Finite Group Databasebeta

← #13148843

Subgroups — #13148843

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721744C211yesmaximal, characteristic, complemented
2C3×#63458C211yesmaximal, characteristic
3C9×#889C211yesmaximal, characteristic, complemented
4#11720996C211yesmaximal, characteristic
5#11720991C211yesmaximal, characteristic
6#11720975C211yesmaximal, characteristic
7#11720974C211yesmaximal, characteristic
8D9×Dic3⋊C4C211yesmaximal, characteristic
9#11720945C211yesmaximal, characteristic
10#11720932C211yesmaximal, characteristic, complemented
11#11720928C211yesmaximal, characteristic, complemented
12#11720918C211yesmaximal, characteristic, complemented
13#11720914C211yesmaximal, characteristic, complemented
14#11720905C211yesmaximal, characteristic, complemented
15#11720904C211yesmaximal, characteristic, complemented
16#10594390—31nomaximal
17#10589791—31nomaximal
18C2×#6226C2211yescharacteristic
19C2×#6226C2211yescharacteristic
20C2×C3×C4×D9C2211yescharacteristic
21C2×C3×Dic18C2211yescharacteristic
22C2×Dic3×D9C2211yescharacteristic
23C2×Dic3×D9C2211yescharacteristic
24C2×#6189C2211yescharacteristic
25C3×C9×C4⋊C4C2211yescharacteristic
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27#88534C2211yescharacteristic
28#88534C2211yescharacteristic
29C4×C3⋊Dic9C2211yescharacteristic
30C3×(C2×D9)⋊C4C2211yescharacteristic
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C9×C4⋊Dic3C2211yescharacteristic
33C9×Dic3⋊C4C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C9×Dic3⋊C4C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39C3×Dic9⋊C4C2211yescharacteristic
40(C2×D9)⋊Dic3C2211yescharacteristic
41(C2×D9)⋊Dic3C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49#88442C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64097—31no—
54C3×#896—31no—
55C3×#889D311yescharacteristic
56#63896—31no—
57#63893—31no—
58#63878—31no—
59#63877—31no—
60D3×Dic3⋊C4—31no—
61#63870—31no—
62#63866—31no—
63#63857—31no—
64#63847—31no—
65#63845—31no—
66#63840—31no—
67#63839—31no—
68C9×#125—31no—
69#63459—31no—
70#63459—31no—
71#63458—31no—
72#63458D311yescharacteristic
73#63454—31no—
74C4⋊C4×D9—31no—
75#63452—31no—
76#63451—31no—
77#63451—31no—
78#63450—31no—
79#63450—31no—
80#63450—31no—
81#63450—61no—
82#63434—31no—
83#63431—31no—
84C22×C3×D9C2311yescharacteristic
85C2×C3×C4×C9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3⋊Dic9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92#6226—41no—
93#6226—41no—
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C3×C4×D9—41no—
102C3×Dic18—41no—
103Dic3×D9—41no—
104Dic3×D9Q814yescomplemented
105#6189—41no—
106#5689—91no—
107#5623—91no—
108C2×#634—31no—
109C2×#634—31no—
110C2×C3×C4×D3—31no—
111C2×C3×Dic6—31no—
112C2×#627—31no—
113C2×D3×Dic3—31no—
114C2×D3×Dic3—31no—
115C32×C4⋊C4C2×D311yescharacteristic
116C4⋊(C3⋊Dic3)—31no—
117#3750—31no—
118#3750—31no—
119C4×C3⋊Dic3—31no—
120C3×(C2×D3)⋊C4—31no—
121C3×(C2×D3)⋊C4—31no—
122C3×C4⋊Dic3C2×D311yescharacteristic
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×Dic3⋊C4C2×D311yescharacteristic
129C3×C4×Dic3C2×D311yescharacteristic
130#3724—31no—
131#3724—31no—
132#3724—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138Dic3⋊Dic3—31no—
139(C2×D3)⋊Dic3—31no—
140(C2×D3)⋊Dic3—31no—
141Dic32—31no—
142Dic32—31no—
143C2×C4×D9—31no—
144C2×C4×D9—31no—
145C2×C4×D9C2×D311yescharacteristic
146C2×Dic18—31no—
147C2×Dic18—31no—
148C2×Dic18—31no—
149C2×Dic18—61no—
150C2×Dic18—61no—
151C2×Dic18C2×D311yescharacteristic
152C9×C4⋊C4—21no—
153C9×C4⋊C4—31no—
154C9×C4⋊C4—31no—
155C9×C4⋊C4—31no—
156C9×C4⋊C4—31no—
157C9×C4⋊C4—31no—
158C9×C4⋊C4C2×D311yescharacteristic
159C42×C9—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4C2×D311yescharacteristic
163(C2×D9)⋊C4C2×D311yescharacteristic
164C4⋊Dic9—31no—
165C4⋊Dic9—31no—
166C4⋊Dic9—31no—
167C4⋊Dic9—31no—
168C4⋊Dic9—31no—
169C4⋊Dic9—31no—
170C4⋊Dic9—61no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—61no—
178Dic9⋊C4—61no—
179Dic9⋊C4C2×D311yescharacteristic
180Dic9⋊C4C2×D311yescharacteristic
181C4×Dic9—31no—
182C4×Dic9—31no—
183C4×Dic9—31no—
184C4×Dic9—61no—
185C22×C3×C9C2411yescharacteristic
186C2×C3×D9—21no—
187C2×C3×D9—21no—
188C2×C3×D9C2×Q812yes—
189C3×C4×C9—21no—
190C3×C4×C9—21no—
191C3×C4×C9—21no—
192C3⋊Dic9—21no—
193C3⋊Dic9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9C2×Q812yes—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3C2×Q812yes—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×#125—31no—
204C3×C22⋊Q8—91no—
205#944—91no—
206#944—91no—
207#924—91no—
208C2×Dic3⋊C4—91no—
209#897—91no—
210#897—91no—
211#896—31no—
212#896—91no—
213#892—91no—
214D3×C4⋊C4—91no—
215#890—91no—
216#889—91no—
217#889—91no—
218#889—91no—
219#889—91no—
220#889—91no—
221#889D911yescharacteristic, complemented
222#888—91no—
223#888—91no—
224#888—91no—
225#888—91no—
226#888—91no—
227#888—181no—
228#878—91no—
229#878—91no—
230#877—91no—
231#877—91no—
232#872—91no—
233#869—91no—
234C22×C3×D3—31no—
235C2×C32×C4C22×D311yescharacteristic
236C2×C32×C4C22×D311yescharacteristic
237C2×C32×C4C22×D311yescharacteristic
238C2×C3⋊Dic3—31no—
239C2×C3⋊Dic3—31no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242#634—121no—
243#634—121no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3C22×D311yescharacteristic
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C3×C4×D3—121no—
252C3×Dic6—121no—
253#627—121no—
254D3×Dic3—34no—
255D3×Dic3—121no—
256C22×D9C22×D311yescharacteristic
257C2×C4×C9—21no—
258C2×C4×C9—21no—
259C2×C4×C9—21no—
260C2×C4×C9—31no—
261C2×C4×C9—31no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9C22×D311yescharacteristic
265C2×C4×C9C22×D311yescharacteristic
266C2×C4×C9C22×D311yescharacteristic
267C2×Dic9—31no—
268C2×Dic9—31no—
269C2×Dic9—31no—
270C2×Dic9—31no—
271C2×Dic9—61no—
272C2×Dic9—61no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9C22×D311yescharacteristic
276C2×Dic9C22×D311yescharacteristic
277C2×Dic9C22×D311yescharacteristic
278C4×D9—34no—
279C4×D9—41no—
280C4×D9—121no—
281Dic18—41no—
282Dic18—121no—
283Dic18—121no—
284Dic18—121no—
285Dic18—122no—
286Dic18—122no—
287#544—271noSylow
288C2×C3×C9C22×Q811yescharacteristic
289C2×C3×C9H(5,𝔽2)11yescharacteristic
290C2×C3×C92−1+411yescharacteristic
291C3×D9—22no—
292C2×C3×Q8—91no—
293C22×C3×C4—91no—
294C22×Dic3—91no—
295C22×Dic3—91no—
296C2×C4×D3—31no—
297C2×C4×D3—91no—
298C2×C4×D3—91no—
299C2×Dic6—31no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6—91no—
306C2×Dic6—181no—
307C2×Dic6—181no—
308C3×C4⋊C4—21no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—31no—
311C3×C4⋊C4—31no—
312C3×C4⋊C4—31no—
313C3×C4⋊C4—31no—
314C3×C4⋊C4—91no—
315C3×C4⋊C4—91no—
316C3×C4⋊C4C2×D911yescharacteristic
317C3×C4⋊C4D3211yescharacteristic
318C3×C22⋊C4—91no—
319C3×C22⋊C4—91no—
320C3×C42—31no—
321#217—91no—
322#217—91no—
323(C2×D3)⋊C4—31no—
324(C2×D3)⋊C4—31no—
325(C2×D3)⋊C4—91no—
326(C2×D3)⋊C4—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—181no—
336C4⋊Dic3C2×D911yescharacteristic
337Dic3⋊C4—31no—
338Dic3⋊C4—31no—
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⋊C4C2×D911yescharacteristic
358Dic3⋊C4C2×D911yescharacteristic
359Dic3⋊C4C2×D911yescharacteristic
360Dic3⋊C4C2×D911yescharacteristic
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—181no—
368C4×Dic3C2×D911yescharacteristic
369C22×C32C23×D311yescharacteristic
370C2×C3×D3—32no—
371C2×C3×D3—61no—
372C2×C3×D3—61no—
373C32×C4—21no—
374C32×C4—21no—
375C32×C4—21no—
376C3⋊Dic3—32no—
377C3⋊Dic3—61no—
378C3⋊Dic3—61no—
379C3⋊Dic3—61no—
380C3×Dic3—21no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—61no—
384C3×Dic3—61no—
385C3×Dic3—61no—
386C3×Dic3D3×Q812yes—
387C22×C9—21no—
388C22×C9C23×D311yescharacteristic
389C2×D9—21no—
390C2×D9—21no—
391C2×D9D3×Q812yes—
392C4×C9—22no—
393C4×C9—21no—
394C4×C9—21no—
395C4×C9—21no—
396C4×C9—32no—
397C4×C9—41no—
398C4×C9—41no—
399C4×C9—61no—
400C4×C9—61no—
401C4×C9—61no—
402Dic9—21no—
403Dic9—21no—
404Dic9—21no—
405Dic9—32no—
406Dic9—62no—
407Dic9—62no—
408Dic9—61no—
409Dic9—61no—
410Dic9—61no—
411Dic9—121no—
412Dic9—121no—
413C4⋊Q8—271no—
414C4⋊Q8—271no—
415C4⋊Q8—271no—
416C4⋊Q8—271no—
417#125—31no—
418#125—271no—
419#125—271no—
420#125—271no—
421C22⋊Q8—91no—
422C22⋊Q8—271no—
423C22⋊Q8—271no—
424C22⋊Q8—271no—
425#117—271no—
426#117—271no—
427C2×C4⋊C4—271no—
428C3×C9#54411yescharacteristic, complemented, Sylow
429C23×C3—91no—
430C22×D3—31no—
431C3×Q8—361no—
432C2×C3×C4—21no—
433C2×C3×C4—21no—
434C2×C3×C4—21no—
435C2×C3×C4—31no—
436C2×C3×C4—31no—
437C2×C3×C4—31no—
438C2×C3×C4—31no—
439C2×C3×C4—91no—
440C2×C3×C4—91no—
441C2×C3×C4—91no—
442C2×C3×C4—361no—
443C2×C3×C4C22×D911yescharacteristic
444C2×C3×C4C22×D911yescharacteristic
445C2×C3×C4C22×D911yescharacteristic
446C2×C3×C4C2×D3211yescharacteristic
447C2×C3×C4C2×D3211yescharacteristic
448C2×C3×C4C2×D3211yescharacteristic
449C2×Dic3—31no—
450C2×Dic3—31no—
451C2×Dic3—31no—
452C2×Dic3—91no—
453C2×Dic3—91no—
454C2×Dic3—94no—
455C2×Dic3—91no—
456C2×Dic3—91no—
457C2×Dic3—91no—
458C2×Dic3—91no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—181no—
462C2×Dic3—181no—
463C2×Dic3—181no—
464C2×Dic3—181no—
465C2×Dic3—361no—
466C2×Dic3C22×D911yescharacteristic
467C2×Dic3C22×D911yescharacteristic
468C2×Dic3C22×D911yescharacteristic
469C2×Dic3C22×D911yescharacteristic
470C4×D3—94no—
471C4×D3—121no—
472C4×D3—361no—
473Dic6—121no—
474Dic6—361no—
475Dic6—361no—
476Dic6—361no—
477Dic6—361no—
478Dic6—361no—
479Dic6—361no—
480Dic6—362no—
481Dic6—362no—
482C2×C32C2×D3×Q811yescharacteristic
483C2×C32#100911yescharacteristic
484C2×C32#101011yescharacteristic
485C3×D3—62no—
486C2×C9—21no—
487C2×C9—21no—
488C2×C9—21no—
489C2×C9#100411yescharacteristic
490C2×C9C2×D3×Q811yescharacteristic
491C2×C9#101011yescharacteristic
492D9—22no—
493C2×Q8—91no—
494C2×Q8—271no—
495C2×Q8—271no—
496C2×Q8—271no—
497C22×C4—91no—
498C22×C4—271no—
499C22×C4—271no—
500C4⋊C4—31no—
501C4⋊C4—31no—
502C4⋊C4—31no—
503C4⋊C4—31no—
504C4⋊C4—31no—
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⋊C4D3×D911yescharacteristic
520C22⋊C4—91no—
521C22⋊C4—91no—
522C22⋊C4—271no—
523C22⋊C4—271no—
524C42—31no—
525C42—271no—
526C42—271no—
527C42—271no—
528C22×C3—21no—
529C22×C3—92no—
530C22×C3—181no—
531C22×C3—181no—
532C22×C3C23×D911yescharacteristic
533C22×C3C22×D3211yescharacteristic
534C2×D3—32no—
535C2×D3—61no—
536C2×D3—61no—
537C3×C4—22no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—32no—
545C3×C4—41no—
546C3×C4—41no—
547C3×C4—61no—
548C3×C4—61no—
549C3×C4—61no—
550C3×C4—181no—
551C3×C4—181no—
552C3×C4—181no—
553Dic3—21no—
554Dic3—21no—
555Dic3—21no—
556Dic3—61no—
557Dic3—61no—
558Dic3—61no—
559Dic3—92no—
560Dic3—92no—
561Dic3—182no—
562Dic3—182no—
563Dic3—181no—
564Dic3—181no—
565Dic3—181no—
566Dic3—181no—
567Dic3—181no—
568Dic3—181no—
569Dic3—361no—
570Dic3—361no—
571Dic3Q8×D912yes—
572C32#568911yescharacteristic
573C9—21no—
574C9#562311yescharacteristic, complemented
575C23—91no—
576Q8—361no—
577Q8—1081no—
578Q8—1081no—
579Q8—1081no—
580C2×C4—31no—
581C2×C4—31no—
582C2×C4—31no—
583C2×C4—31no—
584C2×C4—91no—
585C2×C4—91no—
586C2×C4—91no—
587C2×C4—274no—
588C2×C4—271no—
589C2×C4—271no—
590C2×C4—271no—
591C2×C4—271no—
592C2×C4—361no—
593C2×C4—1081no—
594C2×C4C2×D3×D911yescharacteristic
595C2×C4C2×D3×D911yescharacteristic
596C2×C4C2×D3×D911yescharacteristic
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—182no—
601C2×C3C2×Q8×D911yescharacteristic
602C2×C3#6371511yescharacteristic
603C2×C3#6371611yescharacteristic
604C2×C3#6429811yescharacteristic
605C2×C3#6431411yescharacteristic
606C2×C3D32×Q811yescharacteristic
607D3—62no—
608C22—92no—
609C22—181no—
610C22—181no—
611C22C22×D3×D911yescharacteristic
612C4—21no—
613C4—21no—
614C4—21no—
615C4—32no—
616C4—61no—
617C4—61no—
618C4—61no—
619C4—181no—
620C4—181no—
621C4—181no—
622C4—272no—
623C4—541no—
624C4—541no—
625C4—541no—
626C3—21no—
627C3#1058979111yescharacteristic, complemented
628C3#1059439011yescharacteristic
629C2—182no—
630C2#1172246511yescharacteristic
631C2#1172250111yescharacteristic
632C2D3×Q8×D911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3×D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#544
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. #11721744C2semidirect
3. C9×#889C2semidirect
10. #11720932C2semidirect
11. #11720928C2semidirect
12. #11720918C2semidirect
13. #11720914C2semidirect
14. #11720905C2semidirect
15. #11720904C2semidirect
104. Dic3×D9Q8semidirect
221. #889D9semidirect
428. C3×C9#544semidirect
574. C9#5623semidirect
627. C3#10589791semidirect

Maximal subgroups

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

subgroupindex
#117217442
C3×#634582
C9×#8892
#117209962
#117209912
#117209752
#117209742
D9×Dic3⋊C42
#117209452
#117209322
#117209282
#117209182
#117209142
#117209052
#117209042
#105943903
#105897913