Finite Group Databasebeta

← #10877259

Subgroups — #10877259

Order 768.

882 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#75964C211yesmaximal, characteristic
2D3×#382C211yesmaximal, characteristic, complemented
3C3×#2101C211yesmaximal, characteristic, complemented
4#69613C211yesmaximal, characteristic, complemented
5#69558C211yesmaximal, characteristic, complemented
6#68893C211yesmaximal, characteristic, complemented
7#68893C211yesmaximal, characteristic, complemented
8#68880C211yesmaximal, characteristic
9#68880C211yesmaximal, characteristic
10#68875C211yesmaximal, characteristic, complemented
11#68874C211yesmaximal, characteristic, complemented
12#68798C211yesmaximal, characteristic, complemented
13#68797C211yesmaximal, characteristic, complemented
14#68680C211yesmaximal, characteristic, complemented
15#68662C211yesmaximal, characteristic, complemented
16#27684—31nomaximal, Sylow
17C2×#895C2211yescharacteristic
18C2×#893—21no—
19C2×#893C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×C42×D3C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#382C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33#4990—21no—
34#4987C2211yescharacteristic
35#4977C2211yescharacteristic
36#4977C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4939C2211yescharacteristic
43#4936C2211yescharacteristic
44#4666—21no—
45#4665—21no—
46D3×#95—21no—
47D3×#95—21no—
48#4650—21no—
49#4650—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60C4×#121—31no—
61C2×#382—31no—
62#2101—31no—
63#2101D311yescharacteristic, complemented
64#2092—31no—
65#2092—31no—
66#1949—31no—
67#1949—31no—
68#1874—31no—
69#1874—31no—
70#1853—31no—
71#1852—31no—
72#1804—31no—
73#1804—31no—
74#1797—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3D411yescharacteristic
81C22×C4×D3D411yescharacteristic
82C22×C4×D3C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3D411yescharacteristic
103C2×C4×Dic3D411yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#895—22no—
108#895Q814yes—
109#893—22no—
110#893—22no—
111#893—44no—
112D3×C4⋊C4—41no—
113D3×C4⋊C4—41no—
114D3×C4⋊C4—41no—
115D3×C4⋊C4—41no—
116C42×D3—22no—
117C42×D3—22no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#821—41no—
130#821—41no—
131(C2×D3)⋊C8—22no—
132(C2×D3)⋊C8—22no—
133#818—41no—
134#817—41no—
135Dic3⋊C8—22no—
136Dic3⋊C8—22no—
137#809—41no—
138#809—41no—
139#808—41no—
140#807—41no—
141C2×#121—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C22×C4⋊C4—31no—
145C22×C4⋊C4—31no—
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×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#390—31no—
160#389C2×D311yescharacteristic
161#388—61no—
162#386—61no—
163#382—31no—
164#382—31no—
165#382—31no—
166#382—61no—
167#382—61no—
168#382C2×D311yescharacteristic
169#380—31no—
170C4×C4⋊C4—31no—
171C4×C22⋊C4—61no—
172C2×#95—61no—
173C2×#95—61no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336C2×D311yescharacteristic
184C23×D3—21no—
185C23×D3C2×D411yescharacteristic
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—24no—
204C2×D12—21no—
205C2×D12—42no—
206C2×D12—42no—
207C2×D12C2×Q812yes—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—41no—
217C2×C4×D3—41no—
218C2×C4×D3—42no—
219C2×C4×D3—42no—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3C2×Q812yes—
228C2×C4×D3#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4C2×Q812yes—
244C3×C42—21no—
245C3×C42—21no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259C4×Dic3—22no—
260C4×Dic3—22no—
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—31no—
271C22×D4—61no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—61no—
276C23×C4—61no—
277C23×C4—61no—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282#121—34no—
283#121—62no—
284C4×D4—62no—
285C4×D4—62no—
286C4×D4—124no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—121no—
294C2×C4⋊C4—121no—
295C2×C4⋊C4—121no—
296C2×C4⋊C4—121no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—62no—
310C2×C42—62no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—121no—
314#107—121no—
315#106—121no—
316#106—121no—
317C4⋊C8—62no—
318C4⋊C8—62no—
319#102—121no—
320#102—121no—
321#102—121no—
322#102—121no—
323C22⋊C8—62no—
324C22⋊C8—62no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336C23×C3C22×D411yescharacteristic
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—22no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3C22⋊Q812yes—
349C22×D3#13512yes—
350C22×D3AGL(1,ℤ/8)12yes—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
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×C4C22×Q811yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3C22⋊Q812yes—
391C2×Dic3#12312yes—
392C2×Dic3#12312yes—
393D12—22no—
394D12—22no—
395D12—48no—
396D12—41no—
397D12—41no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—44no—
403C4×D3—44no—
404C4×D3—84no—
405C4×D3—84no—
406C4×D3—84no—
407C3×C8—41no—
408C3×C8—41no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C24—31no—
412C24—61no—
413C2×D4—32no—
414C2×D4—64no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—122no—
421C2×D4—122no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—32no—
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—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4C23×D311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4D3×Q812yes—
485C22⋊C4—64no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C42—21no—
491C42—21no—
492C42—32no—
493C42—32no—
494C42—62no—
495C42—62no—
496C42—122no—
497C22×C3C2×C22⋊Q811yescharacteristic
498C22×C3C2×#12311yescharacteristic
499C22×C3#53511yescharacteristic
500C22×C3C2×#13511yescharacteristic
501C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
502C22×C3#57511yescharacteristic
503C22×C3#57511yescharacteristic
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—22no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3#47814yes—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4—81no—
524C3×C4#55311yescharacteristic
525C3×C4#55311yescharacteristic
526C3×C4#55411yescharacteristic
527C3×C4#55411yescharacteristic
528Dic3—22no—
529Dic3—22no—
530Dic3—22no—
531Dic3—81no—
532Dic3—81no—
533Dic3—81no—
534C23—32no—
535C23—32no—
536C23—32no—
537C23—61no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23—62no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23C2×D3×D411yescharacteristic
549D4—62no—
550D4—62no—
551D4—128no—
552D4—121no—
553D4—121no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—32no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—124no—
586C2×C4—124no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—244no—
597C2×C4—244no—
598C2×C4—244no—
599C2×C4C2×D3×Q811yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C2×C4#100911yescharacteristic
604C2×C4#100911yescharacteristic
605C8—41no—
606C8—41no—
607C8—121no—
608C8—121no—
609C2×C3#248711yescharacteristic
610C2×C3C2×#47811yescharacteristic
611C2×C3#308011yescharacteristic
612C2×C3#309111yescharacteristic
613C2×C3#309411yescharacteristic
614C2×C3#321111yescharacteristic
615C2×C3#321311yescharacteristic
616D3—24no—
617D3—82no—
618C22—34no—
619C22—62no—
620C22—62no—
621C22—62no—
622C22—62no—
623C22—62no—
624C22—62no—
625C22—122no—
626C22—122no—
627C22—122no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22—241no—
632C22D3×C22⋊Q811yescharacteristic
633C22D3×#12311yescharacteristic
634C22#567411yescharacteristic
635C22D3×#13511yescharacteristic
636C22#576811yescharacteristic
637C22D3×AGL(1,ℤ/8)11yescharacteristic
638C22#577711yescharacteristic
639C4—42no—
640C4—41no—
641C4—41no—
642C4—62no—
643C4—62no—
644C4—62no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—241no—
649C4—241no—
650C4—241no—
651C4#569011yescharacteristic
652C4#569011yescharacteristic
653C4#570911yescharacteristic
654C4#570911yescharacteristic
655C3#2768411yescharacteristic, complemented, Sylow
656C2—64no—
657C2—242no—
658C2#7814611yescharacteristic
659C2D3×#47811yescharacteristic
660C2#8123111yescharacteristic
661C2#8137311yescharacteristic
662C2#8151511yescharacteristic
663C2#8164211yescharacteristic
664C2#8168411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2101C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2101D3
hypocenterC3#27684
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
2. D3×#382C2semidirect
3. C3×#2101C2semidirect
4. #69613C2semidirect
5. #69558C2semidirect
6. #68893C2semidirect
7. #68893C2semidirect
10. #68875C2semidirect
11. #68874C2semidirect
12. #68798C2semidirect
13. #68797C2semidirect
14. #68680C2semidirect
15. #68662C2semidirect
63. #2101D3semidirect
655. C3#27684semidirect

Maximal subgroups

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

subgroupindex
#759642
D3×#3822
C3×#21012
#696132
#695582
#688932
#688802
#688752
#688742
#687982
#687972
#686802
#686622
#276843