Finite Group Databasebeta

← #13148953

Subgroups — #13148953

Order 1728.

644 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721745C211yesmaximal, characteristic, complemented
2C3×#63456C211yesmaximal, characteristic
3C9×#888C211yesmaximal, characteristic, complemented
4#11720999C211yesmaximal, characteristic, complemented
5#11720984C211yesmaximal, characteristic, complemented
6#11720981C211yesmaximal, characteristic
7#11720951C211yesmaximal, characteristic
8#11720949C211yesmaximal, characteristic, complemented
9#11720947C211yesmaximal, characteristic, complemented
10#11720947C211yesmaximal, characteristic, complemented
11#11720946C211yesmaximal, characteristic, complemented
12#11720944C211yesmaximal, characteristic, complemented
13#11720939C211yesmaximal, characteristic, complemented
14#11720928C211yesmaximal, characteristic, complemented
15#11720903C211yesmaximal, characteristic, complemented
16#10594495—31nomaximal
17#10589837—31nomaximal
18C2×C3×D36C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×C9×Dic6C2211yescharacteristic
21C2×C9×Dic6C2211yescharacteristic
22C2×Dic3×D9C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C9×C4⋊C4C2211yescharacteristic
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26#88534C2211yescharacteristic
27#88534C2211yescharacteristic
28#88534C2211yescharacteristic
29#88534C2211yescharacteristic
30C4×C3⋊Dic9C2211yescharacteristic
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C3×(C2×D9)⋊C4C2211yescharacteristic
34C9×C4⋊Dic3C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C9×Dic3⋊C4C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39(C2×D9)⋊Dic3C2211yescharacteristic
40(C2×D9)⋊Dic3C2211yescharacteristic
41(C2×D9)⋊Dic3C2211yescharacteristic
42(C2×D9)⋊Dic3C2211yescharacteristic
43(C2×D9)⋊Dic3C2211yescharacteristic
44(C2×D9)⋊Dic3C2211yescharacteristic
45(C2×D9)⋊Dic3C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49#88442C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic9⋊Dic3C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64098—31no—
54C3×#894—31no—
55C3×#888D311yescharacteristic
56#63899—31no—
57#63887—31no—
58#63884—31no—
59#63870—31no—
60#63863—31no—
61#63861—31no—
62#63859—31no—
63#63859—31no—
64#63858—31no—
65#63856—31no—
66#63851—31no—
67#63839—31no—
68C9×C4⋊Q8—31no—
69#63509—31no—
70#63508—31no—
71#63508—31no—
72#63506—31no—
73#63460—31no—
74#63460—31no—
75#63458—31no—
76#63458—31no—
77#63456—31no—
78#63456D311yescharacteristic
79#63451—31no—
80#63451—31no—
81#63451—61no—
82#63438—31no—
83#63438—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
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C3×D36—41no—
100C3×C4×D9—41no—
101C9×Dic6—41no—
102C9×Dic6—41no—
103Dic3×D9—41no—
104#6189—41no—
105#5735—91no—
106#5651—91no—
107C2×C3×D12—31no—
108C2×C3×C4×D3—31no—
109C2×C3×Dic6C2×D311yescharacteristic
110C2×C3×Dic6C2×D311yescharacteristic
111C2×#627—31no—
112C2×D3×Dic3—31no—
113C32×C4⋊C4C2×D311yescharacteristic
114C4⋊(C3⋊Dic3)—31no—
115#3750—31no—
116#3750—31no—
117#3750—31no—
118#3750—31no—
119C4×C3⋊Dic3—31no—
120C3×(C2×D3)⋊C4—31no—
121C3×(C2×D3)⋊C4—31no—
122C3×(C2×D3)⋊C4—31no—
123C3×C4⋊Dic3C2×D311yescharacteristic
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×C4×Dic3C2×D311yescharacteristic
128#3724—31no—
129#3724—31no—
130#3724—31no—
131Dic3⋊Dic3—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134(C2×D3)⋊Dic3—31no—
135(C2×D3)⋊Dic3—31no—
136(C2×D3)⋊Dic3—31no—
137(C2×D3)⋊Dic3—31no—
138(C2×D3)⋊Dic3—31no—
139(C2×D3)⋊Dic3—31no—
140(C2×D3)⋊Dic3—31no—
141Dic32—31no—
142C2×Q8×C9—31no—
143C2×Q8×C9—31no—
144C2×D36C2×D311yescharacteristic
145C2×C4×D9—31no—
146C2×C4×D9C2×D311yescharacteristic
147C2×Dic18—31no—
148C9×C4⋊C4—21no—
149C9×C4⋊C4—31no—
150C9×C4⋊C4—31no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4C2×D311yescharacteristic
153C42×C9—31no—
154(C2×D9)⋊C4—31no—
155(C2×D9)⋊C4—31no—
156(C2×D9)⋊C4—31no—
157(C2×D9)⋊C4—31no—
158(C2×D9)⋊C4—31no—
159(C2×D9)⋊C4—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4C2×D311yescharacteristic
162(C2×D9)⋊C4C2×D311yescharacteristic
163(C2×D9)⋊C4C2×D311yescharacteristic
164C4⋊Dic9—31no—
165C4⋊Dic9—31no—
166C4⋊Dic9—61no—
167Dic9⋊C4—31no—
168Dic9⋊C4—31no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—61no—
177Dic9⋊C4—61no—
178Dic9⋊C4—61no—
179Dic9⋊C4—61no—
180Dic9⋊C4C2×D311yescharacteristic
181C4×Dic9—31no—
182C4×Dic9—31no—
183C4×Dic9—61no—
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—21no—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3×C4×C9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9—21no—
196C3⋊Dic9—21no—
197C3⋊Dic9—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C9×Dic3—21no—
201C9×Dic3—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×C4⋊Q8—31no—
205C3×#123—91no—
206#947—91no—
207#946—91no—
208#946—91no—
209#944—91no—
210#936—91no—
211#935—91no—
212#924—91no—
213#924—91no—
214#898—91no—
215#898—91no—
216#896—91no—
217#896—91no—
218#894—31no—
219#894—91no—
220#889—91no—
221#889—91no—
222#889—91no—
223#889—91no—
224#889—181no—
225#888—91no—
226#888D911yescharacteristic, complemented
227#879—91no—
228#879—91no—
229#879—91no—
230#879—91no—
231#878—91no—
232#878—91no—
233#876—91no—
234#876—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—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—31no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3C22×D311yescharacteristic
247C2×C3×Dic3C22×D311yescharacteristic
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C3×D12—121no—
251C3×C4×D3—121no—
252C3×Dic6—41no—
253C3×Dic6—41no—
254#627—121no—
255D3×Dic3—121no—
256C22×D9C22×D311yescharacteristic
257C22×D9C22×D311yescharacteristic
258Q8×C9—121no—
259Q8×C9—121no—
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—31no—
273C2×Dic9—31no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—61no—
277C2×Dic9—61no—
278C2×Dic9C22×D311yescharacteristic
279C2×Dic9C22×D311yescharacteristic
280D36—41no—
281C4×D9—41no—
282C4×D9—121no—
283Dic18—121no—
284#563—271noSylow
285C2×C3×C9H(5,𝔽2)11yescharacteristic
286C2×C3×C92−1+411yescharacteristic
287C2×C3×C92−1+411yescharacteristic
288C3×D9—41no—
289C3×D9—41no—
290C2×C3×Q8—31no—
291C2×C3×Q8—31no—
292C2×C3×D4—91no—
293C22×C3×C4—91no—
294C22×Dic3—91no—
295C2×D12—31no—
296C2×C4×D3—31no—
297C2×C4×D3—91no—
298C2×Dic6—91no—
299C2×Dic6—91no—
300C2×Dic6C2×D911yescharacteristic
301C2×Dic6C2×D911yescharacteristic
302C3×C4⋊C4—21no—
303C3×C4⋊C4—31no—
304C3×C4⋊C4—31no—
305C3×C4⋊C4—31no—
306C3×C4⋊C4—91no—
307C3×C4⋊C4C2×D911yescharacteristic
308C3×C4⋊C4D3211yescharacteristic
309C3×C22⋊C4—91no—
310C3×C22⋊C4—91no—
311C3×C22⋊C4—91no—
312C3×C42—31no—
313#217—91no—
314#217—91no—
315#217—91no—
316#217—91no—
317#217—91no—
318#217—91no—
319#217—91no—
320(C2×D3)⋊C4—31no—
321(C2×D3)⋊C4—31no—
322(C2×D3)⋊C4—31no—
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—
330C4⋊Dic3—91no—
331C4⋊Dic3—91no—
332C4⋊Dic3—91no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—181no—
336C4⋊Dic3C2×D911yescharacteristic
337Dic3⋊C4—31no—
338Dic3⋊C4—91no—
339Dic3⋊C4—91no—
340Dic3⋊C4—91no—
341Dic3⋊C4—91no—
342Dic3⋊C4—91no—
343Dic3⋊C4—91no—
344Dic3⋊C4—91no—
345Dic3⋊C4—91no—
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—181no—
356Dic3⋊C4—181no—
357Dic3⋊C4—181no—
358Dic3⋊C4—181no—
359Dic3⋊C4C2×D911yescharacteristic
360Dic3⋊C4C2×D911yescharacteristic
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—181no—
366C4×Dic3C2×D911yescharacteristic
367C22×C32C23×D311yescharacteristic
368C2×C3×D3—61no—
369C2×C3×D3—61no—
370C2×C3×D3—61no—
371C2×C3×D3—61no—
372C2×C3×D3—61no—
373C2×C3×D3—61no—
374C32×C4—21no—
375C32×C4—21no—
376C32×C4—21no—
377C3⋊Dic3—61no—
378C3⋊Dic3—61no—
379C3⋊Dic3—61no—
380C3⋊Dic3—61no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—21no—
385C3×Dic3—61no—
386C3×Dic3—61no—
387C22×C9—21no—
388C22×C9C23×D311yescharacteristic
389C2×D9—21no—
390C2×D9—21no—
391C2×D9—21no—
392C2×D9—21no—
393C2×D9—21no—
394C2×D9—21no—
395C4×C9—21no—
396C4×C9—21no—
397C4×C9—21no—
398C4×C9—41no—
399C4×C9—41no—
400C4×C9—41no—
401C4×C9—61no—
402C4×C9—61no—
403C4×C9—61no—
404C4×C9—61no—
405Dic9—21no—
406Dic9—21no—
407Dic9—62no—
408Dic9—61no—
409Dic9—61no—
410Dic9—61no—
411Dic9—61no—
412Dic9—121no—
413Dic9—121no—
414Dic9—121no—
415C4⋊Q8—31no—
416C4⋊Q8—271no—
417#126—271no—
418#126—271no—
419#126—271no—
420#126—271no—
421#125—271no—
422#125—271no—
423#124—271no—
424#123—91no—
425#123—271no—
426C22⋊Q8—271no—
427C22⋊Q8—271no—
428C22⋊Q8—271no—
429C22⋊Q8—271no—
430C3×C9#56311yescharacteristic, complemented, Sylow
431C23×C3—91no—
432C23×C3—91no—
433C22×D3—31no—
434C22×D3—31no—
435C3×Q8—121no—
436C3×Q8—121no—
437C3×D4—361no—
438C2×C3×C4—21no—
439C2×C3×C4—21no—
440C2×C3×C4—21no—
441C2×C3×C4—31no—
442C2×C3×C4—31no—
443C2×C3×C4—31no—
444C2×C3×C4—31no—
445C2×C3×C4—91no—
446C2×C3×C4—91no—
447C2×C3×C4—361no—
448C2×C3×C4C22×D911yescharacteristic
449C2×C3×C4C22×D911yescharacteristic
450C2×C3×C4C22×D911yescharacteristic
451C2×C3×C4C2×D3211yescharacteristic
452C2×C3×C4C2×D3211yescharacteristic
453C2×C3×C4C2×D3211yescharacteristic
454C2×Dic3—31no—
455C2×Dic3—31no—
456C2×Dic3—91no—
457C2×Dic3—91no—
458C2×Dic3—91no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—181no—
465C2×Dic3—181no—
466C2×Dic3—181no—
467C2×Dic3—181no—
468C2×Dic3—361no—
469C2×Dic3C22×D911yescharacteristic
470C2×Dic3C22×D911yescharacteristic
471C2×Dic3C22×D911yescharacteristic
472C2×Dic3C22×D911yescharacteristic
473D12—121no—
474C4×D3—121no—
475C4×D3—361no—
476Dic6—41no—
477Dic6—41no—
478Dic6—361no—
479Dic6—361no—
480C2×C32#100411yescharacteristic
481C2×C32#100711yescharacteristic
482C2×C32#100711yescharacteristic
483C3×D3—121no—
484C3×D3—121no—
485C2×C9—21no—
486C2×C9—21no—
487C2×C9—21no—
488C2×C9#100411yescharacteristic
489C2×C9#101011yescharacteristic
490C2×C9#101011yescharacteristic
491D9—41no—
492D9—41no—
493C2×Q8—31no—
494C2×Q8—31no—
495C2×Q8—271no—
496C2×D4—91no—
497C22×C4—91no—
498C22×C4—271no—
499C4⋊C4—31no—
500C4⋊C4—31no—
501C4⋊C4—31no—
502C4⋊C4—91no—
503C4⋊C4—271no—
504C4⋊C4—271no—
505C4⋊C4—271no—
506C4⋊C4—271no—
507C4⋊C4—271no—
508C4⋊C4—271no—
509C4⋊C4—271no—
510C4⋊C4—271no—
511C4⋊C4—271no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4D3×D911yescharacteristic
515C22⋊C4—91no—
516C22⋊C4—91no—
517C22⋊C4—91no—
518C22⋊C4—271no—
519C22⋊C4—271no—
520C22⋊C4—271no—
521C22⋊C4—271no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C22⋊C4—271no—
525C42—31no—
526C42—271no—
527C42—271no—
528C22×C3—21no—
529C22×C3—181no—
530C22×C3—181no—
531C22×C3—181no—
532C22×C3—181no—
533C22×C3—181no—
534C22×C3—181no—
535C22×C3C23×D911yescharacteristic
536C22×C3C22×D3211yescharacteristic
537C2×D3—61no—
538C2×D3—61no—
539C2×D3—61no—
540C2×D3—61no—
541C2×D3—61no—
542C2×D3—61no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—61no—
553C3×C4—61no—
554C3×C4—61no—
555C3×C4—61no—
556C3×C4—181no—
557C3×C4—181no—
558Dic3—21no—
559Dic3—21no—
560Dic3—21no—
561Dic3—21no—
562Dic3—61no—
563Dic3—61no—
564Dic3—182no—
565Dic3—181no—
566Dic3—181no—
567Dic3—181no—
568Dic3—181no—
569Dic3—181no—
570Dic3—181no—
571Dic3—181no—
572Dic3—181no—
573Dic3—361no—
574Dic3—361no—
575Dic3—361no—
576C32#573511yescharacteristic
577C9—21no—
578C9#565111yescharacteristic, complemented
579C23—91no—
580C23—91no—
581Q8—121no—
582Q8—121no—
583Q8—1081no—
584D4—361no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—31no—
588C2×C4—31no—
589C2×C4—91no—
590C2×C4—91no—
591C2×C4—271no—
592C2×C4—271no—
593C2×C4—271no—
594C2×C4—271no—
595C2×C4—361no—
596C2×C4—1081no—
597C2×C4C2×D3×D911yescharacteristic
598C2×C4C2×D3×D911yescharacteristic
599C2×C4C2×D3×D911yescharacteristic
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—361no—
604C2×C3—361no—
605C2×C3#6371011yescharacteristic
606C2×C3#6371311yescharacteristic
607C2×C3#6371311yescharacteristic
608C2×C3#6429711yescharacteristic
609C2×C3#6431311yescharacteristic
610C2×C3#6431511yescharacteristic
611D3—121no—
612D3—121no—
613C22—181no—
614C22—181no—
615C22—181no—
616C22—181no—
617C22—181no—
618C22—181no—
619C22C22×D3×D911yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—61no—
624C4—61no—
625C4—61no—
626C4—61no—
627C4—181no—
628C4—181no—
629C4—541no—
630C4—541no—
631C4—541no—
632C4—541no—
633C3—21no—
634C3#1058983711yescharacteristic, complemented
635C3#1059449511yescharacteristic
636C2—361no—
637C2—361no—
638C2#1172245811yescharacteristic
639C2#1172249911yescharacteristic
640C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C3×D36C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#563
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721745C2semidirect
3. C9×#888C2semidirect
4. #11720999C2semidirect
5. #11720984C2semidirect
8. #11720949C2semidirect
9. #11720947C2semidirect
10. #11720947C2semidirect
11. #11720946C2semidirect
12. #11720944C2semidirect
13. #11720939C2semidirect
14. #11720928C2semidirect
15. #11720903C2semidirect
226. #888D9semidirect
430. C3×C9#563semidirect
578. C9#5651semidirect
634. C3#10589837semidirect

Maximal subgroups

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

subgroupindex
#117217452
C3×#634562
C9×#8882
#117209992
#117209842
#117209812
#117209512
#117209492
#117209472
#117209462
#117209442
#117209392
#117209282
#117209032
#105944953
#105898373