Finite Group Databasebeta

← #10845257

Subgroups — #10845257

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#463C211yesmaximal, characteristic
2#82955C211yesmaximal, characteristic, complemented
3C2×#5055C211yesmaximal, characteristic, complemented
4#82384C211yesmaximal, characteristic, complemented
5#72051C211yesmaximal, characteristic, complemented
6#72048C211yesmaximal, characteristic, complemented
7#72040C211yesmaximal, characteristic
8#72037C211yesmaximal, characteristic
9#71609C211yesmaximal, characteristic, complemented
10#70469C211yesmaximal, characteristic, complemented
11#70460C211yesmaximal, characteristic, complemented
12#70234C211yesmaximal, characteristic, complemented
13#70222C211yesmaximal, characteristic, complemented
14#70166C211yesmaximal, characteristic, complemented
15#70154C211yesmaximal, characteristic, complemented
16#25335—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×D8C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C3×#463—24no—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5221C2211yescharacteristic
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5212—21no—
39#5212—21no—
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42#5055—24no—
43#4977C2211yescharacteristic
44#4977C2211yescharacteristic
45#4951—21no—
46#4951—21no—
47#4950—21no—
48#4947—21no—
49#4947—21no—
50Dic3×C22⋊C4—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53C2×#803C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#489—31no—
63C2×#463D311yescharacteristic
64#2563—31no—
65#2367—31no—
66#2074—31no—
67#2073—31no—
68#2055—31no—
69#2054—31no—
70#1988—31no—
71#1940—31no—
72#1880—31no—
73#1879—31no—
74#1805—31no—
75#1804—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×C3×C4—21no—
81C23×Dic3—21no—
82C22×Dic6—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—41no—
86C2×C3×D8—22no—
87C2×C3×D8—22no—
88C2×C3×D8—41no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#124—22no—
92C3×#124D414yes—
93C3×C4×D4—22no—
94C3×C4×D4—22no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107Q8×Dic3—22no—
108Q8×Dic3—41no—
109D4×Dic3—22no—
110D4×Dic3—41no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125#869—22no—
126#869—22no—
127C3×C4⋊C8—22no—
128C3×#103—22no—
129C3×#102—22no—
130#835—22no—
131#835—22no—
132#832—22no—
133#832—22no—
134#832—22no—
135#832—22no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95D411yescharacteristic
144C3⋊#95D411yescharacteristic
145Dic3⋊C8—41no—
146Dic3⋊C8—41no—
147#803—22no—
148C22×SD16C2×D311yescharacteristic
149C22×D8C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4C2×D311yescharacteristic
155#489—64no—
156#463—24no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#97—31no—
169#393—61no—
170#393—61no—
171#391—31no—
172#390—31no—
173#390—31no—
174#390—31no—
175#388—61no—
176#388—61no—
177#385—61no—
178#382—31no—
179#382—31no—
180#380—61no—
181#380—61no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8C2×D412yes—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4C2×D412yes—
212C2×C3×D4#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—42no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—42no—
226C22×Dic3—42no—
227C22×Dic3—81no—
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C22×Dic3C2×D411yescharacteristic
231C22×Dic3C2×D411yescharacteristic
232C2×Dic6—42no—
233C2×Dic6—81no—
234C2×Dic6—81no—
235C3×SD16—44no—
236C3×D8—44no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C22⋊C4—24no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—42no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250#217—42no—
251#217—42no—
252#217—81no—
253#217—81no—
254#217—81no—
255#217—81no—
256#217—81no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—41no—
265C4⋊Dic3—41no—
266C4⋊Dic3—42no—
267C4⋊Dic3—42no—
268C4⋊Dic3#4112yes—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—42no—
274C4×Dic3—81no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C22×Q8—61no—
282C22×Q8C22×D311yescharacteristic
283C22×D4C22×D311yescharacteristic
284C22×D4C22×D311yescharacteristic
285C23×C4—21no—
286C23×C4—61no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×SD16—41no—
290C2×D8—22no—
291C2×D8—22no—
292C2×D8—41no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—62no—
298C4⋊Q8—62no—
299#124—22no—
300#124#6714yes—
301C4×Q8—62no—
302C4×Q8—121no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—62no—
306C4×D4—121no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331C4⋊C8—22no—
332C4⋊C8—121no—
333C4⋊C8—121no—
334#103—22no—
335#103—62no—
336#103—62no—
337#102—22no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—62no—
342#97—62no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3C22×D411yescharacteristic
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—41no—
370C3×Q8—41no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—84no—
390C2×C3×C4#12112yes—
391C2×C3×C4#12112yes—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×C3×C42−1+411yescharacteristic
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—84no—
423Dic6—84no—
424C3×C8—41no—
425C3×C8—41no—
426C3⋊C8—41no—
427C3⋊C8—41no—
428C24—21no—
429C24—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—122no—
436C2×Q8—241no—
437C2×Q8—241no—
438C2×Q8C2×#6712yes—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4#23712yes—
450C2×D4C2×#6712yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—241no—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4C23×D311yescharacteristic
470SD16—44no—
471D8—44no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—32no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—42no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C22⋊C4—241no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C42—61no—
509C42—61no—
510C42—61no—
511C42—61no—
512C42—122no—
513C42—241no—
514C42#23712yes—
515C42C2×#6712yes—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3C2×#12111yescharacteristic
531C22×C3#54611yescharacteristic
532C22×C3#54811yescharacteristic
533C22×C3#57511yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57611yescharacteristic
536C22×C3#57711yescharacteristic
537C3×C4—22no—
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—81no—
541C3×C4#52911yescharacteristic
542C3×C4#54211yescharacteristic
543C3×C4#54611yescharacteristic
544C3×C4#54711yescharacteristic
545Dic3—41no—
546Dic3—41no—
547Dic3—42no—
548Dic3—41no—
549Dic3—41no—
550Dic3—81no—
551Dic3—81no—
552Dic3—81no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23C2×D3×D411yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—244no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—84no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—244no—
617C2×C4#93912yes—
618C2×C4#93912yes—
619C2×C4C2×#23711yescharacteristic
620C2×C4C2×#23711yescharacteristic
621C2×C4#100411yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C2×C4#101011yescharacteristic
624C2×C4C22×#6711yescharacteristic
625C8—41no—
626C8—41no—
627C8—121no—
628C8—121no—
629C2×C3—82no—
630C2×C3—82no—
631C2×C3#242711yescharacteristic
632C2×C3#304011yescharacteristic
633C2×C3#304511yescharacteristic
634C2×C3#331911yescharacteristic
635C2×C3#332111yescharacteristic
636C2×C3#337011yescharacteristic
637C2×C3#337611yescharacteristic
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22#559811yescharacteristic
653C22#561211yescharacteristic
654C22#576911yescharacteristic
655C22#577011yescharacteristic
656C22#577311yescharacteristic
657C22#577311yescharacteristic
658C22C2×#93911yescharacteristic
659C4—22no—
660C4—22no—
661C4—42no—
662C4—81no—
663C4—121no—
664C4—121no—
665C4—122no—
666C4—121no—
667C4—121no—
668C4—241no—
669C4—241no—
670C4—241no—
671C4#560011yescharacteristic
672C4#560311yescharacteristic
673C4#580611yescharacteristic
674C4#582411yescharacteristic
675C3#2533511yescharacteristic, complemented, Sylow
676C2—82no—
677C2—82no—
678C2#8052011yescharacteristic
679C2#8052311yescharacteristic
680C2#8056311yescharacteristic
681C2#8056411yescharacteristic
682C2#8264711yescharacteristic
683C2#8390111yescharacteristic
684C2#8393311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#463C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenterC2×#463D3
hypocenterC3#25335
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#463

Complements

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

NH ≅ G/Ntype
2. #82955C2semidirect
3. C2×#5055C2semidirect
4. #82384C2semidirect
5. #72051C2semidirect
6. #72048C2semidirect
9. #71609C2semidirect
10. #70469C2semidirect
11. #70460C2semidirect
12. #70234C2semidirect
13. #70222C2semidirect
14. #70166C2semidirect
15. #70154C2semidirect
675. C3#25335semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4632
#829552
C2×#50552
#823842
#720512
#720482
#720402
#720372
#716092
#704692
#704602
#702342
#702222
#701662
#701542
#253353