Finite Group Databasebeta

← #13153910

Subgroups — #13153910

Order 1728.

1190 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
1C3×C4×D4×C9C211yesmaximal, characteristic, complemented
2#11721839C211yesmaximal, characteristic, complemented
3#11721839C211yesmaximal, characteristic, complemented
4#11721819C211yesmaximal, characteristic, complemented
5#11721818C211yesmaximal, characteristic, complemented
6C4×#6233C211yesmaximal, characteristic, complemented
7C2×#88534C211yesmaximal, characteristic, complemented
8#11721750C211yesmaximal, characteristic, complemented
9#11721745C211yesmaximal, characteristic, complemented
10#11721730C211yesmaximal, characteristic, complemented
11#11721727C211yesmaximal, characteristic, complemented
12#11721727C211yesmaximal, characteristic, complemented
13#11721725C211yesmaximal, characteristic, complemented
14#11721713C211yesmaximal, characteristic, complemented
15#11721709C211yesmaximal, characteristic, complemented
16#10596011—31nomaximal
17#10589642—33nomaximal
18C2×C3×D4×C9C2211yescharacteristic
19C22×C3×C4×C9C2211yescharacteristic
20C22×C3×C4×C9C2211yescharacteristic
21C2×#6233C2211yescharacteristic
22C2×#6233C2211yescharacteristic
23C2×#6233C2211yescharacteristic
24C2×#6233C2211yescharacteristic
25C22×C3⋊Dic9C2211yescharacteristic
26C22×C3⋊Dic9C2211yescharacteristic
27C2×C4⋊(C3⋊D9)C2211yescharacteristic
28C2×C4×C3⋊D9C2211yescharacteristic
29C2×C4×C3⋊D9C2211yescharacteristic
30C3×C9×C4⋊C4C2211yescharacteristic
31C3×C9×C22⋊C4C2211yescharacteristic
32C3×C9×C22⋊C4C2211yescharacteristic
33C3×C42×C9C2211yescharacteristic
34#88551C2211yescharacteristic
35#88551C2211yescharacteristic
36#88536C2211yescharacteristic
37#88536C2211yescharacteristic
38#88536C2211yescharacteristic
39#88536C2211yescharacteristic
40#88536C2211yescharacteristic
41#88536C2211yescharacteristic
42C4⋊(C3⋊Dic9)C2211yescharacteristic
43#88534C2211yescharacteristic
44#88534C2211yescharacteristic
45#88534C2211yescharacteristic
46#88534C2211yescharacteristic
47#88534C2211yescharacteristic
48#88534C2211yescharacteristic
49#88534C2211yescharacteristic
50#88534C2211yescharacteristic
51C4×C3⋊Dic9C2211yescharacteristic
52C4×C3⋊Dic9C2211yescharacteristic
53C32×C4×D4D311yescharacteristic
54#64148—31no—
55#64148—31no—
56#64139—31no—
57#64138—31no—
58C4×#638—31no—
59C2×#3750—31no—
60#64103—31no—
61#64098—31no—
62#64093—31no—
63#64090—31no—
64#64090—31no—
65#64088—31no—
66#64085—31no—
67#64081—31no—
68C4×D4×C9D313yescomplemented
69#63501—33no—
70#63501—33no—
71#63492—33no—
72#63491—33no—
73C4×#611—33no—
74C2×Dic9⋊C4—33no—
75#63456—33no—
76#63451—33no—
77#63446—33no—
78#611⋊C4—33no—
79#611⋊C4—33no—
80#63441—33no—
81#63438—33no—
82#63434—33no—
83C23×C3×C9C2311yescharacteristic
84C23×C3×C9C2311yescharacteristic
85C22×C3⋊D9C2311yescharacteristic
86C22×C3⋊D9C2311yescharacteristic
87C3×D4×C9—22no—
88C2×C3×C4×C9—41no—
89C2×C3×C4×C9—41no—
90C2×C3×C4×C9C2311yescharacteristic
91C2×C3×C4×C9C2311yescharacteristic
92C2×C3×C4×C9C2311yescharacteristic
93C2×C3×C4×C9C2311yescharacteristic
94C2×C3×C4×C9C2311yescharacteristic
95#6233—22no—
96#6233—22no—
97#6233—22no—
98#6233—41no—
99C2×C3⋊Dic9—22no—
100C2×C3⋊Dic9—41no—
101C2×C3⋊Dic9C2311yescharacteristic
102C2×C3⋊Dic9C2311yescharacteristic
103C2×C3⋊Dic9C2311yescharacteristic
104C2×C3⋊Dic9C2311yescharacteristic
105C2×C3⋊Dic9C2311yescharacteristic
106C2×C3⋊Dic9C2311yescharacteristic
107C4⋊(C3⋊D9)—41no—
108C4×C3⋊D9—22no—
109C4×C3⋊D9—41no—
110#5540—93no—
111#5540—91no—
112C2×C32×D4C2×D311yescharacteristic
113C22×C32×C4C2×D311yescharacteristic
114C22×C32×C4C2×D311yescharacteristic
115C2×#638—31no—
116C2×#638—31no—
117C2×#638—31no—
118C2×#638—31no—
119C22×C3⋊Dic3—31no—
120C22×C3⋊Dic3—31no—
121C2×C4⋊(C3⋊D3)—31no—
122C2×C4×C3⋊D3—31no—
123C2×C4×C3⋊D3—31no—
124C32×C4⋊C4C2×D311yescharacteristic
125C32×C22⋊C4C2×D311yescharacteristic
126C32×C22⋊C4C2×D311yescharacteristic
127C32×C42C2×D311yescharacteristic
128#3757—31no—
129#3757—31no—
130#3752—31no—
131#3752—31no—
132#3752—31no—
133#3752—31no—
134#3752—31no—
135#3752—31no—
136C4⋊(C3⋊Dic3)—31no—
137#3750—31no—
138#3750—31no—
139#3750—31no—
140#3750—31no—
141#3750—31no—
142#3750—31no—
143#3750—31no—
144#3750—31no—
145C4×C3⋊Dic3—31no—
146C4×C3⋊Dic3—31no—
147C2×D4×C9C2×D313yes—
148C22×C4×C9C2×D313yes—
149C22×C4×C9C2×D313yes—
150C2×#611—33no—
151C2×#611—33no—
152C2×#611—33no—
153C2×#611—33no—
154C22×Dic9—33no—
155C22×Dic9—33no—
156C2×D36—33no—
157C2×C4×D9—33no—
158C2×C4×D9—33no—
159C9×C4⋊C4C2×D313yes—
160C9×C22⋊C4C2×D313yes—
161C9×C22⋊C4C2×D313yes—
162C42×C9C2×D313yes—
163#3676—33no—
164#3676—33no—
165(C2×D9)⋊C4—33no—
166(C2×D9)⋊C4—33no—
167(C2×D9)⋊C4—33no—
168(C2×D9)⋊C4—33no—
169(C2×D9)⋊C4—33no—
170(C2×D9)⋊C4—33no—
171C4⋊Dic9—33no—
172Dic9⋊C4—33no—
173Dic9⋊C4—33no—
174Dic9⋊C4—33no—
175Dic9⋊C4—33no—
176Dic9⋊C4—33no—
177Dic9⋊C4—33no—
178Dic9⋊C4—33no—
179Dic9⋊C4—33no—
180C4×Dic9—33no—
181C4×Dic9—33no—
182C22×C3×C9—21no—
183C22×C3×C9—21no—
184C22×C3×C9—21no—
185C22×C3×C9—21no—
186C22×C3×C9—21no—
187C22×C3×C9#4112yes—
188C22×C3×C9C2411yescharacteristic
189C2×C3⋊D9—21no—
190C2×C3⋊D9—21no—
191C2×C3⋊D9—21no—
192C2×C3⋊D9—21no—
193C2×C3⋊D9—21no—
194C2×C3⋊D9—21no—
195C3×C4×C9—21no—
196C3×C4×C9—21no—
197C3×C4×C9—21no—
198C3×C4×C9—21no—
199C3×C4×C9—21no—
200C3⋊Dic9—21no—
201C3⋊Dic9—21no—
202C3⋊Dic9—21no—
203C3⋊Dic9—21no—
204C3⋊Dic9—21no—
205C3⋊Dic9#4112yes—
206C3×C4×D4D913yescomplemented
207C3×C4×D4C3⋊D311yescharacteristic
208#939—93no—
209#939—93no—
210#939—91no—
211#939—91no—
212#930—93no—
213#930—91no—
214#929—93no—
215#929—91no—
216C4×#67—93no—
217C4×#67—91no—
218C2×Dic3⋊C4—93no—
219C2×Dic3⋊C4—91no—
220#894—93no—
221#894—91no—
222#889—93no—
223#889—91no—
224#884—93no—
225#884—91no—
226#67⋊C4—93no—
227#67⋊C4—93no—
228#67⋊C4—91no—
229#67⋊C4—91no—
230#879—93no—
231#879—91no—
232#876—93no—
233#876—91no—
234#872—93no—
235#872—91no—
236C23×C32C22×D311yescharacteristic
237C23×C32C22×D311yescharacteristic
238C22×C3⋊D3—31no—
239C22×C3⋊D3—31no—
240C32×D4—22no—
241C2×C32×C4—41no—
242C2×C32×C4—41no—
243C2×C32×C4C22×D311yescharacteristic
244C2×C32×C4C22×D311yescharacteristic
245C2×C32×C4C22×D311yescharacteristic
246C2×C32×C4C22×D311yescharacteristic
247C2×C32×C4C22×D311yescharacteristic
248#638—62no—
249#638—62no—
250#638—62no—
251#638—121no—
252C2×C3⋊Dic3—31no—
253C2×C3⋊Dic3—31no—
254C2×C3⋊Dic3—31no—
255C2×C3⋊Dic3—31no—
256C2×C3⋊Dic3—31no—
257C2×C3⋊Dic3—31no—
258C2×C3⋊Dic3—62no—
259C2×C3⋊Dic3—121no—
260C4⋊(C3⋊D3)—121no—
261C4×C3⋊D3—62no—
262C4×C3⋊D3—121no—
263C23×C9C22×D313yes—
264C23×C9C22×D313yes—
265C22×D9—33no—
266C22×D9—33no—
267D4×C9—26no—
268C2×C4×C9—43no—
269C2×C4×C9—43no—
270C2×C4×C9C22×D313yes—
271C2×C4×C9C22×D313yes—
272C2×C4×C9C22×D313yes—
273C2×C4×C9C22×D313yes—
274C2×C4×C9C22×D313yes—
275#611—66no—
276#611—66no—
277#611—66no—
278#611—123no—
279C2×Dic9—33no—
280C2×Dic9—33no—
281C2×Dic9—33no—
282C2×Dic9—33no—
283C2×Dic9—33no—
284C2×Dic9—33no—
285C2×Dic9—66no—
286C2×Dic9—123no—
287D36—123no—
288C4×D9—66no—
289C4×D9—123no—
290#553—271noSylow
291C2×C3×C9—22no—
292C2×C3×C9—41no—
293C2×C3×C9C2×#4111yescharacteristic
294C2×C3×C9C2×#4111yescharacteristic
295C2×C3×C9H(5,𝔽2)11yescharacteristic
296C3⋊D9—41no—
297C3⋊D9—41no—
298C2×C3×D4C2×D913yes—
299C2×C3×D4C2×C3⋊D311yescharacteristic
300C22×C3×C4C2×D913yes—
301C22×C3×C4C2×D913yes—
302C22×C3×C4C2×C3⋊D311yescharacteristic
303C22×C3×C4C2×C3⋊D311yescharacteristic
304C2×#67—93no—
305C2×#67—93no—
306C2×#67—93no—
307C2×#67—91no—
308C2×#67—91no—
309C2×#67—91no—
310C2×#67—93no—
311C2×#67—91no—
312C22×Dic3—93no—
313C22×Dic3—91no—
314C22×Dic3—93no—
315C22×Dic3—91no—
316C2×D12—93no—
317C2×D12—91no—
318C2×C4×D3—93no—
319C2×C4×D3—91no—
320C2×C4×D3—93no—
321C2×C4×D3—91no—
322C3×C4⋊C4C2×D913yes—
323C3×C4⋊C4C2×C3⋊D311yescharacteristic
324C3×C22⋊C4C2×D913yes—
325C3×C22⋊C4C2×D913yes—
326C3×C22⋊C4C2×C3⋊D311yescharacteristic
327C3×C22⋊C4C2×C3⋊D311yescharacteristic
328C3×C42C2×D913yes—
329C3×C42C2×C3⋊D311yescharacteristic
330#217—93no—
331#217—93no—
332#217—91no—
333#217—91no—
334(C2×D3)⋊C4—93no—
335(C2×D3)⋊C4—93no—
336(C2×D3)⋊C4—93no—
337(C2×D3)⋊C4—93no—
338(C2×D3)⋊C4—93no—
339(C2×D3)⋊C4—93no—
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—
346C4⋊Dic3—93no—
347C4⋊Dic3—91no—
348Dic3⋊C4—93no—
349Dic3⋊C4—93no—
350Dic3⋊C4—93no—
351Dic3⋊C4—93no—
352Dic3⋊C4—93no—
353Dic3⋊C4—93no—
354Dic3⋊C4—93no—
355Dic3⋊C4—93no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—91no—
361Dic3⋊C4—91no—
362Dic3⋊C4—91no—
363Dic3⋊C4—91no—
364C4×Dic3—93no—
365C4×Dic3—93no—
366C4×Dic3—91no—
367C4×Dic3—91no—
368C22×C32—21no—
369C22×C32—21no—
370C22×C32—21no—
371C22×C32—21no—
372C22×C32—21no—
373C22×C32#23512yes—
374C22×C32C23×D311yescharacteristic
375C2×C3⋊D3—61no—
376C2×C3⋊D3—61no—
377C2×C3⋊D3—61no—
378C2×C3⋊D3—61no—
379C2×C3⋊D3—61no—
380C2×C3⋊D3—61no—
381C32×C4—21no—
382C32×C4—21no—
383C32×C4—21no—
384C32×C4—21no—
385C32×C4—21no—
386C3⋊Dic3—32no—
387C3⋊Dic3—61no—
388C3⋊Dic3—61no—
389C3⋊Dic3—61no—
390C3⋊Dic3—61no—
391C3⋊Dic3—61no—
392C22×C9—23no—
393C22×C9—23no—
394C22×C9—23no—
395C22×C9—23no—
396C22×C9—23no—
397C22×C9#23516yes—
398C22×C9C23×D313yes—
399C2×D9—63no—
400C2×D9—63no—
401C2×D9—63no—
402C2×D9—63no—
403C2×D9—63no—
404C2×D9—63no—
405C4×C9—23no—
406C4×C9—23no—
407C4×C9—23no—
408C4×C9—23no—
409C4×C9—23no—
410Dic9—36no—
411Dic9—63no—
412Dic9—63no—
413Dic9—63no—
414Dic9—63no—
415Dic9—63no—
416#126—271no—
417#126—271no—
418#125—271no—
419#123—271no—
420#123—271no—
421#121—271no—
422#121—271no—
423#121—271no—
424#121—271no—
425C4×D4—271no—
426C4×D4—271no—
427C4×D4—271no—
428C4×D4C3⋊D911yescharacteristic, complemented
429#117—271no—
430C2×C4⋊C4—271no—
431C3×C9#55311yescharacteristic, complemented, Sylow
432C23×C3C22×D913yes—
433C23×C3C22×D913yes—
434C23×C3C22×C3⋊D311yescharacteristic
435C23×C3C22×C3⋊D311yescharacteristic
436C22×D3—91no—
437C22×D3—91no—
438C22×D3—93no—
439C22×D3—93no—
440C3×D4—26no—
441C3×D4—22no—
442C2×C3×C4—41no—
443C2×C3×C4—41no—
444C2×C3×C4—43no—
445C2×C3×C4—43no—
446C2×C3×C4C22×D913yes—
447C2×C3×C4C22×D913yes—
448C2×C3×C4C22×D913yes—
449C2×C3×C4C22×D913yes—
450C2×C3×C4C22×D913yes—
451C2×C3×C4C22×C3⋊D311yescharacteristic
452C2×C3×C4C22×C3⋊D311yescharacteristic
453C2×C3×C4C22×C3⋊D311yescharacteristic
454C2×C3×C4C22×C3⋊D311yescharacteristic
455C2×C3×C4C22×C3⋊D311yescharacteristic
456#67—182no—
457#67—186no—
458#67—186no—
459#67—186no—
460#67—182no—
461#67—182no—
462#67—361no—
463#67—363no—
464C2×Dic3—91no—
465C2×Dic3—93no—
466C2×Dic3—91no—
467C2×Dic3—91no—
468C2×Dic3—91no—
469C2×Dic3—91no—
470C2×Dic3—91no—
471C2×Dic3—93no—
472C2×Dic3—93no—
473C2×Dic3—93no—
474C2×Dic3—93no—
475C2×Dic3—93no—
476C2×Dic3—182no—
477C2×Dic3—186no—
478C2×Dic3—361no—
479C2×Dic3—363no—
480D12—361no—
481D12—363no—
482C4×D3—182no—
483C4×D3—186no—
484C4×D3—361no—
485C4×D3—363no—
486C2×C32—22no—
487C2×C32—41no—
488C2×C32C2×#23511yescharacteristic
489C2×C32#100411yescharacteristic
490C2×C32D3×#4111yescharacteristic
491C3⋊D3—121no—
492C3⋊D3—121no—
493C2×C9—26no—
494C2×C9—43no—
495C2×C9C2×#23513yes—
496C2×C9#100413yes—
497C2×C9D3×#4113yes—
498D9—123no—
499D9—123no—
500C2×D4—271no—
501C2×D4—271no—
502C2×D4—271no—
503C2×D4—271no—
504C2×D4—271no—
505C2×D4C2×C3⋊D911yescharacteristic
506C22×C4—271no—
507C22×C4—271no—
508C22×C4—271no—
509C22×C4—271no—
510C22×C4C2×C3⋊D911yescharacteristic
511C22×C4C2×C3⋊D911yescharacteristic
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4C2×C3⋊D911yescharacteristic
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C22⋊C4—271no—
530C22⋊C4C2×C3⋊D911yescharacteristic
531C22⋊C4C2×C3⋊D911yescharacteristic
532C42—271no—
533C42—271no—
534C42C2×C3⋊D911yescharacteristic
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—23no—
538C22×C3—23no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—23no—
543C22×C3—23no—
544C22×C3—23no—
545C22×C3#369716yes—
546C22×C3C23×D913yes—
547C22×C3#382812yes—
548C22×C3C23×C3⋊D311yescharacteristic
549C2×D3—181no—
550C2×D3—181no—
551C2×D3—181no—
552C2×D3—181no—
553C2×D3—181no—
554C2×D3—181no—
555C2×D3—183no—
556C2×D3—183no—
557C2×D3—183no—
558C2×D3—183no—
559C2×D3—183no—
560C2×D3—183no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—21no—
566C3×C4—23no—
567C3×C4—23no—
568C3×C4—23no—
569C3×C4—23no—
570C3×C4—23no—
571Dic3—92no—
572Dic3—96no—
573Dic3—181no—
574Dic3—181no—
575Dic3—181no—
576Dic3—181no—
577Dic3—181no—
578Dic3—183no—
579Dic3—183no—
580Dic3—183no—
581Dic3—183no—
582Dic3—183no—
583C32#554011yescharacteristic
584C9#554013yescomplemented
585C23—271no—
586C23—271no—
587C23C22×C3⋊D911yescharacteristic
588C23C22×C3⋊D911yescharacteristic
589D4—22no—
590D4—542no—
591D4—542no—
592D4—542no—
593D4—1081no—
594D4—1081no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—271no—
598C2×C4—271no—
599C2×C4—271no—
600C2×C4—271no—
601C2×C4—271no—
602C2×C4—271no—
603C2×C4—542no—
604C2×C4—542no—
605C2×C4—1081no—
606C2×C4—1081no—
607C2×C4C22×C3⋊D911yescharacteristic
608C2×C4C22×C3⋊D911yescharacteristic
609C2×C4C22×C3⋊D911yescharacteristic
610C2×C4C22×C3⋊D911yescharacteristic
611C2×C4C22×C3⋊D911yescharacteristic
612C2×C3—22no—
613C2×C3—26no—
614C2×C3—41no—
615C2×C3—43no—
616C2×C3C2×#369713yes—
617C2×C3#6371013yes—
618C2×C3#41×D913yes—
619C2×C3C2×#382811yescharacteristic
620C2×C3#6436111yescharacteristic
621C2×C3#41×C3⋊D311yescharacteristic
622D3—361no—
623D3—361no—
624D3—363no—
625D3—363no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—541no—
632C22—541no—
633C22—541no—
634C22—541no—
635C22—541no—
636C22—541no—
637C22#8873612yes—
638C22C23×C3⋊D911yescharacteristic
639C4—21no—
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—272no—
645C4—541no—
646C4—541no—
647C4—541no—
648C4—541no—
649C4—541no—
650C3#1058964213yescomplemented
651C3#1059601111yescharacteristic
652C2—22no—
653C2—41no—
654C2—1081no—
655C2—1081no—
656C2C2×#8873611yescharacteristic
657C2#1172266111yescharacteristic
658C2#41×C3⋊D911yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C4×D4×C9C2
socleC22×C32C23×D3
hypercenterC4×D4C3⋊D9
hypocenterC3×C9#553
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×C4×D4×C9C2semidirect
2. #11721839C2semidirect
3. #11721839C2semidirect
4. #11721819C2semidirect
5. #11721818C2semidirect
6. C4×#6233C2semidirect
7. C2×#88534C2semidirect
8. #11721750C2semidirect
9. #11721745C2semidirect
10. #11721730C2semidirect
11. #11721727C2semidirect
12. #11721727C2semidirect
13. #11721725C2semidirect
14. #11721713C2semidirect
15. #11721709C2semidirect
68. C4×D4×C9D3semidirect
206. C3×C4×D4D9semidirect
428. C4×D4C3⋊D9semidirect
431. C3×C9#553semidirect
584. C9#5540semidirect
650. C3#10589642semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D4×C92
#117218392
#117218192
#117218182
C4×#62332
C2×#885342
#117217502
#117217452
#117217302
#117217272
#117217252
#117217132
#117217092
#105960113
#105896423