Finite Group Databasebeta

← #13150265

Subgroups — #13150265

Order 1728.

720 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721818C211yesmaximal, characteristic
2C9×#926C211yesmaximal, characteristic, complemented
3C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
4#11721255C211yesmaximal, characteristic
5#11721253C211yesmaximal, characteristic, complemented
6#11721245C211yesmaximal, characteristic, complemented
7#11721241C211yesmaximal, characteristic
8#11720994C211yesmaximal, characteristic, complemented
9#11720926C211yesmaximal, characteristic, complemented
10#11720925C211yesmaximal, characteristic, complemented
11#11720920C211yesmaximal, characteristic, complemented
12#11720918C211yesmaximal, characteristic, complemented
13#11720917C211yesmaximal, characteristic, complemented
14#11720900C211yesmaximal, characteristic, complemented
15#11720898C211yesmaximal, characteristic, complemented
16#10594807—31nomaximal
17#10589633—31nomaximal
18C22×C3×C4×C9C2211yescharacteristic
19C2×#6233C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C22×C3×Dic9C2211yescharacteristic
22C2×(C3⋊D9)⋊C4C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24#88551C2211yescharacteristic
25#88536C2211yescharacteristic
26#88536C2211yescharacteristic
27#88534C2211yescharacteristic
28#88534C2211yescharacteristic
29C9×#217C2211yescharacteristic
30C9×#217C2211yescharacteristic
31C9×C4⋊Dic3C2211yescharacteristic
32C9×C4⋊Dic3C2211yescharacteristic
33C4×C9×Dic3C2211yescharacteristic
34C4×C9×Dic3C2211yescharacteristic
35C3×Dic9⋊C4C2211yescharacteristic
36C3×Dic9⋊C4C2211yescharacteristic
37C3×Dic9⋊C4C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39#88445C2211yescharacteristic
40#88445C2211yescharacteristic
41#88445C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47Dic9⋊Dic3C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64138—31no—
54C3×#926D311yescharacteristic
55C2×C3×Dic3⋊C4—31no—
56#63966—31no—
57#63964—31no—
58#63955—31no—
59#63952—31no—
60#63895—31no—
61#63868—31no—
62#63867—31no—
63#63848—31no—
64#63847—31no—
65#63846—31no—
66#63837—31no—
67#63836—31no—
68C9×#117—31no—
69#63491—31no—
70#63491—61no—
71C2×Dic9⋊C4D311yescharacteristic, complemented
72#63460—31no—
73#63458—31no—
74#63451—31no—
75#63451—31no—
76#63448—31no—
77#611⋊C4—31no—
78#63440—31no—
79#63439—31no—
80#63438—31no—
81#63434—31no—
82#63432—31no—
83C4×Dic18—31no—
84C23×C3×C9C2311yescharacteristic
85C22×C3⋊D9C2311yescharacteristic
86C2×C3×C4×C9—22no—
87C2×C3×C4×C9C2311yescharacteristic
88C2×C3×C4×C9C2311yescharacteristic
89#6233—22no—
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C9×Dic3C2311yescharacteristic
94C2×C9×Dic3C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C3×Dic9—22no—
98C2×C3×Dic9—22no—
99C2×C3×Dic9C2311yescharacteristic
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C2×C3×Dic9C2311yescharacteristic
103(C3⋊D9)⋊C4—22no—
104#6189—22no—
105#5531—91no—
106#5513—91no—
107C22×C32×C4C2×D311yescharacteristic
108C2×#638—31no—
109C22×C3×Dic3—31no—
110C22×C3×Dic3—31no—
111C2×#627—31no—
112C2×#624—31no—
113#3757—31no—
114#3752—31no—
115#3752—31no—
116#3750—31no—
117#3750—31no—
118C3×#217C2×D311yescharacteristic
119C3×#217C2×D311yescharacteristic
120C3×C4⋊Dic3C2×D311yescharacteristic
121C3×C4⋊Dic3C2×D311yescharacteristic
122C3×Dic3⋊C4—31no—
123C3×Dic3⋊C4—31no—
124C3×Dic3⋊C4—31no—
125C3×Dic3⋊C4—31no—
126C3×C4×Dic3C2×D311yescharacteristic
127C3×C4×Dic3C2×D311yescharacteristic
128#3724—31no—
129#3724—31no—
130Dic3⋊Dic3—31no—
131Dic3⋊Dic3—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136#3722—31no—
137#3722—31no—
138#3722—31no—
139Dic32—31no—
140Dic32—31no—
141Dic32—31no—
142C22×C4×C9—21no—
143C22×C4×C9C2×D311yescharacteristic
144C2×#611—31no—
145C2×#611—61no—
146C22×Dic9C2×D311yescharacteristic
147C22×Dic9C2×D311yescharacteristic
148C2×C4×D9—31no—
149C2×Dic18—31no—
150C9×C4⋊C4—31no—
151C9×C4⋊C4—31no—
152C9×C22⋊C4—31no—
153C9×C22⋊C4—31no—
154C42×C9—31no—
155C42×C9—31no—
156#3676—31no—
157#3676—61no—
158(C2×D9)⋊C4—31no—
159(C2×D9)⋊C4—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—31no—
163(C2×D9)⋊C4—61no—
164(C2×D9)⋊C4—61no—
165C4⋊Dic9—31no—
166C4⋊Dic9—31no—
167C4⋊Dic9—31no—
168Dic9⋊C4—31no—
169Dic9⋊C4—31no—
170Dic9⋊C4—31no—
171Dic9⋊C4—31no—
172Dic9⋊C4—31no—
173Dic9⋊C4—31no—
174Dic9⋊C4—31no—
175Dic9⋊C4—61no—
176Dic9⋊C4—61no—
177Dic9⋊C4C2×D311yescharacteristic
178Dic9⋊C4C2×D311yescharacteristic
179Dic9⋊C4C2×D311yescharacteristic
180Dic9⋊C4C2×D311yescharacteristic
181C4×Dic9—31no—
182C4×Dic9—31no—
183C4×Dic9—31no—
184C22×C3×C9—21no—
185C22×C3×C9—21no—
186C22×C3×C9#4112yes—
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—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9#4112yes—
204C3×#117—31no—
205C2×C3×C4⋊C4—91no—
206#929—91no—
207#929—91no—
208#929—91no—
209#929—181no—
210C4×#67—91no—
211#926—91no—
212#926D911yescharacteristic, complemented
213#924—91no—
214C2×Dic3⋊C4—31no—
215#898—91no—
216#898—91no—
217#898—91no—
218#897—91no—
219#896—91no—
220#892—91no—
221#890—91no—
222#889—91no—
223#889—91no—
224#889—91no—
225#889—91no—
226#887—91no—
227#886—91no—
228#67⋊C4—91no—
229#878—91no—
230#877—91no—
231#876—91no—
232#872—91no—
233#870—91no—
234C4×Dic6—91no—
235C23×C32C22×D311yescharacteristic
236C22×C3⋊D3—31no—
237C2×C32×C4—22no—
238C2×C32×C4C22×D311yescharacteristic
239C2×C32×C4C22×D311yescharacteristic
240#638—62no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—31no—
244C2×C3×Dic3—31no—
245C2×C3×Dic3—31no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3—62no—
249C2×C3×Dic3—62no—
250C2×C3×Dic3C22×D311yescharacteristic
251C2×C3×Dic3C22×D311yescharacteristic
252C2×C3×Dic3C22×D311yescharacteristic
253C2×C3×Dic3C22×D311yescharacteristic
254#627—62no—
255#624—62no—
256C23×C9—21no—
257C23×C9C22×D311yescharacteristic
258C22×D9—31no—
259C22×D9—61no—
260C2×C4×C9—22no—
261C2×C4×C9—21no—
262C2×C4×C9—21no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9—31no—
267C2×C4×C9—42no—
268C2×C4×C9C22×D311yescharacteristic
269C2×C4×C9C22×D311yescharacteristic
270#611—62no—
271#611—122no—
272C2×Dic9—22no—
273C2×Dic9—22no—
274C2×Dic9—31no—
275C2×Dic9—31no—
276C2×Dic9—31no—
277C2×Dic9—61no—
278C2×Dic9—61no—
279C2×Dic9—61no—
280C2×Dic9C22×D311yescharacteristic
281C2×Dic9C22×D311yescharacteristic
282C2×Dic9C22×D311yescharacteristic
283C2×Dic9C22×D311yescharacteristic
284C4×D9—62no—
285Dic18—62no—
286#552—271noSylow
287C2×C3×C9—22no—
288C2×C3×C9C2×#4111yescharacteristic
289C2×C3×C9C2×#4111yescharacteristic
290C2×C3×C92−1+411yescharacteristic
291C3⋊D9—41no—
292C22×C3×C4—21no—
293C22×C3×C4—91no—
294C22×C3×C4—91no—
295C22×C3×C4C2×D911yescharacteristic
296C22×C3×C4D3211yescharacteristic
297C2×#67—91no—
298C2×#67—91no—
299C2×#67—181no—
300C22×Dic3—31no—
301C22×Dic3—31no—
302C2×C4×D3—91no—
303C2×C4×D3—91no—
304C2×Dic6—91no—
305C2×Dic6—91no—
306C3×C4⋊C4—31no—
307C3×C4⋊C4—31no—
308C3×C4⋊C4—91no—
309C3×C4⋊C4—91no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C22⋊C4—31no—
313C3×C22⋊C4—31no—
314C3×C42—31no—
315C3×C42—31no—
316#217—91no—
317#217—91no—
318#217—181no—
319#217C2×D911yescharacteristic
320#217C2×D911yescharacteristic
321(C2×D3)⋊C4—91no—
322(C2×D3)⋊C4—91no—
323(C2×D3)⋊C4—91no—
324(C2×D3)⋊C4—91no—
325(C2×D3)⋊C4—91no—
326(C2×D3)⋊C4—91no—
327(C2×D3)⋊C4—91no—
328(C2×D3)⋊C4—91no—
329(C2×D3)⋊C4—91no—
330(C2×D3)⋊C4—91no—
331(C2×D3)⋊C4—181no—
332(C2×D3)⋊C4—181no—
333C4⋊Dic3—91no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3C2×D911yescharacteristic
340C4⋊Dic3C2×D911yescharacteristic
341Dic3⋊C4—31no—
342Dic3⋊C4—31no—
343Dic3⋊C4—31no—
344Dic3⋊C4—31no—
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—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—181no—
360Dic3⋊C4—181no—
361C4×Dic3—91no—
362C4×Dic3—91no—
363C4×Dic3—91no—
364C4×Dic3—91no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3C2×D911yescharacteristic
368C4×Dic3C2×D911yescharacteristic
369C22×C32—21no—
370C22×C32—21no—
371C22×C32#23512yes—
372C22×C32C23×D311yescharacteristic
373C2×C3⋊D3—61no—
374C2×C3⋊D3—61no—
375C2×C3⋊D3—61no—
376C32×C4—21no—
377C32×C4—21no—
378C3⋊Dic3—61no—
379C3⋊Dic3—61no—
380C3⋊Dic3—61no—
381C3×Dic3—21no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—21no—
385C3×Dic3—32no—
386C3×Dic3—61no—
387C3×Dic3—61no—
388C3×Dic3—61no—
389C22×C9—21no—
390C22×C9—21no—
391C22×C9—21no—
392C22×C9—22no—
393C22×C9—41no—
394C22×C9—41no—
395C22×C9#23712yes—
396C22×C9C23×D311yescharacteristic
397C2×D9—61no—
398C2×D9—61no—
399C2×D9—61no—
400C2×D9—121no—
401C2×D9—121no—
402C2×D9—121no—
403C4×C9—21no—
404C4×C9—21no—
405C4×C9—41no—
406C4×C9—41no—
407C4×C9—61no—
408C4×C9—61no—
409C4×C9—61no—
410C4×C9—61no—
411Dic9—21no—
412Dic9—21no—
413Dic9—21no—
414Dic9—61no—
415Dic9—61no—
416Dic9—61no—
417Dic9—121no—
418Dic9—121no—
419Dic9—121no—
420Dic9#23512yes—
421#126—271no—
422#126—271no—
423#125—271no—
424#125—271no—
425#125—271no—
426#123—271no—
427#123—271no—
428C22⋊Q8—271no—
429C22⋊Q8—271no—
430C4×Q8—271no—
431C4×D4—271no—
432#117—31no—
433#117—271no—
434#117—271no—
435C2×C4⋊C4—91no—
436C3×C9#55211yescharacteristic, complemented, Sylow
437C23×C3—21no—
438C23×C3C22×D911yescharacteristic
439C23×C3C2×D3211yescharacteristic
440C22×D3—91no—
441C22×D3—91no—
442C22×D3—181no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—21no—
446C2×C3×C4—21no—
447C2×C3×C4—31no—
448C2×C3×C4—31no—
449C2×C3×C4—31no—
450C2×C3×C4—31no—
451C2×C3×C4—42no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4—91no—
455C2×C3×C4—91no—
456C2×C3×C4—182no—
457C2×C3×C4—182no—
458C2×C3×C4C22×D911yescharacteristic
459C2×C3×C4C22×D911yescharacteristic
460C2×C3×C4C2×D3211yescharacteristic
461C2×C3×C4C2×D3211yescharacteristic
462#67—182no—
463#67—182no—
464#67—362no—
465C2×Dic3—31no—
466C2×Dic3—31no—
467C2×Dic3—31no—
468C2×Dic3—31no—
469C2×Dic3—62no—
470C2×Dic3—62no—
471C2×Dic3—91no—
472C2×Dic3—91no—
473C2×Dic3—91no—
474C2×Dic3—91no—
475C2×Dic3—91no—
476C2×Dic3—91no—
477C2×Dic3—181no—
478C2×Dic3—181no—
479C2×Dic3—181no—
480C2×Dic3C22×D911yescharacteristic
481C2×Dic3C22×D911yescharacteristic
482C2×Dic3C22×D911yescharacteristic
483C2×Dic3C22×D911yescharacteristic
484C4×D3—182no—
485C4×D3—182no—
486Dic6—182no—
487Dic6—182no—
488C2×C32—22no—
489C2×C32C2×#23511yescharacteristic
490C2×C32D3×#4111yescharacteristic
491C2×C32#101011yescharacteristic
492C3⋊D3—121no—
493C2×C9—22no—
494C2×C9—21no—
495C2×C9—21no—
496C2×C9—21no—
497C2×C9—42no—
498C2×C9C2×#23511yescharacteristic
499C2×C9C2×#23711yescharacteristic
500C2×C9#100711yescharacteristic
501D9—121no—
502D9—241no—
503C2×Q8—271no—
504C2×D4—271no—
505C22×C4—91no—
506C22×C4—91no—
507C22×C4—271no—
508C22×C4D3×D911yescharacteristic
509C4⋊C4—31no—
510C4⋊C4—31no—
511C4⋊C4—91no—
512C4⋊C4—91no—
513C4⋊C4—91no—
514C4⋊C4—91no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C4⋊C4—271no—
518C4⋊C4—271no—
519C4⋊C4—271no—
520C4⋊C4—271no—
521C4⋊C4—271no—
522C4⋊C4—271no—
523C4⋊C4—271no—
524C4⋊C4—271no—
525C22⋊C4—31no—
526C22⋊C4—31no—
527C22⋊C4—271no—
528C22⋊C4—271no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4—271no—
533C42—31no—
534C42—31no—
535C42—271no—
536C42—271no—
537C42—271no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—22no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3#369712yes—
547C22×C3C23×D911yescharacteristic
548C22×C3#380412yes—
549C22×C3C22×D3211yescharacteristic
550C2×D3—181no—
551C2×D3—181no—
552C2×D3—181no—
553C2×D3—181no—
554C2×D3—181no—
555C2×D3—181no—
556C2×D3—361no—
557C2×D3—361no—
558C2×D3—361no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—41no—
564C3×C4—41no—
565C3×C4—61no—
566C3×C4—61no—
567C3×C4—61no—
568C3×C4—61no—
569C3×C4—92no—
570C3×C4—181no—
571C3×C4—181no—
572C3×C4—181no—
573Dic3—21no—
574Dic3—21no—
575Dic3—21no—
576Dic3—21no—
577Dic3—32no—
578Dic3—61no—
579Dic3—61no—
580Dic3—61no—
581Dic3—181no—
582Dic3—181no—
583Dic3—181no—
584Dic3—181no—
585Dic3—181no—
586Dic3—181no—
587Dic3—361no—
588Dic3—361no—
589Dic3—361no—
590C32#553111yescharacteristic
591C9—21no—
592C9#551311yescharacteristic, complemented
593C23—271no—
594C23C2×D3×D911yescharacteristic
595Q8—542no—
596D4—542no—
597C2×C4—22no—
598C2×C4—31no—
599C2×C4—31no—
600C2×C4—31no—
601C2×C4—31no—
602C2×C4—91no—
603C2×C4—91no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—182no—
607C2×C4—182no—
608C2×C4—271no—
609C2×C4—271no—
610C2×C4—271no—
611C2×C4—542no—
612C2×C4C2×D3×D911yescharacteristic
613C2×C4C2×D3×D911yescharacteristic
614C2×C3—22no—
615C2×C3—22no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—42no—
620C2×C3C2×#369711yescharacteristic
621C2×C3#41×D911yescharacteristic
622C2×C3#6371611yescharacteristic
623C2×C3#6429711yescharacteristic
624C2×C3D3×#23511yescharacteristic
625C2×C3C2×#380411yescharacteristic
626D3—361no—
627D3—361no—
628D3—721no—
629C22—21no—
630C22—21no—
631C22—541no—
632C22—541no—
633C22—541no—
634C22#8865812yes—
635C22C22×D3×D911yescharacteristic
636C4—21no—
637C4—21no—
638C4—61no—
639C4—61no—
640C4—61no—
641C4—61no—
642C4—92no—
643C4—181no—
644C4—181no—
645C4—181no—
646C4—541no—
647C4—541no—
648C4—541no—
649C3—21no—
650C3#1058963311yescharacteristic, complemented
651C3#1059480711yescharacteristic
652C2—22no—
653C2—1081no—
654C2#1172246411yescharacteristic
655C2D9×#23511yescharacteristic
656C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×#6233C22
hypercenterC22×C4D3×D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#926C2semidirect
3. C2×C3×Dic9⋊C4C2semidirect
5. #11721253C2semidirect
6. #11721245C2semidirect
8. #11720994C2semidirect
9. #11720926C2semidirect
10. #11720925C2semidirect
11. #11720920C2semidirect
12. #11720918C2semidirect
13. #11720917C2semidirect
14. #11720900C2semidirect
15. #11720898C2semidirect
71. C2×Dic9⋊C4D3semidirect
212. #926D9semidirect
436. C3×C9#552semidirect
592. C9#5513semidirect
650. C3#10589633semidirect

Maximal subgroups

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

subgroupindex
#117218182
C9×#9262
C2×C3×Dic9⋊C42
#117212552
#117212532
#117212452
#117212412
#117209942
#117209262
#117209252
#117209202
#117209182
#117209172
#117209002
#117208982
#105948073
#105896333