Finite Group Databasebeta

← #13154246

Subgroups — #13154246

Order 1728.

980 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#11721754C211yesmaximal, characteristic, complemented
3#11721753C211yesmaximal, characteristic, complemented
4#11721752C211yesmaximal, characteristic, complemented
5#11721746C211yesmaximal, characteristic, complemented
6#11721745C211yesmaximal, characteristic, complemented
7#11721744C211yesmaximal, characteristic, complemented
8#11721732C211yesmaximal, characteristic
9#11721731C211yesmaximal, characteristic
10#11721729C211yesmaximal, characteristic
11#11721725C211yesmaximal, characteristic, complemented
12#11721724C211yesmaximal, characteristic, complemented
13#11721724C211yesmaximal, characteristic, complemented
14#11721713C211yesmaximal, characteristic, complemented
15#11721706C211yesmaximal, characteristic, complemented
16#10596183—31nomaximal
17#10589814—33nomaximal
18C2×#6233C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C2×C4×C3⋊D9C2211yescharacteristic
21C2×#6226C2211yescharacteristic
22C2×#6226C2211yescharacteristic
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
36#88536C2211yescharacteristic
37#88536C2211yescharacteristic
38C4⋊(C3⋊Dic9)C2211yescharacteristic
39C4⋊(C3⋊Dic9)C2211yescharacteristic
40C4⋊(C3⋊Dic9)C2211yescharacteristic
41C4⋊(C3⋊Dic9)C2211yescharacteristic
42C4⋊(C3⋊Dic9)C2211yescharacteristic
43C4⋊(C3⋊Dic9)C2211yescharacteristic
44#88534C2211yescharacteristic
45#88534C2211yescharacteristic
46#88534C2211yescharacteristic
47#88534C2211yescharacteristic
48#88534C2211yescharacteristic
49#88534C2211yescharacteristic
50#88534C2211yescharacteristic
51C4×C3⋊Dic9C2211yescharacteristic
52C4×C3⋊Dic9C2211yescharacteristic
53C32×#126D311yescharacteristic
54#64107—31no—
55#64106—31no—
56#64105—31no—
57#64099—31no—
58#64098—31no—
59#64097—31no—
60#64095—31no—
61#64094—31no—
62#64092—31no—
63#64088—31no—
64#64087—31no—
65#64087—31no—
66#64085—31no—
67#64078—31no—
68C9×#126D313yescomplemented
69#63460—33no—
70#63459—33no—
71#63458—33no—
72#63452—33no—
73#63451—33no—
74#63450—33no—
75#63448—33no—
76#63447—33no—
77#63445—33no—
78#63441—33no—
79#63440—33no—
80#63440—33no—
81#63438—33no—
82#63431—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—41no—
92C2×C3⋊Dic9—41no—
93C2×C3⋊Dic9C2311yescharacteristic
94C2×C3⋊Dic9C2311yescharacteristic
95C2×C3⋊Dic9C2311yescharacteristic
96C2×C3⋊Dic9C2311yescharacteristic
97C2×C3⋊Dic9C2311yescharacteristic
98C2×C3⋊Dic9C2311yescharacteristic
99C2×C3⋊Dic9C2311yescharacteristic
100C4×C3⋊D9—41no—
101#6226—41no—
102#6226—41no—
103#6226—41no—
104#5712—93no—
105#5712—91no—
106C2×#638—31no—
107C22×C3⋊Dic3—31no—
108C2×C4×C3⋊D3—31no—
109C2×#634—31no—
110C2×#634—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—
124#3752—31no—
125#3752—31no—
126C4⋊(C3⋊Dic3)—31no—
127C4⋊(C3⋊Dic3)—31no—
128C4⋊(C3⋊Dic3)—31no—
129C4⋊(C3⋊Dic3)—31no—
130C4⋊(C3⋊Dic3)—31no—
131C4⋊(C3⋊Dic3)—31no—
132#3750—31no—
133#3750—31no—
134#3750—31no—
135#3750—31no—
136#3750—31no—
137#3750—31no—
138#3750—31no—
139C4×C3⋊Dic3—31no—
140C4×C3⋊Dic3—31no—
141C2×#611—33no—
142C22×Dic9—33no—
143C2×C4×D9—33no—
144C2×Dic18—33no—
145C2×Dic18—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—
159(C2×D9)⋊C4—33no—
160(C2×D9)⋊C4—33no—
161C4⋊Dic9—33no—
162C4⋊Dic9—33no—
163C4⋊Dic9—33no—
164C4⋊Dic9—33no—
165C4⋊Dic9—33no—
166C4⋊Dic9—33no—
167Dic9⋊C4—33no—
168Dic9⋊C4—33no—
169Dic9⋊C4—33no—
170Dic9⋊C4—33no—
171Dic9⋊C4—33no—
172Dic9⋊C4—33no—
173Dic9⋊C4—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—21no—
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—21no—
196C3×#126D913yescomplemented
197C3×#126C3⋊D311yescharacteristic
198#898—93no—
199#898—91no—
200#897—93no—
201#897—91no—
202#896—93no—
203#896—91no—
204#890—93no—
205#890—91no—
206#889—93no—
207#889—91no—
208#888—93no—
209#888—91no—
210#886—93no—
211#886—91no—
212#885—93no—
213#885—91no—
214#883—93no—
215#883—91no—
216#879—93no—
217#879—91no—
218#878—93no—
219#878—93no—
220#878—91no—
221#878—91no—
222#876—93no—
223#876—91no—
224#869—93no—
225#869—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—121no—
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—121no—
243C4×C3⋊D3—121no—
244#634—121no—
245#634—121no—
246#634—121no—
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—123no—
256C2×Dic9—33no—
257C2×Dic9—33no—
258C2×Dic9—33no—
259C2×Dic9—33no—
260C2×Dic9—33no—
261C2×Dic9—33no—
262C2×Dic9—33no—
263C2×Dic9—123no—
264C4×D9—123no—
265Dic18—123no—
266Dic18—123no—
267Dic18—123no—
268#563—271noSylow
269C2×C3×C9—41no—
270C2×C3×C9H(5,𝔽2)11yescharacteristic
271C2×C3×C92−1+411yescharacteristic
272C2×C3×C92−1+411yescharacteristic
273C3⋊D9—41no—
274C2×#67—93no—
275C2×#67—91no—
276C22×Dic3—93no—
277C22×Dic3—91no—
278C2×C4×D3—93no—
279C2×C4×D3—91no—
280C2×Dic6—93no—
281C2×Dic6—93no—
282C2×Dic6—93no—
283C2×Dic6—91no—
284C2×Dic6—91no—
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—93no—
308(C2×D3)⋊C4—93no—
309(C2×D3)⋊C4—91no—
310(C2×D3)⋊C4—91no—
311(C2×D3)⋊C4—91no—
312(C2×D3)⋊C4—91no—
313(C2×D3)⋊C4—91no—
314C4⋊Dic3—93no—
315C4⋊Dic3—93no—
316C4⋊Dic3—93no—
317C4⋊Dic3—93no—
318C4⋊Dic3—93no—
319C4⋊Dic3—93no—
320C4⋊Dic3—91no—
321C4⋊Dic3—91no—
322C4⋊Dic3—91no—
323C4⋊Dic3—91no—
324C4⋊Dic3—91no—
325C4⋊Dic3—91no—
326Dic3⋊C4—93no—
327Dic3⋊C4—93no—
328Dic3⋊C4—93no—
329Dic3⋊C4—93no—
330Dic3⋊C4—93no—
331Dic3⋊C4—93no—
332Dic3⋊C4—93no—
333Dic3⋊C4—91no—
334Dic3⋊C4—91no—
335Dic3⋊C4—91no—
336Dic3⋊C4—91no—
337Dic3⋊C4—91no—
338Dic3⋊C4—91no—
339Dic3⋊C4—91no—
340C4×Dic3—93no—
341C4×Dic3—93no—
342C4×Dic3—91no—
343C4×Dic3—91no—
344C22×C32—21no—
345C22×C32—21no—
346C22×C32—21no—
347C22×C32C23×D311yescharacteristic
348C2×C3⋊D3—61no—
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—61no—
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—63no—
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—63no—
378Dic9—63no—
379Dic9—63no—
380Dic9—63no—
381Dic9—63no—
382Dic9—63no—
383Dic9—63no—
384C4⋊Q8—271no—
385C4⋊Q8—271no—
386#126—271no—
387#126—271no—
388#126—271no—
389#126C3⋊D911yescharacteristic, complemented
390#125—271no—
391#125—271no—
392#124—271no—
393#123—271no—
394#123—271no—
395C22⋊Q8—271no—
396C22⋊Q8—271no—
397C22⋊Q8—271no—
398C22⋊Q8—271no—
399C3×C9#56311yescharacteristic, 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—361no—
417#67—363no—
418C2×Dic3—91no—
419C2×Dic3—91no—
420C2×Dic3—91no—
421C2×Dic3—91no—
422C2×Dic3—91no—
423C2×Dic3—91no—
424C2×Dic3—91no—
425C2×Dic3—93no—
426C2×Dic3—93no—
427C2×Dic3—93no—
428C2×Dic3—93no—
429C2×Dic3—93no—
430C2×Dic3—93no—
431C2×Dic3—93no—
432C2×Dic3—361no—
433C2×Dic3—363no—
434C4×D3—361no—
435C4×D3—363no—
436Dic6—361no—
437Dic6—361no—
438Dic6—361no—
439Dic6—363no—
440Dic6—363no—
441Dic6—363no—
442C2×C32—41no—
443C2×C32#100911yescharacteristic
444C2×C32#101011yescharacteristic
445C2×C32#101011yescharacteristic
446C3⋊D3—121no—
447C2×C9—43no—
448C2×C9#100913yes—
449C2×C9#101013yes—
450C2×C9#101013yes—
451D9—123no—
452C2×Q8—271no—
453C2×Q8—271no—
454C2×Q8—271no—
455C2×D4—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⋊C4—271no—
480C22⋊C4—271no—
481C22⋊C4C2×C3⋊D911yescharacteristic
482C22⋊C4C2×C3⋊D911yescharacteristic
483C22⋊C4C2×C3⋊D911yescharacteristic
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—181no—
496C2×D3—181no—
497C2×D3—181no—
498C2×D3—183no—
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—181no—
514Dic3—181no—
515Dic3—181no—
516Dic3—181no—
517Dic3—181no—
518Dic3—181no—
519Dic3—181no—
520Dic3—183no—
521Dic3—183no—
522Dic3—183no—
523Dic3—183no—
524Dic3—183no—
525Dic3—183no—
526Dic3—183no—
527C32#571211yescharacteristic
528C9#571213yescomplemented
529C23—271no—
530C23C22×C3⋊D911yescharacteristic
531Q8—1081no—
532Q8—1081no—
533Q8—1081no—
534D4—1081no—
535C2×C4—271no—
536C2×C4—271no—
537C2×C4—271no—
538C2×C4—271no—
539C2×C4—271no—
540C2×C4—271no—
541C2×C4—271no—
542C2×C4—1081no—
543C2×C4—1081no—
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#6371513yes—
553C2×C3#6371613yes—
554C2×C3#6371613yes—
555C2×C3#6436611yescharacteristic
556C2×C3#6436711yescharacteristic
557C2×C3#6436711yescharacteristic
558D3—361no—
559D3—363no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—541no—
564C22—541no—
565C22—541no—
566C22C23×C3⋊D911yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—541no—
574C4—541no—
575C4—541no—
576C4—541no—
577C4—541no—
578C4—541no—
579C4—541no—
580C3#1058981413yescomplemented
581C3#1059618311yescharacteristic
582C2—41no—
583C2—1081no—
584C2#1172267211yescharacteristic
585C2#1172267311yescharacteristic
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#563
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. #11721754C2semidirect
3. #11721753C2semidirect
4. #11721752C2semidirect
5. #11721746C2semidirect
6. #11721745C2semidirect
7. #11721744C2semidirect
11. #11721725C2semidirect
12. #11721724C2semidirect
13. #11721724C2semidirect
14. #11721713C2semidirect
15. #11721706C2semidirect
68. C9×#126D3semidirect
196. C3×#126D9semidirect
389. #126C3⋊D9semidirect
399. C3×C9#563semidirect
528. C9#5712semidirect
580. C3#10589814semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1262
#117217542
#117217532
#117217522
#117217462
#117217452
#117217442
#117217322
#117217312
#117217292
#117217252
#117217242
#117217132
#117217062
#105961833
#105898143