no—
159C9×C22⋊C4—31no—
160C9×C22⋊C4C2×D311yescharacteristic
161C42×C9—31no—
162#3676—31no—
163#3676—31no—
164#3676—31no—
165#3676—61no—
166#3676C2×D311yescharacteristic
167(C2×D9)⋊C4—31no—
168(C2×D9)⋊C4—31no—
169(C2×D9)⋊C4—31no—
170(C2×D9)⋊C4C2×D311yescharacteristic
171C4⋊Dic9—31no—
172C4⋊Dic9—61no—
173C4⋊Dic9C2×D311yescharacteristic
174Dic9⋊C4—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—
183Dic9⋊C4C2×D311yescharacteristic
184C4×Dic9—31no—
185C4×Dic9—31no—
186C4×Dic9—31no—
187C22×C3×C9—21no—
188C22×C3×C9—21no—
189C22×C3×C9—21no—
190C22×C3×C9C2411yescharacteristic
191C2×D3×C9—21no—
192C2×D3×C9—21no—
193C2×D3×C9—21no—
194C2×C3×D9—21no—
195C2×C3×D9—21no—
196C2×C3×D9—21no—
197C3×C4×C9—21no—
198C3×C4×C9—21no—
199C3⋊Dic9—21no—
200C3⋊Dic9—21no—
201C3⋊Dic9—21no—
202C3⋊Dic9#4112yes—
203C9×Dic3—21no—
204C9×Dic3—21no—
205C9×Dic3—21no—
206C3×Dic9—21no—
207C3×Dic9—21no—
208C3×Dic9—21no—
209C3×#124—31no—
210C3×#123—91no—
211#946—91no—
212#944—91no—
213#939—91no—
214#939—91no—
215#936—91no—
216#935—91no—
217C4×#67—91no—
218#924—91no—
219#898—91no—
220#896—91no—
221#888—91no—
222#887—91no—
223#885—91no—
224#885—91no—
225#885—91no—
226#885D911yescharacteristic, complemented
227#883—31no—
228#883—91no—
229#67⋊C4—91no—
230#879—91no—
231#879—91no—
232#878—91no—
233#878—91no—
234#878—91no—
235#878—181no—
236#877—91no—
237#877—91no—
238#876—91no—
239C4×Dic6—91no—
240C23×C32C22×D311yescharacteristic
241C22×C3×D3—31no—
242C22×C3×D3C22×D311yescharacteristic
243C2×C32×C4C22×D311yescharacteristic
244C2×C32×C4C22×D311yescharacteristic
245C2×C3⋊Dic3—31no—
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—31no—
249C2×C3⋊Dic3—62no—
250#634—62no—
251C3×#67—41no—
252C3×#67—121no—
253C2×C3×Dic3—31no—
254C2×C3×Dic3—31no—
255C2×C3×Dic3—31no—
256C2×C3×Dic3C22×D311yescharacteristic
257C2×C3×Dic3C22×D311yescharacteristic
258C2×C3×Dic3C22×D311yescharacteristic
259C3×C4×D3—121no—
260C3×Dic6—41no—
261#627—121no—
262#625—62no—
263D3×Dic3—121no—
264C23×C9—21no—
265C23×C9—31no—
266C23×C9C22×D311yescharacteristic
267C22×D9C22×D311yescharacteristic
268Q8×C9—121no—
269D4×C9—121no—
270C2×C4×C9—21no—
271C2×C4×C9—21no—
272C2×C4×C9—31no—
273C2×C4×C9—31no—
274C2×C4×C9—31no—
275C2×C4×C9C22×D311yescharacteristic
276C2×C4×C9C22×D311yescharacteristic
277#611—41no—
278#611—62no—
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—62no—
288C2×Dic9—122no—
289C2×Dic9—121no—
290C2×Dic9C22×D311yescharacteristic
291C2×Dic9C22×D311yescharacteristic
292C2×Dic9C22×D311yescharacteristic
293C4×D9—41no—
294Dic18—62no—
295Dic18—122no—
296Dic18—121no—
297#542—271noSylow
298C2×C3×C9—41no—
299C2×C3×C9C2×#4111yescharacteristic
300C2×C3×C9H(5,𝔽2)11yescharacteristic
301C2×C3×C92−1+411yescharacteristic
302D3×C9—41no—
303C3×D9—41no—
304C2×C3×Q8—31no—
305C2×C3×D4—31no—
306C2×C3×D4—91no—
307C22×C3×C4—91no—
308C2×#67—31no—
309C2×#67—91no—
310C2×#67—91no—
311C2×#67C2×D911yescharacteristic
312C22×Dic3—91no—
313C22×Dic3—91no—
314C22×Dic3—91no—
315C22×Dic3—181no—
316C2×C4×D3—31no—
317C2×C4×D3—91no—
318C2×Dic6—91no—
319C2×Dic6—91no—
320C2×Dic6—91no—
321C2×Dic6—91no—
322C2×Dic6—181no—
323C2×Dic6C2×D911yescharacteristic
324C3×C4⋊C4—91no—
325C3×C4⋊C4—91no—
326C3×C22⋊C4—21no—
327C3×C22⋊C4—31no—
328C3×C22⋊C4—31no—
329C3×C22⋊C4—31no—
330C3×C22⋊C4—91no—
331C3×C22⋊C4—91no—
332C3×C22⋊C4C2×D911yescharacteristic
333C3×C22⋊C4D3211yescharacteristic
334C3×C42—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—181no—
344#217C2×D911yescharacteristic
345(C2×D3)⋊C4—31no—
346(C2×D3)⋊C4—91no—
347(C2×D3)⋊C4—91no—
348(C2×D3)⋊C4—91no—
349(C2×D3)⋊C4—91no—
350(C2×D3)⋊C4—91no—
351(C2×D3)⋊C4C2×D911yescharacteristic
352(C2×D3)⋊C4C2×D911yescharacteristic
353C4⋊Dic3—31no—
354C4⋊Dic3—91no—
355C4⋊Dic3—91no—
356C4⋊Dic3—91no—
357C4⋊Dic3—181no—
358Dic3⋊C4—31no—
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—91no—
372Dic3⋊C4—181no—
373Dic3⋊C4—181no—
374C4×Dic3—91no—
375C4×Dic3—91no—
376C4×Dic3—91no—
377C4×Dic3—91no—
378C4×Dic3—91no—
379C4×Dic3—91no—
380C4×Dic3C2×D911yescharacteristic
381C22×C32—21no—
382C22×C32—21no—
383C22×C32—21no—
384C22×C32C23×D311yescharacteristic
385C2×C3×D3—21no—
386C2×C3×D3—21no—
387C2×C3×D3—21no—
388C2×C3×D3—61no—
389C2×C3×D3—61no—
390C2×C3×D3—61no—
391C32×C4—21no—
392C32×C4—21no—
393C3⋊Dic3—32no—
394C3⋊Dic3—61no—
395C3⋊Dic3—61no—
396C3⋊Dic3—61no—
397C3×Dic3—21no—
398C3×Dic3—21no—
399C3×Dic3—21no—
400C3×Dic3—61no—
401C3×Dic3—61no—
402C3×Dic3—61no—
403C22×C9—22no—
404C22×C9—21no—
405C22×C9—21no—
406C22×C9—21no—
407C22×C9—21no—
408C22×C9—41no—
409C22×C9—41no—
410C22×C9—61no—
411C22×C9—61no—
412C22×C9—61no—
413C22×C9C23×D311yescharacteristic
414C2×D9—21no—
415C2×D9—21no—
416C2×D9—21no—
417C4×C9—21no—
418C4×C9—21no—
419C4×C9—41no—
420C4×C9—41no—
421C4×C9—61no—
422C4×C9—61no—
423C4×C9—61no—
424Dic9—21no—
425Dic9—21no—
426Dic9—21no—
427Dic9—32no—
428Dic9—62no—
429Dic9—61no—
430Dic9—61no—
431Dic9—61no—
432Dic9—121no—
433Dic9—121no—
434Dic9—121no—
435C4⋊Q8—271no—
436#126—271no—
437#126—271no—
438#124—31no—
439#124—271no—
440#124—271no—
441#123—91no—
442#123—271no—
443C22⋊Q8—271no—
444C22⋊Q8—271no—
445C22⋊Q8—271no—
446#121—271no—
447C4×Q8—271no—
448C4×D4—271no—
449#117—271no—
450C3×C9#54211yescharacteristic, complemented, Sylow
451C23×C3—21no—
452C23×C3—31no—
453C23×C3—91no—
454C23×C3C22×D911yescharacteristic
455C23×C3C2×D3211yescharacteristic
456C22×D3—31no—
457C22×D3C22×D911yescharacteristic
458C3×Q8—121no—
459C3×D4—121no—
460C3×D4—361no—
461C2×C3×C4—21no—
462C2×C3×C4—21no—
463C2×C3×C4—31no—
464C2×C3×C4—31no—
465C2×C3×C4—31no—
466C2×C3×C4—91no—
467C2×C3×C4—91no—
468C2×C3×C4—91no—
469C2×C3×C4—361no—
470C2×C3×C4C22×D911yescharacteristic
471C2×C3×C4C22×D911yescharacteristic
472C2×C3×C4C2×D3211yescharacteristic
473C2×C3×C4C2×D3211yescharacteristic
474#67—41no—
475#67—121no—
476#67—182no—
477#67—182no—
478C2×Dic3—31no—
479C2×Dic3—31no—
480C2×Dic3—31no—
481C2×Dic3—91no—
482C2×Dic3—91no—
483C2×Dic3—91no—
484C2×Dic3—91no—
485C2×Dic3—91no—
486C2×Dic3—91no—
487C2×Dic3—91no—
488C2×Dic3—91no—
489C2×Dic3—181no—
490C2×Dic3—181no—
491C2×Dic3—181no—
492C2×Dic3—181no—
493C2×Dic3—182no—
494C2×Dic3—182no—
495C2×Dic3—362no—
496C2×Dic3—361no—
497C2×Dic3C22×D911yescharacteristic
498C2×Dic3C22×D911yescharacteristic
499C2×Dic3C22×D911yescharacteristic
500C4×D3—121no—
501C4×D3—361no—
502Dic6—41no—
503Dic6—182no—
504Dic6—182no—
505Dic6—362no—
506Dic6—361no—
507Dic6—361no—
508C2×C32—41no—
509C2×C32#100411yescharacteristic
510C2×C32D3×#4111yescharacteristic
511C2×C32#101011yescharacteristic
512C3×D3—41no—
513C3×D3—121no—
514C2×C9—21no—
515C2×C9—21no—
516C2×C9—21no—
517C2×C9—42no—
518C2×C9—41no—
519C2×C9—121no—
520C2×C9#100411yescharacteristic
521C2×C9D3×#4111yescharacteristic
522C2×C9#101011yescharacteristic
523D9—41no—
524C2×Q8—31no—
525C2×Q8—271no—
526C2×Q8—271no—
527C2×D4—31no—
528C2×D4—91no—
529C2×D4—271no—
530C22×C4—91no—
531C22×C4—271no—
532C22×C4—271no—
533C4⋊C4—91no—
534C4⋊C4—91no—
535C4⋊C4—271no—
536C4⋊C4—271no—
537C4⋊C4—271no—
538C4⋊C4—271no—
539C4⋊C4—271no—
540C4⋊C4—271no—
541C4⋊C4—271no—
542C4⋊C4—271no—
543C22⋊C4—31no—
544C22⋊C4—31no—
545C22⋊C4—31no—
546C22⋊C4—91no—
547C22⋊C4—91no—
548C22⋊C4—271no—
549C22⋊C4—271no—
550C22⋊C4—271no—
551C22⋊C4—271no—
552C22⋊C4—271no—
553C22⋊C4—271no—
554C22⋊C4D3×D911yescharacteristic
555C42—31no—
556C42—271no—
557C42—271no—
558C42—271no—
559C22×C3—22no—
560C22×C3—21no—
561C22×C3—21no—
562C22×C3—21no—
563C22×C3—21no—
564C22×C3—21no—
565C22×C3—21no—
566C22×C3—21no—
567C22×C3—41no—
568C22×C3—41no—
569C22×C3—61no—
570C22×C3—61no—
571C22×C3—61no—
572C22×C3—181no—
573C22×C3—181no—
574C22×C3—181no—
575C22×C3C23×D911yescharacteristic
576C22×C3C22×D3211yescharacteristic
577C2×D3—21no—
578C2×D3—21no—
579C2×D3—21no—
580C2×D3—61no—
581C2×D3—61no—
582C2×D3—61no—
583C3×C4—21no—
584C3×C4—21no—
585C3×C4—21no—
586C3×C4—21no—
587C3×C4—41no—
588C3×C4—41no—
589C3×C4—61no—
590C3×C4—61no—
591C3×C4—61no—
592C3×C4—181no—
593C3×C4—181no—
594C3×C4—181no—
595Dic3—21no—
596Dic3—21no—
597Dic3—21no—
598Dic3—61no—
599Dic3—61no—
600Dic3—61no—
601Dic3—92no—
602Dic3—92no—
603Dic3—182no—
604Dic3—181no—
605Dic3—181no—
606Dic3—181no—
607Dic3—181no—
608Dic3—181no—
609Dic3—181no—
610Dic3—361no—
611Dic3—361no—
612Dic3—361no—
613C32#566911yescharacteristic
614C9—21no—
615C9#566211yescharacteristic, complemented
616C23—31no—
617C23—91no—
618C23C2×D3×D911yescharacteristic
619Q8—121no—
620Q8—542no—
621Q8—1081no—
622D4—121no—
623D4—361no—
624D4—542no—
625C2×C4—31no—
626C2×C4—31no—
627C2×C4—31no—
628C2×C4—91no—
629C2×C4—91no—
630C2×C4—91no—
631C2×C4—271no—
632C2×C4—271no—
633C2×C4—271no—
634C2×C4—271no—
635C2×C4—361no—
636C2×C4—542no—
637C2×C4—1081no—
638C2×C4C2×D3×D911yescharacteristic
639C2×C4C2×D3×D911yescharacteristic
640C2×C3—21no—
641C2×C3—21no—
642C2×C3—21no—
643C2×C3—42no—
644C2×C3—41no—
645C2×C3—41no—
646C2×C3—121no—
647C2×C3—361no—
648C2×C3#6371011yescharacteristic
649C2×C3#41×D911yescharacteristic
650C2×C3#6371611yescharacteristic
651C2×C3#6429811yescharacteristic
652C2×C3#6431211yescharacteristic
653C2×C3#6431311yescharacteristic
654D3—41no—
655D3—121no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—61no—
660C22—61no—
661C22—61no—
662C22—181no—
663C22—181no—
664C22—181no—
665C22C22×D3×D911yescharacteristic
666C4—21no—
667C4—21no—
668C4—61no—
669C4—61no—
670C4—61no—
671C4—181no—
672C4—181no—
673C4—181no—
674C4—272no—
675C4—541no—
676C4—541no—
677C4—541no—
678C3—21no—
679C3#1058977111yescharacteristic, complemented
680C3#1059416011yescharacteristic
681C2—41no—
682C2—121no—
683C2—361no—
684C2#1172246511yescharacteristic
685C2#1172249811yescharacteristic
686C2#1172249911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radical#11721290C2
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#542
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. #11721724C2semidirect
2. C9×#885C2semidirect
3. C3×#63445C2semidirect
5. #11721280C2semidirect
6. #11721257C2semidirect
7. #11721248C2semidirect
8. #11721042C2semidirect
9. #11721026C2semidirect
10. #11721012C2semidirect
11. #11720965C2semidirect
12. #11720947C2semidirect
13. #11720944C2semidirect
14. #11720912C2semidirect
15. #11720903C2semidirect
75. #63445D3semidirect
226. #885D9semidirect
450. C3×C9#542semidirect
615. C9#5662semidirect
679. C3#10589771semidirect

Maximal subgroups

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

subgroupindex
#117217242
C9×#8852
C3×#634452
#117212902
#117212802
#117212572
#117212482
#117210422
#117210262
#117210122
#117209652
#117209472
#117209442
#117209122
#117209032
#105941603
#105897713