Finite Group Databasebeta

← #13148043

Subgroups — #13148043

Order 1728.

696 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721725C211yesmaximal, characteristic, complemented
2C3×#63447C211yesmaximal, characteristic
3C9×#877C211yesmaximal, characteristic, complemented
4Dic3×#611C211yesmaximal, characteristic
5#11721257C211yesmaximal, characteristic
6#11721249C211yesmaximal, characteristic, complemented
7#11721246C211yesmaximal, characteristic, complemented
8#11720991C211yesmaximal, characteristic, complemented
9#11720975C211yesmaximal, characteristic, complemented
10C4×#6189C211yesmaximal, characteristic, complemented
11#11720947C211yesmaximal, characteristic, complemented
12#11720947C211yesmaximal, characteristic, complemented
13Dic3×Dic18C211yesmaximal, characteristic, complemented
14#11720904C211yesmaximal, characteristic, complemented
15#11720901C211yesmaximal, characteristic, complemented
16#10594105—31nomaximal
17#10589635—31nomaximal
18C2×C3×#611C2211yescharacteristic
19C22×C9×Dic3C2211yescharacteristic
20C2×C3×Dic18C2211yescharacteristic
21C2×Dic3×D9C2211yescharacteristic
22C2×#6189C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25#88551C2211yescharacteristic
26#88551C2211yescharacteristic
27C4⋊(C3⋊Dic9)C2211yescharacteristic
28#88534C2211yescharacteristic
29#88534C2211yescharacteristic
30C4×C3⋊Dic9C2211yescharacteristic
31C9×#217C2211yescharacteristic
32C3×#3676C2211yescharacteristic
33C3×(C2×D9)⋊C4C2211yescharacteristic
34C3×(C2×D9)⋊C4C2211yescharacteristic
35C9×Dic3⋊C4C2211yescharacteristic
36C9×Dic3⋊C4C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C4×C9×Dic3C2211yescharacteristic
39C3×C4×Dic9C2211yescharacteristic
40(C2×D9)⋊Dic3C2211yescharacteristic
41(C2×D9)⋊Dic3C2211yescharacteristic
42(C2×D9)⋊Dic3C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47Dic9⋊Dic3C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64088—31no—
54C3×#885—31no—
55C3×#877D311yescharacteristic
56Dic3×#67—31no—
57#63964—31no—
58#63959—31no—
59#63956—31no—
60C4×#627—31no—
61#63893—31no—
62#63878—31no—
63#63859—31no—
64#63859—31no—
65Dic3×Dic6—31no—
66#63839—31no—
67#63837—31no—
68C9×#117—31no—
69C4×#611—31no—
70#63486—31no—
71#63459—31no—
72#63454—31no—
73#63450—31no—
74#63449—31no—
75#63447—31no—
76#63447D311yescharacteristic
77#63441—31no—
78#63441—31no—
79#63441—61no—
80#63438—31no—
81#63438—31no—
82C4×Dic18—31no—
83C4×Dic18—31no—
84C23×C3×C9C2311yescharacteristic
85C22×C3×D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3⋊Dic9C2311yescharacteristic
89C2×C3⋊Dic9C2311yescharacteristic
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C3×#611—22no—
93C2×C9×Dic3—22no—
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C3×Dic9C2311yescharacteristic
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C3×Dic18—22no—
102Dic3×D9—22no—
103#6189—22no—
104#6189—22no—
105#5671—91no—
106#5533—91no—
107C2×C3×#67—31no—
108C22×C3×Dic3C2×D311yescharacteristic
109C2×C3×Dic6—31no—
110C2×#627—31no—
111C2×#627—31no—
112C2×D3×Dic3—31no—
113C32×C22⋊C4C2×D311yescharacteristic
114#3757—31no—
115#3757—31no—
116C4⋊(C3⋊Dic3)—31no—
117#3750—31no—
118#3750—31no—
119C4×C3⋊Dic3—31no—
120C3×#217—31no—
121C3×#217C2×D311yescharacteristic
122C3×(C2×D3)⋊C4—31no—
123C3×(C2×D3)⋊C4—31no—
124C3×Dic3⋊C4C2×D311yescharacteristic
125C3×Dic3⋊C4C2×D311yescharacteristic
126C3×C4×Dic3—31no—
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—
136(C2×D3)⋊Dic3—31no—
137(C2×D3)⋊Dic3—31no—
138(C2×D3)⋊Dic3—31no—
139Dic32—31no—
140Dic32—31no—
141Dic32—31no—
142C22×C4×C9—31no—
143C2×#611C2×D311yescharacteristic
144C2×C4×D9—31no—
145C2×Dic18—31no—
146C2×Dic18—31no—
147C2×Dic18C2×D311yescharacteristic
148C9×C4⋊C4—31no—
149C9×C4⋊C4—31no—
150C9×C22⋊C4—21no—
151C9×C22⋊C4—31no—
152C9×C22⋊C4C2×D311yescharacteristic
153C42×C9—31no—
154C42×C9—31no—
155#3676—31no—
156#3676—31no—
157#3676—61no—
158#3676—61no—
159#3676C2×D311yescharacteristic
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—31no—
163(C2×D9)⋊C4C2×D311yescharacteristic
164(C2×D9)⋊C4C2×D311yescharacteristic
165C4⋊Dic9—31no—
166C4⋊Dic9—31no—
167C4⋊Dic9—31no—
168C4⋊Dic9—61no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—61no—
177Dic9⋊C4—61no—
178C4×Dic9—31no—
179C4×Dic9—31no—
180C4×Dic9—31no—
181C4×Dic9—31no—
182C4×Dic9—61no—
183C4×Dic9C2×D311yescharacteristic
184C22×C3×C9—21no—
185C22×C3×C9—21no—
186C22×C3×C9—21no—
187C22×C3×C9C2411yescharacteristic
188C2×C3×D9—21no—
189C2×C3×D9—21no—
190C2×C3×D9—21no—
191C3×C4×C9—21no—
192C3×C4×C9—21no—
193C3⋊Dic9—21no—
194C3⋊Dic9—21no—
195C3⋊Dic9—21no—
196C3⋊Dic9—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C9×Dic3#4112yes—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9#4112yes—
204C3×#124—91no—
205C3×#117—31no—
206Q8×Dic3—91no—
207#944—91no—
208#936—91no—
209D4×Dic3—91no—
210C4×#67—91no—
211#924—91no—
212#897—91no—
213#892—91no—
214#888—91no—
215#887—91no—
216#885—31no—
217#885—91no—
218#879—91no—
219#879—91no—
220#879—91no—
221#879—91no—
222#879—91no—
223#879—91no—
224#879—181no—
225#878—91no—
226#878—91no—
227#877—91no—
228#877D911yescharacteristic, complemented
229#876—91no—
230#876—91no—
231C4×Dic6—91no—
232C4×Dic6—91no—
233C4×Dic6—91no—
234C4×Dic6—91no—
235C23×C32C22×D311yescharacteristic
236C22×C3×D3—31no—
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239C2×C3⋊Dic3—31no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C3×#67—62no—
244C2×C3×Dic3—22no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3C22×D311yescharacteristic
249C2×C3×Dic3C22×D311yescharacteristic
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252C3×Dic6—62no—
253#627—62no—
254#627—62no—
255D3×Dic3—62no—
256C23×C9—21no—
257C23×C9C22×D311yescharacteristic
258C22×D9C22×D311yescharacteristic
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×C9—62no—
266C2×C4×C9C22×D311yescharacteristic
267C2×C4×C9C22×D311yescharacteristic
268#611—22no—
269C2×Dic9—31no—
270C2×Dic9—31no—
271C2×Dic9—31no—
272C2×Dic9—31no—
273C2×Dic9—61no—
274C2×Dic9—61no—
275C2×Dic9—61no—
276C2×Dic9—61no—
277C2×Dic9C22×D311yescharacteristic
278C2×Dic9C22×D311yescharacteristic
279C2×Dic9C22×D311yescharacteristic
280C4×D9—62no—
281Dic18—22no—
282Dic18—62no—
283Dic18—62no—
284#556—271noSylow
285C2×C3×C9—41no—
286C2×C3×C9C2×#4111yescharacteristic
287C2×C3×C9C2×#4111yescharacteristic
288C2×C3×C92−1+411yescharacteristic
289C3×D9—41no—
290C2×C3×Q8—91no—
291C2×C3×D4—91no—
292C22×C3×C4—31no—
293C2×#67—31no—
294C22×Dic3—91no—
295C22×Dic3C2×D911yescharacteristic
296C2×C4×D3—91no—
297C2×Dic6—31no—
298C2×Dic6—91no—
299C2×Dic6—91no—
300C2×Dic6—91no—
301C2×Dic6—91no—
302C3×C4⋊C4—31no—
303C3×C4⋊C4—31no—
304C3×C22⋊C4—21no—
305C3×C22⋊C4—31no—
306C3×C22⋊C4—91no—
307C3×C22⋊C4—91no—
308C3×C22⋊C4—91no—
309C3×C22⋊C4C2×D911yescharacteristic
310C3×C22⋊C4D3211yescharacteristic
311C3×C42—31no—
312C3×C42—31no—
313C3×C42—91no—
314#217—31no—
315#217—91no—
316#217—91no—
317#217—91no—
318#217—91no—
319#217—91no—
320#217—91no—
321#217—91no—
322#217—181no—
323#217—181no—
324#217C2×D911yescharacteristic
325(C2×D3)⋊C4—31no—
326(C2×D3)⋊C4—31no—
327(C2×D3)⋊C4—91no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—91no—
330C4⋊Dic3—91no—
331C4⋊Dic3—91no—
332C4⋊Dic3—91no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—181no—
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⋊C4C2×D911yescharacteristic
354Dic3⋊C4C2×D911yescharacteristic
355C4×Dic3—31no—
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×Dic3C2×D911yescharacteristic
366C4×Dic3C2×D911yescharacteristic
367C22×C32—21no—
368C22×C32—21no—
369C22×C32—21no—
370C22×C32C23×D311yescharacteristic
371C2×C3×D3—61no—
372C2×C3×D3—61no—
373C2×C3×D3—61no—
374C32×C4—21no—
375C32×C4—21no—
376C3⋊Dic3—61no—
377C3⋊Dic3—61no—
378C3⋊Dic3—61no—
379C3⋊Dic3—61no—
380C3×Dic3—21no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—32no—
384C3×Dic3—61no—
385C3×Dic3—61no—
386C3×Dic3#23512yes—
387C22×C9—21no—
388C22×C9—21no—
389C22×C9—21no—
390C22×C9—21no—
391C22×C9—41no—
392C22×C9—41no—
393C22×C9—41no—
394C22×C9C23×D311yescharacteristic
395C2×D9—21no—
396C2×D9—21no—
397C2×D9—21no—
398C4×C9—21no—
399C4×C9—21no—
400C4×C9—32no—
401C4×C9—41no—
402C4×C9—41no—
403C4×C9—61no—
404C4×C9—61no—
405C4×C9—61no—
406Dic9—21no—
407Dic9—21no—
408Dic9—61no—
409Dic9—61no—
410Dic9—61no—
411Dic9—61no—
412Dic9—121no—
413Dic9—121no—
414Dic9—121no—
415Dic9—121no—
416Dic9#23712yes—
417C4⋊Q8—271no—
418#126—271no—
419#126—271no—
420#126—271no—
421#126—271no—
422#124—91no—
423#124—271no—
424C22⋊Q8—271no—
425C22⋊Q8—271no—
426C4×Q8—271no—
427C4×Q8—271no—
428C4×Q8—271no—
429C4×D4—271no—
430#117—31no—
431#117—271no—
432C3×C9#55611yescharacteristic, complemented, Sylow
433C23×C3—21no—
434C23×C3—91no—
435C23×C3C22×D911yescharacteristic
436C23×C3C2×D3211yescharacteristic
437C22×D3—31no—
438C3×Q8—182no—
439C3×D4—182no—
440C2×C3×C4—21no—
441C2×C3×C4—21no—
442C2×C3×C4—31no—
443C2×C3×C4—31no—
444C2×C3×C4—31no—
445C2×C3×C4—31no—
446C2×C3×C4—62no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4C22×D911yescharacteristic
451C2×C3×C4C22×D911yescharacteristic
452C2×C3×C4C2×D3211yescharacteristic
453C2×C3×C4C2×D3211yescharacteristic
454#67—62no—
455C2×Dic3—22no—
456C2×Dic3—31no—
457C2×Dic3—31no—
458C2×Dic3—31no—
459C2×Dic3—91no—
460C2×Dic3—91no—
461C2×Dic3—91no—
462C2×Dic3—91no—
463C2×Dic3—91no—
464C2×Dic3—91no—
465C2×Dic3—91no—
466C2×Dic3—91no—
467C2×Dic3—181no—
468C2×Dic3—181no—
469C2×Dic3—181no—
470C2×Dic3—181no—
471C2×Dic3—182no—
472C2×Dic3C22×D911yescharacteristic
473C2×Dic3C22×D911yescharacteristic
474C2×Dic3C22×D911yescharacteristic
475C2×Dic3C22×D911yescharacteristic
476C4×D3—182no—
477Dic6—62no—
478Dic6—182no—
479Dic6—182no—
480Dic6—182no—
481Dic6—182no—
482C2×C32—41no—
483C2×C32C2×#23511yescharacteristic
484C2×C32D3×#4111yescharacteristic
485C2×C32#101011yescharacteristic
486C3×D3—121no—
487C2×C9—21no—
488C2×C9—21no—
489C2×C9—21no—
490C2×C9—41no—
491C2×C9—81no—
492C2×C9C2×#23711yescharacteristic
493C2×C9D3×#4111yescharacteristic
494C2×C9#101011yescharacteristic
495D9—41no—
496C2×Q8—91no—
497C2×Q8—271no—
498C2×Q8—271no—
499C2×D4—91no—
500C22×C4—31no—
501C22×C4—271no—
502C4⋊C4—31no—
503C4⋊C4—31no—
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—
514C22⋊C4—31no—
515C22⋊C4—91no—
516C22⋊C4—91no—
517C22⋊C4—91no—
518C22⋊C4—271no—
519C22⋊C4—271no—
520C22⋊C4—271no—
521C22⋊C4—271no—
522C22⋊C4—271no—
523C22⋊C4D3×D911yescharacteristic
524C42—31no—
525C42—31no—
526C42—91no—
527C42—271no—
528C42—271no—
529C42—271no—
530C42—271no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—181no—
542C22×C3—181no—
543C22×C3—181no—
544C22×C3C23×D911yescharacteristic
545C22×C3C22×D3211yescharacteristic
546C2×D3—61no—
547C2×D3—61no—
548C2×D3—61no—
549C3×C4—21no—
550C3×C4—21no—
551C3×C4—21no—
552C3×C4—21no—
553C3×C4—32no—
554C3×C4—41no—
555C3×C4—41no—
556C3×C4—61no—
557C3×C4—61no—
558C3×C4—61no—
559C3×C4—92no—
560C3×C4—181no—
561C3×C4—181no—
562Dic3—21no—
563Dic3—21no—
564Dic3—21no—
565Dic3—32no—
566Dic3—61no—
567Dic3—61no—
568Dic3—181no—
569Dic3—181no—
570Dic3—181no—
571Dic3—181no—
572Dic3—181no—
573Dic3—181no—
574Dic3—181no—
575Dic3—181no—
576Dic3—361no—
577Dic3—361no—
578Dic3—361no—
579Dic3—361no—
580Dic3#369712yes—
581C32#553311yescharacteristic
582C9—21no—
583C9#567111yescharacteristic, complemented
584C23—91no—
585C23C2×D3×D911yescharacteristic
586Q8—182no—
587Q8—542no—
588Q8—542no—
589D4—182no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—31no—
593C2×C4—31no—
594C2×C4—62no—
595C2×C4—91no—
596C2×C4—91no—
597C2×C4—91no—
598C2×C4—271no—
599C2×C4—271no—
600C2×C4—271no—
601C2×C4—271no—
602C2×C4—542no—
603C2×C4C2×D3×D911yescharacteristic
604C2×C4C2×D3×D911yescharacteristic
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—41no—
609C2×C3—41no—
610C2×C3—81no—
611C2×C3—361no—
612C2×C3C2×#369711yescharacteristic
613C2×C3#41×D911yescharacteristic
614C2×C3#6371611yescharacteristic
615C2×C3D3×#23511yescharacteristic
616C2×C3#6430911yescharacteristic
617C2×C3D3×#23711yescharacteristic
618D3—121no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—181no—
623C22—181no—
624C22—181no—
625C22C22×D3×D911yescharacteristic
626C4—21no—
627C4—21no—
628C4—32no—
629C4—61no—
630C4—61no—
631C4—61no—
632C4—92no—
633C4—181no—
634C4—181no—
635C4—541no—
636C4—541no—
637C4—541no—
638C4—541no—
639C3—21no—
640C3#1058963511yescharacteristic, complemented
641C3#1059410511yescharacteristic
642C2—41no—
643C2—361no—
644C2D3×#369711yescharacteristic
645C2#1172249411yescharacteristic
646C2D9×#23711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11721725C2semidirect
3. C9×#877C2semidirect
6. #11721249C2semidirect
7. #11721246C2semidirect
8. #11720991C2semidirect
9. #11720975C2semidirect
10. C4×#6189C2semidirect
11. #11720947C2semidirect
12. #11720947C2semidirect
13. Dic3×Dic18C2semidirect
14. #11720904C2semidirect
15. #11720901C2semidirect
228. #877D9semidirect
432. C3×C9#556semidirect
583. C9#5671semidirect
640. C3#10589635semidirect

Maximal subgroups

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

subgroupindex
#117217252
C3×#634472
C9×#8772
Dic3×#6112
#117212572
#117212492
#117212462
#117209912
#117209752
C4×#61892
#117209472
Dic3×Dic182
#117209042
#117209012
#105941053
#105896353