Finite Group Databasebeta

← #224751

Subgroups — #224751

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2374C211yesmaximal, characteristic
2#22627C212yesmaximal, complemented
3#9653C212yesmaximal
4#9652C212yesmaximal, complemented
5#9651C212yesmaximal, complemented
6#8558C211yesmaximal, characteristic, complemented
7#8557C211yesmaximal, characteristic, complemented
8#8557C211yesmaximal, characteristic, complemented
9#8153C211yesmaximal, characteristic, complemented
10#8152C211yesmaximal, characteristic, complemented
11C4×#323C211yesmaximal, characteristic, complemented
12C22×#123C411yescharacteristic
13#2374—22no—
14#2374—22no—
15#2374—22no—
16#2374—22no—
17C2×#390C2211yescharacteristic
18C2×#388C411yescharacteristic
19C2×#388C411yescharacteristic
20C2×#388C2211yescharacteristic
21C2×#388C2211yescharacteristic
22C2×#387C411yescharacteristic
23C2×#387C411yescharacteristic
24C2×#386C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C411yescharacteristic
28C2×C4×C22⋊C4C411yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1940—22no—
32#1917—22no—
33#1916—22no—
34#1797C2212yes—
35#1797C2212yes—
36#1797C2212yes—
37#1771—22no—
38#1749C2212yes—
39#1749C2212yes—
40C4×C22⋊C8C2212yes—
41C2×#324C2212yes—
42C2×#324C2212yes—
43C2×#323C2212yes—
44C2×#323C2212yes—
45C2×#323C2212yes—
46C2×#323C2212yes—
47#1280C2211yescharacteristic
48#1280C2211yescharacteristic
49#1280C2211yescharacteristic
50#1280C2211yescharacteristic
51C23×D4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C2×#123—24no—
55C2×#123—22no—
56C2×#123—22no—
57C2×#123—24no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2311yescharacteristic
72C22×C22⋊C4C2311yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×#97—22no—
88C2×#97—22no—
89C2×C4×C8—22no—
90#390—22no—
91#390—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#386—22no—
105#386—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C2×#95C2×C411yescharacteristic
125C2×#95C2×C411yescharacteristic
126C2×#95C2×C411yescharacteristic
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C411yescharacteristic
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C411yescharacteristic
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C411yescharacteristic
133C2×#95C2311yescharacteristic
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#324—24no—
141#324—42no—
142#323—24no—
143#323—24no—
144#323—42no—
145#323—42no—
146C25C22×C411yescharacteristic
147C25C22×C411yescharacteristic
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—81no—
152C22×D4—81no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—41no—
160C23×C4—42no—
161C23×C4—81no—
162C23×C4—81no—
163C23×C4—81no—
164C23×C4M1614yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2411yescharacteristic
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182#123—216no—
183#123—216no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4M1614yes—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—42no—
242C2×C42—41no—
243C2×C42—81no—
244C2×C42—81no—
245C2×C42—81no—
246C2×C42—81no—
247C2×C42—81no—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C4⋊C8—44no—
255C4⋊C8—82no—
256C4⋊C8—82no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8#4114yes—
265#97—44no—
266#97—82no—
267C4×C8—82no—
268#95—22no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95C22⋊C411yescharacteristic
279#95C22⋊C411yescharacteristic
280#95C22⋊C411yescharacteristic
281#95C22⋊C411yescharacteristic
282#95C22⋊C412yes—
283#95C22⋊C412yes—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24C2×M1612yes—
305C24C2×M1612yes—
306C24C23×C411yescharacteristic
307C2×D4—28no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—84no—
311C2×D4—84no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—84no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×M1612yes—
391C22×C4C2×M1612yes—
392C22×C4C2×M1612yes—
393C22×C4C2×M1612yes—
394C22×C4#13112yes—
395C22×C4C22×D412yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C42−1+411yescharacteristic
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—82no—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—44no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—82no—
435C42—24no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—81no—
444C42—81no—
445C42—81no—
446C42—81no—
447C42—81no—
448C42—81no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23C22×C22⋊C411yescharacteristic
500C23C4×#4111yescharacteristic
501C23#52011yescharacteristic
502C23C22×M1611yescharacteristic
503C23C22×M1611yescharacteristic
504C23C2×#13111yescharacteristic
505C23#56811yescharacteristic
506D4—88no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
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—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—84no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4#40712yes—
584C2×C4#40712yes—
585C2×C4#41012yes—
586C2×C4#41312yes—
587C2×C4#54812yes—
588C2×C4#55012yes—
589C8—44no—
590C8—82no—
591C8—82no—
592C8—82no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22#234111yescharacteristic
628C22C2×#40711yescharacteristic
629C22#288011yescharacteristic
630C22C2×#41011yescharacteristic
631C22#288311yescharacteristic
632C22C2×#41311yescharacteristic
633C22#288811yescharacteristic
634C22#296511yescharacteristic
635C22#297011yescharacteristic
636C22#298911yescharacteristic
637C22#299011yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4—81no—
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—82no—
660C2—82no—
661C2#1736111yescharacteristic
662C2#1736711yescharacteristic
663C2#1741211yescharacteristic
664C2#1742711yescharacteristic
665C2#2253211yescharacteristic
666C2#2253511yescharacteristic
667C2#2253911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22627C2semidirect
4. #9652C2semidirect
5. #9651C2semidirect
6. #8558C2semidirect
7. #8557C2semidirect
8. #8557C2semidirect
9. #8153C2semidirect
10. #8152C2semidirect
11. C4×#323C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23742
#226272
#96532
#96522
#96512
#85582
#85572
#81532
#81522
C4×#3232