Finite Group Databasebeta

← #13147976

Subgroups — #13147976

Order 1728.

692 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721724C211yesmaximal, characteristic
2C9×#877C211yesmaximal, characteristic
3C3×#63440C211yesmaximal, characteristic
4#11721267C211yesmaximal, characteristic
5C2×Dic3⋊Dic9C211yesmaximal, characteristic
6#11721253C211yesmaximal, characteristic
7#11721244C211yesmaximal, characteristic
8#11720970C211yesmaximal, characteristic, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#11720925C211yesmaximal, characteristic, complemented
11#11720925C211yesmaximal, characteristic, complemented
12#11720916C211yesmaximal, characteristic, complemented
13#11720914C211yesmaximal, characteristic, complemented
14#11720905C211yesmaximal, characteristic, complemented
15#11720904C211yesmaximal, characteristic, complemented
16#10594035—31nomaximal
17#10589621—31nomaximal
18C22×C3⋊Dic9C2211yescharacteristic
19C2×#6226C2211yescharacteristic
20C22×C9×Dic3C2211yescharacteristic
21C22×C3×Dic9C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C2×#6189C2211yescharacteristic
25C3×C9×C22⋊C4C2211yescharacteristic
26#88551C2211yescharacteristic
27C4⋊(C3⋊Dic9)C2211yescharacteristic
28#88534C2211yescharacteristic
29#88534C2211yescharacteristic
30C9×#217C2211yescharacteristic
31C3×#3676C2211yescharacteristic
32C9×Dic3⋊C4C2211yescharacteristic
33C9×Dic3⋊C4C2211yescharacteristic
34C4×C9×Dic3C2211yescharacteristic
35C4×C9×Dic3C2211yescharacteristic
36C3×C4⋊Dic9C2211yescharacteristic
37C3×Dic9⋊C4C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39Dic3⋊Dic9C2211yescharacteristic
40Dic3⋊Dic9C2211yescharacteristic
41Dic3⋊Dic9C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64087—31no—
54C3×#878—31no—
55C3×#877D311yescharacteristic
56#63982—31no—
57C2×Dic3⋊Dic3—31no—
58#63964—31no—
59#63955—31no—
60#63918—31no—
61#63867—31no—
62#63867—31no—
63#63866—31no—
64#63846—31no—
65#63845—31no—
66#63840—31no—
67#63839—31no—
68C9×#117—31no—
69#63488—31no—
70C2×C4⋊Dic9—31no—
71#63452—31no—
72#63452—31no—
73#63450—31no—
74#63450—31no—
75#63440—31no—
76#63440—31no—
77#63440—31no—
78#63440—61no—
79#63440D311yescharacteristic
80#63432—31no—
81#63432—31no—
82#63431—31no—
83#63431—31no—
84C23×C3×C9C2311yescharacteristic
85C2×C3×C4×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3⋊Dic9—41no—
88C2×C3⋊Dic9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92#6226—41no—
93C2×C9×Dic3Q814yescomplemented
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C3×Dic9—41no—
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C2×C3×Dic9C2311yescharacteristic
103C3×Dic18—41no—
104#6189—41no—
105#6189—41no—
106#5623—91no—
107#5519—91no—
108C22×C3⋊Dic3—31no—
109C2×#634—31no—
110C22×C3×Dic3—31no—
111C22×C3×Dic3C2×D311yescharacteristic
112C2×C3×Dic6—31no—
113C2×#627—31no—
114C2×#627—31no—
115C32×C22⋊C4C2×D311yescharacteristic
116#3757—31no—
117C4⋊(C3⋊Dic3)—31no—
118#3750—31no—
119#3750—31no—
120C3×#217—31no—
121C3×#217C2×D311yescharacteristic
122C3×C4⋊Dic3—31no—
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×C4×Dic3C2×D311yescharacteristic
128C3×C4×Dic3C2×D311yescharacteristic
129#3724—31no—
130#3724—31no—
131Dic3⋊Dic3—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138Dic3⋊Dic3—31no—
139Dic3⋊Dic3—31no—
140Dic3⋊Dic3—31no—
141Dic32—31no—
142Dic32—31no—
143C22×C4×C9—31no—
144C22×Dic9—31no—
145C22×Dic9—61no—
146C22×Dic9C2×D311yescharacteristic
147C2×Dic18—31no—
148C2×Dic18—31no—
149C2×Dic18—31no—
150C2×Dic18—61no—
151C2×Dic18C2×D311yescharacteristic
152C9×C4⋊C4—31no—
153C9×C4⋊C4—31no—
154C9×C22⋊C4—21no—
155C9×C22⋊C4—31no—
156C9×C22⋊C4C2×D311yescharacteristic
157C42×C9—31no—
158C42×C9—31no—
159#3676—31no—
160#3676—61no—
161#3676C2×D311yescharacteristic
162C4⋊Dic9—31no—
163C4⋊Dic9—31no—
164C4⋊Dic9—31no—
165C4⋊Dic9—31no—
166C4⋊Dic9—31no—
167C4⋊Dic9—31no—
168C4⋊Dic9—31no—
169C4⋊Dic9—31no—
170C4⋊Dic9—31no—
171C4⋊Dic9—61no—
172C4⋊Dic9C2×D311yescharacteristic
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—31no—
179Dic9⋊C4—61no—
180Dic9⋊C4—61no—
181Dic9⋊C4C2×D311yescharacteristic
182Dic9⋊C4C2×D311yescharacteristic
183C4×Dic9—31no—
184C4×Dic9—31no—
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⋊Dic9—21no—
194C3⋊Dic9—21no—
195C9×Dic3—21no—
196C9×Dic3—21no—
197C9×Dic3C2×Q812yes—
198C9×Dic3C2×Q812yes—
199C3×Dic9—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×C22⋊Q8—91no—
204C3×#117—31no—
205#944—91no—
206#944—91no—
207#926—91no—
208C2×C4⋊Dic3—91no—
209#924—91no—
210C2×Dic3⋊C4—91no—
211#890—91no—
212#890—91no—
213#889—91no—
214#889—91no—
215#889—91no—
216#889—91no—
217#888—91no—
218#888—91no—
219#888—91no—
220#888—91no—
221#878—31no—
222#878—91no—
223#878—91no—
224#878—91no—
225#878—91no—
226#878—91no—
227#878—181no—
228#877—91no—
229#877D911yescharacteristic
230#870—91no—
231#870—91no—
232#869—91no—
233#869—91no—
234C23×C32C22×D311yescharacteristic
235C2×C32×C4C22×D311yescharacteristic
236C2×C32×C4C22×D311yescharacteristic
237C2×C3⋊Dic3—31no—
238C2×C3⋊Dic3—31no—
239C2×C3⋊Dic3—31no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—121no—
242#634—121no—
243C2×C3×Dic3—31no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—121no—
248C2×C3×Dic3Dic614yes—
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252C2×C3×Dic3C22×D311yescharacteristic
253C3×Dic6—121no—
254#627—121no—
255#627—121no—
256C23×C9—21no—
257C23×C9C22×D311yescharacteristic
258C2×C4×C9—21no—
259C2×C4×C9—21no—
260C2×C4×C9—34no—
261C2×C4×C9—31no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9C22×D311yescharacteristic
266C2×C4×C9C22×D311yescharacteristic
267C2×Dic9—31no—
268C2×Dic9—31no—
269C2×Dic9—31no—
270C2×Dic9—31no—
271C2×Dic9—41no—
272C2×Dic9—61no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—121no—
277C2×Dic9—122no—
278C2×Dic9C22×D311yescharacteristic
279C2×Dic9C22×D311yescharacteristic
280C2×Dic9C22×D311yescharacteristic
281C2×Dic9C22×D311yescharacteristic
282Dic18—41no—
283Dic18—121no—
284Dic18—121no—
285Dic18—121no—
286Dic18—122no—
287#544—271noSylow
288C2×C3×C9—22no—
289C2×C3×C9C22×Q811yescharacteristic
290C2×C3×C9H(5,𝔽2)11yescharacteristic
291C2×C3×C92−1+411yescharacteristic
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×Dic6—91no—
306C2×Dic6—91no—
307C2×Dic6—181no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C4⋊C4—91no—
313C3×C22⋊C4—21no—
314C3×C22⋊C4—31no—
315C3×C22⋊C4—91no—
316C3×C22⋊C4C2×D911yescharacteristic
317C3×C22⋊C4D3211yescharacteristic
318C3×C42—31no—
319C3×C42—31no—
320#217—31no—
321#217—91no—
322#217—91no—
323#217—181no—
324#217C2×D911yescharacteristic
325C4⋊Dic3—31no—
326C4⋊Dic3—91no—
327C4⋊Dic3—91no—
328C4⋊Dic3—91no—
329C4⋊Dic3—91no—
330C4⋊Dic3—91no—
331C4⋊Dic3—91no—
332C4⋊Dic3—91no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—91no—
338C4⋊Dic3—181no—
339Dic3⋊C4—31no—
340Dic3⋊C4—31no—
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—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—181no—
360Dic3⋊C4—181no—
361Dic3⋊C4C2×D911yescharacteristic
362Dic3⋊C4C2×D911yescharacteristic
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3C2×D911yescharacteristic
368C4×Dic3C2×D911yescharacteristic
369C22×C32—21no—
370C22×C32—21no—
371C22×C32C2×Dic612yes—
372C22×C32C23×D311yescharacteristic
373C32×C4—21no—
374C32×C4—21no—
375C3⋊Dic3—61no—
376C3⋊Dic3—61no—
377C3⋊Dic3—61no—
378C3⋊Dic3—61no—
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—61no—
382C3×Dic3—61no—
383C3×Dic3—61no—
384C3×Dic3—61no—
385C3×Dic3C2×Dic612yes—
386C3×Dic3C2×Dic612yes—
387C22×C9—21no—
388C22×C9—21no—
389C22×C9—21no—
390C22×C9—22no—
391C22×C9—41no—
392C22×C9—41no—
393C22×C9D3×Q812yes—
394C22×C9C23×D311yescharacteristic
395C4×C9—21no—
396C4×C9—21no—
397C4×C9—32no—
398C4×C9—32no—
399C4×C9—41no—
400C4×C9—41no—
401C4×C9—61no—
402C4×C9—61no—
403Dic9—21no—
404Dic9—21no—
405Dic9—21no—
406Dic9—21no—
407Dic9—62no—
408Dic9—61no—
409Dic9—61no—
410Dic9—61no—
411Dic9—61no—
412Dic9—121no—
413Dic9—121no—
414Dic9—121no—
415C4⋊Q8—271no—
416C4⋊Q8—271no—
417C4⋊Q8—271no—
418C4⋊Q8—271no—
419#125—271no—
420#125—271no—
421#125—271no—
422#125—271no—
423C22⋊Q8—91no—
424C22⋊Q8—271no—
425C22⋊Q8—271no—
426C22⋊Q8—271no—
427#117—31no—
428#117—271no—
429C2×C4⋊C4—271no—
430C3×C9#54411yescharacteristic, complemented, Sylow
431C23×C3—21no—
432C23×C3C22×D911yescharacteristic
433C23×C3C2×D3211yescharacteristic
434C3×Q8—361no—
435C2×C3×C4—21no—
436C2×C3×C4—21no—
437C2×C3×C4—31no—
438C2×C3×C4—31no—
439C2×C3×C4—34no—
440C2×C3×C4—31no—
441C2×C3×C4—31no—
442C2×C3×C4—91no—
443C2×C3×C4—91no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
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—91no—
456C2×Dic3—91no—
457C2×Dic3—91no—
458C2×Dic3—91no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—121no—
464C2×Dic3—181no—
465C2×Dic3—181no—
466C2×Dic3—181no—
467C2×Dic3—181no—
468C2×Dic3—361no—
469C2×Dic3—361no—
470C2×Dic3—362no—
471C2×Dic3Dic1814yescomplemented
472C2×Dic3C22×D911yescharacteristic
473C2×Dic3C22×D911yescharacteristic
474C2×Dic3C22×D911yescharacteristic
475C2×Dic3C22×D911yescharacteristic
476Dic6—121no—
477Dic6—361no—
478Dic6—361no—
479Dic6—361no—
480Dic6—361no—
481Dic6—361no—
482Dic6—361no—
483Dic6—362no—
484C2×C32—22no—
485C2×C32C22×Dic611yescharacteristic
486C2×C32#100911yescharacteristic
487C2×C32#101011yescharacteristic
488C2×C9—22no—
489C2×C9—21no—
490C2×C9—21no—
491C2×C9—21no—
492C2×C9—42no—
493C2×C9#100411yescharacteristic
494C2×C9C2×D3×Q811yescharacteristic
495C2×C9#101011yescharacteristic
496C2×Q8—91no—
497C2×Q8—271no—
498C2×Q8—271no—
499C2×Q8—271no—
500C22×C4—31no—
501C22×C4—91no—
502C22×C4—271no—
503C4⋊C4—31no—
504C4⋊C4—31no—
505C4⋊C4—91no—
506C4⋊C4—91no—
507C4⋊C4—91no—
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—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C22⋊C4—31no—
524C22⋊C4—91no—
525C22⋊C4—271no—
526C22⋊C4D3×D911yescharacteristic
527C42—31no—
528C42—31no—
529C42—271no—
530C42—271no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—22no—
536C22×C3—21no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3C2×Dic1812yes—
540C22×C3C23×D911yescharacteristic
541C22×C3D3×Dic612yes—
542C22×C3C22×D3211yescharacteristic
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—32no—
548C3×C4—32no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4—61no—
552C3×C4—61no—
553C3×C4—181no—
554C3×C4—181no—
555C3×C4—181no—
556C3×C4—181no—
557Dic3—21no—
558Dic3—21no—
559Dic3—61no—
560Dic3—61no—
561Dic3—61no—
562Dic3—61no—
563Dic3—182no—
564Dic3—181no—
565Dic3—181no—
566Dic3—181no—
567Dic3—181no—
568Dic3—181no—
569Dic3—181no—
570Dic3—181no—
571Dic3—181no—
572Dic3—361no—
573Dic3—361no—
574Dic3—361no—
575Dic3C2×Dic1812yes—
576Dic3C2×Dic1812yes—
577C32#551911yescharacteristic
578C9—21no—
579C9#562311yescharacteristic, complemented
580C23C2×D3×D911yescharacteristic
581Q8—361no—
582Q8—1081no—
583Q8—1081no—
584Q8—1081no—
585C2×C4—34no—
586C2×C4—31no—
587C2×C4—31no—
588C2×C4—31no—
589C2×C4—31no—
590C2×C4—91no—
591C2×C4—91no—
592C2×C4—91no—
593C2×C4—91no—
594C2×C4—271no—
595C2×C4—271no—
596C2×C4—271no—
597C2×C4—271no—
598C2×C4—361no—
599C2×C4—1081no—
600C2×C4C2×D3×D911yescharacteristic
601C2×C4C2×D3×D911yescharacteristic
602C2×C3—22no—
603C2×C3—22no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—42no—
608C2×C3C22×Dic1811yescharacteristic
609C2×C3#6371511yescharacteristic
610C2×C3#6371611yescharacteristic
611C2×C3C2×D3×Dic611yescharacteristic
612C2×C3#6430911yescharacteristic
613C2×C3#6431411yescharacteristic
614C22—21no—
615C22—21no—
616C22D3×Dic1812yes—
617C22C22×D3×D911yescharacteristic
618C4—21no—
619C4—21no—
620C4—32no—
621C4—32no—
622C4—61no—
623C4—61no—
624C4—181no—
625C4—181no—
626C4—181no—
627C4—181no—
628C4—541no—
629C4—541no—
630C4—541no—
631C4—541no—
632C3—21no—
633C3#1058962111yescharacteristic, complemented
634C3#1059403511yescharacteristic
635C2—22no—
636C2C2×D3×Dic1811yescharacteristic
637C2#1172249411yescharacteristic
638C2#1172250111yescharacteristic
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. #11720970C2semidirect
9. #11720932C2semidirect
10. #11720925C2semidirect
11. #11720925C2semidirect
12. #11720916C2semidirect
13. #11720914C2semidirect
14. #11720905C2semidirect
15. #11720904C2semidirect
93. C2×C9×Dic3Q8semidirect
430. C3×C9#544semidirect
471. C2×Dic3Dic18semidirect
579. C9#5623semidirect
633. C3#10589621semidirect

Maximal subgroups

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

subgroupindex
#117217242
C9×#8772
C3×#634402
#117212672
C2×Dic3⋊Dic92
#117212532
#117212442
#117209702
#117209322
#117209252
#117209162
#117209142
#117209052
#117209042
#105940353
#105896213