Finite Group Databasebeta

← #13148240

Subgroups — #13148240

Order 1728.

718 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721723C211yesmaximal, characteristic, complemented
2C9×#883C211yesmaximal, characteristic
3C3×#63441C211yesmaximal, characteristic, complemented
4#67⋊Dic9C211yesmaximal, characteristic
5#11721280C211yesmaximal, characteristic
6#11721267C211yesmaximal, characteristic, complemented
7#11721247C211yesmaximal, characteristic, complemented
8#11721026C211yesmaximal, characteristic, complemented
9#11721011C211yesmaximal, characteristic, complemented
10D3×Dic9⋊C4C211yesmaximal, characteristic, complemented
11#11720959C211yesmaximal, characteristic, complemented
12#11720937C211yesmaximal, characteristic, complemented
13#11720928C211yesmaximal, characteristic, complemented
14#11720918C211yesmaximal, characteristic, complemented
15#11720901C211yesmaximal, characteristic, complemented
16#10594120—31nomaximal
17#10589752—31nomaximal
18C22×C3⋊Dic9C2211yescharacteristic
19C2×C9×#67C2211yescharacteristic
20C2×C4×D3×C9C2211yescharacteristic
21C2×D3×Dic9C2211yescharacteristic
22C2×D3×Dic9C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25#88551C2211yescharacteristic
26#88534C2211yescharacteristic
27#88534C2211yescharacteristic
28C4×C3⋊Dic9C2211yescharacteristic
29C4×C3⋊Dic9C2211yescharacteristic
30C9×#217C2211yescharacteristic
31C3×#3676C2211yescharacteristic
32C3×#3676C2211yescharacteristic
33C9×(C2×D3)⋊C4C2211yescharacteristic
34C9×C4⋊Dic3C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C3×C4⋊Dic9C2211yescharacteristic
37C3×Dic9⋊C4C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39C3×C4×Dic9C2211yescharacteristic
40(C2×D3)⋊Dic9C2211yescharacteristic
41(C2×D3)⋊Dic9C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47#88442C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64086—31no—
54C3×#883D311yescharacteristic
55C3×#879—31no—
56#63982—31no—
57#67⋊Dic3—31no—
58#63969—31no—
59#63956—31no—
60#63899—31no—
61#63878—31no—
62D3×Dic3⋊C4—31no—
63#63872—31no—
64#63870—31no—
65#63855—31no—
66#63847—31no—
67#63837—31no—
68C9×#123—31no—
69#63497—31no—
70D4×Dic9—31no—
71#63488—31no—
72C2×Dic9⋊C4—31no—
73#63452—31no—
74#63451—31no—
75#63451—31no—
76#63449—31no—
77#63441—31no—
78#63441D311yescharacteristic, complemented
79#63440—31no—
80#63440—31no—
81#63439—31no—
82#63439—31no—
83#63439—61no—
84C23×C3×C9C2311yescharacteristic
85C22×D3×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3⋊Dic9—22no—
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C9×#67—22no—
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C4×D3×C9—22no—
102D3×Dic9—22no—
103D3×Dic9—22no—
104#6189—22no—
105#5708—91no—
106#5650—91no—
107C22×C3⋊Dic3—31no—
108C2×C3×#67C2×D311yescharacteristic
109C2×C3×C4×D3C2×D311yescharacteristic
110C2×#627—31no—
111C2×D3×Dic3—31no—
112C2×D3×Dic3—31no—
113C32×C22⋊C4C2×D311yescharacteristic
114#3757—31no—
115#3750—31no—
116#3750—31no—
117C4×C3⋊Dic3—31no—
118C4×C3⋊Dic3—31no—
119C3×#217—31no—
120C3×#217—31no—
121C3×#217C2×D311yescharacteristic
122C3×(C2×D3)⋊C4C2×D311yescharacteristic
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3C2×D311yescharacteristic
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4—31no—
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×C4×Dic3—31no—
129#3724—31no—
130#3724—31no—
131#3724—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138(C2×D3)⋊Dic3—31no—
139(C2×D3)⋊Dic3—31no—
140Dic32—31no—
141Dic32—31no—
142C2×D4×C9—31no—
143C22×C4×C9—31no—
144C22×Dic9—31no—
145C22×Dic9—31no—
146C22×Dic9—31no—
147C22×Dic9—61no—
148C2×Dic18—31no—
149C9×C4⋊C4—31no—
150C9×C4⋊C4—31no—
151C9×C22⋊C4—21no—
152C9×C22⋊C4—31no—
153C9×C22⋊C4—31no—
154C9×C22⋊C4C2×D311yescharacteristic
155#3676—31no—
156#3676—31no—
157#3676—31no—
158#3676—61no—
159#3676C2×D311yescharacteristic
160#3676C2×D311yescharacteristic
161C4⋊Dic9—31no—
162C4⋊Dic9—31no—
163C4⋊Dic9—31no—
164C4⋊Dic9C2×D311yescharacteristic
165Dic9⋊C4—31no—
166Dic9⋊C4—31no—
167Dic9⋊C4—31no—
168Dic9⋊C4—31no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—61no—
174Dic9⋊C4—61no—
175Dic9⋊C4C2×D311yescharacteristic
176Dic9⋊C4C2×D311yescharacteristic
177C4×Dic9—31no—
178C4×Dic9—31no—
179C4×Dic9—31no—
180C4×Dic9—31no—
181C4×Dic9—61no—
182C4×Dic9—61no—
183C4×Dic9C2×D311yescharacteristic
184C22×C3×C9—21no—
185C22×C3×C9—21no—
186C22×C3×C9—21no—
187C22×C3×C9C2411yescharacteristic
188C2×D3×C9—21no—
189C2×D3×C9—21no—
190C2×D3×C9#4112yes—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3⋊Dic9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9—21no—
196C3⋊Dic9#4112yes—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×#126—91no—
205C3×#123—31no—
206#935—91no—
207D4×Dic3—91no—
208#926—91no—
209C2×Dic3⋊C4—91no—
210#896—91no—
211#892—91no—
212D3×C4⋊C4—91no—
213#890—91no—
214#890—91no—
215#889—91no—
216#889—91no—
217#889—91no—
218#889—91no—
219#887—91no—
220#883—91no—
221#883D911yescharacteristic
222#67⋊C4—91no—
223#879—31no—
224#879—91no—
225#879—91no—
226#878—91no—
227#878—91no—
228#878—91no—
229#877—91no—
230#877—91no—
231#877—91no—
232#877—181no—
233#872—91no—
234C4×Dic6—91no—
235C23×C32C22×D311yescharacteristic
236C22×C3×D3C22×D311yescharacteristic
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239C2×C3⋊Dic3—31no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—62no—
244C3×#67—22no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3—31no—
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252C3×C4×D3—22no—
253#627—62no—
254D3×Dic3—62no—
255D3×Dic3—62no—
256C23×C9—21no—
257C23×C9—31no—
258C23×C9C22×D311yescharacteristic
259D4×C9—62no—
260C2×C4×C9—21no—
261C2×C4×C9—21no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—62no—
266C2×C4×C9C22×D311yescharacteristic
267C2×C4×C9C22×D311yescharacteristic
268C2×Dic9—31no—
269C2×Dic9—31no—
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—61no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—62no—
277C2×Dic9—62no—
278C2×Dic9—62no—
279C2×Dic9—64no—
280C2×Dic9C22×D311yescharacteristic
281C2×Dic9C22×D311yescharacteristic
282C2×Dic9C22×D311yescharacteristic
283C2×Dic9C22×D311yescharacteristic
284Dic18—62no—
285#552—271noSylow
286C2×C3×C9—41no—
287C2×C3×C9C2×#4111yescharacteristic
288C2×C3×C9C2×#4111yescharacteristic
289C2×C3×C92−1+411yescharacteristic
290D3×C9—22no—
291C2×C3×D4—31no—
292C22×C3×C4—31no—
293C2×#67C2×D911yescharacteristic
294C22×Dic3—91no—
295C22×Dic3—91no—
296C22×Dic3—91no—
297C22×Dic3—91no—
298C22×Dic3—181no—
299C2×C4×D3—91no—
300C2×C4×D3—91no—
301C2×C4×D3C2×D911yescharacteristic
302C2×Dic6—91no—
303C2×Dic6—91no—
304C3×C4⋊C4—31no—
305C3×C4⋊C4—31no—
306C3×C4⋊C4—91no—
307C3×C4⋊C4—91no—
308C3×C4⋊C4—91no—
309C3×C22⋊C4—21no—
310C3×C22⋊C4—31no—
311C3×C22⋊C4—31no—
312C3×C22⋊C4—91no—
313C3×C22⋊C4—91no—
314C3×C22⋊C4C2×D911yescharacteristic
315C3×C22⋊C4D3211yescharacteristic
316C3×C42—91no—
317#217—31no—
318#217—31no—
319#217—91no—
320#217—91no—
321#217—91no—
322#217—91no—
323#217—181no—
324#217C2×D911yescharacteristic
325(C2×D3)⋊C4—91no—
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4C2×D911yescharacteristic
328C4⋊Dic3—31no—
329C4⋊Dic3—91no—
330C4⋊Dic3—91no—
331C4⋊Dic3—91no—
332C4⋊Dic3—91no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3C2×D911yescharacteristic
336Dic3⋊C4—31no—
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—181no—
355Dic3⋊C4—181no—
356Dic3⋊C4C2×D911yescharacteristic
357C4×Dic3—31no—
358C4×Dic3—91no—
359C4×Dic3—91no—
360C4×Dic3—91no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—181no—
367C4×Dic3—181no—
368C22×C32—21no—
369C22×C32—21no—
370C22×C32—21no—
371C22×C32C23×D311yescharacteristic
372C2×C3×D3—21no—
373C2×C3×D3—21no—
374C2×C3×D3#23712yes—
375C32×C4—21no—
376C32×C4—21no—
377C3⋊Dic3—32no—
378C3⋊Dic3—61no—
379C3⋊Dic3—61no—
380C3⋊Dic3—61no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—61no—
385C3×Dic3—61no—
386C3×Dic3—61no—
387C3×Dic3—61no—
388C22×C9—22no—
389C22×C9—21no—
390C22×C9—21no—
391C22×C9—21no—
392C22×C9—21no—
393C22×C9—32no—
394C22×C9—41no—
395C22×C9—41no—
396C22×C9—61no—
397C22×C9—61no—
398C22×C9C23×D311yescharacteristic
399C4×C9—21no—
400C4×C9—21no—
401C4×C9—41no—
402C4×C9—41no—
403C4×C9—61no—
404C4×C9—61no—
405C4×C9—61no—
406Dic9—21no—
407Dic9—21no—
408Dic9—21no—
409Dic9—21no—
410Dic9—32no—
411Dic9—62no—
412Dic9—62no—
413Dic9—61no—
414Dic9—61no—
415Dic9—61no—
416Dic9—121no—
417Dic9—121no—
418#126—91no—
419#126—271no—
420#125—271no—
421#125—271no—
422#125—271no—
423#123—31no—
424#123—271no—
425C22⋊Q8—271no—
426C22⋊Q8—271no—
427C4×Q8—271no—
428C4×D4—271no—
429#117—271no—
430#117—271no—
431#117—271no—
432C2×C4⋊C4—271no—
433C3×C9#55211yescharacteristic, complemented, Sylow
434C23×C3—21no—
435C23×C3—31no—
436C23×C3C22×D911yescharacteristic
437C23×C3C2×D3211yescharacteristic
438C22×D3C22×D911yescharacteristic
439C3×D4—62no—
440C2×C3×C4—21no—
441C2×C3×C4—21no—
442C2×C3×C4—31no—
443C2×C3×C4—31no—
444C2×C3×C4—31no—
445C2×C3×C4—62no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4C22×D911yescharacteristic
451C2×C3×C4C22×D911yescharacteristic
452C2×C3×C4C2×D3211yescharacteristic
453C2×C3×C4C2×D3211yescharacteristic
454#67—22no—
455C2×Dic3—31no—
456C2×Dic3—31no—
457C2×Dic3—31no—
458C2×Dic3—31no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—181no—
468C2×Dic3—181no—
469C2×Dic3—181no—
470C2×Dic3—181no—
471C2×Dic3—182no—
472C2×Dic3—182no—
473C2×Dic3—182no—
474C2×Dic3—182no—
475C2×Dic3—184no—
476C2×Dic3C22×D911yescharacteristic
477C2×Dic3C22×D911yescharacteristic
478C2×Dic3C22×D911yescharacteristic
479C4×D3—22no—
480C4×D3—182no—
481C4×D3—182no—
482Dic6—182no—
483Dic6—182no—
484C2×C32—41no—
485C2×C32C2×#23711yescharacteristic
486C2×C32D3×#4111yescharacteristic
487C2×C32#101011yescharacteristic
488C3×D3—22no—
489C2×C9—21no—
490C2×C9—21no—
491C2×C9—21no—
492C2×C9—42no—
493C2×C9—41no—
494C2×C9—62no—
495C2×C9D3×#4111yescharacteristic
496C2×C9D3×#4111yescharacteristic
497C2×C9#101011yescharacteristic
498C2×Q8—271no—
499C2×D4—31no—
500C22×C4—31no—
501C22×C4—271no—
502C22×C4—271no—
503C22×C4—271no—
504C4⋊C4—31no—
505C4⋊C4—31no—
506C4⋊C4—91no—
507C4⋊C4—91no—
508C4⋊C4—91no—
509C4⋊C4—271no—
510C4⋊C4—271no—
511C4⋊C4—271no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C22⋊C4—31no—
521C22⋊C4—31no—
522C22⋊C4—91no—
523C22⋊C4—91no—
524C22⋊C4—271no—
525C22⋊C4—271no—
526C22⋊C4—271no—
527C22⋊C4D3×D911yescharacteristic
528C42—91no—
529C42—271no—
530C42—271no—
531C42—271no—
532C42—271no—
533C22×C3—22no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—32no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—61no—
545C22×C3—61no—
546C22×C3C23×D911yescharacteristic
547C22×C3C22×D3211yescharacteristic
548C2×D3—21no—
549C2×D3—21no—
550C2×D3#369912yes—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—61no—
558C3×C4—61no—
559C3×C4—61no—
560C3×C4—181no—
561C3×C4—181no—
562C3×C4—181no—
563C3×C4—181no—
564Dic3—21no—
565Dic3—21no—
566Dic3—21no—
567Dic3—61no—
568Dic3—61no—
569Dic3—61no—
570Dic3—61no—
571Dic3—92no—
572Dic3—92no—
573Dic3—182no—
574Dic3—182no—
575Dic3—181no—
576Dic3—181no—
577Dic3—181no—
578Dic3—181no—
579Dic3—181no—
580Dic3—181no—
581Dic3—361no—
582Dic3—361no—
583C32#565011yescharacteristic
584C9—21no—
585C9#570811yescharacteristic, complemented
586C23—31no—
587C23C2×D3×D911yescharacteristic
588Q8—542no—
589D4—62no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—31no—
593C2×C4—62no—
594C2×C4—91no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—91no—
598C2×C4—271no—
599C2×C4—271no—
600C2×C4—271no—
601C2×C4—271no—
602C2×C4—542no—
603C2×C4—542no—
604C2×C4—542no—
605C2×C4C2×D3×D911yescharacteristic
606C2×C4C2×D3×D911yescharacteristic
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—42no—
611C2×C3—41no—
612C2×C3—41no—
613C2×C3—62no—
614C2×C3C2×#369911yescharacteristic
615C2×C3#41×D911yescharacteristic
616C2×C3#6371611yescharacteristic
617C2×C3#6430611yescharacteristic
618C2×C3#6430911yescharacteristic
619C2×C3D3×#23711yescharacteristic
620D3—22no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—32no—
625C22—61no—
626C22—61no—
627C22C22×D3×D911yescharacteristic
628C4—21no—
629C4—21no—
630C4—61no—
631C4—61no—
632C4—61no—
633C4—181no—
634C4—181no—
635C4—181no—
636C4—181no—
637C4—272no—
638C4—541no—
639C4—541no—
640C4—541no—
641C3—21no—
642C3#1058975211yescharacteristic, complemented
643C3#1059412011yescharacteristic
644C2—41no—
645C2—62no—
646C2#1172247511yescharacteristic
647C2#1172249411yescharacteristic
648C2D3×#369911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C3×#67C2×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721723C2semidirect
3. C3×#63441C2semidirect
6. #11721267C2semidirect
7. #11721247C2semidirect
8. #11721026C2semidirect
9. #11721011C2semidirect
10. D3×Dic9⋊C4C2semidirect
11. #11720959C2semidirect
12. #11720937C2semidirect
13. #11720928C2semidirect
14. #11720918C2semidirect
15. #11720901C2semidirect
78. #63441D3semidirect
433. C3×C9#552semidirect
585. C9#5708semidirect
642. C3#10589752semidirect

Maximal subgroups

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

subgroupindex
#117217232
C9×#8832
C3×#634412
#67⋊Dic92
#117212802
#117212672
#117212472
#117210262
#117210112
D3×Dic9⋊C42
#117209592
#117209372
#117209282
#117209182
#117209012
#105941203
#105897523