Finite Group Databasebeta

← #13147965

Subgroups — #13147965

Order 1728.

730 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721727C211yesmaximal, characteristic
2C9×#879C211yesmaximal, characteristic, complemented
3C3×#63441C211yesmaximal, characteristic, complemented
4#11721267C211yesmaximal, characteristic, complemented
5#11721248C211yesmaximal, characteristic, complemented
6#11721242C211yesmaximal, characteristic
7#11721242C211yesmaximal, characteristic
8#11720971C211yesmaximal, characteristic, complemented
9#11720931C211yesmaximal, characteristic, complemented
10#11720923C211yesmaximal, characteristic, complemented
11#11720921C211yesmaximal, characteristic, complemented
12#11720920C211yesmaximal, characteristic, complemented
13#11720902C211yesmaximal, characteristic, complemented
14#11720901C211yesmaximal, characteristic, complemented
15#11720900C211yesmaximal, characteristic, complemented
16#10594250—31nomaximal
17#10589804—31nomaximal
18C2×#6233C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C2×C4×C3⋊D9C2211yescharacteristic
21C2×#6189C2211yescharacteristic
22C2×#6189C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25#88536C2211yescharacteristic
26#88534C2211yescharacteristic
27C4×C3⋊Dic9C2211yescharacteristic
28C9×#217C2211yescharacteristic
29C9×#217C2211yescharacteristic
30C3×#3676C2211yescharacteristic
31C3×#3676C2211yescharacteristic
32C9×C4⋊Dic3C2211yescharacteristic
33C9×Dic3⋊C4C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C4×C9×Dic3C2211yescharacteristic
36C3×C4⋊Dic9C2211yescharacteristic
37C3×Dic9⋊C4C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39C3×C4×Dic9C2211yescharacteristic
40#88445C2211yescharacteristic
41#88445C2211yescharacteristic
42#88445C2211yescharacteristic
43#88445C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47#88442C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic3×Dic9C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64090—31no—
54C3×#879—31no—
55C3×#879D311yescharacteristic
56#63982—31no—
57#63957—31no—
58#63953—31no—
59#63953—31no—
60#63919—31no—
61#63874—31no—
62#63865—31no—
63#63848—31no—
64#63848—31no—
65#63838—31no—
66#63837—31no—
67#63837—31no—
68C9×#126—31no—
69#63460—31no—
70#63460—31no—
71#63458—31no—
72#63450—31no—
73#63449—31no—
74#63449—31no—
75#63447—31no—
76#63447—31no—
77#611⋊C4—31no—
78#611⋊C4—61no—
79#63441D311yescharacteristic, complemented
80#63440—31no—
81#63439—31no—
82#63434—31no—
83C4×Dic18—31no—
84C23×C3×C9C2311yescharacteristic
85C22×C3⋊D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88#6233—22no—
89C2×C3⋊Dic9—22no—
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C4×C3⋊D9—22no—
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102#6189—22no—
103#6189—22no—
104#6189—22no—
105#5702—91no—
106#5702—91no—
107C2×#638—31no—
108C22×C3⋊Dic3—31no—
109C2×C4×C3⋊D3—31no—
110C2×#627—31no—
111C2×#627—31no—
112C2×#627—31no—
113C32×C22⋊C4C2×D311yescharacteristic
114#3752—31no—
115#3750—31no—
116C4×C3⋊Dic3—31no—
117C3×#217—31no—
118C3×#217—31no—
119C3×#217C2×D311yescharacteristic
120C3×#217C2×D311yescharacteristic
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3C2×D311yescharacteristic
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×C4×Dic3—31no—
128C3×C4×Dic3C2×D311yescharacteristic
129#3724—31no—
130#3724—31no—
131#3724—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134#3722—31no—
135#3722—31no—
136#3722—31no—
137#3722—31no—
138Dic32—31no—
139Dic32—31no—
140Dic32—31no—
141Dic32—31no—
142C2×#611—31no—
143C2×#611—61no—
144C22×Dic9—31no—
145C22×Dic9—61no—
146C2×C4×D9—31no—
147C2×C4×D9—61no—
148C2×Dic18—31no—
149C2×Dic18—31no—
150C2×Dic18—31no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C4⋊C4—31no—
154C9×C22⋊C4—21no—
155C9×C22⋊C4—31no—
156C9×C22⋊C4—31no—
157C9×C22⋊C4C2×D311yescharacteristic
158C42×C9—31no—
159#3676C2×D311yescharacteristic
160#3676C2×D311yescharacteristic
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—31no—
163(C2×D9)⋊C4—31no—
164(C2×D9)⋊C4—31no—
165(C2×D9)⋊C4—31no—
166(C2×D9)⋊C4—61no—
167C4⋊Dic9—31no—
168C4⋊Dic9C2×D311yescharacteristic
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—61no—
175Dic9⋊C4C2×D311yescharacteristic
176Dic9⋊C4C2×D311yescharacteristic
177C4×Dic9—31no—
178C4×Dic9—31no—
179C4×Dic9—31no—
180C4×Dic9—31no—
181C4×Dic9—31no—
182C4×Dic9—61no—
183C4×Dic9C2×D311yescharacteristic
184C22×C3×C9—21no—
185C22×C3×C9—21no—
186C22×C3×C9—21no—
187C22×C3×C9C2411yescharacteristic
188C2×C3⋊D9—21no—
189C2×C3⋊D9—21no—
190C2×C3⋊D9—21no—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3⋊Dic9—21no—
194C3⋊Dic9#4112yes—
195C3⋊Dic9#4112yes—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×#126—31no—
205C3×#126—91no—
206#898—91no—
207#898—91no—
208#898—91no—
209#898—91no—
210#896—91no—
211#896—91no—
212#888—91no—
213#888—91no—
214#887—91no—
215#887—91no—
216#887—91no—
217#887—91no—
218#885—91no—
219#885—91no—
220#885—91no—
221#885—91no—
222#67⋊C4—91no—
223#67⋊C4—91no—
224#67⋊C4—181no—
225#879—31no—
226#879D911yescharacteristic, complemented
227#878—91no—
228#878—91no—
229#877—91no—
230#877—91no—
231#872—91no—
232#872—91no—
233C4×Dic6—91no—
234C4×Dic6—91no—
235C23×C32C22×D311yescharacteristic
236C22×C3⋊D3—31no—
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239#638—62no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—62no—
244C4×C3⋊D3—62no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3—31no—
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252C2×C3×Dic3C22×D311yescharacteristic
253#627—62no—
254#627—62no—
255#627—62no—
256C23×C9—21no—
257C23×C9C22×D311yescharacteristic
258C22×D9—31no—
259C22×D9—61no—
260C2×C4×C9—21no—
261C2×C4×C9—21no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9C22×D311yescharacteristic
267C2×C4×C9C22×D311yescharacteristic
268#611—62no—
269#611—64no—
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—31no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—62no—
277C2×Dic9—122no—
278C2×Dic9C22×D311yescharacteristic
279C2×Dic9C22×D311yescharacteristic
280C2×Dic9C22×D311yescharacteristic
281C2×Dic9C22×D311yescharacteristic
282C4×D9—62no—
283C4×D9—122no—
284Dic18—62no—
285Dic18—62no—
286Dic18—62no—
287#556—271noSylow
288C2×C3×C9—41no—
289C2×C3×C9C2×#4111yescharacteristic
290C2×C3×C9C2×#4111yescharacteristic
291C2×C3×C92−1+411yescharacteristic
292C3⋊D9—41no—
293C2×#67—91no—
294C2×#67—91no—
295C2×#67—181no—
296C22×Dic3—91no—
297C22×Dic3—91no—
298C22×Dic3—181no—
299C2×C4×D3—91no—
300C2×C4×D3—91no—
301C2×C4×D3—181no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6—91no—
306C2×Dic6—91no—
307C2×Dic6—91no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—31no—
311C3×C4⋊C4—91no—
312C3×C4⋊C4—91no—
313C3×C4⋊C4—91no—
314C3×C22⋊C4—21no—
315C3×C22⋊C4—31no—
316C3×C22⋊C4—31no—
317C3×C22⋊C4—91no—
318C3×C22⋊C4—91no—
319C3×C22⋊C4C2×D911yescharacteristic
320C3×C22⋊C4D3211yescharacteristic
321C3×C42—31no—
322C3×C42—91no—
323#217—31no—
324#217—31no—
325#217C2×D911yescharacteristic
326#217C2×D911yescharacteristic
327(C2×D3)⋊C4—91no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—91no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4—91no—
334(C2×D3)⋊C4—91no—
335(C2×D3)⋊C4—91no—
336(C2×D3)⋊C4—91no—
337(C2×D3)⋊C4—181no—
338C4⋊Dic3—31no—
339C4⋊Dic3—91no—
340C4⋊Dic3—91no—
341C4⋊Dic3C2×D911yescharacteristic
342Dic3⋊C4—31no—
343Dic3⋊C4—31no—
344Dic3⋊C4—91no—
345Dic3⋊C4—91no—
346Dic3⋊C4—91no—
347Dic3⋊C4—91no—
348Dic3⋊C4—91no—
349Dic3⋊C4—91no—
350Dic3⋊C4—91no—
351Dic3⋊C4—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—181no—
355Dic3⋊C4C2×D911yescharacteristic
356Dic3⋊C4C2×D911yescharacteristic
357C4×Dic3—31no—
358C4×Dic3—91no—
359C4×Dic3—91no—
360C4×Dic3—91no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3—181no—
369C4×Dic3C2×D911yescharacteristic
370C22×C32—21no—
371C22×C32—21no—
372C22×C32—21no—
373C22×C32C23×D311yescharacteristic
374C2×C3⋊D3—61no—
375C2×C3⋊D3—61no—
376C2×C3⋊D3—61no—
377C32×C4—21no—
378C32×C4—21no—
379C3⋊Dic3—32no—
380C3⋊Dic3—32no—
381C3⋊Dic3—61no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—21no—
385C3×Dic3—21no—
386C3×Dic3—61no—
387C3×Dic3—61no—
388C3×Dic3—61no—
389C3×Dic3—61no—
390C22×C9—22no—
391C22×C9—21no—
392C22×C9—21no—
393C22×C9—21no—
394C22×C9—21no—
395C22×C9—41no—
396C22×C9—41no—
397C22×C9C23×D311yescharacteristic
398C2×D9—62no—
399C2×D9—61no—
400C2×D9—61no—
401C2×D9—61no—
402C2×D9—121no—
403C2×D9—121no—
404C4×C9—21no—
405C4×C9—21no—
406C4×C9—41no—
407C4×C9—41no—
408C4×C9—61no—
409C4×C9—61no—
410C4×C9—61no—
411C4×C9—61no—
412Dic9—21no—
413Dic9—21no—
414Dic9—21no—
415Dic9—21no—
416Dic9—32no—
417Dic9—32no—
418Dic9—62no—
419Dic9—62no—
420Dic9—61no—
421Dic9—121no—
422C4⋊Q8—271no—
423#126—31no—
424#126—91no—
425#126—271no—
426#126—271no—
427#124—271no—
428#124—271no—
429C22⋊Q8—271no—
430C22⋊Q8—271no—
431C4×Q8—271no—
432C4×Q8—271no—
433C4×Q8—271no—
434C4×D4—271no—
435#117—271no—
436#117—271no—
437C3×C9#55611yescharacteristic, complemented, Sylow
438C23×C3—21no—
439C23×C3C22×D911yescharacteristic
440C23×C3C2×D3211yescharacteristic
441C22×D3—91no—
442C22×D3—91no—
443C22×D3—181no—
444C2×C3×C4—21no—
445C2×C3×C4—21no—
446C2×C3×C4—31no—
447C2×C3×C4—31no—
448C2×C3×C4—31no—
449C2×C3×C4—31no—
450C2×C3×C4—91no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4C22×D911yescharacteristic
455C2×C3×C4C22×D911yescharacteristic
456C2×C3×C4C2×D3211yescharacteristic
457C2×C3×C4C2×D3211yescharacteristic
458#67—182no—
459#67—182no—
460#67—184no—
461C2×Dic3—31no—
462C2×Dic3—31no—
463C2×Dic3—31no—
464C2×Dic3—31no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—91no—
468C2×Dic3—91no—
469C2×Dic3—91no—
470C2×Dic3—91no—
471C2×Dic3—181no—
472C2×Dic3—181no—
473C2×Dic3—181no—
474C2×Dic3—182no—
475C2×Dic3—182no—
476C2×Dic3—362no—
477C2×Dic3C22×D911yescharacteristic
478C2×Dic3C22×D911yescharacteristic
479C2×Dic3C22×D911yescharacteristic
480C2×Dic3C22×D911yescharacteristic
481C4×D3—182no—
482C4×D3—182no—
483C4×D3—362no—
484Dic6—182no—
485Dic6—182no—
486Dic6—182no—
487Dic6—182no—
488Dic6—182no—
489Dic6—182no—
490C2×C32—41no—
491C2×C32D3×#4111yescharacteristic
492C2×C32D3×#4111yescharacteristic
493C2×C32#101011yescharacteristic
494C3⋊D3—121no—
495C2×C9—21no—
496C2×C9—21no—
497C2×C9—21no—
498C2×C9—42no—
499C2×C9—41no—
500C2×C9D3×#4111yescharacteristic
501C2×C9D3×#4111yescharacteristic
502C2×C9#101011yescharacteristic
503D9—122no—
504D9—121no—
505C2×Q8—271no—
506C2×Q8—271no—
507C2×Q8—271no—
508C2×D4—271no—
509C22×C4—271no—
510C22×C4—271no—
511C4⋊C4—31no—
512C4⋊C4—31no—
513C4⋊C4—31no—
514C4⋊C4—91no—
515C4⋊C4—91no—
516C4⋊C4—91no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C22⋊C4—31no—
524C22⋊C4—31no—
525C22⋊C4—91no—
526C22⋊C4—91no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4D3×D911yescharacteristic
533C42—31no—
534C42—91no—
535C42—271no—
536C42—271no—
537C42—271no—
538C42—271no—
539C42—271no—
540C22×C3—22no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—21no—
544C22×C3—21no—
545C22×C3—21no—
546C22×C3—21no—
547C22×C3—21no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3C23×D911yescharacteristic
551C22×C3C22×D3211yescharacteristic
552C2×D3—182no—
553C2×D3—181no—
554C2×D3—181no—
555C2×D3—181no—
556C2×D3—181no—
557C2×D3—181no—
558C2×D3—181no—
559C2×D3—361no—
560C2×D3—361no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—61no—
568C3×C4—61no—
569C3×C4—61no—
570C3×C4—61no—
571C3×C4—181no—
572C3×C4—181no—
573C3×C4—181no—
574C3×C4—181no—
575Dic3—21no—
576Dic3—21no—
577Dic3—21no—
578Dic3—21no—
579Dic3—61no—
580Dic3—61no—
581Dic3—61no—
582Dic3—61no—
583Dic3—92no—
584Dic3—92no—
585Dic3—92no—
586Dic3—92no—
587Dic3—182no—
588Dic3—182no—
589Dic3—181no—
590Dic3—181no—
591Dic3—361no—
592C32#570211yescharacteristic
593C9—21no—
594C9#570211yescharacteristic, complemented
595C23—271no—
596C23C2×D3×D911yescharacteristic
597Q8—542no—
598Q8—542no—
599Q8—542no—
600D4—542no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—31no—
604C2×C4—31no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—91no—
609C2×C4—271no—
610C2×C4—271no—
611C2×C4—271no—
612C2×C4—542no—
613C2×C4—542no—
614C2×C4C2×D3×D911yescharacteristic
615C2×C4C2×D3×D911yescharacteristic
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—42no—
620C2×C3—41no—
621C2×C3—41no—
622C2×C3#41×D911yescharacteristic
623C2×C3#41×D911yescharacteristic
624C2×C3#6371611yescharacteristic
625C2×C3#6430611yescharacteristic
626C2×C3#6430911yescharacteristic
627C2×C3#6431211yescharacteristic
628D3—362no—
629D3—361no—
630D3—361no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—541no—
635C22—541no—
636C22—541no—
637C22C22×D3×D911yescharacteristic
638C4—21no—
639C4—21no—
640C4—61no—
641C4—61no—
642C4—61no—
643C4—61no—
644C4—181no—
645C4—181no—
646C4—181no—
647C4—181no—
648C4—272no—
649C4—272no—
650C4—541no—
651C3—21no—
652C3#1058980411yescharacteristic, complemented
653C3#1059425011yescharacteristic
654C2—41no—
655C2—1081no—
656C2#1172247511yescharacteristic
657C2#1172249411yescharacteristic
658C2#1172249811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6233C22
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C9×#879C2semidirect
3. C3×#63441C2semidirect
4. #11721267C2semidirect
5. #11721248C2semidirect
8. #11720971C2semidirect
9. #11720931C2semidirect
10. #11720923C2semidirect
11. #11720921C2semidirect
12. #11720920C2semidirect
13. #11720902C2semidirect
14. #11720901C2semidirect
15. #11720900C2semidirect
79. #63441D3semidirect
226. #879D9semidirect
437. C3×C9#556semidirect
594. C9#5702semidirect
652. C3#10589804semidirect

Maximal subgroups

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

subgroupindex
#117217272
C9×#8792
C3×#634412
#117212672
#117212482
#117212422
#117209712
#117209312
#117209232
#117209212
#117209202
#117209022
#117209012
#117209002
#105942503
#105898043