Finite Group Databasebeta

← #13148839

Subgroups — #13148839

Order 1728.

722 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×#63458C211yesmaximal, characteristic
3C9×#887C211yesmaximal, characteristic, complemented
4#11720991C211yesmaximal, characteristic
5#11720984C211yesmaximal, characteristic
6#11720974C211yesmaximal, characteristic
7#11720974C211yesmaximal, characteristic
8C4×Dic3×D9C211yesmaximal, characteristic
9#11720939C211yesmaximal, characteristic
10#11720928C211yesmaximal, characteristic, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#11720904C211yesmaximal, characteristic, complemented
13#11720903C211yesmaximal, characteristic, complemented
14#11720900C211yesmaximal, characteristic, complemented
15#11720900C211yesmaximal, characteristic, complemented
16#10594386—31nomaximal
17#10589680—31nomaximal
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic
20C2×C3×Dic18C2211yescharacteristic
21C2×Dic3×D9C2211yescharacteristic
22C2×Dic3×D9C2211yescharacteristic
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×Dic3⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C4×C9×Dic3C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C4×C9×Dic3C2211yescharacteristic
39C3×Dic9⋊C4C2211yescharacteristic
40C3×Dic9⋊C4C2211yescharacteristic
41(C2×D9)⋊Dic3C2211yescharacteristic
42(C2×D9)⋊Dic3C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47Dic9⋊Dic3C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic3×Dic9C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64098—31no—
54C3×#896—31no—
55C3×#887D311yescharacteristic
56#63893—31no—
57#63887—31no—
58#63877—31no—
59#63877—31no—
60C4×D3×Dic3—31no—
61#63870—31no—
62#63851—31no—
63Dic3×Dic6—31no—
64#63839—31no—
65#63839—31no—
66#63837—31no—
67#63837—31no—
68C4×Q8×C9—31no—
69#63508—31no—
70#63506—31no—
71#63459—31no—
72#63458—31no—
73#63458D311yescharacteristic
74#63451—31no—
75#63451—31no—
76#63451—61no—
77#63450—31no—
78#63434—31no—
79#63434—31no—
80C42×D9—31no—
81C4×Dic18—31no—
82C4×Dic18—31no—
83C4×Dic18—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—22no—
100C9×Dic6—22no—
101C3×Dic18—22no—
102Dic3×D9—22no—
103Dic3×D9Q814yescomplemented
104#6189—22no—
105#6189—22no—
106#5622—91no—
107#5578—91no—
108C2×C3×C4×D3—31no—
109C2×C3×Dic6—31no—
110C2×C3×Dic6C2×D311yescharacteristic
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×Dic3⋊C4—31no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×C4×Dic3C2×D311yescharacteristic
129C3×C4×Dic3C2×D311yescharacteristic
130C3×C4×Dic3C2×D311yescharacteristic
131#3724—31no—
132#3724—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137(C2×D3)⋊Dic3—31no—
138(C2×D3)⋊Dic3—31no—
139Dic32—31no—
140Dic32—31no—
141Dic32—31no—
142Dic32—31no—
143C2×Q8×C9—31no—
144C2×C4×D9—31no—
145C2×C4×D9—31no—
146C2×C4×D9C2×D311yescharacteristic
147C2×Dic18—31no—
148C2×Dic18—31no—
149C2×Dic18C2×D311yescharacteristic
150C9×C4⋊C4—21no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C4⋊C4C2×D311yescharacteristic
154C42×C9—31no—
155C42×C9—31no—
156C42×C9—31no—
157(C2×D9)⋊C4—31no—
158(C2×D9)⋊C4—31no—
159(C2×D9)⋊C4C2×D311yescharacteristic
160(C2×D9)⋊C4C2×D311yescharacteristic
161C4⋊Dic9—31no—
162C4⋊Dic9—31no—
163C4⋊Dic9—31no—
164C4⋊Dic9—61no—
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⋊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—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×Dic3C2×Q812yes—
199C9×Dic3#4112yes—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9#4112yes—
203C3×C22⋊Q8—91no—
204C3×C4×Q8—31no—
205#946—91no—
206Q8×Dic3—91no—
207#944—91no—
208#944—91no—
209#924—91no—
210C2×C4×Dic3—91no—
211#897—91no—
212#896—31no—
213#896—91no—
214#889—91no—
215#889—91no—
216#889—91no—
217#889—91no—
218#889—181no—
219#888—91no—
220#888—91no—
221#887—91no—
222#887—91no—
223#887D911yescharacteristic, complemented
224#878—91no—
225#878—91no—
226#877—91no—
227#877—91no—
228#872—91no—
229#872—91no—
230C42×D3—91no—
231C4×Dic6—91no—
232C4×Dic6—91no—
233C4×Dic6—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—62no—
250C3×Dic6—22no—
251C3×Dic6—62no—
252#627—62no—
253#627—62no—
254D3×Dic3—34no—
255D3×Dic3—62no—
256C22×D9C22×D311yescharacteristic
257Q8×C9—62no—
258C2×C4×C9—21no—
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×C9C22×D311yescharacteristic
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×Dic9C22×D311yescharacteristic
277C2×Dic9C22×D311yescharacteristic
278C2×Dic9C22×D311yescharacteristic
279C4×D9—22no—
280C4×D9—34no—
281C4×D9—62no—
282Dic18—22no—
283Dic18—62no—
284Dic18—62no—
285#533—271noSylow
286C2×C3×C9C22×Q811yescharacteristic
287C2×C3×C9C2×#4111yescharacteristic
288C2×C3×C9C2×#4111yescharacteristic
289C3×D9—22no—
290C2×C3×Q8—31no—
291C2×C3×Q8—91no—
292C22×C3×C4—91no—
293C22×Dic3—91no—
294C22×Dic3—91no—
295C2×C4×D3—31no—
296C2×C4×D3—91no—
297C2×C4×D3—91no—
298C2×Dic6—31no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C2×Dic6—91no—
303C2×Dic6C2×D911yescharacteristic
304C3×C4⋊C4—21no—
305C3×C4⋊C4—31no—
306C3×C4⋊C4—31no—
307C3×C4⋊C4—91no—
308C3×C4⋊C4—91no—
309C3×C4⋊C4C2×D911yescharacteristic
310C3×C4⋊C4D3211yescharacteristic
311C3×C22⋊C4—91no—
312C3×C22⋊C4—91no—
313C3×C42—31no—
314C3×C42—31no—
315C3×C42—31no—
316#217—91no—
317#217—91no—
318(C2×D3)⋊C4—31no—
319(C2×D3)⋊C4—31no—
320(C2×D3)⋊C4—91no—
321(C2×D3)⋊C4—91no—
322C4⋊Dic3—91no—
323C4⋊Dic3—91no—
324C4⋊Dic3—91no—
325C4⋊Dic3—91no—
326C4⋊Dic3—91no—
327C4⋊Dic3—91no—
328C4⋊Dic3—181no—
329Dic3⋊C4—31no—
330Dic3⋊C4—31no—
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—181no—
348Dic3⋊C4—181no—
349Dic3⋊C4—181no—
350Dic3⋊C4—181no—
351Dic3⋊C4C2×D911yescharacteristic
352Dic3⋊C4C2×D911yescharacteristic
353C4×Dic3—91no—
354C4×Dic3—91no—
355C4×Dic3—91no—
356C4×Dic3—91no—
357C4×Dic3—91no—
358C4×Dic3—91no—
359C4×Dic3—91no—
360C4×Dic3—91no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—181no—
364C4×Dic3C2×D911yescharacteristic
365C4×Dic3C2×D911yescharacteristic
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—32no—
381C3×Dic3—61no—
382C3×Dic3—61no—
383C3×Dic3#23512yes—
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—32no—
395C4×C9—41no—
396C4×C9—41no—
397C4×C9—41no—
398C4×C9—61no—
399C4×C9—61no—
400Dic9—21no—
401Dic9—21no—
402Dic9—32no—
403Dic9—62no—
404Dic9—61no—
405Dic9—61no—
406Dic9—61no—
407Dic9—121no—
408Dic9—121no—
409Dic9—121no—
410Dic9#23712yes—
411C4⋊Q8—271no—
412C4⋊Q8—271no—
413#125—271no—
414#125—271no—
415C22⋊Q8—91no—
416C22⋊Q8—271no—
417C22⋊Q8—271no—
418C22⋊Q8—271no—
419C4×Q8—31no—
420C4×Q8—271no—
421C4×Q8—271no—
422C4×Q8—271no—
423#117—271no—
424#117—271no—
425C2×C42—271no—
426C3×C9#53311yescharacteristic, complemented, Sylow
427C23×C3—91no—
428C22×D3—31no—
429C3×Q8—62no—
430C3×Q8—182no—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—21no—
434C2×C3×C4—31no—
435C2×C3×C4—31no—
436C2×C3×C4—31no—
437C2×C3×C4—31no—
438C2×C3×C4—91no—
439C2×C3×C4—91no—
440C2×C3×C4—91no—
441C2×C3×C4—182no—
442C2×C3×C4C22×D911yescharacteristic
443C2×C3×C4C22×D911yescharacteristic
444C2×C3×C4C22×D911yescharacteristic
445C2×C3×C4C2×D3211yescharacteristic
446C2×C3×C4C2×D3211yescharacteristic
447C2×C3×C4C2×D3211yescharacteristic
448C2×Dic3—31no—
449C2×Dic3—31no—
450C2×Dic3—31no—
451C2×Dic3—91no—
452C2×Dic3—91no—
453C2×Dic3—94no—
454C2×Dic3—91no—
455C2×Dic3—91no—
456C2×Dic3—91no—
457C2×Dic3—91no—
458C2×Dic3—91no—
459C2×Dic3—91no—
460C2×Dic3—181no—
461C2×Dic3—181no—
462C2×Dic3—181no—
463C2×Dic3—181no—
464C2×Dic3—182no—
465C2×Dic3C22×D911yescharacteristic
466C2×Dic3C22×D911yescharacteristic
467C2×Dic3C22×D911yescharacteristic
468C2×Dic3C22×D911yescharacteristic
469C4×D3—62no—
470C4×D3—94no—
471C4×D3—182no—
472Dic6—22no—
473Dic6—62no—
474Dic6—182no—
475Dic6—182no—
476Dic6—182no—
477Dic6—182no—
478C2×C32C2×#23511yescharacteristic
479C2×C32C2×D3×Q811yescharacteristic
480C2×C32D3×#4111yescharacteristic
481C3×D3—62no—
482C2×C9—21no—
483C2×C9—21no—
484C2×C9—21no—
485C2×C9C2×#23711yescharacteristic
486C2×C9C2×D3×Q811yescharacteristic
487C2×C9D3×#4111yescharacteristic
488D9—22no—
489C2×Q8—31no—
490C2×Q8—91no—
491C2×Q8—271no—
492C2×Q8—271no—
493C22×C4—91no—
494C22×C4—271no—
495C22×C4—271no—
496C4⋊C4—31no—
497C4⋊C4—31no—
498C4⋊C4—91no—
499C4⋊C4—91no—
500C4⋊C4—271no—
501C4⋊C4—271no—
502C4⋊C4—271no—
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⋊C4D3×D911yescharacteristic
512C22⋊C4—91no—
513C22⋊C4—91no—
514C22⋊C4—271no—
515C22⋊C4—271no—
516C42—31no—
517C42—31no—
518C42—31no—
519C42—271no—
520C42—271no—
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—32no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—61no—
545C3×C4—61no—
546C3×C4—92no—
547C3×C4—181no—
548C3×C4—181no—
549Dic3—21no—
550Dic3—21no—
551Dic3—32no—
552Dic3—61no—
553Dic3—61no—
554Dic3—92no—
555Dic3—92no—
556Dic3—182no—
557Dic3—181no—
558Dic3—181no—
559Dic3—181no—
560Dic3—181no—
561Dic3—181no—
562Dic3—181no—
563Dic3—361no—
564Dic3—361no—
565Dic3—361no—
566Dic3#369712yes—
567Dic3Q8×D912yes—
568C32#557811yescharacteristic
569C9—21no—
570C9#562211yescharacteristic, complemented
571C23—91no—
572Q8—62no—
573Q8—182no—
574Q8—542no—
575Q8—542no—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—31no—
579C2×C4—31no—
580C2×C4—91no—
581C2×C4—91no—
582C2×C4—91no—
583C2×C4—182no—
584C2×C4—274no—
585C2×C4—271no—
586C2×C4—271no—
587C2×C4—271no—
588C2×C4—271no—
589C2×C4—542no—
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×#369711yescharacteristic
598C2×C3C2×Q8×D911yescharacteristic
599C2×C3#41×D911yescharacteristic
600C2×C3D3×#23511yescharacteristic
601C2×C3D3×#23711yescharacteristic
602C2×C3D32×Q811yescharacteristic
603D3—62no—
604C22—92no—
605C22—181no—
606C22—181no—
607C22C22×D3×D911yescharacteristic
608C4—21no—
609C4—21no—
610C4—21no—
611C4—32no—
612C4—32no—
613C4—61no—
614C4—61no—
615C4—92no—
616C4—181no—
617C4—181no—
618C4—272no—
619C4—541no—
620C4—541no—
621C4—541no—
622C3—21no—
623C3#1058968011yescharacteristic, complemented
624C3#1059438611yescharacteristic
625C2—182no—
626C2D3×#369711yescharacteristic
627C2D9×#23711yescharacteristic
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#533
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×#887C2semidirect
10. #11720928C2semidirect
11. Dic3×Dic18C2semidirect
12. #11720904C2semidirect
13. #11720903C2semidirect
14. #11720900C2semidirect
15. #11720900C2semidirect
103. Dic3×D9Q8semidirect
223. #887D9semidirect
426. C3×C9#533semidirect
570. C9#5622semidirect
623. C3#10589680semidirect

Maximal subgroups

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

subgroupindex
#117217452
C3×#634582
C9×#8872
#117209912
#117209842
#117209742
C4×Dic3×D92
#117209392
#117209282
Dic3×Dic182
#117209042
#117209032
#117209002
#105943863
#105896803