Finite Group Databasebeta

← #13148881

Subgroups — #13148881

Order 1728.

720 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721750C211yesmaximal, characteristic
2C9×#889C211yesmaximal, characteristic, complemented
3C3×#63449C211yesmaximal, characteristic, complemented
4#11720997C211yesmaximal, characteristic
5#11720992C211yesmaximal, characteristic
6#11720985C211yesmaximal, characteristic, complemented
7#11720985C211yesmaximal, characteristic, complemented
8#11720980C211yesmaximal, characteristic, complemented
9#11720934C211yesmaximal, characteristic, complemented
10#11720934C211yesmaximal, characteristic, complemented
11#11720933C211yesmaximal, characteristic, complemented
12#11720931C211yesmaximal, characteristic, complemented
13#11720918C211yesmaximal, characteristic, complemented
14#11720899C211yesmaximal, characteristic, complemented
15#11720898C211yesmaximal, characteristic, complemented
16#10594349—31nomaximal
17#10589795—31nomaximal
18C2×C4⋊(C3⋊D9)C2211yescharacteristic
19C2×C4×C3⋊D9C2211yescharacteristic
20C2×C3×Dic18C2211yescharacteristic
21C2×(C3⋊D9)⋊C4C2211yescharacteristic
22C2×(C3⋊D9)⋊C4C2211yescharacteristic
23C2×(C3⋊D9)⋊C4C2211yescharacteristic
24C3×C9×C4⋊C4C2211yescharacteristic
25#88536C2211yescharacteristic
26#88536C2211yescharacteristic
27#88536C2211yescharacteristic
28#88534C2211yescharacteristic
29C9×C4⋊Dic3C2211yescharacteristic
30C9×Dic3⋊C4C2211yescharacteristic
31C9×Dic3⋊C4C2211yescharacteristic
32C9×Dic3⋊C4C2211yescharacteristic
33C9×Dic3⋊C4C2211yescharacteristic
34C4×C9×Dic3C2211yescharacteristic
35C3×Dic9⋊C4C2211yescharacteristic
36C3×Dic9⋊C4C2211yescharacteristic
37C3×C4×Dic9C2211yescharacteristic
38C3×C4×Dic9C2211yescharacteristic
39C3×C4×Dic9C2211yescharacteristic
40#88445C2211yescharacteristic
41#88445C2211yescharacteristic
42#88445C2211yescharacteristic
43#88445C2211yescharacteristic
44#88445C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49#88442C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic9⋊Dic3C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64103—31no—
54C3×#889D311yescharacteristic
55C3×#887—31no—
56#63897—31no—
57#63894—31no—
58#63888—31no—
59#63888—31no—
60#63883—31no—
61#63874—31no—
62#63868—31no—
63#63868—31no—
64#63867—31no—
65#63847—31no—
66#63836—31no—
67#63836—31no—
68C9×#125—31no—
69#63460—31no—
70#63460—31no—
71#63458—31no—
72#63458—31no—
73#63456—31no—
74#63456—31no—
75#63456—61no—
76#63455—31no—
77#63454—31no—
78C4⋊C4×D9—31no—
79#63451—31no—
80#63451—31no—
81#63449D311yescharacteristic, complemented
82#63434—31no—
83#63434—31no—
84C22×C3⋊D9C2311yescharacteristic
85C22×C3⋊D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C4⋊(C3⋊D9)—22no—
92C4×C3⋊D9—22no—
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C3×Dic18—22no—
102(C3⋊D9)⋊C4—22no—
103(C3⋊D9)⋊C4—22no—
104(C3⋊D9)⋊C4—22no—
105#5693—91no—
106#5581—91no—
107C2×C4⋊(C3⋊D3)—31no—
108C2×C4×C3⋊D3—31no—
109C2×C3×Dic6—31no—
110C2×#624—31no—
111C2×#624—31no—
112C2×#624—31no—
113C32×C4⋊C4C2×D311yescharacteristic
114#3752—31no—
115#3752—31no—
116#3752—31no—
117#3750—31no—
118C3×C4⋊Dic3C2×D311yescharacteristic
119C3×Dic3⋊C4—31no—
120C3×Dic3⋊C4—31no—
121C3×Dic3⋊C4C2×D311yescharacteristic
122C3×Dic3⋊C4C2×D311yescharacteristic
123C3×Dic3⋊C4C2×D311yescharacteristic
124C3×Dic3⋊C4C2×D311yescharacteristic
125C3×C4×Dic3—31no—
126C3×C4×Dic3—31no—
127C3×C4×Dic3—31no—
128C3×C4×Dic3C2×D311yescharacteristic
129#3724—31no—
130#3724—31no—
131#3724—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136#3722—31no—
137#3722—31no—
138#3722—31no—
139#3722—31no—
140#3722—31no—
141Dic32—31no—
142C2×D36—31no—
143C2×D36—61no—
144C2×C4×D9—31no—
145C2×C4×D9—31no—
146C2×C4×D9—31no—
147C2×C4×D9—31no—
148C2×C4×D9—61no—
149C2×Dic18C2×D311yescharacteristic
150C9×C4⋊C4—21no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C4⋊C4—31no—
154C9×C4⋊C4—31no—
155C9×C4⋊C4—31no—
156C9×C4⋊C4C2×D311yescharacteristic
157C42×C9—31no—
158(C2×D9)⋊C4—31no—
159(C2×D9)⋊C4—31no—
160(C2×D9)⋊C4—31no—
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—
167(C2×D9)⋊C4—61no—
168(C2×D9)⋊C4—61no—
169C4⋊Dic9—31no—
170C4⋊Dic9—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—61no—
178Dic9⋊C4C2×D311yescharacteristic
179Dic9⋊C4C2×D311yescharacteristic
180C4×Dic9—31no—
181C4×Dic9C2×D311yescharacteristic
182C4×Dic9C2×D311yescharacteristic
183C4×Dic9C2×D311yescharacteristic
184C22×C3×C9C2411yescharacteristic
185C2×C3⋊D9—21no—
186C2×C3⋊D9—21no—
187C2×C3⋊D9—21no—
188C2×C3⋊D9—21no—
189C2×C3⋊D9—21no—
190C2×C3⋊D9#4112yes—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3×C4×C9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9—21no—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9#4112yes—
204C3×#125—31no—
205C3×C4×Q8—91no—
206#946—91no—
207#946—91no—
208#898—91no—
209#898—91no—
210#896—91no—
211#896—91no—
212#894—91no—
213#894—91no—
214#894—91no—
215#894—91no—
216#894—181no—
217#893—91no—
218#892—91no—
219#892—91no—
220D3×C4⋊C4—91no—
221D3×C4⋊C4—91no—
222#889—91no—
223#889—91no—
224#889—91no—
225#889D911yescharacteristic, complemented
226#887—31no—
227#876—91no—
228#876—91no—
229C4×D12—91no—
230#872—91no—
231#872—91no—
232#872—91no—
233#872—91no—
234#870—91no—
235C22×C3⋊D3—31no—
236C22×C3⋊D3—31no—
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239C2×C32×C4C22×D311yescharacteristic
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C4⋊(C3⋊D3)—62no—
243C4×C3⋊D3—62no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252C3×Dic6—62no—
253#624—62no—
254#624—62no—
255#624—62no—
256C22×D9—31no—
257C22×D9—31no—
258C22×D9—61no—
259C22×D9—61no—
260C2×C4×C9—21no—
261C2×C4×C9—21no—
262C2×C4×C9—21no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9—31no—
267C2×C4×C9C22×D311yescharacteristic
268C2×C4×C9C22×D311yescharacteristic
269C2×C4×C9C22×D311yescharacteristic
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—61no—
273C2×Dic9—61no—
274C2×Dic9C22×D311yescharacteristic
275C2×Dic9C22×D311yescharacteristic
276C2×Dic9C22×D311yescharacteristic
277C2×Dic9C22×D311yescharacteristic
278D36—62no—
279D36—122no—
280C4×D9—62no—
281C4×D9—62no—
282C4×D9—62no—
283C4×D9—62no—
284C4×D9—122no—
285Dic18—22no—
286#552—271noSylow
287C2×C3×C9C2×#4111yescharacteristic
288C2×C3×C9C2×#4111yescharacteristic
289C2×C3×C92−1+411yescharacteristic
290C3⋊D9—22no—
291C3⋊D9—41no—
292C2×C3×Q8—91no—
293C2×D12—91no—
294C2×D12—91no—
295C2×D12—181no—
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×C4×D3—91no—
299C2×C4×D3—91no—
300C2×C4×D3—91no—
301C2×C4×D3—91no—
302C2×C4×D3—91no—
303C2×C4×D3—91no—
304C2×C4×D3—181no—
305C2×Dic6—31no—
306C3×C4⋊C4—21no—
307C3×C4⋊C4—31no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—31no—
311C3×C4⋊C4—31no—
312C3×C4⋊C4—91no—
313C3×C4⋊C4—91no—
314C3×C4⋊C4C2×D911yescharacteristic
315C3×C4⋊C4D3211yescharacteristic
316C3×C42—31no—
317C3×C42—91no—
318C3×C42—91no—
319C3×C42—91no—
320(C2×D3)⋊C4—91no—
321(C2×D3)⋊C4—91no—
322(C2×D3)⋊C4—91no—
323(C2×D3)⋊C4—91no—
324(C2×D3)⋊C4—91no—
325(C2×D3)⋊C4—91no—
326(C2×D3)⋊C4—91no—
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—181no—
337(C2×D3)⋊C4—181no—
338(C2×D3)⋊C4—181no—
339C4⋊Dic3—91no—
340C4⋊Dic3—91no—
341C4⋊Dic3—91no—
342C4⋊Dic3—91no—
343C4⋊Dic3C2×D911yescharacteristic
344Dic3⋊C4—31no—
345Dic3⋊C4—31no—
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—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—181no—
359Dic3⋊C4C2×D911yescharacteristic
360Dic3⋊C4C2×D911yescharacteristic
361Dic3⋊C4C2×D911yescharacteristic
362Dic3⋊C4C2×D911yescharacteristic
363C4×Dic3—31no—
364C4×Dic3—31no—
365C4×Dic3—31no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C4×Dic3C2×D911yescharacteristic
369C22×C32C23×D311yescharacteristic
370C2×C3⋊D3—32no—
371C2×C3⋊D3—61no—
372C2×C3⋊D3—61no—
373C2×C3⋊D3—61no—
374C2×C3⋊D3—61no—
375C2×C3⋊D3—61no—
376C32×C4—21no—
377C32×C4—21no—
378C32×C4—21no—
379C3⋊Dic3—61no—
380C3⋊Dic3—61no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—21no—
385C3×Dic3—32no—
386C3×Dic3—61no—
387C3×Dic3—61no—
388C3×Dic3—61no—
389C22×C9—21no—
390C22×C9C23×D311yescharacteristic
391C2×D9—32no—
392C2×D9—62no—
393C2×D9—61no—
394C2×D9—61no—
395C2×D9—61no—
396C2×D9—61no—
397C2×D9—61no—
398C2×D9—121no—
399C2×D9—121no—
400C2×D9—121no—
401C2×D9—121no—
402C2×D9—121no—
403C4×C9—21no—
404C4×C9—21no—
405C4×C9—21no—
406C4×C9—41no—
407C4×C9—41no—
408C4×C9—41no—
409C4×C9—61no—
410C4×C9—61no—
411C4×C9—61no—
412C4×C9—61no—
413Dic9—21no—
414Dic9—21no—
415Dic9—21no—
416Dic9—61no—
417Dic9—61no—
418Dic9—121no—
419Dic9—121no—
420Dic9#23512yes—
421#126—271no—
422#126—271no—
423#125—31no—
424#125—271no—
425#125—271no—
426#123—271no—
427#123—271no—
428C22⋊Q8—271no—
429C22⋊Q8—271no—
430C4×Q8—91no—
431C4×D4—271no—
432#117—271no—
433#117—271no—
434#117—271no—
435C2×C4⋊C4—271no—
436C3×C9#55211yescharacteristic, complemented, Sylow
437C22×D3—91no—
438C22×D3—91no—
439C22×D3—91no—
440C22×D3—91no—
441C22×D3—181no—
442C22×D3—181no—
443C3×Q8—182no—
444C2×C3×C4—21no—
445C2×C3×C4—21no—
446C2×C3×C4—21no—
447C2×C3×C4—31no—
448C2×C3×C4—31no—
449C2×C3×C4—31no—
450C2×C3×C4—31no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4—91no—
455C2×C3×C4C22×D911yescharacteristic
456C2×C3×C4C22×D911yescharacteristic
457C2×C3×C4C22×D911yescharacteristic
458C2×C3×C4C2×D3211yescharacteristic
459C2×C3×C4C2×D3211yescharacteristic
460C2×C3×C4C2×D3211yescharacteristic
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—181no—
470C2×Dic3—181no—
471C2×Dic3C22×D911yescharacteristic
472C2×Dic3C22×D911yescharacteristic
473C2×Dic3C22×D911yescharacteristic
474C2×Dic3C22×D911yescharacteristic
475D12—182no—
476D12—182no—
477D12—362no—
478C4×D3—182no—
479C4×D3—182no—
480C4×D3—182no—
481C4×D3—182no—
482C4×D3—182no—
483C4×D3—182no—
484C4×D3—182no—
485C4×D3—182no—
486C4×D3—362no—
487Dic6—62no—
488C2×C32#100711yescharacteristic
489C2×C32D3×#4111yescharacteristic
490C2×C32D3×#4111yescharacteristic
491C3⋊D3—62no—
492C3⋊D3—121no—
493C2×C9—21no—
494C2×C9—21no—
495C2×C9—21no—
496C2×C9C2×#23511yescharacteristic
497C2×C9#100711yescharacteristic
498C2×C9D3×#4111yescharacteristic
499D9—62no—
500D9—122no—
501D9—121no—
502D9—241no—
503C2×Q8—91no—
504C2×D4—271no—
505C22×C4—271no—
506C22×C4—271no—
507C22×C4—271no—
508C22×C4—271no—
509C4⋊C4—31no—
510C4⋊C4—31no—
511C4⋊C4—31no—
512C4⋊C4—31no—
513C4⋊C4—31no—
514C4⋊C4—91no—
515C4⋊C4—91no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C4⋊C4—271no—
524C4⋊C4D3×D911yescharacteristic
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4—271no—
533C42—31no—
534C42—91no—
535C42—91no—
536C42—91no—
537C42—271no—
538C22×C3—21no—
539C22×C3C23×D911yescharacteristic
540C22×C3C22×D3211yescharacteristic
541C2×D3—92no—
542C2×D3—92no—
543C2×D3—182no—
544C2×D3—181no—
545C2×D3—181no—
546C2×D3—181no—
547C2×D3—181no—
548C2×D3—181no—
549C2×D3—181no—
550C2×D3—181no—
551C2×D3—181no—
552C2×D3—181no—
553C2×D3—181no—
554C2×D3—361no—
555C2×D3—361no—
556C2×D3—361no—
557C2×D3—361no—
558C2×D3—361no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—61no—
569C3×C4—61no—
570C3×C4—61no—
571C3×C4—61no—
572C3×C4—92no—
573C3×C4—181no—
574C3×C4—181no—
575C3×C4—181no—
576Dic3—21no—
577Dic3—21no—
578Dic3—21no—
579Dic3—21no—
580Dic3—32no—
581Dic3—61no—
582Dic3—61no—
583Dic3—61no—
584Dic3—181no—
585Dic3—181no—
586Dic3—181no—
587Dic3—181no—
588Dic3—361no—
589Dic3—361no—
590C32#569311yescharacteristic
591C9—21no—
592C9#558111yescharacteristic, complemented
593C23—271no—
594C23—271no—
595Q8—182no—
596D4—542no—
597C2×C4—31no—
598C2×C4—31no—
599C2×C4—31no—
600C2×C4—31no—
601C2×C4—91no—
602C2×C4—91no—
603C2×C4—91no—
604C2×C4—91no—
605C2×C4—271no—
606C2×C4—271no—
607C2×C4—542no—
608C2×C4—542no—
609C2×C4—542no—
610C2×C4—542no—
611C2×C4C2×D3×D911yescharacteristic
612C2×C4C2×D3×D911yescharacteristic
613C2×C4C2×D3×D911yescharacteristic
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3#6371311yescharacteristic
618C2×C3#41×D911yescharacteristic
619C2×C3#41×D911yescharacteristic
620C2×C3D3×#23511yescharacteristic
621C2×C3#6431211yescharacteristic
622C2×C3#6431611yescharacteristic
623D3—182no—
624D3—182no—
625D3—362no—
626D3—361no—
627D3—361no—
628D3—721no—
629C22—272no—
630C22—541no—
631C22—541no—
632C22—541no—
633C22—541no—
634C22—541no—
635C22C22×D3×D911yescharacteristic
636C4—21no—
637C4—21no—
638C4—21no—
639C4—61no—
640C4—61no—
641C4—61no—
642C4—61no—
643C4—92no—
644C4—181no—
645C4—181no—
646C4—181no—
647C4—541no—
648C4—541no—
649C3—21no—
650C3#1058979511yescharacteristic, complemented
651C3#1059434911yescharacteristic
652C2—542no—
653C2—1081no—
654C2D9×#23511yescharacteristic
655C2#1172249811yescharacteristic
656C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#889C2semidirect
3. C3×#63449C2semidirect
6. #11720985C2semidirect
7. #11720985C2semidirect
8. #11720980C2semidirect
9. #11720934C2semidirect
10. #11720934C2semidirect
11. #11720933C2semidirect
12. #11720931C2semidirect
13. #11720918C2semidirect
14. #11720899C2semidirect
15. #11720898C2semidirect
81. #63449D3semidirect
225. #889D9semidirect
436. C3×C9#552semidirect
592. C9#5581semidirect
650. C3#10589795semidirect

Maximal subgroups

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

subgroupindex
#117217502
C9×#8892
C3×#634492
#117209972
#117209922
#117209852
#117209802
#117209342
#117209332
#117209312
#117209182
#117208992
#117208982
#105943493
#105897953