Finite Group Databasebeta

← #385136

Subgroups — #385136

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1850C212yesmaximal, complemented
4C2×#1797C212yesmaximal, complemented
5C4×#386C211yesmaximal, characteristic
6#11322C212yesmaximal, complemented
7#11042C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10876C212yesmaximal, complemented
10C24×C8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#386C411yescharacteristic
23C2×#386C411yescharacteristic
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1954—24no—
27#1869—24no—
28#1850—28no—
29#1797—28no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1437C412yes—
37#1436C412yes—
38C22⋊#95C411yescharacteristic
39C22⋊#95C411yescharacteristic
40C4×#95C2212yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—24no—
48C23×C8—42no—
49C23×C8—41no—
50C23×C8—41no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4—41no—
58C2×C4×D4—41no—
59C2×C4×D4D414yes—
60C2×C4×D4Q814yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C42—24no—
69C22×C42—24no—
70C22×C42—41no—
71C22×C42—41no—
72C22×C42C2×C412yes—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8—42no—
84C2×C4⋊C8—41no—
85C2×C4⋊C8—41no—
86C2×C4⋊C8—42no—
87C2×C4⋊C8—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—41no—
99C2×C22⋊C8—41no—
100C2×C4×C8—22no—
101C2×C4×C8—41no—
102C2×C4×C8—41no—
103#386—24no—
104#386C818yescomplemented
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—41no—
108C4×C22⋊C4—41no—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116#336—24no—
117#336—24no—
118#336—24no—
119#336—42no—
120#336—42no—
121#336D418yescomplemented
122C25C22×C411yescharacteristic
123C25C22×C411yescharacteristic
124C22×D4—41no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4—41no—
128C22×D4C4⋊C414yes—
129C22×D4C2×C814yes—
130C22×D4M1614yes—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—41no—
152C23×C4—42no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4C2×C814yescomplemented
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×Q812yes—
168C23×C4C2×Q812yes—
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—44no—
197C22×C8—48no—
198C22×C8—48no—
199C22×C8—44no—
200C22×C8—44no—
201C22×C8C2×D414yes—
202C22×C8C2×D414yes—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—44no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C4⋊C414yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4C4⋊C414yes—
225C2×C22⋊C4C4⋊C414yes—
226C2×C22⋊C4C2×C818yes—
227C2×C22⋊C4C2×C814yescomplemented
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42—48no—
246C2×C42—44no—
247C2×C42—42no—
248C2×C42C2×D412yes—
249C2×C42C2×D414yes—
250C2×C42C2×Q812yes—
251C2×C42#4114yes—
252C4⋊C8—48no—
253C4⋊C8—48no—
254C4⋊C8—44no—
255C4⋊C8—48no—
256C22⋊C8—48no—
257C22⋊C8—48no—
258C22⋊C8—48no—
259C22⋊C8—44no—
260C4×C8—44no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—42no—
266#95—42no—
267#95C2×C818yescomplemented
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—21no—
283C24—21no—
284C24C2×C4⋊C412yes—
285C24C2×C4⋊C412yes—
286C24C22×C812yes—
287C24C22×C812yes—
288C24C2×M1612yes—
289C24C2×M1612yes—
290C24C23×C411yescharacteristic
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4C4⋊C8116yescomplemented
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
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—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—216no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4C2×C4⋊C412yes—
384C22×C4C2×C4⋊C412yes—
385C22×C4C2×C4⋊C412yes—
386C22×C4C2×C4⋊C412yes—
387C22×C4C4×D414yes—
388C22×C4C4×D414yes—
389C22×C4C22×C814yes—
390C22×C4C22×C812yes—
391C22×C4C22×C814yes—
392C22×C4C2×M1612yes—
393C22×C4#13114yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C22×Q811yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)11yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—416no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C4⋊C4—44no—
434C22⋊C4—216no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—48no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23C2×C4⋊C814yes—
498C23C2×C4⋊C814yes—
499C23C22×C4⋊C411yescharacteristic
500C23C2×C4×D412yes—
501C23#51811yescharacteristic
502C23#51811yescharacteristic
503C23#52012yes—
504C23C23×C811yescharacteristic
505C23C22×M1611yescharacteristic
506C23C2×#13112yes—
507C23#56811yescharacteristic
508C23#56811yescharacteristic
509C23#56812yes—
510D4—416no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4C2×C4⋊C814yes—
574C2×C4C8×D418yes—
575C2×C4#53411yescharacteristic
576C2×C4#53411yescharacteristic
577C2×C4#53512yes—
578C2×C4#53712yes—
579C2×C4#54012yes—
580C2×C4#54212yes—
581C2×C4#54311yescharacteristic
582C2×C4#54311yescharacteristic
583C2×C4D4212yes—
584C2×C4D4211yescharacteristic
585C2×C4#54612yes—
586C2×C4#54712yes—
587C2×C4D4×Q812yes—
588C2×C4#55111yescharacteristic
589C2×C4#55412yes—
590C2×C4#56011yescharacteristic
591C2×C4#56212yes—
592C2×C4#56311yescharacteristic
593C8—44no—
594C8—44no—
595C8—82no—
596C8—82no—
597C22—28no—
598C22—28no—
599C22—24no—
600C22—24no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22#231511yescharacteristic
620C22#232412yes—
621C22D4×C4⋊C412yes—
622C22#235811yescharacteristic
623C22#237611yescharacteristic
624C22C22×C4⋊C811yescharacteristic
625C22#290511yescharacteristic
626C22#290612yes—
627C22C2×C8×D412yes—
628C22#293012yes—
629C22#293312yes—
630C22D4×M1612yes—
631C22#297211yescharacteristic
632C22#297211yescharacteristic
633C22#297311yescharacteristic
634C22#297311yescharacteristic
635C22#297711yescharacteristic
636C22#297811yescharacteristic
637C22#298112yes—
638C22#298812yes—
639C22#299012yes—
640C22#299312yes—
641C22#299512yes—
642C4—28no—
643C4—24no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—44no—
648C4—42no—
649C4—82no—
650C4#266911yescharacteristic
651C4#267112yes—
652C4#268311yescharacteristic
653C4#269212yes—
654C4#271711yescharacteristic
655C4#274811yescharacteristic
656C2—44no—
657C2—44no—
658C2#1440111yescharacteristic
659C2#2193011yescharacteristic
660C2#2193711yescharacteristic
661C2#2213312yes—
662C2#2214312yes—
663C2D4×C4⋊C812yes—
664C2#2264912yes—
665C2#2284211yescharacteristic
666C2#2285011yescharacteristic
667C2#2331411yescharacteristic
668C2#2332211yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1850C2semidirect
4. C2×#1797C2semidirect
6. #11322C2semidirect
7. #11042C2semidirect
8. #11042C2semidirect
9. #10876C2semidirect
104. #386C8semidirect
121. #336D4semidirect
156. C23×C4C2×C8semidirect
227. C2×C22⋊C4C2×C8semidirect
267. #95C2×C8semidirect
297. C2×D4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18692
C2×#18502
C2×#17972
C4×#3862
#113222
#110422
#108762