Finite Group Databasebeta

← #13147664

Subgroups — #13147664

Order 1728.

762 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#11721713C211yesmaximal, characteristic, complemented
2C42×D3×C9C211yesmaximal, characteristic, complemented
3C3×#63438C211yesmaximal, characteristic, complemented
4#11721034C211yesmaximal, characteristic
5#11721032C211yesmaximal, characteristic, complemented
6#11721030C211yesmaximal, characteristic, complemented
7#11720977C211yesmaximal, characteristic, complemented
8#11720976C211yesmaximal, characteristic, complemented
9#11720974C211yesmaximal, characteristic, complemented
10#11720955C211yesmaximal, characteristic, complemented
11#11720954C211yesmaximal, characteristic, complemented
12#11720940C211yesmaximal, characteristic, complemented
13#11720925C211yesmaximal, characteristic, complemented
14#11720900C211yesmaximal, characteristic, complemented
15#11720898C211yesmaximal, characteristic, complemented
16#10593975—31nomaximal
17#10589581—31nomaximal
18C2×C4×D3×C9C2211yescharacteristic
19C2×C4×D3×C9C2211yescharacteristic
20C2×C4×D3×C9C2211yescharacteristic
21C2×#6195C2211yescharacteristic
22C2×#6194C2211yescharacteristic
23C2×#6192C2211yescharacteristic
24C2×(C3⋊D9)⋊C4C2211yescharacteristic
25C2×Dic3×D9C2211yescharacteristic
26C2×#6189C2211yescharacteristic
27C3×C42×C9C2211yescharacteristic
28#88536C2211yescharacteristic
29#88536C2211yescharacteristic
30#88536C2211yescharacteristic
31#88534C2211yescharacteristic
32#88534C2211yescharacteristic
33#88534C2211yescharacteristic
34C3×(C2×D9)⋊C4C2211yescharacteristic
35C3×(C2×D9)⋊C4C2211yescharacteristic
36C3×(C2×D9)⋊C4C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C4×C9×Dic3C2211yescharacteristic
39C4×C9×Dic3C2211yescharacteristic
40C3×Dic9⋊C4C2211yescharacteristic
41C3×Dic9⋊C4C2211yescharacteristic
42C3×Dic9⋊C4C2211yescharacteristic
43(C2×D3)⋊Dic9C2211yescharacteristic
44(C2×D3)⋊Dic9C2211yescharacteristic
45#88445C2211yescharacteristic
46(C2×D9)⋊Dic3C2211yescharacteristic
47Dic3⋊Dic9C2211yescharacteristic
48Dic3⋊Dic9C2211yescharacteristic
49#88442C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64085—31no—
54C3×#876—31no—
55C3×C42×D3D311yescharacteristic
56#63906—31no—
57#63905—31no—
58#63904—31no—
59#63880—31no—
60#63879—31no—
61#63877—31no—
62#63867—31no—
63#63852—31no—
64#63851—31no—
65#63850—31no—
66#63837—31no—
67#63836—31no—
68C2×C42×C9—31no—
69#63492—31no—
70#63491—31no—
71#63491—31no—
72C4×#611—31no—
73C4×#611—31no—
74#63486—31no—
75#63438—31no—
76#63438—61no—
77#63438D311yescharacteristic, complemented
78#63437—31no—
79C4×D36—31no—
80#63434—31no—
81#63434—31no—
82#63432—31no—
83C4×Dic18—31no—
84C22×C3⋊D9C2311yescharacteristic
85C22×D3×C9C2311yescharacteristic
86C22×C3×D9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3×C4×C9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C3×Dic9C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C4×D3×C9—22no—
100C4×D3×C9—22no—
101C4×D3×C9—22no—
102#6195—22no—
103#6194—22no—
104#6192—22no—
105(C3⋊D9)⋊C4—22no—
106Dic3×D9—22no—
107#6189—22no—
108#5706—91no—
109#5479—91no—
110C2×C3×C4×D3C2×D311yescharacteristic
111C2×C3×C4×D3C2×D311yescharacteristic
112C2×C3×C4×D3C2×D311yescharacteristic
113C2×#627—31no—
114C2×#626—31no—
115C2×#626—31no—
116C2×#625—31no—
117C2×#624—31no—
118C2×D3×Dic3—31no—
119C32×C42C2×D311yescharacteristic
120#3752—31no—
121#3752—31no—
122#3752—31no—
123#3750—31no—
124#3750—31no—
125#3750—31no—
126C3×(C2×D3)⋊C4—31no—
127C3×(C2×D3)⋊C4—31no—
128C3×(C2×D3)⋊C4—31no—
129C3×Dic3⋊C4—31no—
130C3×Dic3⋊C4—31no—
131C3×Dic3⋊C4—31no—
132C3×C4×Dic3C2×D311yescharacteristic
133C3×C4×Dic3C2×D311yescharacteristic
134C3×C4×Dic3C2×D311yescharacteristic
135#3724—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138Dic3⋊Dic3—31no—
139#3722—31no—
140(C2×D3)⋊Dic3—31no—
141(C2×D3)⋊Dic3—31no—
142(C2×D3)⋊Dic3—31no—
143Dic32—31no—
144Dic32—31no—
145C22×C4×C9—31no—
146C22×C4×C9—31no—
147C22×C4×C9—31no—
148C2×#611—31no—
149C2×#611—31no—
150C2×D36—31no—
151C2×C4×D9—31no—
152C2×C4×D9—31no—
153C2×Dic18—31no—
154C42×C9—21no—
155C42×C9—31no—
156C42×C9—31no—
157C42×C9—31no—
158C42×C9C2×D311yescharacteristic
159#3676—31no—
160#3676—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—
169(C2×D9)⋊C4C2×D311yescharacteristic
170(C2×D9)⋊C4C2×D311yescharacteristic
171(C2×D9)⋊C4C2×D311yescharacteristic
172C4⋊Dic9—31no—
173C4⋊Dic9—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—31no—
179Dic9⋊C4—61no—
180Dic9⋊C4—61no—
181Dic9⋊C4—61no—
182Dic9⋊C4C2×D311yescharacteristic
183Dic9⋊C4C2×D311yescharacteristic
184Dic9⋊C4C2×D311yescharacteristic
185C4×Dic9—31no—
186C4×Dic9—31no—
187C22×C3×C9C2411yescharacteristic
188C2×C3⋊D9—21no—
189C2×C3⋊D9—21no—
190C2×C3⋊D9—21no—
191C2×D3×C9—21no—
192C2×D3×C9—21no—
193C2×D3×C9#4112yes—
194C2×C3×D9—21no—
195C2×C3×D9—21no—
196C2×C3×D9—21no—
197C3×C4×C9—21no—
198C3×C4×C9—21no—
199C3×C4×C9—21no—
200C3⋊Dic9—21no—
201C3⋊Dic9—21no—
202C3⋊Dic9—21no—
203C9×Dic3—21no—
204C9×Dic3#4112yes—
205C9×Dic3#4112yes—
206C3×Dic9—21no—
207C3×Dic9—21no—
208C3×Dic9—21no—
209C3×#126—91no—
210C2×C3×C42—31no—
211#930—91no—
212#929—91no—
213#929—91no—
214C4×#67—91no—
215C4×#67—91no—
216#924—91no—
217#896—91no—
218#894—91no—
219#893—91no—
220#892—91no—
221#889—91no—
222#887—91no—
223#885—91no—
224#884—91no—
225#883—91no—
226#67⋊C4—91no—
227#67⋊C4—91no—
228#877—91no—
229#876—31no—
230#876—91no—
231#876—91no—
232#876—181no—
233#875—91no—
234C4×D12—91no—
235#872—91no—
236#872—91no—
237C42×D3D911yescharacteristic, complemented
238#870—91no—
239C4×Dic6—91no—
240C22×C3⋊D3—31no—
241C22×C3×D3—31no—
242C22×C3×D3C22×D311yescharacteristic
243C2×C32×C4C22×D311yescharacteristic
244C2×C32×C4C22×D311yescharacteristic
245C2×C32×C4C22×D311yescharacteristic
246C2×C3⋊Dic3—31no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—31no—
249C2×C3×Dic3—31no—
250C2×C3×Dic3—31no—
251C2×C3×Dic3—31no—
252C2×C3×Dic3C22×D311yescharacteristic
253C2×C3×Dic3C22×D311yescharacteristic
254C2×C3×Dic3C22×D311yescharacteristic
255C3×C4×D3—22no—
256C3×C4×D3—22no—
257C3×C4×D3—22no—
258#627—62no—
259#626—62no—
260#626—62no—
261#625—62no—
262#624—62no—
263D3×Dic3—62no—
264C23×C9—31no—
265C22×D9—31no—
266C22×D9—61no—
267C22×D9C22×D311yescharacteristic
268C2×C4×C9—21no—
269C2×C4×C9—21no—
270C2×C4×C9—21no—
271C2×C4×C9—31no—
272C2×C4×C9—31no—
273C2×C4×C9—31no—
274C2×C4×C9—62no—
275C2×C4×C9—62no—
276C2×C4×C9—62no—
277C2×C4×C9C22×D311yescharacteristic
278C2×C4×C9C22×D311yescharacteristic
279C2×C4×C9C22×D311yescharacteristic
280#611—62no—
281#611—62no—
282C2×Dic9—31no—
283C2×Dic9—31no—
284C2×Dic9—31no—
285C2×Dic9—61no—
286C2×Dic9—61no—
287C2×Dic9—61no—
288C2×Dic9C22×D311yescharacteristic
289C2×Dic9C22×D311yescharacteristic
290C2×Dic9C22×D311yescharacteristic
291D36—62no—
292C4×D9—62no—
293C4×D9—62no—
294Dic18—62no—
295#529—271noSylow
296C2×C3×C9C2×#4111yescharacteristic
297C2×C3×C9C2×#4111yescharacteristic
298C2×C3×C9C2×#4111yescharacteristic
299C3⋊D9—41no—
300D3×C9—22no—
301C3×D9—41no—
302C22×C3×C4—31no—
303C22×C3×C4—31no—
304C22×C3×C4—31no—
305C2×#67—91no—
306C2×#67—91no—
307C2×#67—91no—
308C2×#67—91no—
309C22×Dic3—91no—
310C2×D12—91no—
311C2×D12—91no—
312C2×C4×D3—91no—
313C2×C4×D3—91no—
314C2×C4×D3—91no—
315C2×C4×D3C2×D911yescharacteristic
316C2×C4×D3C2×D911yescharacteristic
317C2×C4×D3C2×D911yescharacteristic
318C2×Dic6—91no—
319C2×Dic6—91no—
320C3×C4⋊C4—91no—
321C3×C4⋊C4—91no—
322C3×C4⋊C4—91no—
323C3×C22⋊C4—91no—
324C3×C22⋊C4—91no—
325C3×C22⋊C4—91no—
326C3×C42—21no—
327C3×C42—31no—
328C3×C42—31no—
329C3×C42—31no—
330C3×C42C2×D911yescharacteristic
331C3×C42D3211yescharacteristic
332#217—91no—
333#217—91no—
334#217—91no—
335(C2×D3)⋊C4—31no—
336(C2×D3)⋊C4—31no—
337(C2×D3)⋊C4—31no—
338(C2×D3)⋊C4—91no—
339(C2×D3)⋊C4—91no—
340(C2×D3)⋊C4—91no—
341(C2×D3)⋊C4—91no—
342(C2×D3)⋊C4—91no—
343(C2×D3)⋊C4—91no—
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—181no—
350(C2×D3)⋊C4—181no—
351(C2×D3)⋊C4—181no—
352C4⋊Dic3—91no—
353C4⋊Dic3—91no—
354C4⋊Dic3—91no—
355Dic3⋊C4—31no—
356Dic3⋊C4—31no—
357Dic3⋊C4—31no—
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—181no—
370Dic3⋊C4—181no—
371Dic3⋊C4—181no—
372C4×Dic3—91no—
373C4×Dic3—91no—
374C4×Dic3—91no—
375C4×Dic3—91no—
376C4×Dic3C2×D911yescharacteristic
377C4×Dic3C2×D911yescharacteristic
378C4×Dic3C2×D911yescharacteristic
379C22×C32C23×D311yescharacteristic
380C2×C3⋊D3—61no—
381C2×C3⋊D3—61no—
382C2×C3⋊D3—61no—
383C2×C3×D3—21no—
384C2×C3×D3—21no—
385C2×C3×D3—61no—
386C2×C3×D3—61no—
387C2×C3×D3—61no—
388C2×C3×D3#23512yes—
389C32×C4—21no—
390C32×C4—21no—
391C32×C4—21no—
392C3⋊Dic3—61no—
393C3⋊Dic3—61no—
394C3⋊Dic3—61no—
395C3×Dic3—21no—
396C3×Dic3—61no—
397C3×Dic3—61no—
398C3×Dic3—61no—
399C3×Dic3#23512yes—
400C3×Dic3#23512yes—
401C22×C9—21no—
402C22×C9—32no—
403C22×C9—61no—
404C22×C9—61no—
405C22×C9C23×D311yescharacteristic
406C2×D9—21no—
407C2×D9—21no—
408C2×D9—21no—
409C2×D9—61no—
410C2×D9—61no—
411C2×D9—61no—
412C2×D9—121no—
413C2×D9—121no—
414C2×D9—121no—
415C4×C9—21no—
416C4×C9—21no—
417C4×C9—21no—
418C4×C9—32no—
419C4×C9—32no—
420C4×C9—41no—
421C4×C9—41no—
422C4×C9—41no—
423C4×C9—61no—
424Dic9—21no—
425Dic9—21no—
426Dic9—21no—
427Dic9—61no—
428Dic9—61no—
429Dic9—61no—
430Dic9—121no—
431Dic9—121no—
432Dic9—121no—
433#126—91no—
434#126—271no—
435#125—271no—
436#124—271no—
437#123—271no—
438#123—271no—
439C22⋊Q8—271no—
440#121—271no—
441C4×Q8—271no—
442C4×D4—271no—
443C4×D4—271no—
444C4×D4—271no—
445#117—271no—
446#117—271no—
447C2×C42—31no—
448C3×C9#52911yescharacteristic, complemented, Sylow
449C23×C3—31no—
450C23×C3—91no—
451C22×D3—31no—
452C22×D3—91no—
453C22×D3—91no—
454C22×D3—181no—
455C22×D3C22×D911yescharacteristic
456C2×C3×C4—21no—
457C2×C3×C4—21no—
458C2×C3×C4—21no—
459C2×C3×C4—31no—
460C2×C3×C4—31no—
461C2×C3×C4—31no—
462C2×C3×C4—62no—
463C2×C3×C4—62no—
464C2×C3×C4—62no—
465C2×C3×C4—91no—
466C2×C3×C4—91no—
467C2×C3×C4—91no—
468C2×C3×C4C22×D911yescharacteristic
469C2×C3×C4C22×D911yescharacteristic
470C2×C3×C4C22×D911yescharacteristic
471C2×C3×C4C2×D3211yescharacteristic
472C2×C3×C4C2×D3211yescharacteristic
473C2×C3×C4C2×D3211yescharacteristic
474#67—182no—
475#67—182no—
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—181no—
488C2×Dic3—181no—
489C2×Dic3—181no—
490C2×Dic3—182no—
491C2×Dic3C22×D911yescharacteristic
492C2×Dic3C22×D911yescharacteristic
493C2×Dic3C22×D911yescharacteristic
494D12—182no—
495D12—182no—
496C4×D3—22no—
497C4×D3—22no—
498C4×D3—22no—
499C4×D3—182no—
500C4×D3—182no—
501C4×D3—182no—
502Dic6—182no—
503Dic6—182no—
504C2×C32C2×#23511yescharacteristic
505C2×C32C2×#23511yescharacteristic
506C2×C32C2×#23511yescharacteristic
507C3⋊D3—121no—
508C3×D3—22no—
509C3×D3—121no—
510C2×C9—21no—
511C2×C9—21no—
512C2×C9—21no—
513C2×C9—62no—
514C2×C9D3×#4111yescharacteristic
515C2×C9D3×#4111yescharacteristic
516C2×C9D3×#4111yescharacteristic
517D9—41no—
518D9—121no—
519D9—241no—
520C2×Q8—271no—
521C2×D4—271no—
522C2×D4—271no—
523C2×D4—271no—
524C22×C4—31no—
525C22×C4—31no—
526C22×C4—31no—
527C22×C4—271no—
528C22×C4—271no—
529C4⋊C4—91no—
530C4⋊C4—91no—
531C4⋊C4—91no—
532C4⋊C4—271no—
533C4⋊C4—271no—
534C4⋊C4—271no—
535C4⋊C4—271no—
536C4⋊C4—271no—
537C4⋊C4—271no—
538C4⋊C4—271no—
539C22⋊C4—91no—
540C22⋊C4—91no—
541C22⋊C4—91no—
542C22⋊C4—271no—
543C22⋊C4—271no—
544C22⋊C4—271no—
545C22⋊C4—271no—
546C22⋊C4—271no—
547C22⋊C4—271no—
548C22⋊C4—271no—
549C42—31no—
550C42—31no—
551C42—31no—
552C42—271no—
553C42—271no—
554C42D3×D911yescharacteristic
555C22×C3—21no—
556C22×C3—32no—
557C22×C3—61no—
558C22×C3—61no—
559C22×C3—181no—
560C22×C3—181no—
561C22×C3—181no—
562C22×C3C23×D911yescharacteristic
563C22×C3C22×D3211yescharacteristic
564C2×D3—21no—
565C2×D3—21no—
566C2×D3—61no—
567C2×D3—61no—
568C2×D3—61no—
569C2×D3—181no—
570C2×D3—181no—
571C2×D3—181no—
572C2×D3—181no—
573C2×D3—181no—
574C2×D3—181no—
575C2×D3—361no—
576C2×D3—361no—
577C2×D3—361no—
578C2×D3#369712yes—
579C3×C4—21no—
580C3×C4—21no—
581C3×C4—21no—
582C3×C4—21no—
583C3×C4—21no—
584C3×C4—21no—
585C3×C4—32no—
586C3×C4—32no—
587C3×C4—41no—
588C3×C4—41no—
589C3×C4—41no—
590C3×C4—61no—
591C3×C4—181no—
592C3×C4—181no—
593C3×C4—181no—
594Dic3—21no—
595Dic3—61no—
596Dic3—61no—
597Dic3—61no—
598Dic3—181no—
599Dic3—181no—
600Dic3—181no—
601Dic3—181no—
602Dic3—181no—
603Dic3—181no—
604Dic3—361no—
605Dic3—361no—
606Dic3—361no—
607Dic3#369712yes—
608Dic3#369712yes—
609C32#547911yescharacteristic
610C9—21no—
611C9#570611yescharacteristic, complemented
612C23—31no—
613C23—91no—
614C23—271no—
615Q8—542no—
616D4—542no—
617D4—542no—
618D4—542no—
619C2×C4—31no—
620C2×C4—31no—
621C2×C4—31no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—91no—
626C2×C4—91no—
627C2×C4—91no—
628C2×C4—271no—
629C2×C4—271no—
630C2×C4—271no—
631C2×C4—542no—
632C2×C4—542no—
633C2×C4C2×D3×D911yescharacteristic
634C2×C4C2×D3×D911yescharacteristic
635C2×C4C2×D3×D911yescharacteristic
636C2×C3—21no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—62no—
640C2×C3—361no—
641C2×C3C2×#369711yescharacteristic
642C2×C3C2×#369711yescharacteristic
643C2×C3C2×#369711yescharacteristic
644C2×C3D3×#23511yescharacteristic
645C2×C3D3×#23511yescharacteristic
646C2×C3D3×#23511yescharacteristic
647D3—22no—
648D3—121no—
649D3—361no—
650D3—361no—
651D3—721no—
652C22—32no—
653C22—61no—
654C22—61no—
655C22—181no—
656C22—181no—
657C22—181no—
658C22—541no—
659C22—541no—
660C22—541no—
661C22C22×D3×D911yescharacteristic
662C4—21no—
663C4—21no—
664C4—21no—
665C4—32no—
666C4—32no—
667C4—61no—
668C4—181no—
669C4—181no—
670C4—181no—
671C4—541no—
672C4—541no—
673C4—541no—
674C3—21no—
675C3#1058958111yescharacteristic, complemented
676C3#1059397511yescharacteristic
677C2—62no—
678C2—361no—
679C2—1081no—
680C2D3×#369711yescharacteristic
681C2D3×#369711yescharacteristic
682C2D3×#369711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721034C2
hypercenterC42D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721713C2semidirect
2. C42×D3×C9C2semidirect
3. C3×#63438C2semidirect
5. #11721032C2semidirect
6. #11721030C2semidirect
7. #11720977C2semidirect
8. #11720976C2semidirect
9. #11720974C2semidirect
10. #11720955C2semidirect
11. #11720954C2semidirect
12. #11720940C2semidirect
13. #11720925C2semidirect
14. #11720900C2semidirect
15. #11720898C2semidirect
77. #63438D3semidirect
237. C42×D3D9semidirect
448. C3×C9#529semidirect
611. C9#5706semidirect
675. C3#10589581semidirect

Maximal subgroups

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

subgroupindex
#117217132
C42×D3×C92
C3×#634382
#117210342
#117210322
#117210302
#117209772
#117209762
#117209742
#117209552
#117209542
#117209402
#117209252
#117209002
#117208982
#105939753
#105895813