Finite Group Databasebeta

← #10690575

Subgroups — #10690575

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4956C211yesmaximal, characteristic
2C2×#4726C211yesmaximal, characteristic, complemented
3C2×C3×#323C211yesmaximal, characteristic, complemented
4C2×#4525C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic, complemented
6#67674C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9083—31nomaximal, Sylow
9C23×#67C411yescharacteristic
10C23×C4×D3C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#217C411yescharacteristic
13C22×(C2×D3)⋊C4C411yescharacteristic
14C22×ΓL(1,𝔽25)—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×C3⋊#95C411yescharacteristic
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×(C2×D3)⋊C8—21no—
22C2×(C2×D3)⋊C8—21no—
23C2×(C2×D3)⋊C8—21no—
24#4956—21no—
25#4956—21no—
26#4956—21no—
27#4956—21no—
28#4726—22no—
29#4726—22no—
30C3×#323—22no—
31#4525—22no—
32C2×#407—31no—
33C2×#386—31no—
34#1779—31no—
35C2×#323—31no—
36C2×#323D311yescharacteristic, complemented
37#1280—31no—
38#1280—31no—
39C25×C3C2×C411yescharacteristic
40C24×D3C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—22no—
44C22×#67—21no—
45C22×#67—21no—
46C22×#67—22no—
47C22×#67—42no—
48C22×#67—42no—
49C23×Dic3C2×C411yescharacteristic
50C23×Dic3C2×C411yescharacteristic
51C23×Dic3C2×C411yescharacteristic
52C22×C4×D3—21no—
53C22×C4×D3—21no—
54C22×C4×D3—21no—
55C22×C4×D3—21no—
56C22×C4×D3—42no—
57C22×C4×D3—42no—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3D411yescharacteristic
62C22×C3×C8—21no—
63C22×C3×C8—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71C2×#217—21no—
72C2×#217—42no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—42no—
76C22×C3⋊C8—21no—
77C22×C3⋊C8—21no—
78C2×ΓL(1,𝔽25)—24no—
79C2×ΓL(1,𝔽25)—42no—
80C2×ΓL(1,𝔽25)—42no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—21no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8D412yescomplemented
85C3⋊#95—21no—
86C3⋊#95—21no—
87C3⋊#95—21no—
88C3⋊#95—21no—
89#830—21no—
90#830—21no—
91#830—21no—
92#830D412yescomplemented
93(C2×D3)⋊C8—24no—
94(C2×D3)⋊C8—42no—
95(C2×D3)⋊C8—42no—
96C23×D4—31no—
97C24×C4—31no—
98C22×M16—61no—
99C22×C22⋊C4—31no—
100C22×C22⋊C4—31no—
101C22×C22⋊C4C2×D311yescharacteristic
102#407—62no—
103#407—62no—
104C2×C22⋊C8—31no—
105C2×C22⋊C8—31no—
106C2×C22⋊C8—61no—
107C2×C22⋊C8—61no—
108C2×C22⋊C8—61no—
109C2×C22⋊C8C2×D311yescharacteristic
110C2×C22⋊C8C2×D311yescharacteristic
111#386—61no—
112#386—61no—
113#386—61no—
114#386—61no—
115C2×#95—31no—
116C2×#95—31no—
117#323—22no—
118#323—62no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—21no—
125C24×C3—41no—
126C24×C3—41no—
127C24×C3—41no—
128C24×C3—41no—
129C24×C3M1612yescomplemented
130C24×C3C22×C411yescharacteristic
131C23×D3—21no—
132C23×D3—21no—
133C23×D3—22no—
134C23×D3—22no—
135C23×D3—21no—
136C23×D3—21no—
137C23×D3—21no—
138C23×D3—21no—
139C23×D3—41no—
140C23×D3—41no—
141C23×D3C22⋊C411yescharacteristic
142C23×D3C22⋊C411yescharacteristic
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4#4111yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C2×#67—24no—
157C2×#67—24no—
158C2×#67—48no—
159C2×#67—42no—
160C2×#67—42no—
161C2×#67—48no—
162C2×#67—42no—
163C2×#67—42no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—41no—
173C22×Dic3—41no—
174C22×Dic3—41no—
175C22×Dic3—41no—
176C22×Dic3—41no—
177C22×Dic3C22⋊C411yescharacteristic
178C22×Dic3C22⋊C411yescharacteristic
179C22×Dic3M1612yescomplemented
180C2×C4×D3—22no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—21no—
188C2×C4×D3—21no—
189C2×C4×D3—21no—
190C2×C4×D3—21no—
191C2×C4×D3—22no—
192C2×C4×D3—44no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×C4×D3—41no—
197C2×C4×D3—41no—
198C2×C4×D3—41no—
199C2×C4×D3—41no—
200C2×C3×C8—22no—
201C2×C3×C8—41no—
202C2×C3×C8—41no—
203C2×C3×C8—41no—
204C2×C3×C8—41no—
205C2×C3×C8—41no—
206C3×C22⋊C4—41no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—41no—
210#217—42no—
211#217—42no—
212(C2×D3)⋊C4—42no—
213(C2×D3)⋊C4—42no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—41no—
216C2×C3⋊C8—41no—
217C2×C3⋊C8—41no—
218C2×C3⋊C8—41no—
219C2×C3⋊C8—41no—
220ΓL(1,𝔽25)—48no—
221C25—31no—
222C25C4×D311yescharacteristic
223C22×D4—62no—
224C22×D4—61no—
225C22×D4—61no—
226C22×D4—62no—
227C22×D4—122no—
228C22×D4—122no—
229C23×C4—31no—
230C23×C4—31no—
231C23×C4—31no—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—61no—
237C23×C4—61no—
238C23×C4—61no—
239C23×C4—61no—
240C23×C4—122no—
241C23×C4—122no—
242C23×C4C4×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C2×M16—64no—
245C2×M16—122no—
246C2×M16—122no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—61no—
250C22×C8—61no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4—122no—
262C2×C22⋊C4—122no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—32no—
267C22⋊C8—64no—
268C22⋊C8—61no—
269C22⋊C8—61no—
270C22⋊C8—61no—
271C22⋊C8—122no—
272C22⋊C8—122no—
273C22⋊C8#6712yescomplemented
274#95—61no—
275#95—61no—
276#95—61no—
277#95—61no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—81no—
298C23×C3—81no—
299C23×C3—81no—
300C23×C3—81no—
301C23×C3—81no—
302C23×C3—81no—
303C23×C3—81no—
304C23×C3—81no—
305C23×C3C2×C22⋊C411yescharacteristic
306C23×C3C4×D411yescharacteristic
307C23×C3C4×D411yescharacteristic
308C23×C3C2×M1611yescharacteristic
309C23×C3C2×M1611yescharacteristic
310C23×C3#13111yescharacteristic
311C23×C3#13111yescharacteristic
312C22×D3—24no—
313C22×D3—24no—
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—21no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—42no—
329C22×D3—42no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—41no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3#9912yes—
341C22×D3#10012yes—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—81no—
354C2×C3×C4—81no—
355C2×C3×C4#12011yescharacteristic
356C2×C3×C4#12111yescharacteristic
357C2×C3×C4#12111yescharacteristic
358C2×C3×C4#12311yescharacteristic
359#67—48no—
360#67—48no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—81no—
377C2×Dic3—81no—
378C2×Dic3#9912yes—
379C4×D3—24no—
380C4×D3—24no—
381C4×D3—42no—
382C4×D3—42no—
383C4×D3—42no—
384C4×D3—42no—
385C4×D3—42no—
386C4×D3—42no—
387C3×C8—42no—
388C3×C8—81no—
389C3⋊C8—42no—
390C3⋊C8—81no—
391C24—21no—
392C24—21no—
393C24—21no—
394C24—21no—
395C24—21no—
396C24—21no—
397C24—31no—
398C24—31no—
399C24—41no—
400C24—41no—
401C24—41no—
402C24—41no—
403C24—62no—
404C24—62no—
405C24—61no—
406C24—61no—
407C24—61no—
408C24—61no—
409C24—61no—
410C24—61no—
411C24—121no—
412C24—121no—
413C24ΓL(1,𝔽25)12yescomplemented
414C24C2×C4×D311yescharacteristic
415C2×D4—64no—
416C2×D4—64no—
417C2×D4—122no—
418C2×D4—122no—
419C2×D4—128no—
420C2×D4—122no—
421C2×D4—122no—
422C2×D4—128no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—32no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—121no—
454C22×C4—121no—
455C22×C4—121no—
456C22×C4—121no—
457C22×C4—124no—
458C22×C4—124no—
459C22×C4—124no—
460C22×C4—124no—
461C22×C4—121no—
462C22×C4—121no—
463C22×C4—121no—
464C22×C4—121no—
465C22×C4—121no—
466C22×C4#23511yescharacteristic
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4#23711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472M16—128no—
473C2×C8—22no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—62no—
480C2×C8—121no—
481C2×C8—121no—
482C2×C8—121no—
483C2×C8—121no—
484C2×C8—121no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—122no—
490C22⋊C4—122no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—21no—
499C22×C3—21no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—82no—
509C22×C3—82no—
510C22×C3—82no—
511C22×C3—82no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3#38611yescharacteristic
517C22×C3#40711yescharacteristic
518C22×C3#40811yescharacteristic
519C22×C3C2×#9911yescharacteristic
520C22×C3C2×#9911yescharacteristic
521C22×C3C2×#10011yescharacteristic
522C22×C3#41311yescharacteristic
523C22×C3#43511yescharacteristic
524C22×C3#43511yescharacteristic
525C22×C3#43611yescharacteristic
526C22×C3#43611yescharacteristic
527C2×D3—22no—
528C2×D3—22no—
529C2×D3—22no—
530C2×D3—22no—
531C2×D3—22no—
532C2×D3—22no—
533C2×D3—44no—
534C2×D3—44no—
535C2×D3—44no—
536C2×D3—44no—
537C2×D3—41no—
538C2×D3—41no—
539C2×D3—41no—
540C2×D3—41no—
541C2×D3—41no—
542C2×D3—41no—
543C2×D3—41no—
544C2×D3—41no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—81no—
550Dic3—22no—
551Dic3—41no—
552Dic3—42no—
553Dic3—81no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—32no—
562C23—32no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—64no—
576C23—64no—
577C23—61no—
578C23—61no—
579C23—61no—
580C23—61no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—81no—
590C23—81no—
591C23—81no—
592C23—81no—
593C23—81no—
594C23—81no—
595C23—81no—
596C23—81no—
597C23—122no—
598C23—122no—
599C23—122no—
600C23—122no—
601C23—122no—
602C23—122no—
603C23—122no—
604C23—122no—
605C23—122no—
606C23—122no—
607C23—121no—
608C23—121no—
609C23—121no—
610C23—121no—
611C23D3×C22⋊C411yescharacteristic
612C23#67⋊C411yescharacteristic
613C23C2×ΓL(1,𝔽25)11yescharacteristic
614C23#90111yescharacteristic
615C23D3×M1611yescharacteristic
616C23#90711yescharacteristic
617C23C4×#6711yescharacteristic
618D4—128no—
619D4—128no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—32no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—64no—
633C2×C4—64no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—62no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—122no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—122no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—241no—
658C2×C4—241no—
659C2×C4#88311yescharacteristic
660C2×C4#88411yescharacteristic
661C2×C4#93711yescharacteristic
662C2×C4#93811yescharacteristic
663C8—42no—
664C8—81no—
665C8—122no—
666C8—241no—
667C2×C3—21no—
668C2×C3—21no—
669C2×C3—21no—
670C2×C3—21no—
671C2×C3—82no—
672C2×C3#146211yescharacteristic
673C2×C3#146511yescharacteristic
674C2×C3#146811yescharacteristic
675C2×C3#147111yescharacteristic
676C2×C3#187011yescharacteristic
677C2×C3#189811yescharacteristic
678C2×C3#190011yescharacteristic
679D3—42no—
680D3—42no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—21no—
688C22—21no—
689C22—21no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—62no—
697C22—62no—
698C22—62no—
699C22—62no—
700C22—62no—
701C22—62no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—81no—
707C22—81no—
708C22—81no—
709C22—81no—
710C22—124no—
711C22—124no—
712C22—124no—
713C22—124no—
714C22—121no—
715C22—121no—
716C22—121no—
717C22—121no—
718C22—121no—
719C22—121no—
720C22—121no—
721C22—121no—
722C22#472611yescharacteristic
723C22#472711yescharacteristic
724C22#472911yescharacteristic
725C22#473011yescharacteristic
726C22D3×#9911yescharacteristic
727C22D3×#9911yescharacteristic
728C22D3×#10011yescharacteristic
729C22#474511yescharacteristic
730C22#495611yescharacteristic
731C22#511111yescharacteristic
732C22#512811yescharacteristic
733C4—21no—
734C4—21no—
735C4—21no—
736C4—21no—
737C4—62no—
738C4—81no—
739C4—122no—
740C4—121no—
741C4—241no—
742C3#908311yescharacteristic, complemented, Sylow
743C2—21no—
744C2—21no—
745C2—21no—
746C2—21no—
747C2—82no—
748C2—122no—
749C2—122no—
750C2#6812011yescharacteristic
751C2#6812111yescharacteristic
752C2#6812211yescharacteristic
753C2#6812311yescharacteristic
754C2#6995811yescharacteristic
755C2#7003611yescharacteristic
756C2#7008111yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#323C2
socleC23×C3C2×C22⋊C4
prime radicalC23×#67C4
cosocleC23×C3×C4C23
hypercenterC2×#323D3
hypocenterC3#9083
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C2×#4726C2semidirect
3. C2×C3×#323C2semidirect
4. C2×#4525C2semidirect
5. #69955C2semidirect
6. #67674C2semidirect
7. #67674C2semidirect
36. C2×#323D3semidirect
84. C3×C22⋊C8D4semidirect
92. #830D4semidirect
129. C24×C3M16semidirect
179. C22×Dic3M16semidirect
273. C22⋊C8#67semidirect
413. C24ΓL(1,𝔽25)semidirect
742. C3#9083semidirect

Maximal subgroups

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

subgroupindex
C2×#49562
C2×#47262
C2×C3×#3232
C2×#45252
#699552
#676742
#90833