Finite Group Databasebeta

← #13147971

Subgroups — #13147971

Order 1728.

694 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721723C211yesmaximal, characteristic
2C9×#878C211yesmaximal, characteristic
3C3×#63440C211yesmaximal, characteristic
4#11721253C211yesmaximal, characteristic
5C2×#88442C211yesmaximal, characteristic
6#11721249C211yesmaximal, characteristic
7#11721248C211yesmaximal, characteristic
8#11720971C211yesmaximal, characteristic, complemented
9#11720930C211yesmaximal, characteristic, complemented
10#11720928C211yesmaximal, characteristic, complemented
11#11720928C211yesmaximal, characteristic, complemented
12#11720919C211yesmaximal, characteristic, complemented
13#11720918C211yesmaximal, characteristic, complemented
14#11720904C211yesmaximal, characteristic, complemented
15#11720903C211yesmaximal, characteristic, complemented
16#10594033—31nomaximal
17#10589725—31nomaximal
18C22×C3⋊Dic9C2211yescharacteristic
19C22×C9×Dic3C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C2×C9×Dic6C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C2×#6189C2211yescharacteristic
25C3×C9×C22⋊C4C2211yescharacteristic
26#88551C2211yescharacteristic
27#88534C2211yescharacteristic
28#88534C2211yescharacteristic
29C4×C3⋊Dic9C2211yescharacteristic
30C4×C3⋊Dic9C2211yescharacteristic
31C9×#217C2211yescharacteristic
32C3×#3676C2211yescharacteristic
33C9×C4⋊Dic3C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C3×C4⋊Dic9C2211yescharacteristic
37C3×Dic9⋊C4C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39Dic3⋊Dic9C2211yescharacteristic
40Dic3⋊Dic9C2211yescharacteristic
41#88442C2211yescharacteristic
42#88442C2211yescharacteristic
43#88442C2211yescharacteristic
44#88442C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64086—31no—
54C3×#878—31no—
55C3×#878D311yescharacteristic
56C2×#3724—31no—
57#63964—31no—
58#63964—31no—
59#63957—31no—
60#63919—31no—
61#63873—31no—
62#63870—31no—
63#63870—31no—
64#63847—31no—
65#63847—31no—
66#63839—31no—
67#63839—31no—
68C9×C22⋊Q8—31no—
69#63506—31no—
70#63506—31no—
71#63486—31no—
72C2×Dic9⋊C4—31no—
73#63451—31no—
74#63451—31no—
75#63451—31no—
76#63451—31no—
77#63450—31no—
78#63450—31no—
79#63440—31no—
80#63440D311yescharacteristic
81#63439—31no—
82#63439—31no—
83#63439—61no—
84C23×C3×C9C2311yescharacteristic
85C2×C3×C4×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3⋊Dic9Q814yescomplemented
88C2×C3⋊Dic9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C9×Dic3—41no—
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9—41no—
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C9×Dic6—41no—
103C3×Dic18—41no—
104#6189—41no—
105#6189—41no—
106#5623—91no—
107#5623—91no—
108C22×C3⋊Dic3—31no—
109C22×C3×Dic3—31no—
110C22×C3×Dic3C2×D311yescharacteristic
111C2×C3×Dic6—31no—
112C2×C3×Dic6C2×D311yescharacteristic
113C2×#627—31no—
114C2×#627—31no—
115C32×C22⋊C4C2×D311yescharacteristic
116#3757—31no—
117#3750—31no—
118#3750—31no—
119C4×C3⋊Dic3—31no—
120C4×C3⋊Dic3—31no—
121C3×#217—31no—
122C3×#217C2×D311yescharacteristic
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3C2×D311yescharacteristic
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4—31no—
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×Dic3⋊C4C2×D311yescharacteristic
129#3724—31no—
130#3724—31no—
131#3724—31no—
132#3724—31no—
133#3724—31no—
134#3724—31no—
135#3724—31no—
136#3724—31no—
137Dic3⋊Dic3—31no—
138Dic3⋊Dic3—31no—
139Dic3⋊Dic3—31no—
140Dic3⋊Dic3—31no—
141Dic32—31no—
142Dic32—31no—
143C2×Q8×C9—31no—
144C22×C4×C9—31no—
145C22×Dic9—31no—
146C22×Dic9—61no—
147C22×Dic9C2×D311yescharacteristic
148C2×Dic18—31no—
149C2×Dic18—31no—
150C2×Dic18C2×D311yescharacteristic
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C4⋊C4—31no—
154C9×C22⋊C4—21no—
155C9×C22⋊C4—31no—
156C9×C22⋊C4C2×D311yescharacteristic
157#3676—31no—
158#3676—61no—
159#3676C2×D311yescharacteristic
160C4⋊Dic9—31no—
161C4⋊Dic9—31no—
162C4⋊Dic9C2×D311yescharacteristic
163Dic9⋊C4—31no—
164Dic9⋊C4—31no—
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—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—61no—
176Dic9⋊C4—61no—
177Dic9⋊C4C2×D311yescharacteristic
178Dic9⋊C4C2×D311yescharacteristic
179C4×Dic9—31no—
180C4×Dic9—31no—
181C4×Dic9—31no—
182C4×Dic9—31no—
183C4×Dic9—61no—
184C4×Dic9—61no—
185C22×C3×C9—21no—
186C22×C3×C9—21no—
187C22×C3×C9C2×Q812yes—
188C22×C3×C9C2411yescharacteristic
189C3×C4×C9—21no—
190C3×C4×C9—21no—
191C3⋊Dic9—21no—
192C3⋊Dic9—21no—
193C3⋊Dic9C2×Q812yes—
194C3⋊Dic9C2×Q812yes—
195C9×Dic3—21no—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C3×Dic9—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×C22⋊Q8—31no—
204C3×C22⋊Q8—91no—
205#944—91no—
206#944—91no—
207#944—91no—
208#944—91no—
209#924—91no—
210#924—91no—
211C2×Dic3⋊C4—91no—
212C2×Dic3⋊C4—91no—
213#889—91no—
214#889—91no—
215#889—91no—
216#889—91no—
217#889—91no—
218#889—91no—
219#889—91no—
220#889—91no—
221#888—91no—
222#888—91no—
223#888—91no—
224#888—91no—
225#878—31no—
226#878—91no—
227#878—91no—
228#878D911yescharacteristic
229#877—91no—
230#877—91no—
231#877—91no—
232#877—91no—
233#877—181no—
234C23×C32C22×D311yescharacteristic
235C2×C32×C4C22×D311yescharacteristic
236C2×C32×C4C22×D311yescharacteristic
237C2×C3⋊Dic3—34no—
238C2×C3⋊Dic3—31no—
239C2×C3⋊Dic3—31no—
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×Dic3—41no—
247C2×C3×Dic3—121no—
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252C3×Dic6—41no—
253C3×Dic6—121no—
254#627—121no—
255#627—121no—
256C23×C9—21no—
257C23×C9C22×D311yescharacteristic
258Q8×C9—121no—
259C2×C4×C9—21no—
260C2×C4×C9—21no—
261C2×C4×C9—31no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—121no—
266C2×C4×C9C22×D311yescharacteristic
267C2×C4×C9C22×D311yescharacteristic
268C2×Dic9—34no—
269C2×Dic9—31no—
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—31no—
273C2×Dic9—41no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—64no—
277C2×Dic9—61no—
278C2×Dic9—61no—
279C2×Dic9C22×D311yescharacteristic
280C2×Dic9C22×D311yescharacteristic
281C2×Dic9C22×D311yescharacteristic
282C2×Dic9C22×D311yescharacteristic
283Dic18—41no—
284Dic18—121no—
285Dic18—121no—
286#544—271noSylow
287C2×C3×C9—22no—
288C2×C3×C9C22×Q811yescharacteristic
289C2×C3×C9H(5,𝔽2)11yescharacteristic
290C2×C3×C92−1+411yescharacteristic
291C2×C3×Q8—31no—
292C2×C3×Q8—91no—
293C22×C3×C4—31no—
294C22×C3×C4—91no—
295C22×Dic3—31no—
296C22×Dic3—91no—
297C22×Dic3—91no—
298C22×Dic3—181no—
299C22×Dic3C2×D911yescharacteristic
300C2×Dic6—31no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6C2×D911yescharacteristic
306C3×C4⋊C4—31no—
307C3×C4⋊C4—31no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—91no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C22⋊C4—21no—
313C3×C22⋊C4—31no—
314C3×C22⋊C4—91no—
315C3×C22⋊C4C2×D911yescharacteristic
316C3×C22⋊C4D3211yescharacteristic
317#217—31no—
318#217—91no—
319#217—91no—
320#217—181no—
321#217C2×D911yescharacteristic
322C4⋊Dic3—31no—
323C4⋊Dic3—91no—
324C4⋊Dic3—91no—
325C4⋊Dic3—91no—
326C4⋊Dic3—91no—
327C4⋊Dic3C2×D911yescharacteristic
328Dic3⋊C4—31no—
329Dic3⋊C4—31no—
330Dic3⋊C4—91no—
331Dic3⋊C4—91no—
332Dic3⋊C4—91no—
333Dic3⋊C4—91no—
334Dic3⋊C4—91no—
335Dic3⋊C4—91no—
336Dic3⋊C4—91no—
337Dic3⋊C4—91no—
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
357Dic3⋊C4C2×D911yescharacteristic
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×C32D3×Q812yes—
371C22×C32C23×D311yescharacteristic
372C32×C4—21no—
373C32×C4—21no—
374C3⋊Dic3—32no—
375C3⋊Dic3—32no—
376C3⋊Dic3—61no—
377C3⋊Dic3—61no—
378C3×Dic3—21no—
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—21no—
382C3×Dic3—61no—
383C3×Dic3—61no—
384C3×Dic3—61no—
385C3×Dic3—61no—
386C22×C9—21no—
387C22×C9—21no—
388C22×C9—21no—
389C22×C9—22no—
390C22×C9—41no—
391C22×C9—41no—
392C22×C9D3×Q812yes—
393C22×C9C23×D311yescharacteristic
394C4×C9—21no—
395C4×C9—21no—
396C4×C9—41no—
397C4×C9—41no—
398C4×C9—61no—
399C4×C9—61no—
400C4×C9—61no—
401C4×C9—61no—
402Dic9—21no—
403Dic9—21no—
404Dic9—21no—
405Dic9—21no—
406Dic9—32no—
407Dic9—32no—
408Dic9—62no—
409Dic9—62no—
410Dic9—61no—
411Dic9—61no—
412Dic9—121no—
413Dic9—121no—
414C4⋊Q8—271no—
415C4⋊Q8—271no—
416C4⋊Q8—271no—
417C4⋊Q8—271no—
418#125—271no—
419#125—271no—
420#125—271no—
421#125—271no—
422C22⋊Q8—31no—
423C22⋊Q8—91no—
424C22⋊Q8—271no—
425C22⋊Q8—271no—
426#117—271no—
427#117—271no—
428C2×C4⋊C4—271no—
429C3×C9#54411yescharacteristic, complemented, Sylow
430C23×C3—21no—
431C23×C3C22×D911yescharacteristic
432C23×C3C2×D3211yescharacteristic
433C3×Q8—121no—
434C3×Q8—361no—
435C2×C3×C4—21no—
436C2×C3×C4—21no—
437C2×C3×C4—31no—
438C2×C3×C4—31no—
439C2×C3×C4—31no—
440C2×C3×C4—31no—
441C2×C3×C4—91no—
442C2×C3×C4—91no—
443C2×C3×C4—91no—
444C2×C3×C4—91no—
445C2×C3×C4—121no—
446C2×C3×C4—361no—
447C2×C3×C4C22×D911yescharacteristic
448C2×C3×C4C22×D911yescharacteristic
449C2×C3×C4C2×D3211yescharacteristic
450C2×C3×C4C2×D3211yescharacteristic
451C2×Dic3—31no—
452C2×Dic3—31no—
453C2×Dic3—31no—
454C2×Dic3—31no—
455C2×Dic3—41no—
456C2×Dic3—91no—
457C2×Dic3—91no—
458C2×Dic3—94no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—94no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—121no—
467C2×Dic3—181no—
468C2×Dic3—181no—
469C2×Dic3—184no—
470C2×Dic3—181no—
471C2×Dic3—181no—
472C2×Dic3C22×D911yescharacteristic
473C2×Dic3C22×D911yescharacteristic
474C2×Dic3C22×D911yescharacteristic
475C2×Dic3C22×D911yescharacteristic
476Dic6—41no—
477Dic6—121no—
478Dic6—361no—
479Dic6—361no—
480Dic6—361no—
481Dic6—361no—
482C2×C32—22no—
483C2×C32#100411yescharacteristic
484C2×C32C2×D3×Q811yescharacteristic
485C2×C32#101011yescharacteristic
486C2×C9—22no—
487C2×C9—21no—
488C2×C9—21no—
489C2×C9—21no—
490C2×C9—42no—
491C2×C9#100411yescharacteristic
492C2×C9C2×D3×Q811yescharacteristic
493C2×C9#101011yescharacteristic
494C2×Q8—31no—
495C2×Q8—91no—
496C2×Q8—271no—
497C2×Q8—271no—
498C22×C4—31no—
499C22×C4—91no—
500C22×C4—271no—
501C4⋊C4—31no—
502C4⋊C4—31no—
503C4⋊C4—31no—
504C4⋊C4—91no—
505C4⋊C4—91no—
506C4⋊C4—91no—
507C4⋊C4—271no—
508C4⋊C4—271no—
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—
520C4⋊C4—271no—
521C22⋊C4—31no—
522C22⋊C4—91no—
523C22⋊C4—271no—
524C22⋊C4D3×D911yescharacteristic
525C42—271no—
526C42—271no—
527C42—271no—
528C42—271no—
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—22no—
534C22×C3—21no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3Q8×D912yes—
538C22×C3C23×D911yescharacteristic
539C22×C3#379712yes—
540C22×C3C22×D3211yescharacteristic
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—41no—
546C3×C4—41no—
547C3×C4—61no—
548C3×C4—61no—
549C3×C4—61no—
550C3×C4—61no—
551C3×C4—181no—
552C3×C4—181no—
553C3×C4—181no—
554C3×C4—181no—
555Dic3—21no—
556Dic3—21no—
557Dic3—21no—
558Dic3—21no—
559Dic3—61no—
560Dic3—61no—
561Dic3—61no—
562Dic3—61no—
563Dic3—92no—
564Dic3—92no—
565Dic3—92no—
566Dic3—92no—
567Dic3—182no—
568Dic3—182no—
569Dic3—181no—
570Dic3—181no—
571Dic3—181no—
572Dic3—181no—
573Dic3—361no—
574Dic3—361no—
575C32#562311yescharacteristic
576C9—21no—
577C9#562311yescharacteristic, complemented
578C23C2×D3×D911yescharacteristic
579Q8—121no—
580Q8—361no—
581Q8—1081no—
582Q8—1081no—
583C2×C4—31no—
584C2×C4—31no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—91no—
588C2×C4—91no—
589C2×C4—91no—
590C2×C4—91no—
591C2×C4—121no—
592C2×C4—274no—
593C2×C4—271no—
594C2×C4—271no—
595C2×C4—271no—
596C2×C4—271no—
597C2×C4—361no—
598C2×C4C2×D3×D911yescharacteristic
599C2×C4C2×D3×D911yescharacteristic
600C2×C3—22no—
601C2×C3—22no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—42no—
606C2×C3#6371011yescharacteristic
607C2×C3C2×Q8×D911yescharacteristic
608C2×C3#6371611yescharacteristic
609C2×C3C2×#379711yescharacteristic
610C2×C3#6430911yescharacteristic
611C2×C3#6431311yescharacteristic
612C22—21no—
613C22—21no—
614C22(C3⋊D9)⋊Q812yes—
615C22C22×D3×D911yescharacteristic
616C4—21no—
617C4—21no—
618C4—61no—
619C4—61no—
620C4—61no—
621C4—61no—
622C4—181no—
623C4—181no—
624C4—181no—
625C4—181no—
626C4—272no—
627C4—272no—
628C4—541no—
629C4—541no—
630C3—21no—
631C3#1058972511yescharacteristic, complemented
632C3#1059403311yescharacteristic
633C2—22no—
634C2C2×(C3⋊D9)⋊Q811yescharacteristic
635C2#1172249411yescharacteristic
636C2#1172249911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC23×C32C22×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#544
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #11720971C2semidirect
9. #11720930C2semidirect
10. #11720928C2semidirect
11. #11720928C2semidirect
12. #11720919C2semidirect
13. #11720918C2semidirect
14. #11720904C2semidirect
15. #11720903C2semidirect
87. C2×C3⋊Dic9Q8semidirect
429. C3×C9#544semidirect
577. C9#5623semidirect
631. C3#10589725semidirect

Maximal subgroups

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

subgroupindex
#117217232
C9×#8782
C3×#634402
#117212532
C2×#884422
#117212492
#117212482
#117209712
#117209302
#117209282
#117209192
#117209182
#117209042
#117209032
#105940333
#105897253