Finite Group Databasebeta

← #13148845

Subgroups — #13148845

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721745C211yesmaximal, characteristic, complemented
2C3×#63459C211yesmaximal, characteristic
3C9×#889C211yesmaximal, characteristic, complemented
4#11720999C211yesmaximal, characteristic
5#11720991C211yesmaximal, characteristic
6#11720975C211yesmaximal, characteristic
7#11720974C211yesmaximal, characteristic
8D9×Dic3⋊C4C211yesmaximal, characteristic
9#11720971C211yesmaximal, characteristic, complemented
10#11720970C211yesmaximal, characteristic, complemented
11#11720939C211yesmaximal, characteristic
12#11720919C211yesmaximal, characteristic, complemented
13#11720916C211yesmaximal, characteristic, complemented
14#11720904C211yesmaximal, characteristic, complemented
15#11720904C211yesmaximal, characteristic, complemented
16#10594389—31nomaximal
17#10589791—31nomaximal
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×Dic3×D9C2211yescharacteristic
21C2×Dic3×D9C2211yescharacteristic
22C2×#6189C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C2×#6189C2211yescharacteristic
25C3×C9×C4⋊C4C2211yescharacteristic
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27#88534C2211yescharacteristic
28#88534C2211yescharacteristic
29#88534C2211yescharacteristic
30#88534C2211yescharacteristic
31C4×C3⋊Dic9C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C3×(C2×D9)⋊C4C2211yescharacteristic
34C9×C4⋊Dic3C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C9×Dic3⋊C4C2211yescharacteristic
37C9×Dic3⋊C4C2211yescharacteristic
38C9×Dic3⋊C4C2211yescharacteristic
39C4×C9×Dic3C2211yescharacteristic
40C3×C4⋊Dic9C2211yescharacteristic
41C3×C4⋊Dic9C2211yescharacteristic
42(C2×D9)⋊Dic3C2211yescharacteristic
43(C2×D9)⋊Dic3C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45Dic3⋊Dic9C2211yescharacteristic
46Dic3⋊Dic9C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49#88442C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64098—31no—
54C3×#897—31no—
55C3×#889D311yescharacteristic
56#63919—31no—
57#63918—31no—
58#63899—31no—
59#63893—31no—
60#63878—31no—
61#63877—31no—
62D3×Dic3⋊C4—31no—
63#63851—31no—
64#63847—31no—
65#63846—31no—
66#63839—31no—
67#63839—31no—
68C9×#125—31no—
69#63459—31no—
70#63459D311yescharacteristic
71#63458—31no—
72#63458—31no—
73#63454—31no—
74C4⋊C4×D9—31no—
75#63452—31no—
76#63451—31no—
77#63451—31no—
78#63451—61no—
79#63450—31no—
80#63450—31no—
81#63450—31no—
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
92C2×C9×Dic3C2311yescharacteristic
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C3×Dic9C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C3×C4×D9—41no—
100C3×Dic18—41no—
101Dic3×D9—41no—
102Dic3×D9Q814yescomplemented
103#6189—41no—
104#6189—41no—
105#6189—41no—
106#5689—91no—
107#5623—91no—
108C2×C3×C4×D3—31no—
109C2×C3×Dic6—31no—
110C2×#627—31no—
111C2×#627—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—
119#3750—31no—
120#3750—31no—
121C4×C3⋊Dic3—31no—
122C3×(C2×D3)⋊C4—31no—
123C3×(C2×D3)⋊C4—31no—
124C3×C4⋊Dic3—31no—
125C3×C4⋊Dic3—31no—
126C3×C4⋊Dic3C2×D311yescharacteristic
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×Dic3⋊C4C2×D311yescharacteristic
129C3×Dic3⋊C4C2×D311yescharacteristic
130C3×Dic3⋊C4C2×D311yescharacteristic
131C3×C4×Dic3C2×D311yescharacteristic
132#3724—31no—
133#3724—31no—
134#3724—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×Dic18C2×D311yescharacteristic
150C9×C4⋊C4—21no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C4⋊C4—31no—
154C9×C4⋊C4—31no—
155C9×C4⋊C4—31no—
156C9×C4⋊C4C2×D311yescharacteristic
157C42×C9—31no—
158(C2×D9)⋊C4—31no—
159(C2×D9)⋊C4—31no—
160(C2×D9)⋊C4C2×D311yescharacteristic
161(C2×D9)⋊C4C2×D311yescharacteristic
162C4⋊Dic9—31no—
163C4⋊Dic9—31no—
164C4⋊Dic9—31no—
165C4⋊Dic9—31no—
166C4⋊Dic9—61no—
167C4⋊Dic9C2×D311yescharacteristic
168C4⋊Dic9C2×D311yescharacteristic
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
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⋊C4—61no—
180Dic9⋊C4—61no—
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—31no—
210#897—91no—
211#896—91no—
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#889—181no—
222#889D911yescharacteristic, complemented
223#888—91no—
224#888—91no—
225#888—91no—
226#888—91no—
227#888—91no—
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—
242C2×C3×Dic3—31no—
243C2×C3×Dic3—31no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3C22×D311yescharacteristic
246C2×C3×Dic3C22×D311yescharacteristic
247C2×C3×Dic3C22×D311yescharacteristic
248C2×C3×Dic3C22×D311yescharacteristic
249C3×C4×D3—121no—
250C3×Dic6—121no—
251#627—121no—
252#627—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—
285#544—271noSylow
286C2×C3×C9C22×Q811yescharacteristic
287C2×C3×C9H(5,𝔽2)11yescharacteristic
288C2×C3×C92−1+411yescharacteristic
289C3×D9—22no—
290C2×C3×Q8—91no—
291C22×C3×C4—91no—
292C22×Dic3—91no—
293C22×Dic3—91no—
294C2×C4×D3—31no—
295C2×C4×D3—91no—
296C2×C4×D3—91no—
297C2×Dic6—31no—
298C2×Dic6—91no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6—91no—
304C3×C4⋊C4—21no—
305C3×C4⋊C4—31no—
306C3×C4⋊C4—31no—
307C3×C4⋊C4—31no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C4⋊C4C2×D911yescharacteristic
313C3×C4⋊C4D3211yescharacteristic
314C3×C22⋊C4—91no—
315C3×C22⋊C4—91no—
316C3×C42—31no—
317#217—91no—
318#217—91no—
319(C2×D3)⋊C4—31no—
320(C2×D3)⋊C4—31no—
321(C2×D3)⋊C4—91no—
322(C2×D3)⋊C4—91no—
323C4⋊Dic3—31no—
324C4⋊Dic3—31no—
325C4⋊Dic3—91no—
326C4⋊Dic3—91no—
327C4⋊Dic3—91no—
328C4⋊Dic3—91no—
329C4⋊Dic3—91no—
330C4⋊Dic3—91no—
331C4⋊Dic3—181no—
332C4⋊Dic3C2×D911yescharacteristic
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—181no—
352Dic3⋊C4—181no—
353Dic3⋊C4—181no—
354Dic3⋊C4—181no—
355Dic3⋊C4C2×D911yescharacteristic
356Dic3⋊C4C2×D911yescharacteristic
357Dic3⋊C4C2×D911yescharacteristic
358Dic3⋊C4C2×D911yescharacteristic
359C4×Dic3—91no—
360C4×Dic3—91no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—181no—
366C4×Dic3C2×D911yescharacteristic
367C22×C32C23×D311yescharacteristic
368C2×C3×D3—32no—
369C2×C3×D3—61no—
370C2×C3×D3—61no—
371C32×C4—21no—
372C32×C4—21no—
373C32×C4—21no—
374C3⋊Dic3—32no—
375C3⋊Dic3—61no—
376C3⋊Dic3—61no—
377C3⋊Dic3—61no—
378C3×Dic3—21no—
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—61no—
382C3×Dic3—61no—
383C3×Dic3—61no—
384C3×Dic3D3×Q812yes—
385C22×C9—21no—
386C22×C9C23×D311yescharacteristic
387C2×D9—21no—
388C2×D9—21no—
389C2×D9D3×Q812yes—
390C4×C9—21no—
391C4×C9—21no—
392C4×C9—21no—
393C4×C9—32no—
394C4×C9—41no—
395C4×C9—41no—
396C4×C9—41no—
397C4×C9—61no—
398C4×C9—61no—
399C4×C9—61no—
400Dic9—21no—
401Dic9—21no—
402Dic9—21no—
403Dic9—32no—
404Dic9—62no—
405Dic9—61no—
406Dic9—61no—
407Dic9—61no—
408Dic9—121no—
409Dic9—121no—
410Dic9—121no—
411C4⋊Q8—271no—
412C4⋊Q8—271no—
413C4⋊Q8—271no—
414C4⋊Q8—271no—
415#125—31no—
416#125—271no—
417#125—271no—
418#125—271no—
419C22⋊Q8—91no—
420C22⋊Q8—271no—
421C22⋊Q8—271no—
422C22⋊Q8—271no—
423#117—271no—
424#117—271no—
425C2×C4⋊C4—271no—
426C3×C9#54411yescharacteristic, complemented, Sylow
427C23×C3—91no—
428C22×D3—31no—
429C3×Q8—361no—
430C2×C3×C4—21no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—31no—
434C2×C3×C4—31no—
435C2×C3×C4—31no—
436C2×C3×C4—31no—
437C2×C3×C4—91no—
438C2×C3×C4—91no—
439C2×C3×C4—91no—
440C2×C3×C4—361no—
441C2×C3×C4C22×D911yescharacteristic
442C2×C3×C4C22×D911yescharacteristic
443C2×C3×C4C22×D911yescharacteristic
444C2×C3×C4C2×D3211yescharacteristic
445C2×C3×C4C2×D3211yescharacteristic
446C2×C3×C4C2×D3211yescharacteristic
447C2×Dic3—31no—
448C2×Dic3—31no—
449C2×Dic3—31no—
450C2×Dic3—91no—
451C2×Dic3—91no—
452C2×Dic3—94no—
453C2×Dic3—91no—
454C2×Dic3—91no—
455C2×Dic3—91no—
456C2×Dic3—91no—
457C2×Dic3—91no—
458C2×Dic3—91no—
459C2×Dic3—181no—
460C2×Dic3—181no—
461C2×Dic3—181no—
462C2×Dic3—181no—
463C2×Dic3—361no—
464C2×Dic3C22×D911yescharacteristic
465C2×Dic3C22×D911yescharacteristic
466C2×Dic3C22×D911yescharacteristic
467C2×Dic3C22×D911yescharacteristic
468C4×D3—94no—
469C4×D3—121no—
470C4×D3—361no—
471Dic6—121no—
472Dic6—361no—
473Dic6—361no—
474Dic6—361no—
475Dic6—361no—
476Dic6—361no—
477Dic6—361no—
478C2×C32C2×D3×Q811yescharacteristic
479C2×C32#100911yescharacteristic
480C2×C32#101011yescharacteristic
481C3×D3—62no—
482C2×C9—21no—
483C2×C9—21no—
484C2×C9—21no—
485C2×C9#100411yescharacteristic
486C2×C9C2×D3×Q811yescharacteristic
487C2×C9#101011yescharacteristic
488D9—22no—
489C2×Q8—91no—
490C2×Q8—271no—
491C2×Q8—271no—
492C2×Q8—271no—
493C22×C4—91no—
494C22×C4—271no—
495C22×C4—271no—
496C4⋊C4—31no—
497C4⋊C4—31no—
498C4⋊C4—31no—
499C4⋊C4—31no—
500C4⋊C4—31no—
501C4⋊C4—91no—
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⋊C4—271no—
515C4⋊C4D3×D911yescharacteristic
516C22⋊C4—91no—
517C22⋊C4—91no—
518C22⋊C4—271no—
519C22⋊C4—271no—
520C42—31no—
521C42—271no—
522C42—271no—
523C42—271no—
524C22×C3—21no—
525C22×C3—92no—
526C22×C3—181no—
527C22×C3—181no—
528C22×C3C23×D911yescharacteristic
529C22×C3C22×D3211yescharacteristic
530C2×D3—32no—
531C2×D3—61no—
532C2×D3—61no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—32no—
540C3×C4—41no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4—61no—
544C3×C4—61no—
545C3×C4—61no—
546C3×C4—181no—
547C3×C4—181no—
548C3×C4—181no—
549Dic3—21no—
550Dic3—21no—
551Dic3—21no—
552Dic3—61no—
553Dic3—61no—
554Dic3—61no—
555Dic3—92no—
556Dic3—92no—
557Dic3—182no—
558Dic3—181no—
559Dic3—181no—
560Dic3—181no—
561Dic3—181no—
562Dic3—181no—
563Dic3—181no—
564Dic3—361no—
565Dic3—361no—
566Dic3—361no—
567Dic3Q8×D912yes—
568C32#568911yescharacteristic
569C9—21no—
570C9#562311yescharacteristic, complemented
571C23—91no—
572Q8—361no—
573Q8—1081no—
574Q8—1081no—
575Q8—1081no—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—31no—
579C2×C4—31no—
580C2×C4—91no—
581C2×C4—91no—
582C2×C4—91no—
583C2×C4—274no—
584C2×C4—271no—
585C2×C4—271no—
586C2×C4—271no—
587C2×C4—271no—
588C2×C4—361no—
589C2×C4—1081no—
590C2×C4C2×D3×D911yescharacteristic
591C2×C4C2×D3×D911yescharacteristic
592C2×C4C2×D3×D911yescharacteristic
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—182no—
597C2×C3C2×Q8×D911yescharacteristic
598C2×C3#6371511yescharacteristic
599C2×C3#6371611yescharacteristic
600C2×C3#6430711yescharacteristic
601C2×C3#6430911yescharacteristic
602C2×C3D32×Q811yescharacteristic
603D3—62no—
604C22—92no—
605C22—181no—
606C22—181no—
607C22C22×D3×D911yescharacteristic
608C4—21no—
609C4—21no—
610C4—21no—
611C4—32no—
612C4—61no—
613C4—61no—
614C4—61no—
615C4—181no—
616C4—181no—
617C4—181no—
618C4—272no—
619C4—541no—
620C4—541no—
621C4—541no—
622C3—21no—
623C3#1058979111yescharacteristic, complemented
624C3#1059438911yescharacteristic
625C2—182no—
626C2#1172247711yescharacteristic
627C2#1172249411yescharacteristic
628C2D3×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. #11721745C2semidirect
3. C9×#889C2semidirect
9. #11720971C2semidirect
10. #11720970C2semidirect
12. #11720919C2semidirect
13. #11720916C2semidirect
14. #11720904C2semidirect
15. #11720904C2semidirect
102. Dic3×D9Q8semidirect
222. #889D9semidirect
426. C3×C9#544semidirect
570. C9#5623semidirect
623. C3#10589791semidirect

Maximal subgroups

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

subgroupindex
#117217452
C3×#634592
C9×#8892
#117209992
#117209912
#117209752
#117209742
D9×Dic3⋊C42
#117209712
#117209702
#117209392
#117209192
#117209162
#117209042
#105943893
#105897913