Finite Group Databasebeta

← #13147966

Subgroups — #13147966

Order 1728.

736 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721727C211yesmaximal, characteristic
2C9×#877C211yesmaximal, characteristic, complemented
3C3×#63439C211yesmaximal, characteristic, complemented
4#11721267C211yesmaximal, characteristic, complemented
5#11721255C211yesmaximal, characteristic
6C2×#88442C211yesmaximal, characteristic, complemented
7#11721251C211yesmaximal, characteristic
8C4×#6189C211yesmaximal, characteristic, complemented
9#11720934C211yesmaximal, characteristic, complemented
10#11720933C211yesmaximal, characteristic, complemented
11#11720931C211yesmaximal, characteristic, complemented
12#11720928C211yesmaximal, characteristic, complemented
13#11720926C211yesmaximal, characteristic, complemented
14#11720925C211yesmaximal, characteristic, complemented
15#11720923C211yesmaximal, characteristic, complemented
16#10594046—31nomaximal
17#10589633—31nomaximal
18C2×#6233C2211yescharacteristic
19C22×C3⋊Dic9C2211yescharacteristic
20C2×C4×C3⋊D9C2211yescharacteristic
21C22×C9×Dic3C2211yescharacteristic
22C22×C3×Dic9C2211yescharacteristic
23C2×#6189C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25#88536C2211yescharacteristic
26#88534C2211yescharacteristic
27C4×C3⋊Dic9C2211yescharacteristic
28C9×#217C2211yescharacteristic
29C3×#3676C2211yescharacteristic
30C9×Dic3⋊C4C2211yescharacteristic
31C9×Dic3⋊C4C2211yescharacteristic
32C4×C9×Dic3C2211yescharacteristic
33C4×C9×Dic3C2211yescharacteristic
34C3×Dic9⋊C4C2211yescharacteristic
35C3×Dic9⋊C4C2211yescharacteristic
36C3×C4×Dic9C2211yescharacteristic
37C3×C4×Dic9C2211yescharacteristic
38#88445C2211yescharacteristic
39#88445C2211yescharacteristic
40#88445C2211yescharacteristic
41#88445C2211yescharacteristic
42Dic3⋊Dic9C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47#88442C2211yescharacteristic
48#88442C2211yescharacteristic
49#88442C2211yescharacteristic
50Dic9⋊Dic3C2211yescharacteristic
51Dic9⋊Dic3C2211yescharacteristic
52Dic9⋊Dic3C2211yescharacteristic
53#64090—31no—
54C3×#877—31no—
55C3×#877D311yescharacteristic
56#63982—31no—
57C2×#3724—31no—
58#63966—31no—
59#63966—31no—
60C4×#627—31no—
61#63874—31no—
62#63870—31no—
63#63868—31no—
64#63868—31no—
65#63867—31no—
66#63867—31no—
67#63865—31no—
68C9×#117—31no—
69#63491—31no—
70C2×Dic9⋊C4—31no—
71#63460—31no—
72#63458—31no—
73#63451—31no—
74#63451—31no—
75#63448—31no—
76#611⋊C4—31no—
77#611⋊C4—61no—
78#63440—31no—
79#63439D311yescharacteristic, complemented
80#63438—31no—
81#63434—31no—
82#63432—31no—
83C4×Dic18—31no—
84C23×C3×C9C2311yescharacteristic
85C22×C3⋊D9C2311yescharacteristic
86C2×C3×C4×C9C2311yescharacteristic
87C2×C3×C4×C9C2311yescharacteristic
88#6233—22no—
89C2×C3⋊Dic9—22no—
90C2×C3⋊Dic9C2311yescharacteristic
91C2×C3⋊Dic9C2311yescharacteristic
92C2×C3⋊Dic9C2311yescharacteristic
93C4×C3⋊D9—22no—
94C2×C9×Dic3—22no—
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C9×Dic3C2311yescharacteristic
99C2×C3×Dic9—22no—
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C2×C3×Dic9C2311yescharacteristic
103C2×C3×Dic9C2311yescharacteristic
104#6189—22no—
105#5531—91no—
106#5531—91no—
107C2×#638—31no—
108C22×C3⋊Dic3—31no—
109C2×C4×C3⋊D3—31no—
110C22×C3×Dic3—31no—
111C22×C3×Dic3C2×D311yescharacteristic
112C2×#627—31no—
113C32×C22⋊C4C2×D311yescharacteristic
114#3752—31no—
115#3750—31no—
116C4×C3⋊Dic3—31no—
117C3×#217—31no—
118C3×#217C2×D311yescharacteristic
119C3×Dic3⋊C4—31no—
120C3×Dic3⋊C4—31no—
121C3×Dic3⋊C4C2×D311yescharacteristic
122C3×Dic3⋊C4C2×D311yescharacteristic
123C3×C4×Dic3—31no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3C2×D311yescharacteristic
126C3×C4×Dic3C2×D311yescharacteristic
127#3724—31no—
128#3724—31no—
129#3724—31no—
130#3724—31no—
131#3724—31no—
132Dic3⋊Dic3—31no—
133Dic3⋊Dic3—31no—
134Dic3⋊Dic3—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138#3722—31no—
139#3722—31no—
140#3722—31no—
141#3722—31no—
142C22×C4×C9—31no—
143C2×#611—31no—
144C2×#611—61no—
145C22×Dic9—31no—
146C22×Dic9—61no—
147C22×Dic9C2×D311yescharacteristic
148C2×C4×D9—31no—
149C2×C4×D9—61no—
150C2×Dic18—31no—
151C9×C4⋊C4—31no—
152C9×C4⋊C4—31no—
153C9×C22⋊C4—21no—
154C9×C22⋊C4—31no—
155C9×C22⋊C4C2×D311yescharacteristic
156C42×C9—31no—
157C42×C9—31no—
158#3676C2×D311yescharacteristic
159(C2×D9)⋊C4—31no—
160(C2×D9)⋊C4—31no—
161(C2×D9)⋊C4—31no—
162(C2×D9)⋊C4—31no—
163(C2×D9)⋊C4—31no—
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—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—61no—
178Dic9⋊C4C2×D311yescharacteristic
179Dic9⋊C4C2×D311yescharacteristic
180C4×Dic9—31no—
181C4×Dic9—61no—
182C4×Dic9C2×D311yescharacteristic
183C4×Dic9C2×D311yescharacteristic
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#4112yes—
196C9×Dic3—21no—
197C9×Dic3—21no—
198C9×Dic3—21no—
199C9×Dic3—21no—
200C3×Dic9—21no—
201C3×Dic9—21no—
202C3×Dic9—21no—
203C3×Dic9—21no—
204C3×#117—31no—
205C3×#117—91no—
206#929—91no—
207#929—91no—
208C2×Dic3⋊C4—91no—
209C2×Dic3⋊C4—91no—
210#898—91no—
211#898—91no—
212#896—91no—
213#896—91no—
214#889—91no—
215#889—91no—
216#889—91no—
217#889—91no—
218#886—91no—
219#886—91no—
220#67⋊C4—91no—
221#67⋊C4—91no—
222#67⋊C4—181no—
223#878—91no—
224#878—91no—
225#877—31no—
226#877D911yescharacteristic, complemented
227#876—91no—
228#876—91no—
229#872—91no—
230#872—91no—
231#870—91no—
232#870—91no—
233C4×Dic6—91no—
234C4×Dic6—91no—
235C23×C32C22×D311yescharacteristic
236C22×C3⋊D3—31no—
237C2×C32×C4C22×D311yescharacteristic
238C2×C32×C4C22×D311yescharacteristic
239#638—62no—
240C2×C3⋊Dic3—31no—
241C2×C3⋊Dic3—31no—
242C2×C3⋊Dic3—31no—
243C2×C3⋊Dic3—62no—
244C4×C3⋊D3—62no—
245C2×C3×Dic3—22no—
246C2×C3×Dic3—31no—
247C2×C3×Dic3—31no—
248C2×C3×Dic3—31no—
249C2×C3×Dic3—31no—
250C2×C3×Dic3—62no—
251C2×C3×Dic3C22×D311yescharacteristic
252C2×C3×Dic3C22×D311yescharacteristic
253C2×C3×Dic3C22×D311yescharacteristic
254C2×C3×Dic3C22×D311yescharacteristic
255#627—62no—
256C23×C9—21no—
257C23×C9C22×D311yescharacteristic
258C22×D9—31no—
259C22×D9—61no—
260C2×C4×C9—21no—
261C2×C4×C9—21no—
262C2×C4×C9—31no—
263C2×C4×C9—31no—
264C2×C4×C9—31no—
265C2×C4×C9—31no—
266C2×C4×C9—62no—
267C2×C4×C9C22×D311yescharacteristic
268C2×C4×C9C22×D311yescharacteristic
269#611—64no—
270#611—62no—
271C2×Dic9—22no—
272C2×Dic9—31no—
273C2×Dic9—31no—
274C2×Dic9—31no—
275C2×Dic9—61no—
276C2×Dic9—61no—
277C2×Dic9—61no—
278C2×Dic9—62no—
279C2×Dic9—122no—
280C2×Dic9C22×D311yescharacteristic
281C2×Dic9C22×D311yescharacteristic
282C2×Dic9C22×D311yescharacteristic
283C2×Dic9C22×D311yescharacteristic
284C4×D9—62no—
285C4×D9—122no—
286Dic18—62no—
287#552—271noSylow
288C2×C3×C9—22no—
289C2×C3×C9C2×#4111yescharacteristic
290C2×C3×C9C2×#4111yescharacteristic
291C2×C3×C92−1+411yescharacteristic
292C3⋊D9—41no—
293C22×C3×C4—31no—
294C22×C3×C4—91no—
295C2×#67—91no—
296C2×#67—91no—
297C2×#67—181no—
298C22×Dic3—31no—
299C22×Dic3—91no—
300C22×Dic3—91no—
301C22×Dic3—181no—
302C22×Dic3C2×D911yescharacteristic
303C2×C4×D3—91no—
304C2×C4×D3—91no—
305C2×C4×D3—181no—
306C2×Dic6—91no—
307C2×Dic6—91no—
308C3×C4⋊C4—31no—
309C3×C4⋊C4—31no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C22⋊C4—21no—
313C3×C22⋊C4—31no—
314C3×C22⋊C4—91no—
315C3×C22⋊C4C2×D911yescharacteristic
316C3×C22⋊C4D3211yescharacteristic
317C3×C42—31no—
318C3×C42—31no—
319C3×C42—91no—
320C3×C42—91no—
321#217—31no—
322#217C2×D911yescharacteristic
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—91no—
332(C2×D3)⋊C4—91no—
333(C2×D3)⋊C4—181no—
334C4⋊Dic3—91no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—91no—
338C4⋊Dic3—91no—
339C4⋊Dic3—91no—
340Dic3⋊C4—31no—
341Dic3⋊C4—31no—
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—91no—
352Dic3⋊C4—91no—
353Dic3⋊C4—91no—
354Dic3⋊C4—91no—
355Dic3⋊C4—91no—
356Dic3⋊C4—91no—
357Dic3⋊C4—91no—
358Dic3⋊C4—91no—
359Dic3⋊C4—91no—
360Dic3⋊C4—181no—
361Dic3⋊C4C2×D911yescharacteristic
362Dic3⋊C4C2×D911yescharacteristic
363C4×Dic3—31no—
364C4×Dic3—31no—
365C4×Dic3—91no—
366C4×Dic3—91no—
367C4×Dic3—181no—
368C4×Dic3C2×D911yescharacteristic
369C4×Dic3C2×D911yescharacteristic
370C22×C32—21no—
371C22×C32—21no—
372C22×C32#23512yes—
373C22×C32C23×D311yescharacteristic
374C2×C3⋊D3—61no—
375C2×C3⋊D3—61no—
376C2×C3⋊D3—61no—
377C32×C4—21no—
378C32×C4—21no—
379C3⋊Dic3—32no—
380C3⋊Dic3—61no—
381C3⋊Dic3—61no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—21no—
385C3×Dic3—21no—
386C3×Dic3—61no—
387C3×Dic3—61no—
388C3×Dic3—61no—
389C3×Dic3—61no—
390C22×C9—21no—
391C22×C9—21no—
392C22×C9—21no—
393C22×C9—22no—
394C22×C9—41no—
395C22×C9—41no—
396C22×C9#23512yes—
397C22×C9C23×D311yescharacteristic
398C2×D9—62no—
399C2×D9—61no—
400C2×D9—61no—
401C2×D9—61no—
402C2×D9—121no—
403C2×D9—121no—
404C4×C9—21no—
405C4×C9—21no—
406C4×C9—41no—
407C4×C9—41no—
408C4×C9—61no—
409C4×C9—61no—
410C4×C9—61no—
411C4×C9—61no—
412Dic9—21no—
413Dic9—21no—
414Dic9—21no—
415Dic9—21no—
416Dic9—32no—
417Dic9—62no—
418Dic9—62no—
419Dic9—61no—
420Dic9—61no—
421Dic9—121no—
422#126—271no—
423#126—271no—
424#125—271no—
425#125—271no—
426#125—271no—
427#123—271no—
428#123—271no—
429C22⋊Q8—271no—
430C22⋊Q8—271no—
431C4×Q8—271no—
432C4×D4—271no—
433#117—31no—
434#117—91no—
435#117—271no—
436C2×C4⋊C4—271no—
437C3×C9#55211yescharacteristic, complemented, Sylow
438C23×C3—21no—
439C23×C3C22×D911yescharacteristic
440C23×C3C2×D3211yescharacteristic
441C22×D3—91no—
442C22×D3—91no—
443C22×D3—181no—
444C2×C3×C4—21no—
445C2×C3×C4—21no—
446C2×C3×C4—31no—
447C2×C3×C4—31no—
448C2×C3×C4—31no—
449C2×C3×C4—31no—
450C2×C3×C4—62no—
451C2×C3×C4—91no—
452C2×C3×C4—91no—
453C2×C3×C4—91no—
454C2×C3×C4—91no—
455C2×C3×C4—182no—
456C2×C3×C4C22×D911yescharacteristic
457C2×C3×C4C22×D911yescharacteristic
458C2×C3×C4C2×D3211yescharacteristic
459C2×C3×C4C2×D3211yescharacteristic
460#67—184no—
461#67—182no—
462#67—182no—
463C2×Dic3—22no—
464C2×Dic3—31no—
465C2×Dic3—31no—
466C2×Dic3—31no—
467C2×Dic3—31no—
468C2×Dic3—62no—
469C2×Dic3—91no—
470C2×Dic3—91no—
471C2×Dic3—91no—
472C2×Dic3—91no—
473C2×Dic3—91no—
474C2×Dic3—91no—
475C2×Dic3—181no—
476C2×Dic3—181no—
477C2×Dic3—181no—
478C2×Dic3—182no—
479C2×Dic3—182no—
480C2×Dic3—362no—
481C2×Dic3C22×D911yescharacteristic
482C2×Dic3C22×D911yescharacteristic
483C2×Dic3C22×D911yescharacteristic
484C2×Dic3C22×D911yescharacteristic
485C4×D3—182no—
486C4×D3—182no—
487C4×D3—362no—
488Dic6—182no—
489Dic6—182no—
490C2×C32—22no—
491C2×C32C2×#23511yescharacteristic
492C2×C32D3×#4111yescharacteristic
493C2×C32#101011yescharacteristic
494C3⋊D3—121no—
495C2×C9—22no—
496C2×C9—21no—
497C2×C9—21no—
498C2×C9—21no—
499C2×C9—42no—
500C2×C9C2×#23511yescharacteristic
501C2×C9D3×#4111yescharacteristic
502C2×C9#101011yescharacteristic
503D9—122no—
504D9—121no—
505C2×Q8—271no—
506C2×D4—271no—
507C22×C4—31no—
508C22×C4—91no—
509C22×C4—271no—
510C22×C4—271no—
511C4⋊C4—31no—
512C4⋊C4—31no—
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—
525C4⋊C4—271no—
526C4⋊C4—271no—
527C22⋊C4—31no—
528C22⋊C4—91no—
529C22⋊C4—271no—
530C22⋊C4—271no—
531C22⋊C4—271no—
532C22⋊C4—271no—
533C22⋊C4—271no—
534C22⋊C4D3×D911yescharacteristic
535C42—31no—
536C42—31no—
537C42—91no—
538C42—91no—
539C42—271no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—21no—
544C22×C3—22no—
545C22×C3—21no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3#369712yes—
549C22×C3C23×D911yescharacteristic
550C22×C3#379812yes—
551C22×C3C22×D3211yescharacteristic
552C2×D3—182no—
553C2×D3—181no—
554C2×D3—181no—
555C2×D3—181no—
556C2×D3—181no—
557C2×D3—181no—
558C2×D3—181no—
559C2×D3—361no—
560C2×D3—361no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—41no—
566C3×C4—41no—
567C3×C4—61no—
568C3×C4—61no—
569C3×C4—61no—
570C3×C4—61no—
571C3×C4—181no—
572C3×C4—181no—
573C3×C4—181no—
574C3×C4—181no—
575Dic3—21no—
576Dic3—21no—
577Dic3—21no—
578Dic3—21no—
579Dic3—61no—
580Dic3—61no—
581Dic3—61no—
582Dic3—61no—
583Dic3—92no—
584Dic3—92no—
585Dic3—182no—
586Dic3—182no—
587Dic3—181no—
588Dic3—181no—
589Dic3—181no—
590Dic3—181no—
591Dic3—361no—
592C32#553111yescharacteristic
593C9—21no—
594C9#553111yescharacteristic, complemented
595C23—271no—
596C23C2×D3×D911yescharacteristic
597Q8—542no—
598D4—542no—
599C2×C4—31no—
600C2×C4—31no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—62no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—182no—
609C2×C4—271no—
610C2×C4—271no—
611C2×C4—271no—
612C2×C4—542no—
613C2×C4—542no—
614C2×C4C2×D3×D911yescharacteristic
615C2×C4C2×D3×D911yescharacteristic
616C2×C3—22no—
617C2×C3—22no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—42no—
622C2×C3C2×#369711yescharacteristic
623C2×C3#41×D911yescharacteristic
624C2×C3#6371611yescharacteristic
625C2×C3C2×#379811yescharacteristic
626C2×C3#6430911yescharacteristic
627C2×C3#6431211yescharacteristic
628D3—362no—
629D3—361no—
630D3—361no—
631C22—21no—
632C22—21no—
633C22—541no—
634C22—541no—
635C22—541no—
636C22#8864012yes—
637C22C22×D3×D911yescharacteristic
638C4—21no—
639C4—21no—
640C4—61no—
641C4—61no—
642C4—61no—
643C4—61no—
644C4—181no—
645C4—181no—
646C4—181no—
647C4—181no—
648C4—272no—
649C4—541no—
650C4—541no—
651C3—21no—
652C3#1058963311yescharacteristic, complemented
653C3#1059404611yescharacteristic
654C2—22no—
655C2—1081no—
656C2C2×#8864011yescharacteristic
657C2#1172249411yescharacteristic
658C2#1172249811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
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×#877C2semidirect
3. C3×#63439C2semidirect
4. #11721267C2semidirect
6. C2×#88442C2semidirect
8. C4×#6189C2semidirect
9. #11720934C2semidirect
10. #11720933C2semidirect
11. #11720931C2semidirect
12. #11720928C2semidirect
13. #11720926C2semidirect
14. #11720925C2semidirect
15. #11720923C2semidirect
79. #63439D3semidirect
226. #877D9semidirect
437. C3×C9#552semidirect
594. C9#5531semidirect
652. C3#10589633semidirect

Maximal subgroups

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

subgroupindex
#117217272
C9×#8772
C3×#634392
#117212672
#117212552
C2×#884422
#117212512
C4×#61892
#117209342
#117209332
#117209312
#117209282
#117209262
#117209252
#117209232
#105940463
#105896333