Finite Group Databasebeta

← #365301

Subgroups — #365301

Order 512.

1338 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
1C2×#1942C211yesmaximal, characteristic, complemented
2C2×#1808C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C2×C4×#107C211yesmaximal, characteristic, complemented
5C2×C8×C22⋊C4C211yesmaximal, characteristic
6#13974C212yesmaximal
7#11315C212yesmaximal, complemented
8#11199C212yesmaximal
9#10884C212yesmaximal, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10862C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#389C2212yes—
21C2×#388—22no—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1942—24no—
27#1808—24no—
28#1808C418yescomplemented
29C4×#107—24no—
30C8×C22⋊C4—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1443—22no—
40#1440—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8C2×C414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107C2×C414yes—
78C2×#107C2×C414yes—
79C2×#107D414yes—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8D414yes—
89#389—24no—
90#389—24no—
91#388—48no—
92#380—24no—
93#380—24no—
94#380—42no—
95#380C2×C418yes—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—42no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—42no—
115#340C2×C418yes—
116#336—22no—
117#336—22no—
118C25C2×D411yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C22×C412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—28no—
162C22×C8—28no—
163C22×C8—28no—
164C22×C8—44no—
165C22×C8—44no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8C22×C412yes—
169C22×C8C22×C412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8#4112yes—
174C22×C8#4112yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177C22×C8#4112yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C22×C414yes—
203C2×C4⋊C4C22×C414yes—
204C2×C4⋊C4C2×D414yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4D814yes—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—82no—
230C2×C42C2×D412yes—
231C2×C42#4112yes—
232#107—28no—
233#107—28no—
234#107—28no—
235#107—44no—
236C22⋊C8—28no—
237C4×C8—28no—
238C4×C8—44no—
239#95—28no—
240#95—22no—
241#95—22no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—82no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24C4×D412yes—
262C24C2×D812yes—
263C24#13512yes—
264C24C22×D411yescharacteristic
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4C4×D412yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4C2×D812yes—
353C22×C4C2×D812yes—
354C22×C4#13512yes—
355C22×C4#13512yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C42−1+411yescharacteristic
368C22×C42−1+411yescharacteristic
369C22×C42−1+411yescharacteristic
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—416no—
392C2×C8#11714yes—
393C2×C8C4×D414yes—
394C2×C8C4×D414yes—
395C2×C8#12114yes—
396C2×C8#12314yes—
397C2×C8#12414yes—
398C2×C8#12614yes—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—88no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C22⋊C4—416no—
419C22⋊C4—88no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—88no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23C4×D814yes—
464C23C2×C4×D411yescharacteristic
465C23C2×#12111yescharacteristic
466C23#53211yescharacteristic
467C23D4211yescharacteristic
468C23#54711yescharacteristic
469C23#54711yescharacteristic
470C23#54711yescharacteristic
471C23C22×D811yescharacteristic
472C23C2×#13511yescharacteristic
473C23C2×#13511yescharacteristic
474C23C2×#13511yescharacteristic
475C23#57611yescharacteristic
476C23#57811yescharacteristic
477C23#57811yescharacteristic
478C23#57811yescharacteristic
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4#46614yes—
550C2×C4C2×#11711yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C4×#4111yescharacteristic
554C2×C4#51911yescharacteristic
555C2×C4#52011yescharacteristic
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12311yescharacteristic
558C2×C4#52511yescharacteristic
559C2×C4C2×#12411yescharacteristic
560C2×C4C2×#12611yescharacteristic
561C2×C4#52911yescharacteristic
562C2×C4#53611yescharacteristic
563C2×C4#53711yescharacteristic
564C2×C4#54111yescharacteristic
565C2×C4#54211yescharacteristic
566C2×C4#54711yescharacteristic
567C2×C4#54811yescharacteristic
568C2×C4#54811yescharacteristic
569C2×C4#55211yescharacteristic
570C2×C4#55211yescharacteristic
571C2×C4#55211yescharacteristic
572C2×C4#55311yescharacteristic
573C2×C4#55311yescharacteristic
574C2×C4#55311yescharacteristic
575C2×C4#55611yescharacteristic
576C2×C4#55611yescharacteristic
577C2×C4#55611yescharacteristic
578C8—42no—
579C8—42no—
580C8—42no—
581C8#38818yescomplemented
582C22—24no—
583C22—24no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22C4×#12111yescharacteristic
598C22#234411yescharacteristic
599C22#236111yescharacteristic
600C22#241611yescharacteristic
601C22#242211yescharacteristic
602C22#249611yescharacteristic
603C22#254711yescharacteristic
604C22C2×C4×D811yescharacteristic
605C22C4×#13511yescharacteristic
606C22#295011yescharacteristic
607C22#295111yescharacteristic
608C22C2×#46611yescharacteristic
609C22#305011yescharacteristic
610C22#305911yescharacteristic
611C22#306111yescharacteristic
612C22#316811yescharacteristic
613C22#316911yescharacteristic
614C22#317511yescharacteristic
615C22#317711yescharacteristic
616C22D4×D811yescharacteristic
617C22#328311yescharacteristic
618C22#328511yescharacteristic
619C22#328511yescharacteristic
620C22#333411yescharacteristic
621C22#333411yescharacteristic
622C22#333411yescharacteristic
623C22#333611yescharacteristic
624C22#333611yescharacteristic
625C22#333611yescharacteristic
626C22#333911yescharacteristic
627C22#333911yescharacteristic
628C22#333911yescharacteristic
629C22#334211yescharacteristic
630C22#334211yescharacteristic
631C22#334211yescharacteristic
632C4—24no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—84no—
638C4—84no—
639C4—84no—
640C4—162no—
641C4C2×#38811yescharacteristic
642C4#232311yescharacteristic
643C4#234711yescharacteristic
644C4#236211yescharacteristic
645C4#241511yescharacteristic
646C4#242111yescharacteristic
647C4#249411yescharacteristic
648C4#256211yescharacteristic
649C2—44no—
650C2#1397411yescharacteristic
651C2C4×#46611yescharacteristic
652C2#2152511yescharacteristic
653C2#2261111yescharacteristic
654C2#2261411yescharacteristic
655C2#2302511yescharacteristic
656C2#2302611yescharacteristic
657C2#2477911yescharacteristic
658C2#2478011yescharacteristic
659C2#2503311yescharacteristic
660C2#2503411yescharacteristic
661C2#2783711yescharacteristic
662C2#2783911yescharacteristic
663C2#2870411yescharacteristic
664C2#2870811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1942C2semidirect
4. C2×C4×#107C2semidirect
7. #11315C2semidirect
9. #10884C2semidirect
10. #10862C2semidirect
11. #10862C2semidirect
28. #1808C4semidirect
581. C8#388semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×#18082
C2×C4×#1072
C2×C8×C22⋊C42
#139742
#113152
#111992
#108842
#108622