Finite Group Databasebeta

← #13154243

Subgroups — #13154243

Order 1728.

1070 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#126C211yesmaximal, characteristic, complemented
2#11721752C211yesmaximal, characteristic, complemented
3#11721748C211yesmaximal, characteristic, complemented
4C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
5#11721746C211yesmaximal, characteristic, complemented
6#11721745C211yesmaximal, characteristic, complemented
7#11721745C211yesmaximal, characteristic, complemented
8#11721729C211yesmaximal, characteristic
9#11721729C211yesmaximal, characteristic
10#11721727C211yesmaximal, characteristic
11#11721725C211yesmaximal, characteristic, complemented
12#11721724C211yesmaximal, characteristic, complemented
13#11721723C211yesmaximal, characteristic, complemented
14#11721709C211yesmaximal, characteristic, complemented
15C4×#6226C211yesmaximal, characteristic, complemented
16#10596179—31nomaximal
17#10589810—33nomaximal
18C2×#6233C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C2×C4×C3⋊D9C2211yescharacteristic
21C2×C4×C3⋊D9C2211yescharacteristic
22C2×C4×C3⋊D9C2211yescharacteristic
23C2×#6226C2211yescharacteristic
24C3×C9×C4⋊C4C2211yescharacteristic
25C3×C9×C4⋊C4C2211yescharacteristic
26C3×C9×C4⋊C4C2211yescharacteristic
27C3×C9×C22⋊C4C2211yescharacteristic
28C3×C9×C22⋊C4C2211yescharacteristic
29C3×C9×C22⋊C4C2211yescharacteristic
30C3×C42×C9C2211yescharacteristic
31#88551C2211yescharacteristic
32#88551C2211yescharacteristic
33#88536C2211yescharacteristic
34#88536C2211yescharacteristic
35#88536C2211yescharacteristic
36C4⋊(C3⋊Dic9)C2211yescharacteristic
37C4⋊(C3⋊Dic9)C2211yescharacteristic
38C4⋊(C3⋊Dic9)C2211yescharacteristic
39C4⋊(C3⋊Dic9)C2211yescharacteristic
40#88534C2211yescharacteristic
41#88534C2211yescharacteristic
42#88534C2211yescharacteristic
43#88534C2211yescharacteristic
44#88534C2211yescharacteristic
45#88534C2211yescharacteristic
46#88534C2211yescharacteristic
47#88534C2211yescharacteristic
48#88534C2211yescharacteristic
49C4×C3⋊Dic9C2211yescharacteristic
50C4×C3⋊Dic9C2211yescharacteristic
51C4×C3⋊Dic9C2211yescharacteristic
52C4×C3⋊Dic9C2211yescharacteristic
53C32×#126D311yescharacteristic
54#64105—31no—
55#64101—31no—
56C4⋊C4×C3⋊D3—31no—
57#64099—31no—
58#64098—31no—
59#64098—31no—
60#64092—31no—
61#64092—31no—
62#64090—31no—
63#64088—31no—
64#64087—31no—
65#64086—31no—
66#64081—31no—
67C4×#634—31no—
68C9×#126D313yescomplemented
69#63458—33no—
70#63454—33no—
71C4⋊C4×D9—33no—
72#63452—33no—
73#63451—33no—
74#63451—33no—
75#63445—33no—
76#63445—33no—
77#611⋊C4—33no—
78#63441—33no—
79#63440—33no—
80#63439—33no—
81#63434—33no—
82C4×Dic18—33no—
83C23×C3×C9C2311yescharacteristic
84C22×C3⋊D9C2311yescharacteristic
85C2×C3×C4×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89C2×C3×C4×C9C2311yescharacteristic
90C2×C3×C4×C9C2311yescharacteristic
91#6233—22no—
92C2×C3⋊Dic9—22no—
93C2×C3⋊Dic9C2311yescharacteristic
94C2×C3⋊Dic9C2311yescharacteristic
95C2×C3⋊Dic9C2311yescharacteristic
96C2×C3⋊Dic9C2311yescharacteristic
97C2×C3⋊Dic9C2311yescharacteristic
98C2×C3⋊Dic9C2311yescharacteristic
99C2×C3⋊Dic9C2311yescharacteristic
100C4×C3⋊D9—22no—
101C4×C3⋊D9—22no—
102C4×C3⋊D9—22no—
103#6226—22no—
104#5708—93no—
105#5708—91no—
106C2×#638—31no—
107C22×C3⋊Dic3—31no—
108C2×C4×C3⋊D3—31no—
109C2×C4×C3⋊D3—31no—
110C2×C4×C3⋊D3—31no—
111C2×#634—31no—
112C32×C4⋊C4C2×D311yescharacteristic
113C32×C4⋊C4C2×D311yescharacteristic
114C32×C4⋊C4C2×D311yescharacteristic
115C32×C22⋊C4C2×D311yescharacteristic
116C32×C22⋊C4C2×D311yescharacteristic
117C32×C22⋊C4C2×D311yescharacteristic
118C32×C42C2×D311yescharacteristic
119#3757—31no—
120#3757—31no—
121#3752—31no—
122#3752—31no—
123#3752—31no—
124C4⋊(C3⋊Dic3)—31no—
125C4⋊(C3⋊Dic3)—31no—
126C4⋊(C3⋊Dic3)—31no—
127C4⋊(C3⋊Dic3)—31no—
128#3750—31no—
129#3750—31no—
130#3750—31no—
131#3750—31no—
132#3750—31no—
133#3750—31no—
134#3750—31no—
135#3750—31no—
136#3750—31no—
137C4×C3⋊Dic3—31no—
138C4×C3⋊Dic3—31no—
139C4×C3⋊Dic3—31no—
140C4×C3⋊Dic3—31no—
141C2×#611—33no—
142C22×Dic9—33no—
143C2×C4×D9—33no—
144C2×C4×D9—33no—
145C2×C4×D9—33no—
146C2×Dic18—33no—
147C9×C4⋊C4C2×D313yes—
148C9×C4⋊C4C2×D313yes—
149C9×C4⋊C4C2×D313yes—
150C9×C22⋊C4C2×D313yes—
151C9×C22⋊C4C2×D313yes—
152C9×C22⋊C4C2×D313yes—
153C42×C9C2×D313yes—
154#3676—33no—
155#3676—33no—
156(C2×D9)⋊C4—33no—
157(C2×D9)⋊C4—33no—
158(C2×D9)⋊C4—33no—
159C4⋊Dic9—33no—
160C4⋊Dic9—33no—
161C4⋊Dic9—33no—
162C4⋊Dic9—33no—
163Dic9⋊C4—33no—
164Dic9⋊C4—33no—
165Dic9⋊C4—33no—
166Dic9⋊C4—33no—
167Dic9⋊C4—33no—
168Dic9⋊C4—33no—
169Dic9⋊C4—33no—
170Dic9⋊C4—33no—
171Dic9⋊C4—33no—
172C4×Dic9—33no—
173C4×Dic9—33no—
174C4×Dic9—33no—
175C4×Dic9—33no—
176C22×C3×C9—21no—
177C22×C3×C9—21no—
178C22×C3×C9—21no—
179C22×C3×C9C2411yescharacteristic
180C2×C3⋊D9—21no—
181C2×C3⋊D9—21no—
182C2×C3⋊D9#4112yes—
183C3×C4×C9—21no—
184C3×C4×C9—21no—
185C3×C4×C9—21no—
186C3×C4×C9—21no—
187C3×C4×C9—21no—
188C3×C4×C9—21no—
189C3⋊Dic9—21no—
190C3⋊Dic9—21no—
191C3⋊Dic9—21no—
192C3⋊Dic9—21no—
193C3⋊Dic9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9#4112yes—
196C3×#126D913yescomplemented
197C3×#126C3⋊D311yescharacteristic
198#896—93no—
199#896—91no—
200#892—93no—
201#892—91no—
202D3×C4⋊C4—93no—
203D3×C4⋊C4—91no—
204#890—93no—
205#890—91no—
206#889—93no—
207#889—93no—
208#889—91no—
209#889—91no—
210#883—93no—
211#883—93no—
212#883—91no—
213#883—91no—
214#67⋊C4—93no—
215#67⋊C4—91no—
216#879—93no—
217#879—91no—
218#878—93no—
219#878—91no—
220#877—93no—
221#877—91no—
222#872—93no—
223#872—91no—
224C4×Dic6—93no—
225C4×Dic6—91no—
226C23×C32C22×D311yescharacteristic
227C22×C3⋊D3—31no—
228C2×C32×C4C22×D311yescharacteristic
229C2×C32×C4C22×D311yescharacteristic
230C2×C32×C4C22×D311yescharacteristic
231C2×C32×C4C22×D311yescharacteristic
232C2×C32×C4C22×D311yescharacteristic
233C2×C32×C4C22×D311yescharacteristic
234#638—62no—
235C2×C3⋊Dic3—31no—
236C2×C3⋊Dic3—31no—
237C2×C3⋊Dic3—31no—
238C2×C3⋊Dic3—31no—
239C2×C3⋊Dic3—31no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—62no—
243C4×C3⋊D3—62no—
244C4×C3⋊D3—62no—
245C4×C3⋊D3—62no—
246#634—62no—
247C23×C9C22×D313yes—
248C22×D9—33no—
249C2×C4×C9C22×D313yes—
250C2×C4×C9C22×D313yes—
251C2×C4×C9C22×D313yes—
252C2×C4×C9C22×D313yes—
253C2×C4×C9C22×D313yes—
254C2×C4×C9C22×D313yes—
255#611—66no—
256C2×Dic9—33no—
257C2×Dic9—33no—
258C2×Dic9—33no—
259C2×Dic9—33no—
260C2×Dic9—33no—
261C2×Dic9—33no—
262C2×Dic9—33no—
263C2×Dic9—66no—
264C4×D9—66no—
265C4×D9—66no—
266C4×D9—66no—
267Dic18—66no—
268#552—271noSylow
269C2×C3×C9—41no—
270C2×C3×C9C2×#4111yescharacteristic
271C2×C3×C9C2×#4111yescharacteristic
272C2×C3×C92−1+411yescharacteristic
273C3⋊D9—22no—
274C2×#67—93no—
275C2×#67—91no—
276C22×Dic3—93no—
277C22×Dic3—91no—
278C2×C4×D3—93no—
279C2×C4×D3—93no—
280C2×C4×D3—93no—
281C2×C4×D3—91no—
282C2×C4×D3—91no—
283C2×C4×D3—91no—
284C2×Dic6—93no—
285C2×Dic6—91no—
286C3×C4⋊C4C2×D913yes—
287C3×C4⋊C4C2×D913yes—
288C3×C4⋊C4C2×D913yes—
289C3×C4⋊C4C2×C3⋊D311yescharacteristic
290C3×C4⋊C4C2×C3⋊D311yescharacteristic
291C3×C4⋊C4C2×C3⋊D311yescharacteristic
292C3×C22⋊C4C2×D913yes—
293C3×C22⋊C4C2×D913yes—
294C3×C22⋊C4C2×D913yes—
295C3×C22⋊C4C2×C3⋊D311yescharacteristic
296C3×C22⋊C4C2×C3⋊D311yescharacteristic
297C3×C22⋊C4C2×C3⋊D311yescharacteristic
298C3×C42C2×D913yes—
299C3×C42C2×C3⋊D311yescharacteristic
300#217—93no—
301#217—93no—
302#217—91no—
303#217—91no—
304(C2×D3)⋊C4—93no—
305(C2×D3)⋊C4—93no—
306(C2×D3)⋊C4—93no—
307(C2×D3)⋊C4—91no—
308(C2×D3)⋊C4—91no—
309(C2×D3)⋊C4—91no—
310C4⋊Dic3—93no—
311C4⋊Dic3—93no—
312C4⋊Dic3—93no—
313C4⋊Dic3—93no—
314C4⋊Dic3—91no—
315C4⋊Dic3—91no—
316C4⋊Dic3—91no—
317C4⋊Dic3—91no—
318Dic3⋊C4—93no—
319Dic3⋊C4—93no—
320Dic3⋊C4—93no—
321Dic3⋊C4—93no—
322Dic3⋊C4—93no—
323Dic3⋊C4—93no—
324Dic3⋊C4—93no—
325Dic3⋊C4—93no—
326Dic3⋊C4—93no—
327Dic3⋊C4—91no—
328Dic3⋊C4—91no—
329Dic3⋊C4—91no—
330Dic3⋊C4—91no—
331Dic3⋊C4—91no—
332Dic3⋊C4—91no—
333Dic3⋊C4—91no—
334Dic3⋊C4—91no—
335Dic3⋊C4—91no—
336C4×Dic3—93no—
337C4×Dic3—93no—
338C4×Dic3—93no—
339C4×Dic3—93no—
340C4×Dic3—91no—
341C4×Dic3—91no—
342C4×Dic3—91no—
343C4×Dic3—91no—
344C22×C32—21no—
345C22×C32—21no—
346C22×C32—21no—
347C22×C32C23×D311yescharacteristic
348C2×C3⋊D3—32no—
349C2×C3⋊D3—61no—
350C2×C3⋊D3—61no—
351C32×C4—21no—
352C32×C4—21no—
353C32×C4—21no—
354C32×C4—21no—
355C32×C4—21no—
356C32×C4—21no—
357C3⋊Dic3—32no—
358C3⋊Dic3—61no—
359C3⋊Dic3—61no—
360C3⋊Dic3—61no—
361C3⋊Dic3—61no—
362C3⋊Dic3—61no—
363C3⋊Dic3—61no—
364C22×C9—23no—
365C22×C9—23no—
366C22×C9—23no—
367C22×C9C23×D313yes—
368C2×D9—36no—
369C2×D9—63no—
370C2×D9—63no—
371C4×C9—23no—
372C4×C9—23no—
373C4×C9—23no—
374C4×C9—23no—
375C4×C9—23no—
376C4×C9—23no—
377Dic9—36no—
378Dic9—63no—
379Dic9—63no—
380Dic9—63no—
381Dic9—63no—
382Dic9—63no—
383Dic9—63no—
384#126—271no—
385#126C3⋊D911yescharacteristic, complemented
386#125—271no—
387#125—271no—
388#125—271no—
389#123—271no—
390#123—271no—
391C22⋊Q8—271no—
392C22⋊Q8—271no—
393C4×Q8—271no—
394C4×D4—271no—
395#117—271no—
396#117—271no—
397#117—271no—
398C2×C4⋊C4—271no—
399C3×C9#55211yescharacteristic, complemented, Sylow
400C23×C3C22×D913yes—
401C23×C3C22×C3⋊D311yescharacteristic
402C22×D3—91no—
403C22×D3—93no—
404C2×C3×C4C22×D913yes—
405C2×C3×C4C22×D913yes—
406C2×C3×C4C22×D913yes—
407C2×C3×C4C22×D913yes—
408C2×C3×C4C22×D913yes—
409C2×C3×C4C22×D913yes—
410C2×C3×C4C22×C3⋊D311yescharacteristic
411C2×C3×C4C22×C3⋊D311yescharacteristic
412C2×C3×C4C22×C3⋊D311yescharacteristic
413C2×C3×C4C22×C3⋊D311yescharacteristic
414C2×C3×C4C22×C3⋊D311yescharacteristic
415C2×C3×C4C22×C3⋊D311yescharacteristic
416#67—182no—
417#67—186no—
418C2×Dic3—91no—
419C2×Dic3—93no—
420C2×Dic3—91no—
421C2×Dic3—91no—
422C2×Dic3—91no—
423C2×Dic3—91no—
424C2×Dic3—91no—
425C2×Dic3—91no—
426C2×Dic3—93no—
427C2×Dic3—93no—
428C2×Dic3—93no—
429C2×Dic3—93no—
430C2×Dic3—93no—
431C2×Dic3—93no—
432C2×Dic3—182no—
433C2×Dic3—186no—
434C4×D3—182no—
435C4×D3—186no—
436C4×D3—182no—
437C4×D3—182no—
438C4×D3—186no—
439C4×D3—186no—
440Dic6—182no—
441Dic6—186no—
442C2×C32—41no—
443C2×C32D3×#4111yescharacteristic
444C2×C32D3×#4111yescharacteristic
445C2×C32#101011yescharacteristic
446C3⋊D3—62no—
447C2×C9—43no—
448C2×C9D3×#4113yes—
449C2×C9D3×#4113yes—
450C2×C9#101013yes—
451D9—66no—
452C2×Q8—271no—
453C2×D4—271no—
454C22×C4—271no—
455C22×C4—271no—
456C22×C4—271no—
457C22×C4—271no—
458C4⋊C4—271no—
459C4⋊C4—271no—
460C4⋊C4—271no—
461C4⋊C4—271no—
462C4⋊C4—271no—
463C4⋊C4—271no—
464C4⋊C4—271no—
465C4⋊C4—271no—
466C4⋊C4—271no—
467C4⋊C4—271no—
468C4⋊C4—271no—
469C4⋊C4—271no—
470C4⋊C4—271no—
471C4⋊C4C2×C3⋊D911yescharacteristic
472C4⋊C4C2×C3⋊D911yescharacteristic
473C4⋊C4C2×C3⋊D911yescharacteristic
474C22⋊C4—271no—
475C22⋊C4—271no—
476C22⋊C4—271no—
477C22⋊C4—271no—
478C22⋊C4—271no—
479C22⋊C4C2×C3⋊D911yescharacteristic
480C22⋊C4C2×C3⋊D911yescharacteristic
481C22⋊C4C2×C3⋊D911yescharacteristic
482C42—271no—
483C42—271no—
484C42—271no—
485C42—271no—
486C42C2×C3⋊D911yescharacteristic
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—23no—
491C22×C3—23no—
492C22×C3—23no—
493C22×C3C23×D913yes—
494C22×C3C23×C3⋊D311yescharacteristic
495C2×D3—92no—
496C2×D3—96no—
497C2×D3—181no—
498C2×D3—181no—
499C2×D3—183no—
500C2×D3—183no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—23no—
508C3×C4—23no—
509C3×C4—23no—
510C3×C4—23no—
511C3×C4—23no—
512C3×C4—23no—
513Dic3—92no—
514Dic3—96no—
515Dic3—181no—
516Dic3—181no—
517Dic3—181no—
518Dic3—181no—
519Dic3—181no—
520Dic3—181no—
521Dic3—183no—
522Dic3—183no—
523Dic3—183no—
524Dic3—183no—
525Dic3—183no—
526Dic3—183no—
527C32#570811yescharacteristic
528C9#570813yescomplemented
529C23—271no—
530C23C22×C3⋊D911yescharacteristic
531Q8—542no—
532D4—542no—
533C2×C4—271no—
534C2×C4—271no—
535C2×C4—271no—
536C2×C4—271no—
537C2×C4—271no—
538C2×C4—271no—
539C2×C4—271no—
540C2×C4—542no—
541C2×C4—542no—
542C2×C4—542no—
543C2×C4—542no—
544C2×C4C22×C3⋊D911yescharacteristic
545C2×C4C22×C3⋊D911yescharacteristic
546C2×C4C22×C3⋊D911yescharacteristic
547C2×C4C22×C3⋊D911yescharacteristic
548C2×C4C22×C3⋊D911yescharacteristic
549C2×C4C22×C3⋊D911yescharacteristic
550C2×C3—41no—
551C2×C3—43no—
552C2×C3#41×D913yes—
553C2×C3#41×D913yes—
554C2×C3#6371613yes—
555C2×C3#41×C3⋊D311yescharacteristic
556C2×C3#41×C3⋊D311yescharacteristic
557C2×C3#6436711yescharacteristic
558D3—182no—
559D3—186no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—272no—
564C22—541no—
565C22—541no—
566C22C23×C3⋊D911yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—272no—
574C4—541no—
575C4—541no—
576C4—541no—
577C4—541no—
578C4—541no—
579C4—541no—
580C3#1058981013yescomplemented
581C3#1059617911yescharacteristic
582C2—41no—
583C2—542no—
584C2#41×C3⋊D911yescharacteristic
585C2#41×C3⋊D911yescharacteristic
586C2#1172267311yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#126C2
socleC22×C32C23×D3
prime radicalC2×#6233C22
hypercenter#126C3⋊D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C3×C9×#126C2semidirect
2. #11721752C2semidirect
3. #11721748C2semidirect
4. C4⋊C4×C3⋊D9C2semidirect
5. #11721746C2semidirect
6. #11721745C2semidirect
7. #11721745C2semidirect
11. #11721725C2semidirect
12. #11721724C2semidirect
13. #11721723C2semidirect
14. #11721709C2semidirect
15. C4×#6226C2semidirect
68. C9×#126D3semidirect
196. C3×#126D9semidirect
385. #126C3⋊D9semidirect
399. C3×C9#552semidirect
528. C9#5708semidirect
580. C3#10589810semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1262
#117217522
#117217482
C4⋊C4×C3⋊D92
#117217462
#117217452
#117217292
#117217272
#117217252
#117217242
#117217232
#117217092
C4×#62262
#105961793
#105898103