Finite Group Databasebeta

← #10900967

Subgroups — #10900967

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5688C211yesmaximal, characteristic, complemented
2C2×#4784C211yesmaximal, characteristic
3C2×#4779C211yesmaximal, characteristic
4#76475C211yesmaximal, characteristic
5C3×#2025C211yesmaximal, characteristic, complemented
6#71739C211yesmaximal, characteristic, complemented
7#70163C211yesmaximal, characteristic, complemented
8#70163C211yesmaximal, characteristic, complemented
9#69639C211yesmaximal, characteristic, complemented
10#68873C211yesmaximal, characteristic, complemented
11#68827C211yesmaximal, characteristic, complemented
12#68826C211yesmaximal, characteristic, complemented
13#68813C211yesmaximal, characteristic, complemented
14#68664C211yesmaximal, characteristic, complemented
15#68634C211yesmaximal, characteristic, complemented
16#29644—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×#938C2211yescharacteristic
19#5688—24no—
20#5688—24no—
21C2×#896—21no—
22C2×#896—21no—
23C2×#892C2211yescharacteristic
24C2×#892C2211yescharacteristic
25C2×#888C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#887—21no—
28C2×#887—21no—
29C2×#883—21no—
30C2×C42×D3C2211yescharacteristic
31C2×#870C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C3×#390C2211yescharacteristic
35#5221C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×(C2×D3)⋊C8C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#817C2211yescharacteristic
42#4987C2211yescharacteristic
43#4984C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46#4956—21no—
47#4951—21no—
48#4945—21no—
49#4928C2211yescharacteristic
50#4784—24no—
51#4779—24no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#533—31no—
65C2×#482—31no—
66C2×#482—31no—
67#2854—31no—
68#2059—31no—
69#2059—31no—
70#2058—31no—
71#2057—31no—
72#2025—31no—
73#2025D311yescharacteristic, complemented
74#2023—31no—
75#2023—31no—
76#1825—31no—
77#1825—31no—
78#1817—31no—
79C22×C3×D4C2311yescharacteristic
80C22×#67—21no—
81C22×C4×D3D411yescharacteristic
82C22×C4×D3D411yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×Dic6—21no—
85C22×Dic6—21no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#125—22no—
89C3×#125—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101#938—22no—
102#938—22no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#896—28no—
124#896—28no—
125#892—22no—
126#892—22no—
127#892D414yes—
128#892D414yes—
129#888—22no—
130#888—22no—
131#888—22no—
132#888—22no—
133#887—28no—
134#887—28no—
135#883—44no—
136C42×D3—22no—
137C42×D3—22no—
138#870—22no—
139#870—22no—
140C3×#102—22no—
141C3×#102—22no—
142#832—22no—
143#832—22no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147(C2×D3)⋊C8—22no—
148(C2×D3)⋊C8—22no—
149#818—22no—
150#817—22no—
151#808—22no—
152#807—22no—
153#533—64no—
154#533—64no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8—31no—
157C2×#125—31no—
158C2×#125C2×D311yescharacteristic
159C2×#123—61no—
160C2×C22⋊Q8—61no—
161C2×C22⋊Q8—61no—
162C2×#121—31no—
163C2×C4×Q8—61no—
164C2×C4×Q8—61no—
165C2×#117—31no—
166C2×#117—31no—
167C22×C42—31no—
168#482—64no—
169#482—64no—
170C2×#107—31no—
171C2×#107—31no—
172C2×#106—31no—
173C2×#106—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×#102C2×D311yescharacteristic
177C2×#102C2×D311yescharacteristic
178C2×C22⋊C8—31no—
179C2×C22⋊C8—31no—
180#390—31no—
181#390C2×D311yescharacteristic
182#389—31no—
183#389—31no—
184#388—61no—
185#388—61no—
186#386—61no—
187#380—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195#340C2×D311yescharacteristic
196#336—31no—
197#336—31no—
198#336—31no—
199#336C2×D311yescharacteristic
200C24×C3—21no—
201C23×D3C2×D411yescharacteristic
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—42no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—22no—
231C2×C4×D3—22no—
232C2×C4×D3—41no—
233C2×C4×D3—41no—
234C2×C4×D3C2×D412yes—
235C2×C4×D3C2×D412yes—
236C2×Dic6—24no—
237C2×Dic6—24no—
238C2×Dic6—42no—
239C2×Dic6—42no—
240C2×Dic6—42no—
241C2×Dic6—42no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4C2×D412yes—
255C3×C4⋊C4C2×D412yes—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—42no—
258C3×C42—21no—
259C3×C42—21no—
260#217—42no—
261#217—42no—
262#217—42no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—42no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273C4⋊Dic3C2×D412yes—
274C4⋊Dic3C2×D412yes—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279Dic3⋊C4—42no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—22no—
283C4×Dic3—22no—
284C4×Dic3—24no—
285C4×Dic3—24no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C22×Q8—61no—
293C22×Q8—61no—
294C22×D4—61no—
295C22×D4C22×D311yescharacteristic
296C23×C4—31no—
297C23×C4—31no—
298C23×C4—31no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C4⋊Q8—62no—
304C4⋊Q8—62no—
305C4⋊Q8—62no—
306C4⋊Q8—62no—
307#125—22no—
308#125—22no—
309#125—62no—
310#125—62no—
311#123—124no—
312C22⋊Q8—68no—
313C22⋊Q8—68no—
314#121—62no—
315#121—62no—
316C4×Q8—68no—
317C4×Q8—68no—
318#117—34no—
319#117—34no—
320#117—62no—
321#117—62no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C42—31no—
346C2×C42—31no—
347C2×C42—31no—
348C2×C42—62no—
349C2×C42—62no—
350C2×C42—61no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353#107—62no—
354#107—62no—
355#106—62no—
356#106—62no—
357#102—22no—
358#102—22no—
359#102—62no—
360#102—62no—
361C22⋊C8—62no—
362C22⋊C8—62no—
363#95—61no—
364#95—61no—
365#95—61no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3C22×D411yescharacteristic
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—21no—
380C22×D3#12012yes—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C42−1+411yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×C3×C42−1+411yescharacteristic
406#67—84no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3#12012yes—
426C2×Dic3#13512yes—
427C2×Dic3AGL(1,ℤ/8)12yes—
428C4×D3—22no—
429C4×D3—22no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—44no—
433C4×D3—44no—
434Dic6—48no—
435Dic6—48no—
436C3×C8—41no—
437C3×C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C24—21no—
441C24—31no—
442C2×Q8—64no—
443C2×Q8—64no—
444C2×Q8—122no—
445C2×Q8—122no—
446C2×Q8—122no—
447C2×Q8—122no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—32no—
464C22×C4—32no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—121no—
478C22×C4—121no—
479C22×C4D3×D411yescharacteristic
480C22×C4D3×D411yescharacteristic
481C22×C4C23×D311yescharacteristic
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—32no—
501C4⋊C4—32no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—64no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4D3×D412yes—
515C4⋊C4D3×D412yes—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—64no—
519C22⋊C4—64no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C42—21no—
525C42—21no—
526C42—62no—
527C42—62no—
528C42—64no—
529C42—64no—
530C42—61no—
531C42—61no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3C2×#12011yescharacteristic
540C22×C3#53611yescharacteristic
541C22×C3#53611yescharacteristic
542C22×C3C2×#13511yescharacteristic
543C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57511yescharacteristic
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—22no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—41no—
553C3×C4—42no—
554C3×C4—41no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—81no—
558C3×C4#53611yescharacteristic
559C3×C4#53611yescharacteristic
560C3×C4#53611yescharacteristic
561C3×C4#53611yescharacteristic
562Dic3—22no—
563Dic3—22no—
564Dic3—22no—
565Dic3—42no—
566Dic3—42no—
567Dic3—81no—
568Dic3—81no—
569C23—32no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—61no—
578C23—61no—
579C23—61no—
580C23—61no—
581C23—61no—
582C23—61no—
583C23C2×D3×D411yescharacteristic
584Q8—128no—
585Q8—128no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—244no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—32no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—124no—
622C2×C4—124no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4C2×D3×D411yescharacteristic
634C2×C4C2×D3×D411yescharacteristic
635C2×C4#100711yescharacteristic
636C2×C4#101011yescharacteristic
637C2×C4#101011yescharacteristic
638C2×C4#101011yescharacteristic
639C8—41no—
640C8—41no—
641C8—121no—
642C8—121no—
643C2×C3—161no—
644C2×C3#261411yescharacteristic
645C2×C3#300211yescharacteristic
646C2×C3#300611yescharacteristic
647C2×C3#322811yescharacteristic
648C2×C3#323311yescharacteristic
649C2×C3#323511yescharacteristic
650C2×C3#324011yescharacteristic
651D3—42no—
652C22—62no—
653C22—62no—
654C22—62no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—121no—
663C22—121no—
664C22—121no—
665C22—121no—
666C22D3×#12011yescharacteristic
667C22#562811yescharacteristic
668C22#567511yescharacteristic
669C22D3×#13511yescharacteristic
670C22#576811yescharacteristic
671C22D3×AGL(1,ℤ/8)11yescharacteristic
672C22#577311yescharacteristic
673C4—41no—
674C4—42no—
675C4—41no—
676C4—42no—
677C4—62no—
678C4—62no—
679C4—62no—
680C4—81no—
681C4—122no—
682C4—122no—
683C4—241no—
684C4—241no—
685C4#562811yescharacteristic
686C4#562811yescharacteristic
687C4#567511yescharacteristic
688C4#567511yescharacteristic
689C3#2964411yescharacteristic, complemented, Sylow
690C2—122no—
691C2—161no—
692C2#7687011yescharacteristic
693C2#8008211yescharacteristic
694C2#8008411yescharacteristic
695C2#8117211yescharacteristic
696C2#8117511yescharacteristic
697C2#8164911yescharacteristic
698C2#8172211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2025C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2025D3
hypocenterC3#29644
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2025

Complements

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

NH ≅ G/Ntype
1. C2×#5688C2semidirect
5. C3×#2025C2semidirect
6. #71739C2semidirect
7. #70163C2semidirect
8. #70163C2semidirect
9. #69639C2semidirect
10. #68873C2semidirect
11. #68827C2semidirect
12. #68826C2semidirect
13. #68813C2semidirect
14. #68664C2semidirect
15. #68634C2semidirect
73. #2025D3semidirect
689. C3#29644semidirect

Maximal subgroups

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

subgroupindex
C2×#56882
C2×#47842
C2×#47792
#764752
C3×#20252
#717392
#701632
#696392
#688732
#688272
#688262
#688132
#686642
#686342
#296443