Finite Group Databasebeta

← #13150945

Subgroups — #13150945

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721839C211yesmaximal, characteristic
2C9×#936C211yesmaximal, characteristic, complemented
3C3×#63497C211yesmaximal, characteristic, complemented
4#11721275C211yesmaximal, characteristic, complemented
5#11721267C211yesmaximal, characteristic, complemented
6#11721253C211yesmaximal, characteristic, complemented
7#11721248C211yesmaximal, characteristic, complemented
8#11721247C211yesmaximal, characteristic, complemented
9#11721242C211yesmaximal, characteristic, complemented
10#11721242C211yesmaximal, characteristic, complemented
11#11721241C211yesmaximal, characteristic, complemented
12#11720986C211yesmaximal, characteristic, complemented
13#11720926C211yesmaximal, characteristic, complemented
14#11720903C211yesmaximal, characteristic, complemented
15#11720900C211yesmaximal, characteristic, complemented
16#10595211—31nomaximal
17#10589771—31nomaximal
18C2×C3×D4×C9C2211yescharacteristic
19C2×#6233C2211yescharacteristic
20C2×#6233C2211yescharacteristic
21C22×C3⋊Dic9C2211yescharacteristic
22C22×C3×Dic9C2211yescharacteristic
23C2×C9×Dic6C2211yescharacteristic
24C2×(C3⋊D9)⋊C4C2211yescharacteristic
25C2×#6189C2211yescharacteristic
26C2×#6189C2211yescharacteristic
27#88551C2211yescharacteristic
28#88536C2211yescharacteristic
29#88534C2211yescharacteristic
30C9×#217C2211yescharacteristic
31C9×#217C2211yescharacteristic
32C9×#217C2211yescharacteristic
33C9×#217C2211yescharacteristic
34C3×#3676C2211yescharacteristic
35C3×#3676C2211yescharacteristic
36C3×#3676C2211yescharacteristic
37C4×C9×Dic3C2211yescharacteristic
38C3×Dic9⋊C4C2211yescharacteristic
39C3×Dic9⋊C4C2211yescharacteristic
40#88445C2211yescharacteristic
41#88445C2211yescharacteristic
42#88445C2211yescharacteristic
43Dic3⋊Dic9C2211yescharacteristic
44Dic3⋊Dic9C2211yescharacteristic
45#88442C2211yescharacteristic
46#88442C2211yescharacteristic
47#88442C2211yescharacteristic
48Dic9⋊Dic3C2211yescharacteristic
49Dic9⋊Dic3C2211yescharacteristic
50Dic3×Dic9C2211yescharacteristic
51Dic3×Dic9C2211yescharacteristic
52Dic3×Dic9C2211yescharacteristic
53#64148—31no—
54C3×#936D311yescharacteristic
55C3×#935—31no—
56#63982—31no—
57#63975—31no—
58#63964—31no—
59#63957—31no—
60#63956—31no—
61#63953—31no—
62#63953—31no—
63#63952—31no—
64#63888—31no—
65#63868—31no—
66#63839—31no—
67#63837—31no—
68C9×#124—31no—
69#63508—31no—
70#63506—31no—
71#63501—31no—
72#63501—61no—
73#63497D311yescharacteristic, complemented
74#63447—31no—
75#63447—31no—
76#63445—31no—
77#611⋊C4—31no—
78#63441—31no—
79#63440—31no—
80#63440—31no—
81#63439—31no—
82#63438—31no—
83C4×Dic18—31no—
84C23×C3×C9C2311yescharacteristic
85C23×C3×C9C2311yescharacteristic
86C22×C3⋊D9C2311yescharacteristic
87C3×D4×C9—41no—
88C2×C3×C4×C9C2311yescharacteristic
89#6233—22no—
90#6233—41no—
91C2×C3⋊Dic9—22no—
92C2×C3⋊Dic9C2311yescharacteristic
93C2×C3⋊Dic9C2311yescharacteristic
94C2×C3⋊Dic9C2311yescharacteristic
95C2×C9×Dic3C2311yescharacteristic
96C2×C9×Dic3C2311yescharacteristic
97C2×C9×Dic3C2311yescharacteristic
98C2×C9×Dic3C2311yescharacteristic
99C2×C3×Dic9—41no—
100C2×C3×Dic9C2311yescharacteristic
101C2×C3×Dic9C2311yescharacteristic
102C2×C3×Dic9C2311yescharacteristic
103C2×C3×Dic9C2311yescharacteristic
104C9×Dic6—41no—
105(C3⋊D9)⋊C4—41no—
106#6189—22no—
107#6189—41no—
108#5669—91no—
109#5662—91no—
110C2×C32×D4C2×D311yescharacteristic
111C2×#638—31no—
112C2×#638—31no—
113C22×C3⋊Dic3—31no—
114C22×C3×Dic3—31no—
115C2×C3×Dic6C2×D311yescharacteristic
116C2×#627—31no—
117C2×#627—31no—
118C2×#624—31no—
119#3757—31no—
120#3752—31no—
121#3750—31no—
122C3×#217—31no—
123C3×#217—31no—
124C3×#217—31no—
125C3×#217C2×D311yescharacteristic
126C3×#217C2×D311yescharacteristic
127C3×#217C2×D311yescharacteristic
128C3×#217C2×D311yescharacteristic
129C3×Dic3⋊C4—31no—
130C3×Dic3⋊C4—31no—
131C3×C4×Dic3C2×D311yescharacteristic
132#3724—31no—
133#3724—31no—
134#3724—31no—
135Dic3⋊Dic3—31no—
136Dic3⋊Dic3—31no—
137Dic3⋊Dic3—31no—
138Dic3⋊Dic3—31no—
139#3722—31no—
140#3722—31no—
141#3722—31no—
142Dic32—31no—
143Dic32—31no—
144Dic32—31no—
145C2×Q8×C9—31no—
146C2×D4×C9—21no—
147C2×D4×C9C2×D311yescharacteristic
148C2×#611—31no—
149C2×#611—31no—
150C2×#611—61no—
151C2×#611—61no—
152C22×Dic9—31no—
153C22×Dic9—61no—
154C22×Dic9C2×D311yescharacteristic
155C2×C4×D9—31no—
156C2×Dic18—31no—
157C2×Dic18—31no—
158C9×C22⋊C4—31no—
159C9×C22⋊C4—31no—
160C9×C22⋊C4—31no—
161C9×C22⋊C4—31no—
162C42×C9—31no—
163#3676—31no—
164#3676—61no—
165#3676C2×D311yescharacteristic
166#3676C2×D311yescharacteristic
167#3676C2×D311yescharacteristic
168(C2×D9)⋊C4—31no—
169(C2×D9)⋊C4—31no—
170(C2×D9)⋊C4—31no—
171(C2×D9)⋊C4—31no—
172(C2×D9)⋊C4—61no—
173C4⋊Dic9—31no—
174C4⋊Dic9—31no—
175Dic9⋊C4—31no—
176Dic9⋊C4—31no—
177Dic9⋊C4—31no—
178Dic9⋊C4—31no—
179Dic9⋊C4—31no—
180Dic9⋊C4—31no—
181Dic9⋊C4—61no—
182Dic9⋊C4C2×D311yescharacteristic
183Dic9⋊C4C2×D311yescharacteristic
184C4×Dic9—31no—
185C4×Dic9—31no—
186C4×Dic9—31no—
187C22×C3×C9—21no—
188C22×C3×C9—21no—
189C22×C3×C9—21no—
190C22×C3×C9—21no—
191C22×C3×C9—21no—
192C22×C3×C9—21no—
193C22×C3×C9C2411yescharacteristic
194C2×C3⋊D9—21no—
195C2×C3⋊D9—21no—
196C2×C3⋊D9—21no—
197C3×C4×C9—21no—
198C3⋊Dic9—21no—
199C3⋊Dic9—21no—
200C3⋊Dic9#4112yes—
201C9×Dic3—21no—
202C9×Dic3—21no—
203C9×Dic3—21no—
204C9×Dic3—21no—
205C3×Dic9—21no—
206C3×Dic9—21no—
207C3×Dic9—21no—
208C3×Dic9—21no—
209C3×#124—31no—
210C3×#123—91no—
211#946—91no—
212#944—91no—
213#939—91no—
214#939—91no—
215#939—181no—
216#936D911yescharacteristic, complemented
217#935—31no—
218C4×#67—91no—
219#924—91no—
220#898—91no—
221#896—91no—
222#888—91no—
223#887—91no—
224#885—91no—
225#885—91no—
226#885—91no—
227#885—91no—
228#883—91no—
229#883—91no—
230#67⋊C4—91no—
231#879—91no—
232#879—91no—
233#878—91no—
234#878—91no—
235#878—91no—
236#877—91no—
237#877—91no—
238#876—91no—
239C4×Dic6—91no—
240C23×C32C22×D311yescharacteristic
241C23×C32C22×D311yescharacteristic
242C22×C3⋊D3—31no—
243C32×D4—41no—
244C2×C32×C4C22×D311yescharacteristic
245#638—62no—
246#638—121no—
247C2×C3⋊Dic3—31no—
248C2×C3⋊Dic3—31no—
249C2×C3⋊Dic3—31no—
250C2×C3⋊Dic3—62no—
251C2×C3×Dic3—31no—
252C2×C3×Dic3—31no—
253C2×C3×Dic3—31no—
254C2×C3×Dic3—31no—
255C2×C3×Dic3—121no—
256C2×C3×Dic3C22×D311yescharacteristic
257C2×C3×Dic3C22×D311yescharacteristic
258C2×C3×Dic3C22×D311yescharacteristic
259C2×C3×Dic3C22×D311yescharacteristic
260C3×Dic6—41no—
261#627—62no—
262#627—121no—
263#624—121no—
264C23×C9—21no—
265C23×C9—21no—
266C23×C9C22×D311yescharacteristic
267C23×C9C22×D311yescharacteristic
268C22×D9—31no—
269C22×D9—61no—
270Q8×C9—121no—
271D4×C9—41no—
272D4×C9—42no—
273C2×C4×C9—21no—
274C2×C4×C9—31no—
275C2×C4×C9—31no—
276C2×C4×C9—31no—
277C2×C4×C9—31no—
278C2×C4×C9C22×D311yescharacteristic
279#611—62no—
280#611—122no—
281#611—121no—
282#611—122no—
283C2×Dic9—31no—
284C2×Dic9—31no—
285C2×Dic9—31no—
286C2×Dic9—41no—
287C2×Dic9—61no—
288C2×Dic9—61no—
289C2×Dic9—61no—
290C2×Dic9—62no—
291C2×Dic9—122no—
292C2×Dic9C22×D311yescharacteristic
293C2×Dic9C22×D311yescharacteristic
294C2×Dic9C22×D311yescharacteristic
295C2×Dic9C22×D311yescharacteristic
296C4×D9—121no—
297Dic18—62no—
298Dic18—121no—
299#542—271noSylow
300C2×C3×C9—41no—
301C2×C3×C9—41no—
302C2×C3×C9C2×#4111yescharacteristic
303C2×C3×C9H(5,𝔽2)11yescharacteristic
304C2×C3×C92−1+411yescharacteristic
305C3⋊D9—41no—
306C2×C3×Q8—31no—
307C2×C3×D4—21no—
308C2×C3×D4C2×D911yescharacteristic
309C2×C3×D4D3211yescharacteristic
310C22×C3×C4—91no—
311C2×#67—91no—
312C2×#67—91no—
313C2×#67—91no—
314C2×#67—91no—
315C2×#67—181no—
316C2×#67—181no—
317C22×Dic3—31no—
318C22×Dic3—91no—
319C22×Dic3—91no—
320C22×Dic3—181no—
321C2×C4×D3—91no—
322C2×C4×D3—91no—
323C2×Dic6—91no—
324C2×Dic6—91no—
325C2×Dic6—91no—
326C2×Dic6—91no—
327C2×Dic6C2×D911yescharacteristic
328C3×C4⋊C4—91no—
329C3×C4⋊C4—91no—
330C3×C22⋊C4—31no—
331C3×C22⋊C4—31no—
332C3×C22⋊C4—31no—
333C3×C22⋊C4—31no—
334C3×C22⋊C4—91no—
335C3×C22⋊C4—91no—
336C3×C22⋊C4—91no—
337C3×C42—31no—
338#217—31no—
339#217—31no—
340#217—31no—
341#217—91no—
342#217—91no—
343#217—181no—
344#217C2×D911yescharacteristic
345#217C2×D911yescharacteristic
346#217C2×D911yescharacteristic
347#217C2×D911yescharacteristic
348(C2×D3)⋊C4—91no—
349(C2×D3)⋊C4—91no—
350(C2×D3)⋊C4—91no—
351(C2×D3)⋊C4—91no—
352(C2×D3)⋊C4—91no—
353(C2×D3)⋊C4—91no—
354(C2×D3)⋊C4—91no—
355(C2×D3)⋊C4—91no—
356(C2×D3)⋊C4—181no—
357C4⋊Dic3—91no—
358C4⋊Dic3—91no—
359C4⋊Dic3—91no—
360C4⋊Dic3—91no—
361Dic3⋊C4—31no—
362Dic3⋊C4—31no—
363Dic3⋊C4—91no—
364Dic3⋊C4—91no—
365Dic3⋊C4—91no—
366Dic3⋊C4—91no—
367Dic3⋊C4—91no—
368Dic3⋊C4—91no—
369Dic3⋊C4—91no—
370Dic3⋊C4—91no—
371Dic3⋊C4—91no—
372Dic3⋊C4—91no—
373Dic3⋊C4—91no—
374Dic3⋊C4—91no—
375Dic3⋊C4—181no—
376C4×Dic3—91no—
377C4×Dic3—91no—
378C4×Dic3—91no—
379C4×Dic3—91no—
380C4×Dic3—91no—
381C4×Dic3—91no—
382C4×Dic3C2×D911yescharacteristic
383C22×C32—21no—
384C22×C32—21no—
385C22×C32—21no—
386C22×C32—21no—
387C22×C32—21no—
388C22×C32—21no—
389C22×C32C23×D311yescharacteristic
390C2×C3⋊D3—61no—
391C2×C3⋊D3—61no—
392C2×C3⋊D3—61no—
393C32×C4—21no—
394C3⋊Dic3—32no—
395C3⋊Dic3—61no—
396C3⋊Dic3—61no—
397C3×Dic3—21no—
398C3×Dic3—21no—
399C3×Dic3—21no—
400C3×Dic3—21no—
401C3×Dic3—61no—
402C3×Dic3—61no—
403C3×Dic3—61no—
404C3×Dic3—61no—
405C22×C9—22no—
406C22×C9—21no—
407C22×C9—21no—
408C22×C9—21no—
409C22×C9—21no—
410C22×C9—21no—
411C22×C9—21no—
412C22×C9—21no—
413C22×C9—41no—
414C22×C9—41no—
415C22×C9—41no—
416C22×C9—41no—
417C22×C9—41no—
418C22×C9C23×D311yescharacteristic
419C2×D9—61no—
420C2×D9—61no—
421C2×D9—61no—
422C2×D9—121no—
423C2×D9—121no—
424C2×D9—121no—
425C4×C9—21no—
426C4×C9—41no—
427C4×C9—61no—
428C4×C9—61no—
429C4×C9—61no—
430C4×C9—61no—
431Dic9—21no—
432Dic9—21no—
433Dic9—21no—
434Dic9—21no—
435Dic9—32no—
436Dic9—62no—
437Dic9—61no—
438Dic9—61no—
439Dic9—121no—
440Dic9—121no—
441C4⋊Q8—271no—
442#126—271no—
443#126—271no—
444#124—31no—
445#124—271no—
446#124—271no—
447#123—91no—
448#123—271no—
449C22⋊Q8—271no—
450C22⋊Q8—271no—
451C22⋊Q8—271no—
452#121—271no—
453C4×Q8—271no—
454C4×D4—271no—
455#117—271no—
456C3×C9#54211yescharacteristic, complemented, Sylow
457C23×C3—21no—
458C23×C3—21no—
459C23×C3C22×D911yescharacteristic
460C23×C3C22×D911yescharacteristic
461C23×C3C2×D3211yescharacteristic
462C23×C3C2×D3211yescharacteristic
463C22×D3—91no—
464C22×D3—91no—
465C22×D3—181no—
466C3×Q8—121no—
467C3×D4—41no—
468C3×D4—41no—
469C3×D4—42no—
470C2×C3×C4—21no—
471C2×C3×C4—31no—
472C2×C3×C4—31no—
473C2×C3×C4—31no—
474C2×C3×C4—31no—
475C2×C3×C4—91no—
476C2×C3×C4—91no—
477C2×C3×C4—91no—
478C2×C3×C4—91no—
479C2×C3×C4—361no—
480C2×C3×C4C22×D911yescharacteristic
481C2×C3×C4C2×D3211yescharacteristic
482#67—182no—
483#67—182no—
484#67—362no—
485#67—361no—
486#67—361no—
487#67—362no—
488C2×Dic3—31no—
489C2×Dic3—31no—
490C2×Dic3—31no—
491C2×Dic3—31no—
492C2×Dic3—91no—
493C2×Dic3—91no—
494C2×Dic3—91no—
495C2×Dic3—91no—
496C2×Dic3—91no—
497C2×Dic3—91no—
498C2×Dic3—121no—
499C2×Dic3—181no—
500C2×Dic3—181no—
501C2×Dic3—181no—
502C2×Dic3—182no—
503C2×Dic3—182no—
504C2×Dic3—362no—
505C2×Dic3C22×D911yescharacteristic
506C2×Dic3C22×D911yescharacteristic
507C2×Dic3C22×D911yescharacteristic
508C2×Dic3C22×D911yescharacteristic
509C4×D3—361no—
510C4×D3—361no—
511Dic6—41no—
512Dic6—182no—
513Dic6—182no—
514Dic6—361no—
515Dic6—361no—
516C2×C32—41no—
517C2×C32—41no—
518C2×C32#100411yescharacteristic
519C2×C32D3×#4111yescharacteristic
520C2×C32#101011yescharacteristic
521C3⋊D3—121no—
522C2×C9—21no—
523C2×C9—21no—
524C2×C9—21no—
525C2×C9—42no—
526C2×C9—41no—
527C2×C9—41no—
528C2×C9—81no—
529C2×C9#100411yescharacteristic
530C2×C9D3×#4111yescharacteristic
531C2×C9#101011yescharacteristic
532D9—121no—
533D9—241no—
534C2×Q8—31no—
535C2×Q8—271no—
536C2×Q8—271no—
537C2×D4—271no—
538C2×D4—271no—
539C2×D4D3×D911yescharacteristic
540C22×C4—91no—
541C22×C4—271no—
542C22×C4—271no—
543C4⋊C4—91no—
544C4⋊C4—91no—
545C4⋊C4—271no—
546C4⋊C4—271no—
547C4⋊C4—271no—
548C4⋊C4—271no—
549C4⋊C4—271no—
550C4⋊C4—271no—
551C4⋊C4—271no—
552C4⋊C4—271no—
553C22⋊C4—31no—
554C22⋊C4—31no—
555C22⋊C4—31no—
556C22⋊C4—31no—
557C22⋊C4—91no—
558C22⋊C4—91no—
559C22⋊C4—91no—
560C22⋊C4—271no—
561C22⋊C4—271no—
562C22⋊C4—271no—
563C22⋊C4—271no—
564C22⋊C4—271no—
565C42—31no—
566C42—271no—
567C42—271no—
568C42—271no—
569C22×C3—22no—
570C22×C3—21no—
571C22×C3—21no—
572C22×C3—21no—
573C22×C3—21no—
574C22×C3—21no—
575C22×C3—21no—
576C22×C3—21no—
577C22×C3—21no—
578C22×C3—21no—
579C22×C3—21no—
580C22×C3—21no—
581C22×C3—21no—
582C22×C3—21no—
583C22×C3—41no—
584C22×C3—41no—
585C22×C3—41no—
586C22×C3—41no—
587C22×C3—41no—
588C22×C3C23×D911yescharacteristic
589C22×C3C22×D3211yescharacteristic
590C2×D3—181no—
591C2×D3—181no—
592C2×D3—181no—
593C2×D3—181no—
594C2×D3—181no—
595C2×D3—181no—
596C2×D3—361no—
597C2×D3—361no—
598C2×D3—361no—
599C3×C4—21no—
600C3×C4—21no—
601C3×C4—41no—
602C3×C4—61no—
603C3×C4—61no—
604C3×C4—61no—
605C3×C4—61no—
606C3×C4—181no—
607C3×C4—181no—
608C3×C4—181no—
609C3×C4—181no—
610Dic3—21no—
611Dic3—21no—
612Dic3—21no—
613Dic3—21no—
614Dic3—61no—
615Dic3—61no—
616Dic3—61no—
617Dic3—61no—
618Dic3—92no—
619Dic3—92no—
620Dic3—182no—
621Dic3—181no—
622Dic3—181no—
623Dic3—181no—
624Dic3—181no—
625Dic3—361no—
626Dic3—361no—
627C32#566911yescharacteristic
628C9—21no—
629C9#566211yescharacteristic, complemented
630C23—271no—
631C23C2×D3×D911yescharacteristic
632C23C2×D3×D911yescharacteristic
633Q8—121no—
634Q8—542no—
635Q8—1081no—
636D4—41no—
637D4—542no—
638D4—1081no—
639C2×C4—31no—
640C2×C4—31no—
641C2×C4—31no—
642C2×C4—31no—
643C2×C4—91no—
644C2×C4—91no—
645C2×C4—91no—
646C2×C4—91no—
647C2×C4—271no—
648C2×C4—271no—
649C2×C4—271no—
650C2×C4—361no—
651C2×C4—542no—
652C2×C4—1081no—
653C2×C4C2×D3×D911yescharacteristic
654C2×C3—21no—
655C2×C3—21no—
656C2×C3—21no—
657C2×C3—42no—
658C2×C3—41no—
659C2×C3—41no—
660C2×C3—41no—
661C2×C3—41no—
662C2×C3—81no—
663C2×C3#6371011yescharacteristic
664C2×C3#41×D911yescharacteristic
665C2×C3#6371611yescharacteristic
666C2×C3#6430911yescharacteristic
667C2×C3#6431211yescharacteristic
668C2×C3#6433011yescharacteristic
669D3—361no—
670D3—361no—
671D3—721no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—541no—
679C22—541no—
680C22—541no—
681C22C22×D3×D911yescharacteristic
682C4—21no—
683C4—61no—
684C4—61no—
685C4—61no—
686C4—61no—
687C4—181no—
688C4—181no—
689C4—181no—
690C4—181no—
691C4—272no—
692C4—541no—
693C4—541no—
694C3—21no—
695C3#1058977111yescharacteristic, complemented
696C3#1059521111yescharacteristic
697C2—41no—
698C2—41no—
699C2—1081no—
700C2#1172249411yescharacteristic
701C2#1172249811yescharacteristic
702C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radical#11721839C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C9×#936C2semidirect
3. C3×#63497C2semidirect
4. #11721275C2semidirect
5. #11721267C2semidirect
6. #11721253C2semidirect
7. #11721248C2semidirect
8. #11721247C2semidirect
9. #11721242C2semidirect
10. #11721242C2semidirect
11. #11721241C2semidirect
12. #11720986C2semidirect
13. #11720926C2semidirect
14. #11720903C2semidirect
15. #11720900C2semidirect
73. #63497D3semidirect
216. #936D9semidirect
456. C3×C9#542semidirect
629. C9#5662semidirect
695. C3#10589771semidirect

Maximal subgroups

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

subgroupindex
#117218392
C9×#9362
C3×#634972
#117212752
#117212672
#117212532
#117212482
#117212472
#117212422
#117212412
#117209862
#117209262
#117209032
#117209002
#105952113
#105897713