Finite Group Databasebeta

← #13147962

Subgroups — #13147962

Order 1728.

708 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721725C211yesmaximal, characteristic, complemented
2C3×#63446C211yesmaximal, characteristic
3C9×#878C211yesmaximal, characteristic, complemented
4Dic3×#611C211yesmaximal, characteristic, complemented
5#11721259C211yesmaximal, characteristic, complemented
6#11721257C211yesmaximal, characteristic, complemented
7#11721257C211yesmaximal, characteristic, complemented
8#11720999C211yesmaximal, characteristic, complemented
9#11720981C211yesmaximal, characteristic, complemented
10#11720951C211yesmaximal, characteristic, complemented
11#11720949C211yesmaximal, characteristic, complemented
12#11720939C211yesmaximal, characteristic, complemented
13#11720907C211yesmaximal, characteristic, complemented
14#11720903C211yesmaximal, characteristic, complemented
15#11720902C211yesmaximal, characteristic, complemented
16#10594149—31nomaximal
17#10589742—31nomaximal
18C2×C3×#611C2211yescharacteristic
19C2×C3×#611C2211yescharacteristic
20C22×C9×Dic3C2211yescharacteristic
21C2×C3×D36C2211yescharacteristic
22C2×C3×C4×D9C2211yescharacteristic
23C2×C9×Dic6C2211yescharacteristic
24C2×Dic3×D9C2211yescharacteristic
25C2×#6189C2211yescharacteristic
26C2×#6189C2211yescharacteristic
27C3×C9×C22⋊C4C2211yescharacteristic
28#88551C2211yescharacteristic
29#88551C2211yescharacteristic
30C4⋊(C3⋊Dic9)C2211yescharacteristic
31#88534C2211yescharacteristic
32#88534C2211yescharacteristic
33C4×C3⋊Dic9C2211yescharacteristic
34C9×#217C2211yescharacteristic
35C3×(C2×D9)⋊C4C2211yescharacteristic
36C9×C4⋊Dic3C2211yescharacteristic
37C9×Dic3⋊C4C2211yescharacteristic
38C9×Dic3⋊C4C2211yescharacteristic
39C3×Dic9⋊C4C2211yescharacteristic
40(C2×D9)⋊Dic3C2211yescharacteristic
41(C2×D9)⋊Dic3C2211yescharacteristic
42(C2×D9)⋊Dic3C2211yescharacteristic
43(C2×D9)⋊Dic3C2211yescharacteristic
44(C2×D9)⋊Dic3C2211yescharacteristic
45(C2×D9)⋊Dic3C2211yescharacteristic
46(C2×D9)⋊Dic3C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64088—31no—
54C3×#884—31no—
55C3×#878D311yescharacteristic
56Dic3×#67—31no—
57#63961—31no—
58#63959—31no—
59#63959—31no—
60#63899—31no—
61#63884—31no—
62#63863—31no—
63#63861—31no—
64#63851—31no—
65#63841—31no—
66#63839—31no—
67#63838—31no—
68C9×C22⋊Q8—31no—
69#63509—31no—
70#63506—31no—
71#63491—31no—
72C4×#611—31no—
73#63460—31no—
74#63458—31no—
75#63458—31no—
76#63456—31no—
77#63454—31no—
78#63449—31no—
79#63447—31no—
80#63447—31no—
81#63446D311yescharacteristic
82#63441—31no—
83#63441—61no—
84C23×C3×C9C2311yescharacteristic
85C22×C3×D9C2311yescharacteristic
86C22×C3×D9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C3×#611—22no—
94C3×#611—41no—
95C2×C9×Dic3—41no—
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C9×Dic3C2311yescharacteristic
99C2×C9×Dic3C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C3×D36—41no—
103C3×C4×D9—22no—
104C9×Dic6—41no—
105Dic3×D9—41no—
106#6189—22no—
107#6189—41no—
108#5640—91no—
109#5603—91no—
110C2×C3×#67—31no—
111C2×C3×#67—31no—
112C22×C3×Dic3C2×D311yescharacteristic
113C2×C3×D12—31no—
114C2×C3×C4×D3—31no—
115C2×C3×Dic6C2×D311yescharacteristic
116C2×#627—31no—
117C2×#627—31no—
118C2×D3×Dic3—31no—
119C32×C22⋊C4C2×D311yescharacteristic
120#3757—31no—
121#3757—31no—
122C4⋊(C3⋊Dic3)—31no—
123#3750—31no—
124#3750—31no—
125C4×C3⋊Dic3—31no—
126C3×#217C2×D311yescharacteristic
127C3×(C2×D3)⋊C4—31no—
128C3×C4⋊Dic3C2×D311yescharacteristic
129C3×Dic3⋊C4—31no—
130C3×Dic3⋊C4C2×D311yescharacteristic
131C3×Dic3⋊C4C2×D311yescharacteristic
132#3724—31no—
133#3724—31no—
134Dic3⋊Dic3—31no—
135(C2×D3)⋊Dic3—31no—
136(C2×D3)⋊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—
146C22×C4×C9—31no—
147C2×#611C2×D311yescharacteristic
148C2×#611C2×D311yescharacteristic
149C2×D36C2×D311yescharacteristic
150C2×C4×D9—31no—
151C2×C4×D9C2×D311yescharacteristic
152C2×Dic18—31no—
153C2×Dic18—31no—
154C9×C4⋊C4—31no—
155C9×C4⋊C4—31no—
156C9×C4⋊C4—31no—
157C9×C22⋊C4—21no—
158C9×C22⋊C4—31no—
159C9×C22⋊C4C2×D311yescharacteristic
160#3676—31no—
161#3676—31no—
162#3676—61no—
163#3676—61no—
164(C2×D9)⋊C4—31no—
165(C2×D9)⋊C4—31no—
166(C2×D9)⋊C4—31no—
167(C2×D9)⋊C4—31no—
168(C2×D9)⋊C4—31no—
169(C2×D9)⋊C4—31no—
170(C2×D9)⋊C4—31no—
171(C2×D9)⋊C4C2×D311yescharacteristic
172C4⋊Dic9—31no—
173C4⋊Dic9—61no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—31no—
179Dic9⋊C4—61no—
180Dic9⋊C4—61no—
181Dic9⋊C4C2×D311yescharacteristic
182C4×Dic9—31no—
183C4×Dic9—31no—
184C4×Dic9—31no—
185C4×Dic9—31no—
186C4×Dic9—61no—
187C22×C3×C9—21no—
188C22×C3×C9—21no—
189C22×C3×C9—21no—
190C22×C3×C9C2411yescharacteristic
191C2×C3×D9—21no—
192C2×C3×D9—21no—
193C2×C3×D9—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—21no—
203C9×Dic3—21no—
204C9×Dic3—21no—
205C9×Dic3—21no—
206C9×Dic3—21no—
207C3×Dic9—21no—
208C3×Dic9#4112yes—
209C3×C22⋊Q8—31no—
210C3×#121—91no—
211#947—91no—
212#944—91no—
213#936—91no—
214#936—91no—
215#936—91no—
216#935—91no—
217#935—91no—
218D4×Dic3—91no—
219#929—91no—
220C4×#67—91no—
221#926—91no—
222#924—91no—
223#898—91no—
224#896—91no—
225#896—91no—
226#894—91no—
227#892—91no—
228#888—91no—
229#887—91no—
230#887—91no—
231#885—91no—
232#885—91no—
233#884—31no—
234#879—91no—
235#879—91no—
236#879—91no—
237#879—181no—
238#878—91no—
239#878D911yescharacteristic, complemented
240C23×C32C22×D311yescharacteristic
241C22×C3×D3—31no—
242C22×C3×D3—31no—
243C2×C32×C4C22×D311yescharacteristic
244C2×C32×C4C22×D311yescharacteristic
245C2×C3⋊Dic3—31no—
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—31no—
249C3×#67—62no—
250C3×#67—121no—
251C2×C3×Dic3—31no—
252C2×C3×Dic3—31no—
253C2×C3×Dic3—41no—
254C2×C3×Dic3C22×D311yescharacteristic
255C2×C3×Dic3C22×D311yescharacteristic
256C2×C3×Dic3C22×D311yescharacteristic
257C2×C3×Dic3C22×D311yescharacteristic
258C3×D12—121no—
259C3×C4×D3—62no—
260C3×Dic6—41no—
261#627—62no—
262#627—121no—
263D3×Dic3—121no—
264C23×C9—21no—
265C23×C9C22×D311yescharacteristic
266C22×D9C22×D311yescharacteristic
267C22×D9C22×D311yescharacteristic
268Q8×C9—121no—
269C2×C4×C9—21no—
270C2×C4×C9—21no—
271C2×C4×C9—31no—
272C2×C4×C9—31no—
273C2×C4×C9—31no—
274C2×C4×C9—31no—
275C2×C4×C9—121no—
276C2×C4×C9C22×D311yescharacteristic
277C2×C4×C9C22×D311yescharacteristic
278#611—22no—
279#611—41no—
280C2×Dic9—31no—
281C2×Dic9—31no—
282C2×Dic9—31no—
283C2×Dic9—31no—
284C2×Dic9—61no—
285C2×Dic9—61no—
286C2×Dic9—61no—
287C2×Dic9—61no—
288C2×Dic9C22×D311yescharacteristic
289C2×Dic9C22×D311yescharacteristic
290D36—41no—
291C4×D9—22no—
292C4×D9—121no—
293Dic18—62no—
294Dic18—121no—
295#542—271noSylow
296C2×C3×C9—41no—
297C2×C3×C9C2×#4111yescharacteristic
298C2×C3×C9H(5,𝔽2)11yescharacteristic
299C2×C3×C92−1+411yescharacteristic
300C3×D9—41no—
301C3×D9—41no—
302C2×C3×Q8—31no—
303C2×C3×D4—91no—
304C2×C3×D4—91no—
305C2×C3×D4—91no—
306C22×C3×C4—31no—
307C22×C3×C4—91no—
308C2×#67—31no—
309C2×#67—31no—
310C22×Dic3—91no—
311C22×Dic3C2×D911yescharacteristic
312C2×D12—31no—
313C2×C4×D3—31no—
314C2×C4×D3—91no—
315C2×Dic6—91no—
316C2×Dic6—91no—
317C2×Dic6—91no—
318C2×Dic6—91no—
319C2×Dic6C2×D911yescharacteristic
320C3×C4⋊C4—31no—
321C3×C4⋊C4—31no—
322C3×C4⋊C4—31no—
323C3×C4⋊C4—91no—
324C3×C22⋊C4—21no—
325C3×C22⋊C4—31no—
326C3×C22⋊C4—91no—
327C3×C22⋊C4C2×D911yescharacteristic
328C3×C22⋊C4D3211yescharacteristic
329#217—91no—
330#217—91no—
331#217—91no—
332#217—91no—
333#217—91no—
334#217—91no—
335#217—91no—
336#217—91no—
337#217—91no—
338#217—91no—
339#217—91no—
340#217—181no—
341#217—181no—
342#217C2×D911yescharacteristic
343(C2×D3)⋊C4—31no—
344(C2×D3)⋊C4—91no—
345(C2×D3)⋊C4—91no—
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—
351C4⋊Dic3—91no—
352C4⋊Dic3—91no—
353C4⋊Dic3—91no—
354C4⋊Dic3—181no—
355C4⋊Dic3C2×D911yescharacteristic
356Dic3⋊C4—31no—
357Dic3⋊C4—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—181no—
367Dic3⋊C4—181no—
368Dic3⋊C4C2×D911yescharacteristic
369Dic3⋊C4C2×D911yescharacteristic
370C4×Dic3—91no—
371C4×Dic3—91no—
372C4×Dic3—91no—
373C4×Dic3—91no—
374C4×Dic3—91no—
375C4×Dic3—91no—
376C4×Dic3—91no—
377C4×Dic3—91no—
378C4×Dic3—181no—
379C22×C32—21no—
380C22×C32—21no—
381C22×C32—21no—
382C22×C32C23×D311yescharacteristic
383C2×C3×D3—61no—
384C2×C3×D3—61no—
385C2×C3×D3—61no—
386C2×C3×D3—61no—
387C2×C3×D3—61no—
388C2×C3×D3—61no—
389C32×C4—21no—
390C32×C4—21no—
391C3⋊Dic3—61no—
392C3⋊Dic3—61no—
393C3⋊Dic3—61no—
394C3⋊Dic3—61no—
395C3×Dic3—21no—
396C3×Dic3—21no—
397C3×Dic3—21no—
398C3×Dic3—21no—
399C3×Dic3—32no—
400C3×Dic3—61no—
401C22×C9—21no—
402C22×C9—21no—
403C22×C9—21no—
404C22×C9—21no—
405C22×C9—41no—
406C22×C9—41no—
407C22×C9—41no—
408C22×C9C23×D311yescharacteristic
409C2×D9—21no—
410C2×D9—21no—
411C2×D9—21no—
412C2×D9—21no—
413C2×D9—21no—
414C2×D9—21no—
415C4×C9—21no—
416C4×C9—21no—
417C4×C9—41no—
418C4×C9—41no—
419C4×C9—61no—
420C4×C9—61no—
421C4×C9—61no—
422C4×C9—61no—
423Dic9—21no—
424Dic9—61no—
425Dic9—61no—
426Dic9—61no—
427Dic9—61no—
428Dic9—121no—
429Dic9—121no—
430Dic9—121no—
431Dic9—121no—
432Dic9#23712yes—
433C4⋊Q8—271no—
434#126—271no—
435#126—271no—
436#124—271no—
437#124—271no—
438#124—271no—
439#123—271no—
440#123—271no—
441C22⋊Q8—31no—
442C22⋊Q8—271no—
443C22⋊Q8—271no—
444#121—91no—
445C4×Q8—271no—
446C4×D4—271no—
447#117—271no—
448C3×C9#54211yescharacteristic, complemented, Sylow
449C23×C3—21no—
450C23×C3—91no—
451C23×C3—91no—
452C23×C3C22×D911yescharacteristic
453C23×C3C2×D3211yescharacteristic
454C22×D3—31no—
455C22×D3—31no—
456C3×Q8—121no—
457C3×D4—182no—
458C3×D4—361no—
459C3×D4—361no—
460C2×C3×C4—21no—
461C2×C3×C4—21no—
462C2×C3×C4—31no—
463C2×C3×C4—31no—
464C2×C3×C4—31no—
465C2×C3×C4—31no—
466C2×C3×C4—91no—
467C2×C3×C4—91no—
468C2×C3×C4—121no—
469C2×C3×C4—182no—
470C2×C3×C4C22×D911yescharacteristic
471C2×C3×C4C22×D911yescharacteristic
472C2×C3×C4C2×D3211yescharacteristic
473C2×C3×C4C2×D3211yescharacteristic
474#67—62no—
475#67—121no—
476C2×Dic3—31no—
477C2×Dic3—31no—
478C2×Dic3—41no—
479C2×Dic3—91no—
480C2×Dic3—91no—
481C2×Dic3—91no—
482C2×Dic3—91no—
483C2×Dic3—91no—
484C2×Dic3—91no—
485C2×Dic3—91no—
486C2×Dic3—91no—
487C2×Dic3—181no—
488C2×Dic3—181no—
489C2×Dic3—181no—
490C2×Dic3—181no—
491C2×Dic3—361no—
492C2×Dic3C22×D911yescharacteristic
493C2×Dic3C22×D911yescharacteristic
494C2×Dic3C22×D911yescharacteristic
495C2×Dic3C22×D911yescharacteristic
496D12—121no—
497C4×D3—62no—
498C4×D3—361no—
499Dic6—41no—
500Dic6—182no—
501Dic6—182no—
502Dic6—361no—
503Dic6—361no—
504C2×C32—41no—
505C2×C32#100411yescharacteristic
506C2×C32#100711yescharacteristic
507C2×C32D3×#4111yescharacteristic
508C3×D3—121no—
509C3×D3—121no—
510C2×C9—21no—
511C2×C9—21no—
512C2×C9—21no—
513C2×C9—41no—
514C2×C9—81no—
515C2×C9C2×#23711yescharacteristic
516C2×C9#100411yescharacteristic
517C2×C9#101011yescharacteristic
518D9—41no—
519D9—41no—
520C2×Q8—31no—
521C2×Q8—271no—
522C2×Q8—271no—
523C2×D4—91no—
524C2×D4—91no—
525C2×D4—91no—
526C22×C4—31no—
527C22×C4—91no—
528C22×C4—271no—
529C4⋊C4—31no—
530C4⋊C4—31no—
531C4⋊C4—31no—
532C4⋊C4—91no—
533C4⋊C4—271no—
534C4⋊C4—271no—
535C4⋊C4—271no—
536C4⋊C4—271no—
537C4⋊C4—271no—
538C4⋊C4—271no—
539C22⋊C4—31no—
540C22⋊C4—91no—
541C22⋊C4—271no—
542C22⋊C4—271no—
543C22⋊C4—271no—
544C22⋊C4—271no—
545C22⋊C4—271no—
546C22⋊C4—271no—
547C22⋊C4—271no—
548C22⋊C4—271no—
549C22⋊C4—271no—
550C22⋊C4D3×D911yescharacteristic
551C42—271no—
552C42—271no—
553C42—271no—
554C42—271no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—21no—
559C22×C3—21no—
560C22×C3—21no—
561C22×C3—21no—
562C22×C3—41no—
563C22×C3—41no—
564C22×C3—41no—
565C22×C3—181no—
566C22×C3—181no—
567C22×C3—181no—
568C22×C3—181no—
569C22×C3—181no—
570C22×C3—181no—
571C22×C3C23×D911yescharacteristic
572C22×C3C22×D3211yescharacteristic
573C2×D3—61no—
574C2×D3—61no—
575C2×D3—61no—
576C2×D3—61no—
577C2×D3—61no—
578C2×D3—61no—
579C3×C4—21no—
580C3×C4—21no—
581C3×C4—21no—
582C3×C4—21no—
583C3×C4—41no—
584C3×C4—41no—
585C3×C4—61no—
586C3×C4—61no—
587C3×C4—61no—
588C3×C4—61no—
589C3×C4—92no—
590C3×C4—181no—
591Dic3—21no—
592Dic3—21no—
593Dic3—21no—
594Dic3—21no—
595Dic3—32no—
596Dic3—61no—
597Dic3—181no—
598Dic3—181no—
599Dic3—181no—
600Dic3—181no—
601Dic3—181no—
602Dic3—181no—
603Dic3—181no—
604Dic3—181no—
605Dic3—361no—
606Dic3—361no—
607Dic3—361no—
608Dic3—361no—
609C32#564011yescharacteristic
610C9—21no—
611C9#560311yescharacteristic, complemented
612C23—91no—
613C23—91no—
614C23C2×D3×D911yescharacteristic
615Q8—121no—
616Q8—542no—
617Q8—1081no—
618D4—182no—
619D4—361no—
620D4—361no—
621C2×C4—31no—
622C2×C4—31no—
623C2×C4—31no—
624C2×C4—31no—
625C2×C4—91no—
626C2×C4—91no—
627C2×C4—121no—
628C2×C4—182no—
629C2×C4—271no—
630C2×C4—271no—
631C2×C4—271no—
632C2×C4—271no—
633C2×C4—1081no—
634C2×C4C2×D3×D911yescharacteristic
635C2×C4C2×D3×D911yescharacteristic
636C2×C3—21no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—41no—
640C2×C3—41no—
641C2×C3—81no—
642C2×C3—361no—
643C2×C3—361no—
644C2×C3#6371011yescharacteristic
645C2×C3#6371311yescharacteristic
646C2×C3#41×D911yescharacteristic
647C2×C3#6429711yescharacteristic
648C2×C3D3×#23711yescharacteristic
649C2×C3#6431311yescharacteristic
650D3—121no—
651D3—121no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—181no—
656C22—181no—
657C22—181no—
658C22—181no—
659C22—181no—
660C22—181no—
661C22C22×D3×D911yescharacteristic
662C4—21no—
663C4—21no—
664C4—61no—
665C4—61no—
666C4—61no—
667C4—61no—
668C4—92no—
669C4—181no—
670C4—541no—
671C4—541no—
672C4—541no—
673C4—541no—
674C3—21no—
675C3#1058974211yescharacteristic, complemented
676C3#1059414911yescharacteristic
677C2—41no—
678C2—361no—
679C2—361no—
680C2#1172245811yescharacteristic
681C2D9×#23711yescharacteristic
682C2#1172249911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC3×#63446C2
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. #11721725C2semidirect
3. C9×#878C2semidirect
4. Dic3×#611C2semidirect
5. #11721259C2semidirect
6. #11721257C2semidirect
7. #11721257C2semidirect
8. #11720999C2semidirect
9. #11720981C2semidirect
10. #11720951C2semidirect
11. #11720949C2semidirect
12. #11720939C2semidirect
13. #11720907C2semidirect
14. #11720903C2semidirect
15. #11720902C2semidirect
239. #878D9semidirect
448. C3×C9#542semidirect
611. C9#5603semidirect
675. C3#10589742semidirect

Maximal subgroups

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

subgroupindex
#117217252
C3×#634462
C9×#8782
Dic3×#6112
#117212592
#117212572
#117209992
#117209812
#117209512
#117209492
#117209392
#117209072
#117209032
#117209022
#105941493
#105897423