Finite Group Databasebeta

← #10877210

Subgroups — #10877210

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#121C211yesmaximal, characteristic
2#82224C211yesmaximal, characteristic, complemented
3C3×#1875C211yesmaximal, characteristic
4#71678C211yesmaximal, characteristic
5#71576C211yesmaximal, characteristic, complemented
6#70182C211yesmaximal, characteristic, complemented
7#70181C211yesmaximal, characteristic, complemented
8#70153C211yesmaximal, characteristic, complemented
9#70153C211yesmaximal, characteristic, complemented
10#69929C211yesmaximal, characteristic, complemented
11#69908C211yesmaximal, characteristic, complemented
12#68846C211yesmaximal, characteristic, complemented
13#68845C211yesmaximal, characteristic, complemented
14#68796C211yesmaximal, characteristic, complemented
15#68795C211yesmaximal, characteristic, complemented
16#27679—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×D4×Dic3—21no—
20C2×D4×Dic3C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4×Dic3C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26#5221C2211yescharacteristic
27#5220—21no—
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4984C2211yescharacteristic
37#4977C2211yescharacteristic
38#4977C2211yescharacteristic
39#4975C2211yescharacteristic
40#4975C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4951—21no—
45#4942—21no—
46#4942—21no—
47Dic3×C22⋊C4—21no—
48C4×Dic3⋊C4—21no—
49C4×Dic3⋊C4—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60C4×#121—31no—
61#2285—31no—
62#2101—31no—
63#2101—31no—
64#2092—31no—
65#2092—31no—
66#1948—31no—
67#1947—31no—
68#1875—31no—
69#1875D311yescharacteristic
70#1854—31no—
71#1854—31no—
72#1802—31no—
73#1802—31no—
74#1797—31no—
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C23×Dic3D411yescharacteristic
80C23×Dic3D411yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#121—22no—
84C3×#121Q814yes—
85C3×#117—41no—
86C3×#117—41no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100D4×Dic3—44no—
101#926—41no—
102#926—41no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—22no—
113C2×C4×Dic3—22no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C3×#102—41no—
123C3×#102—41no—
124C3×C22⋊C8—22no—
125#832—41no—
126#832—41no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#830—22no—
135#818—41no—
136#817—41no—
137Dic3⋊C8—22no—
138Dic3⋊C8—22no—
139#808—41no—
140#807—41no—
141C2×#121C2×D311yescharacteristic
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C2×#117—31no—
145C2×#117C2×D311yescharacteristic
146C22×C42—31no—
147C2×#107—31no—
148C2×#107—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×C4⋊C8—31no—
152C2×C4⋊C8—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102C2×D311yescharacteristic
156C2×#102C2×D311yescharacteristic
157C2×C22⋊C8—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#390—31no—
160#389—31no—
161#388—61no—
162#386—61no—
163#384—31no—
164#384—31no—
165#382—31no—
166#382—31no—
167#380—31no—
168#380—61no—
169#380—61no—
170C4×C4⋊C4—31no—
171C4×C4⋊C4—61no—
172C4×C4⋊C4—61no—
173C4×C22⋊C4—61no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184C24×C3—21no—
185C24×C3C2×D411yescharacteristic
186C2×C3×D4—24no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—42no—
193C2×C3×D4—42no—
194C2×C3×D4C2×Q812yes—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×Q812yes—
204C22×C3×C4#4112yes—
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—41no—
215C22×Dic3—41no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4C2×Q812yes—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—42no—
237C3×C42—42no—
238#217—42no—
239#217—42no—
240#217—42no—
241#217—42no—
242#217—42no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252Dic3⋊C4—42no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—22no—
259C4×Dic3—22no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3#4112yes—
263C4×Dic3#4112yes—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C22×D4—21no—
271C22×D4C22×D311yescharacteristic
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—61no—
275C23×C4C22×D311yescharacteristic
276C22×C8—31no—
277C22×C8—31no—
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280#121—22no—
281#121Dic614yes—
282C4×D4—62no—
283C4×D4—62no—
284C4×D4—124no—
285#117—41no—
286#117—41no—
287#117—121no—
288#117—121no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—21no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—62no—
314C2×C42—62no—
315C2×C42—61no—
316C2×C42—61no—
317#107—121no—
318#107—121no—
319#106—121no—
320#106—121no—
321C4⋊C8—62no—
322C4⋊C8—62no—
323#102—41no—
324#102—41no—
325#102—121no—
326#102—121no—
327C22⋊C8—22no—
328C22⋊C8—62no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—22no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3C22⋊Q812yes—
350C23×C3C22×D411yescharacteristic
351C3×D4—22no—
352C3×D4—22no—
353C3×D4—48no—
354C3×D4—41no—
355C3×D4—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4C22⋊Q812yes—
372C2×C3×C4C22×Q811yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—44no—
387C2×Dic3—44no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—84no—
401C2×Dic3#12312yes—
402C2×Dic3#12312yes—
403C2×Dic3#13512yes—
404C2×Dic3AGL(1,ℤ/8)12yes—
405C3×C8—41no—
406C3×C8—41no—
407C3⋊C8—41no—
408C3⋊C8—41no—
409C24—21no—
410C24D3×D411yescharacteristic
411C2×D4—24no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×Dic612yes—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—122no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4C2×Dic612yes—
445C22×C4D3×D411yescharacteristic
446C22×C4#23712yes—
447C22×C4C23×D311yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—122no—
474C4⋊C4—122no—
475C4⋊C4—122no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4C2×Dic612yes—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—122no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C42—32no—
487C42—32no—
488C42—42no—
489C42—62no—
490C42—62no—
491C42—61no—
492C42—61no—
493C42—122no—
494C42—122no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3—81no—
509C22×C3C2×C22⋊Q811yescharacteristic
510C22×C3C2×#12311yescharacteristic
511C22×C3#53511yescharacteristic
512C22×C3C2×#13511yescharacteristic
513C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
514C22×C3#57511yescharacteristic
515C22×C3#57511yescharacteristic
516C3×C4—22no—
517C3×C4—42no—
518C3×C4—81no—
519C3×C4—81no—
520C3×C4#55311yescharacteristic
521C3×C4#55311yescharacteristic
522C3×C4#55411yescharacteristic
523C3×C4#55411yescharacteristic
524Dic3—22no—
525Dic3—41no—
526Dic3—41no—
527Dic3—81no—
528Dic3—81no—
529Dic3—81no—
530Dic3—81no—
531Dic3#48214yes—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23#87812yes—
546C23C2×D3×D411yescharacteristic
547D4—22no—
548D4—22no—
549D4—48no—
550D4—41no—
551D4—41no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—32no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—124no—
580C2×C4—124no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—244no—
594C2×C4#87812yes—
595C2×C4C22×Dic611yescharacteristic
596C2×C4C2×#23711yescharacteristic
597C2×C4#100411yescharacteristic
598C2×C4#100411yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C8—41no—
602C8—41no—
603C8—121no—
604C8—121no—
605C2×C3—42no—
606C2×C3—82no—
607C2×C3#248711yescharacteristic
608C2×C3#307611yescharacteristic
609C2×C3#307711yescharacteristic
610C2×C3C2×#48211yescharacteristic
611C2×C3#309311yescharacteristic
612C2×C3#321211yescharacteristic
613C2×C3#321411yescharacteristic
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22C2×#87811yescharacteristic
629C22#559211yescharacteristic
630C22D3×#12311yescharacteristic
631C22D3×#13511yescharacteristic
632C22#576811yescharacteristic
633C22D3×AGL(1,ℤ/8)11yescharacteristic
634C22#577311yescharacteristic
635C4—22no—
636C4—34no—
637C4—42no—
638C4—62no—
639C4—81no—
640C4—81no—
641C4—121no—
642C4—121no—
643C4—241no—
644C4—241no—
645C4—241no—
646C4—241no—
647C4#553911yescharacteristic
648C4#553911yescharacteristic
649C4#560111yescharacteristic
650C4#560111yescharacteristic
651C3#2767911yescharacteristic, complemented, Sylow
652C2—42no—
653C2—82no—
654C2#7473811yescharacteristic
655C2#7882211yescharacteristic
656C2#7891711yescharacteristic
657C2#8007711yescharacteristic
658C2#8007811yescharacteristic
659C2D3×#48211yescharacteristic
660C2#8137211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1875C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#1875D3
hypocenterC3#27679
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. #82224C2semidirect
5. #71576C2semidirect
6. #70182C2semidirect
7. #70181C2semidirect
8. #70153C2semidirect
9. #70153C2semidirect
10. #69929C2semidirect
11. #69908C2semidirect
12. #68846C2semidirect
13. #68845C2semidirect
14. #68796C2semidirect
15. #68795C2semidirect
651. C3#27679semidirect

Maximal subgroups

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

subgroupindex
Dic3×#1212
#822242
C3×#18752
#716782
#715762
#701822
#701812
#701532
#699292
#699082
#688462
#688452
#687962
#687952
#276793