Finite Group Databasebeta

← #13148840

Subgroups — #13148840

Order 1728.

736 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721743C211yesmaximal, characteristic, complemented
2C3×#63458C211yesmaximal, characteristic
3C9×#889C211yesmaximal, characteristic, complemented
4#11720999C211yesmaximal, characteristic
5#11720996C211yesmaximal, characteristic
6#11720975C211yesmaximal, characteristic
7#11720975C211yesmaximal, characteristic
8C4×Dic3×D9C211yesmaximal, characteristic
9#11720939C211yesmaximal, characteristic
10#11720925C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#11720905C211yesmaximal, characteristic, complemented
13#11720904C211yesmaximal, characteristic, complemented
14#11720902C211yesmaximal, characteristic, complemented
15#11720902C211yesmaximal, characteristic, complemented
16#10594388—31nomaximal
17#10589790—31nomaximal
18C2×#6226C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×Dic18C2211yescharacteristic
21C2×Dic3×D9C2211yescharacteristic
22C2×Dic3×D9C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C2×#6189C2211yescharacteristic
25C3×C9×C4⋊C4C2211yescharacteristic
26#88534C2211yescharacteristic
27#88534C2211yescharacteristic
28C4×C3⋊Dic9C2211yescharacteristic
29C4×C3⋊Dic9C2211yescharacteristic
30C4×C3⋊Dic9C2211yescharacteristic
31C3×(C2×D9)⋊C4C2211yescharacteristic
32C3×(C2×D9)⋊C4C2211yescharacteristic
33C9×C4⋊Dic3C2211yescharacteristic
34C9×Dic3⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C9×Dic3⋊C4C2211yescharacteristic
37C9×Dic3⋊C4C2211yescharacteristic
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#64096—31no—
54C3×#896—31no—
55C3×#889D311yescharacteristic
56#63899—31no—
57#63896—31no—
58#63878—31no—
59#63878—31no—
60C4×D3×Dic3—31no—
61#63867—31no—
62#63851—31no—
63#63843—31no—
64#63840—31no—
65#63839—31no—
66#63838—31no—
67#63838—31no—
68C9×#125—31no—
69#63458—31no—
70#63458—31no—
71#63458—31no—
72#63458D311yescharacteristic
73#63454—31no—
74#63454—31no—
75#63450—31no—
76#63450—31no—
77#63449—31no—
78#63449—31no—
79#63449—31no—
80#63449—31no—
81#63449—61no—
82C42×D9—31no—
83#63432—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—22no—
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C3×C4×D9—22no—
101C3×Dic18—22no—
102Dic3×D9—22no—
103Dic3×D9Q814yescomplemented
104#6189—22no—
105#6189—22no—
106#5688—91no—
107#5622—91no—
108C2×#634—31no—
109C2×C3×C4×D3—31no—
110C2×C3×Dic6—31no—
111C2×#627—31no—
112C2×#627—31no—
113C2×D3×Dic3—31no—
114C2×D3×Dic3—31no—
115C32×C4⋊C4C2×D311yescharacteristic
116#3750—31no—
117#3750—31no—
118C4×C3⋊Dic3—31no—
119C4×C3⋊Dic3—31no—
120C4×C3⋊Dic3—31no—
121C3×(C2×D3)⋊C4—31no—
122C3×(C2×D3)⋊C4—31no—
123C3×C4⋊Dic3C2×D311yescharacteristic
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4C2×D311yescharacteristic
127C3×Dic3⋊C4C2×D311yescharacteristic
128C3×Dic3⋊C4C2×D311yescharacteristic
129C3×Dic3⋊C4C2×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×C4×D9—31no—
144C2×C4×D9—31no—
145C2×C4×D9C2×D311yescharacteristic
146C2×Dic18—31no—
147C2×Dic18—31no—
148C2×Dic18—31no—
149C2×Dic18—61no—
150C2×Dic18C2×D311yescharacteristic
151C9×C4⋊C4—21no—
152C9×C4⋊C4—31no—
153C9×C4⋊C4—31no—
154C9×C4⋊C4—31no—
155C9×C4⋊C4—31no—
156C9×C4⋊C4—31no—
157C9×C4⋊C4C2×D311yescharacteristic
158C42×C9—31no—
159(C2×D9)⋊C4—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4C2×D311yescharacteristic
162(C2×D9)⋊C4C2×D311yescharacteristic
163C4⋊Dic9—31no—
164C4⋊Dic9—31no—
165Dic9⋊C4—31no—
166Dic9⋊C4—31no—
167Dic9⋊C4—31no—
168Dic9⋊C4—31no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—61no—
172Dic9⋊C4—61no—
173Dic9⋊C4C2×D311yescharacteristic
174Dic9⋊C4C2×D311yescharacteristic
175C4×Dic9—31no—
176C4×Dic9—31no—
177C4×Dic9—31no—
178C4×Dic9—31no—
179C4×Dic9—31no—
180C4×Dic9—31no—
181C4×Dic9—31no—
182C4×Dic9—61no—
183C4×Dic9—61no—
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⋊Dic9C2×Q812yes—
195C3⋊Dic9#4112yes—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3C2×Q812yes—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9#4112yes—
203C3×#125—31no—
204C3×C22⋊Q8—91no—
205Q8×Dic3—91no—
206#944—91no—
207#924—91no—
208C2×C4×Dic3—91no—
209#896—31no—
210#896—91no—
211#896—91no—
212#896—91no—
213#892—91no—
214#892—91no—
215#889—91no—
216#889D911yescharacteristic, complemented
217#888—91no—
218#888—91no—
219#888—91no—
220#887—91no—
221#887—91no—
222#887—91no—
223#887—91no—
224#887—91no—
225#887—91no—
226#887—91no—
227#887—181no—
228#878—91no—
229#878—91no—
230#877—91no—
231#877—91no—
232C42×D3—91no—
233#870—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—62no—
243C2×C3×Dic3—31no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3C22×D311yescharacteristic
247C2×C3×Dic3C22×D311yescharacteristic
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C3×C4×D3—62no—
251C3×Dic6—62no—
252#627—62no—
253#627—62no—
254D3×Dic3—34no—
255D3×Dic3—62no—
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—22no—
279C4×D9—34no—
280C4×D9—62no—
281Dic18—22no—
282Dic18—62no—
283Dic18—62no—
284Dic18—64no—
285Dic18—62no—
286#533—271noSylow
287C2×C3×C9C22×Q811yescharacteristic
288C2×C3×C9C2×#4111yescharacteristic
289C2×C3×C9C2×#4111yescharacteristic
290C3×D9—22no—
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×Dic6—91no—
304C2×Dic6—91no—
305C2×Dic6—181no—
306C3×C4⋊C4—21no—
307C3×C4⋊C4—31no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—31no—
311C3×C4⋊C4—31no—
312C3×C4⋊C4—91no—
313C3×C4⋊C4—91no—
314C3×C4⋊C4C2×D911yescharacteristic
315C3×C4⋊C4D3211yescharacteristic
316C3×C22⋊C4—91no—
317C3×C22⋊C4—91no—
318C3×C42—31no—
319#217—91no—
320#217—91no—
321(C2×D3)⋊C4—31no—
322(C2×D3)⋊C4—31no—
323(C2×D3)⋊C4—91no—
324(C2×D3)⋊C4—91no—
325C4⋊Dic3—91no—
326C4⋊Dic3—91no—
327C4⋊Dic3—91no—
328C4⋊Dic3—91no—
329C4⋊Dic3C2×D911yescharacteristic
330Dic3⋊C4—31no—
331Dic3⋊C4—31no—
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—181no—
345Dic3⋊C4—181no—
346Dic3⋊C4C2×D911yescharacteristic
347Dic3⋊C4C2×D911yescharacteristic
348Dic3⋊C4C2×D911yescharacteristic
349Dic3⋊C4C2×D911yescharacteristic
350C4×Dic3—91no—
351C4×Dic3—91no—
352C4×Dic3—91no—
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—91no—
364C4×Dic3—181no—
365C4×Dic3—181no—
366C4×Dic3—181no—
367C4×Dic3C2×D911yescharacteristic
368C22×C32C23×D311yescharacteristic
369C2×C3×D3—32no—
370C2×C3×D3—61no—
371C2×C3×D3—61no—
372C32×C4—21no—
373C32×C4—21no—
374C32×C4—21no—
375C3⋊Dic3—32no—
376C3⋊Dic3—32no—
377C3⋊Dic3—61no—
378C3⋊Dic3—61no—
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—21no—
382C3×Dic3—32no—
383C3×Dic3—61no—
384C3×Dic3—61no—
385C3×Dic3D3×Q812yes—
386C22×C9—21no—
387C22×C9C23×D311yescharacteristic
388C2×D9—21no—
389C2×D9—21no—
390C2×D9D3×Q812yes—
391C4×C9—22no—
392C4×C9—21no—
393C4×C9—21no—
394C4×C9—21no—
395C4×C9—32no—
396C4×C9—41no—
397C4×C9—41no—
398C4×C9—61no—
399C4×C9—61no—
400C4×C9—61no—
401Dic9—21no—
402Dic9—21no—
403Dic9—32no—
404Dic9—32no—
405Dic9—62no—
406Dic9—62no—
407Dic9—62no—
408Dic9—61no—
409Dic9—61no—
410Dic9—121no—
411Dic9#23712yes—
412C4⋊Q8—271no—
413C4⋊Q8—271no—
414#125—31no—
415#125—271no—
416C22⋊Q8—91no—
417C22⋊Q8—271no—
418C22⋊Q8—271no—
419C22⋊Q8—271no—
420C4×Q8—271no—
421C4×Q8—271no—
422C4×Q8—271no—
423C4×Q8—271no—
424#117—271no—
425#117—271no—
426C2×C42—271no—
427C3×C9#53311yescharacteristic, complemented, Sylow
428C23×C3—91no—
429C22×D3—31no—
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—91no—
454C2×Dic3—91no—
455C2×Dic3—94no—
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—62no—
473Dic6—182no—
474Dic6—182no—
475Dic6—182no—
476Dic6—182no—
477Dic6—182no—
478Dic6—182no—
479Dic6—184no—
480C2×C32C2×D3×Q811yescharacteristic
481C2×C32D3×#4111yescharacteristic
482C2×C32D3×#4111yescharacteristic
483C3×D3—62no—
484C2×C9—21no—
485C2×C9—21no—
486C2×C9—21no—
487C2×C9C2×#23711yescharacteristic
488C2×C9C2×D3×Q811yescharacteristic
489C2×C9D3×#4111yescharacteristic
490D9—22no—
491C2×Q8—91no—
492C2×Q8—271no—
493C2×Q8—271no—
494C2×Q8—271no—
495C22×C4—91no—
496C22×C4—271no—
497C22×C4—271no—
498C4⋊C4—31no—
499C4⋊C4—31no—
500C4⋊C4—31no—
501C4⋊C4—31no—
502C4⋊C4—31no—
503C4⋊C4—91no—
504C4⋊C4—91no—
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⋊C4D3×D911yescharacteristic
514C22⋊C4—91no—
515C22⋊C4—91no—
516C22⋊C4—271no—
517C22⋊C4—271no—
518C42—31no—
519C42—271no—
520C42—271no—
521C42—271no—
522C42—271no—
523C42—271no—
524C42—271no—
525C42—271no—
526C22×C3—21no—
527C22×C3—92no—
528C22×C3—181no—
529C22×C3—181no—
530C22×C3C23×D911yescharacteristic
531C22×C3C22×D3211yescharacteristic
532C2×D3—32no—
533C2×D3—61no—
534C2×D3—61no—
535C3×C4—22no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—32no—
543C3×C4—41no—
544C3×C4—41no—
545C3×C4—61no—
546C3×C4—61no—
547C3×C4—61no—
548C3×C4—92no—
549C3×C4—181no—
550C3×C4—181no—
551Dic3—21no—
552Dic3—21no—
553Dic3—21no—
554Dic3—32no—
555Dic3—61no—
556Dic3—61no—
557Dic3—92no—
558Dic3—92no—
559Dic3—92no—
560Dic3—92no—
561Dic3—182no—
562Dic3—182no—
563Dic3—182no—
564Dic3—181no—
565Dic3—181no—
566Dic3—181no—
567Dic3—181no—
568Dic3—361no—
569Dic3Q8×D912yes—
570C32#568811yescharacteristic
571C9—21no—
572C9#562211yescharacteristic, complemented
573C23—91no—
574Q8—182no—
575Q8—542no—
576Q8—542no—
577Q8—542no—
578C2×C4—31no—
579C2×C4—31no—
580C2×C4—31no—
581C2×C4—31no—
582C2×C4—91no—
583C2×C4—91no—
584C2×C4—91no—
585C2×C4—182no—
586C2×C4—274no—
587C2×C4—271no—
588C2×C4—271no—
589C2×C4—271no—
590C2×C4—271no—
591C2×C4—542no—
592C2×C4C2×D3×D911yescharacteristic
593C2×C4C2×D3×D911yescharacteristic
594C2×C4C2×D3×D911yescharacteristic
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—182no—
599C2×C3C2×Q8×D911yescharacteristic
600C2×C3#41×D911yescharacteristic
601C2×C3#41×D911yescharacteristic
602C2×C3#6430611yescharacteristic
603C2×C3D3×#23711yescharacteristic
604C2×C3D32×Q811yescharacteristic
605D3—62no—
606C22—92no—
607C22—181no—
608C22—181no—
609C22C22×D3×D911yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—32no—
614C4—61no—
615C4—61no—
616C4—61no—
617C4—92no—
618C4—181no—
619C4—181no—
620C4—272no—
621C4—272no—
622C4—541no—
623C4—541no—
624C3—21no—
625C3#1058979011yescharacteristic, complemented
626C3#1059438811yescharacteristic
627C2—182no—
628C2#1172247511yescharacteristic
629C2D9×#23711yescharacteristic
630C2D3×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. #11721743C2semidirect
3. C9×#889C2semidirect
10. #11720925C2semidirect
11. #11720910C2semidirect
12. #11720905C2semidirect
13. #11720904C2semidirect
14. #11720902C2semidirect
15. #11720902C2semidirect
103. Dic3×D9Q8semidirect
216. #889D9semidirect
427. C3×C9#533semidirect
572. C9#5622semidirect
625. C3#10589790semidirect

Maximal subgroups

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

subgroupindex
#117217432
C3×#634582
C9×#8892
#117209992
#117209962
#117209752
C4×Dic3×D92
#117209392
#117209252
#117209102
#117209052
#117209042
#117209022
#105943883
#105897903