Finite Group Databasebeta

← #366932

Subgroups — #366932

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1940C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6C2×#1762C211yesmaximal, characteristic
7#13804C211yesmaximal, characteristic, complemented
8#11357C212yesmaximal, complemented
9#11076C212yesmaximal
10#10931C212yesmaximal
11#10849C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—22no—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2542—24no—
22#2542C418yescomplemented
23C2×#399—21no—
24C2×#399—21no—
25C2×#390C2212yes—
26C2×#384C2212yes—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1993—24no—
32#1987—24no—
33#1940—24no—
34#1804—24no—
35#1762—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1436—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47C4×#95—21no—
48C23×Q8—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C22⋊Q8—216no—
61C2×C22⋊Q8—48no—
62C2×C22⋊Q8—48no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102—42no—
99C2×#102Q814yescomplemented
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103#399—28no—
104#399—28no—
105#390—24no—
106#390C2×C418yes—
107#384—24no—
108#384C2×C418yes—
109#382—22no—
110#382D414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—42no—
114C4×C4⋊C4C2×C414yes—
115C4×C4⋊C4D414yes—
116C4×C4⋊C4D414yes—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—22no—
131#340—22no—
132#336—24no—
133C25—21no—
134C22×Q8—24no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4C22×C412yes—
151C22×D4C2×Q812yes—
152C22×D4#4112yes—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C2×Q812yes—
188C22×C8#4112yes—
189C4⋊Q8—28no—
190C4⋊Q8—28no—
191C4⋊Q8—44no—
192C4⋊Q8—44no—
193C22⋊Q8—464no—
194C4×D4—28no—
195C4×D4—28no—
196C4×D4—44no—
197C4×D4—44no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22⋊C418yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×Q812yes—
239C2×C4⋊C4#4112yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—82no—
257C2×C42—82no—
258C2×C42C22×C414yes—
259C2×C42C22×C412yes—
260C2×C42C2×D414yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C4⋊C8—48no—
267C4⋊C8—44no—
268#102—28no—
269#102—28no—
270#102—48no—
271#97—44no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—44no—
276#95—44no—
277#95—82no—
278#95C22⋊C414yes—
279#95C22⋊C414yes—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24C22×D411yescharacteristic
292C2×Q8—416no—
293C2×Q8—48no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—88no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4C4×Q814yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—416no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—48no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—88no—
373C22×C4C2×C22⋊C414yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C414yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4AGL(1,ℤ/8)14yes—
382C22×C4AGL(1,ℤ/8)14yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×Q811yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C42−1+411yescharacteristic
395C22×C42−1+411yescharacteristic
396C22×C42−1+411yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—216no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—416no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4C22⋊Q814yescomplemented
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—416no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—84no—
450C42—88no—
451C42C4×D414yes—
452C42C4×D414yes—
453C42#12414yes—
454C42#12714yes—
455C23—24no—
456C23—42no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—82no—
469C23—82no—
470C23—81no—
471C23—81no—
472C23—82no—
473C23—81no—
474C23—81no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23C22×C22⋊C411yescharacteristic
479C23C2×#12111yescharacteristic
480C23C2×#12111yescharacteristic
481C23#54611yescharacteristic
482C23#54711yescharacteristic
483C23#54811yescharacteristic
484C23D4×Q811yescharacteristic
485C23C2×AGL(1,ℤ/8)12yes—
486C23C2×AGL(1,ℤ/8)12yes—
487C23#57712yes—
488C23#57812yes—
489Q8—816no—
490D4—24no—
491D4—24no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—816no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4#39014yes—
550C2×C4#39014yes—
551C2×C4#41818yes—
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×Q811yescharacteristic
555C2×C4C4×#4111yescharacteristic
556C2×C4#51911yescharacteristic
557C2×C4#52011yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4#52511yescharacteristic
560C2×C4C2×#12411yescharacteristic
561C2×C4#52911yescharacteristic
562C2×C4C2×#12711yescharacteristic
563C2×C4#53311yescharacteristic
564C2×C4#53611yescharacteristic
565C2×C4#53711yescharacteristic
566C2×C4#54111yescharacteristic
567C2×C4#54211yescharacteristic
568C2×C4#54611yescharacteristic
569C2×C4#54611yescharacteristic
570C2×C4#54711yescharacteristic
571C2×C4#54711yescharacteristic
572C2×C4#54811yescharacteristic
573C2×C4#54811yescharacteristic
574C2×C4D4×Q811yescharacteristic
575C2×C4D4×Q811yescharacteristic
576C2×C4#55211yescharacteristic
577C2×C4#55211yescharacteristic
578C2×C4#55411yescharacteristic
579C2×C4#55611yescharacteristic
580C8—42no—
581C8—42no—
582C8—82no—
583C22—44no—
584C22—48no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22C2×#39011yescharacteristic
595C22#233911yescharacteristic
596C22Q8×C22⋊C411yescharacteristic
597C22#241511yescharacteristic
598C22#241711yescharacteristic
599C22#242711yescharacteristic
600C22#242911yescharacteristic
601C22C2×#41812yes—
602C22#289812yes—
603C22#303811yescharacteristic
604C22#303811yescharacteristic
605C22#303812yes—
606C22#304211yescharacteristic
607C22#304411yescharacteristic
608C22#304612yes—
609C22#330612yes—
610C22#330812yes—
611C22#334312yes—
612C22#334512yes—
613C22#336712yes—
614C22#337312yes—
615C22#338412yes—
616C22#338612yes—
617C4—24no—
618C4—24no—
619C4—28no—
620C4—44no—
621C4—44no—
622C4—82no—
623C4—84no—
624C4—82no—
625C4—84no—
626C4C4×C22⋊Q811yescharacteristic
627C4#232311yescharacteristic
628C4#236011yescharacteristic
629C4#236211yescharacteristic
630C4#248911yescharacteristic
631C4#249211yescharacteristic
632C4#254011yescharacteristic
633C4#254211yescharacteristic
634C2—84no—
635C2#1399711yescharacteristic
636C2#2126011yescharacteristic
637C2#2126211yescharacteristic
638C2#2249512yes—
639C2#2252312yes—
640C2#2469211yescharacteristic
641C2#2469411yescharacteristic
642C2#2482711yescharacteristic
643C2#2482911yescharacteristic
644C2#2533412yes—
645C2#2546512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1993C2semidirect
3. C2×#1987C2semidirect
4. C2×#1940C2semidirect
7. #13804C2semidirect
8. #11357C2semidirect
11. #10849C2semidirect
22. #2542C4semidirect
99. C2×#102Q8semidirect
441. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#19932
C2×#19872
C2×#19402
C2×#18042
C2×#17622
#138042
#113572
#110762
#109312
#108492