Finite Group Databasebeta

← #356970

Subgroups — #356970

Order 512.

2266 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×C42×D4C211yesmaximal, characteristic
2C2×#1955C212yesmaximal
3C2×#1771C212yesmaximal, complemented
4C2×#1749C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
6#10922C212yesmaximal, complemented
7#10921C212yesmaximal, complemented
8#10832C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic, complemented
10#10712C211yesmaximal, characteristic, complemented
11#10693C211yesmaximal, characteristic, complemented
12C22×C4×D4C412yes—
13C22×C4×D4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2212yes—
22C22×#97C2212yes—
23C22×C4×C8C2212yes—
24C42×D4—28no—
25C2×C4×C4⋊C4C412yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C412yes—
28C2×C4×C22⋊C4C412yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1955C4116yescomplemented
33#1771—28no—
34#1749—24no—
35C4×C22⋊C8—24no—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C412yes—
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—28no—
54C2×C4×D4—28no—
55C2×C4×D4—28no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C418yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—24no—
69C22×C42—24no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—42no—
73C22×C42—42no—
74C22×C42C2×C412yes—
75C22×C42C2×C412yes—
76C22×C42C2×C412yes—
77C22×C42C2×C412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—24no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8C2×C418yescomplemented
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—42no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8C2×C418yescomplemented
95C2×C22⋊C8C2×C418yescomplemented
96C2×C22⋊C8C2×C418yescomplemented
97C2×C22⋊C8C2×C418yescomplemented
98C2×#97—24no—
99C2×#97—24no—
100C2×#97—24no—
101C2×#97—24no—
102C2×#97—42no—
103C2×#97C2×C418yescomplemented
104C2×C4×C8—24no—
105C2×C4×C8—24no—
106C2×C4×C8—42no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4D418yes—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—28no—
112C4×C22⋊C4—28no—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C43—24no—
117#336—24no—
118#336—22no—
119#336—22no—
120#336—24no—
121#336—24no—
122#336—22no—
123#336—42no—
124#336D414yescomplemented
125C25C22×C411yescharacteristic
126C25C22×C411yescharacteristic
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4C4214yes—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4M1618yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C414yes—
167C23×C4C22×C414yes—
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C2411yescharacteristic
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—22no—
197C22×C8C22×C414yes—
198C22×C8C22×C414yes—
199C22×C8C22×C414yes—
200C22×C8C22×C414yes—
201C22×C8C2×D414yes—
202C22×C8#4114yes—
203C4×D4—264no—
204C4×D4—216no—
205C4×D4—216no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4C4214yes—
211C2×C4⋊C4C22⋊C418yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4C4214yes—
222C2×C22⋊C4C4214yes—
223C2×C22⋊C4M1618yes—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—48no—
246C2×C42—48no—
247C2×C42—48no—
248C2×C42—48no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C22×C414yes—
253C2×C42C2×D414yes—
254C2×C42C2×D414yes—
255C2×C42C2×D412yes—
256C2×C42C2×D414yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C4⋊C8—216no—
261C22⋊C8—216no—
262C22⋊C8—216no—
263C22⋊C8—48no—
264C22⋊C8—48no—
265#97—216no—
266#97—48no—
267C4×C8—48no—
268#95—24no—
269#95—24no—
270#95C22⋊C418yes—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24C2×C4212yes—
290C24C2×C4212yes—
291C24C2×M1614yes—
292C24C2×M1614yes—
293C24C23×C411yescharacteristic
294C2×D4—216no—
295C2×D4—216no—
296C2×D4—28no—
297C2×D4—28no—
298C2×D4—28no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—216no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
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—24no—
347C22×C4—24no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—48no—
381C22×C4C2×C4212yes—
382C22×C4C2×C4212yes—
383C22×C4C2×C4212yes—
384C22×C4C2×C4212yes—
385C22×C4C2×C22⋊C414yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C414yes—
389C22×C4#11712yes—
390C22×C4#11712yes—
391C22×C4C4×D414yes—
392C22×C4C2×M1614yes—
393C22×C4C2×M1614yes—
394C22×C4C2×M1614yes—
395C22×C4C2×M1614yes—
396C22×C4#13114yes—
397C22×C4C23×C412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4112yes—
403C22×C4C2×#4111yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+411yescharacteristic
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
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—
433C2×C8—42no—
434C2×C8C4×D418yescomplemented
435C4⋊C4—28no—
436C4⋊C4—216no—
437C22⋊C4—216no—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—216no—
441C42—216no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—48no—
449C42—48no—
450C42—48no—
451C42—42no—
452C42—42no—
453C42C4×D418yes—
454C42C4×D418yes—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C4×M1614yes—
500C23C4×M1614yes—
501C23C22×C4211yescharacteristic
502C23C22×C22⋊C411yescharacteristic
503C23C2×#11711yescharacteristic
504C23C2×C4×D411yescharacteristic
505C23C4×#4111yescharacteristic
506C23#51811yescharacteristic
507C23#52011yescharacteristic
508C23C22×M1612yes—
509C23C22×M1612yes—
510C23C2×#13112yes—
511C23#56812yes—
512D4—232no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
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—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—432no—
546C2×C4—432no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4C4×C22⋊C414yes—
572C2×C4C4×C22⋊C414yes—
573C2×C4#40718yes—
574C2×C4#43518yes—
575C2×C4C2×C4×D412yes—
576C2×C4C2×C4×D412yes—
577C2×C4C2×C4×D412yes—
578C2×C4C4×#4112yes—
579C2×C4#51812yes—
580C2×C4#52012yes—
581C2×C4D4212yes—
582C2×C4#54612yes—
583C2×C4#54711yescharacteristic
584C2×C4#54812yes—
585C2×C4#55012yes—
586C2×C4#55011yescharacteristic
587C2×C4#55312yes—
588C2×C4#55511yescharacteristic
589C2×C4#55611yescharacteristic
590C2×C4#55912yes—
591C8—28no—
592C8—44no—
593C8—44no—
594C8—44no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—44no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C2×C4×C22⋊C411yescharacteristic
618C22C42×D411yescharacteristic
619C22#227511yescharacteristic
620C22D4×C22⋊C411yescharacteristic
621C22#234111yescharacteristic
622C22#235411yescharacteristic
623C22#235711yescharacteristic
624C22C2×C4×M1611yescharacteristic
625C22C2×C4×M1611yescharacteristic
626C22C4×#13111yescharacteristic
627C22#287511yescharacteristic
628C22C2×#40712yes—
629C22#288012yes—
630C22#292012yes—
631C22#292512yes—
632C22C2×#43512yes—
633C22D4×M1612yes—
634C22#296512yes—
635C22#297012yes—
636C22#297312yes—
637C22#297412yes—
638C22#298912yes—
639C22#299012yes—
640C4—28no—
641C4—28no—
642C4—24no—
643C4—24no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4#236112yes—
649C4#236312yes—
650C4#236612yes—
651C4#236712yes—
652C2—44no—
653C2—44no—
654C2#1382011yescharacteristic
655C2C4×#40711yescharacteristic
656C2#2037811yescharacteristic
657C2C4×#43511yescharacteristic
658C2#2049111yescharacteristic
659C2#2055611yescharacteristic
660C2#2055911yescharacteristic
661C2#2239912yes—
662C2#2253212yes—
663C2#2279712yes—
664C2#2283712yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
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×#1771C2semidirect
4. C2×#1749C2semidirect
5. C2×C4×C22⋊C8C2semidirect
6. #10922C2semidirect
7. #10921C2semidirect
8. #10832C2semidirect
9. #10712C2semidirect
10. #10712C2semidirect
11. #10693C2semidirect
32. #1955C4semidirect
85. C2×C4⋊C8C2×C4semidirect
94. C2×C22⋊C8C2×C4semidirect
95. C2×C22⋊C8C2×C4semidirect
96. C2×C22⋊C8C2×C4semidirect
97. C2×C22⋊C8C2×C4semidirect
103. C2×#97C2×C4semidirect
124. #336D4semidirect
434. C2×C8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D42
C2×#19552
C2×#17712
C2×#17492
C2×C4×C22⋊C82
#109222
#109212
#108322
#107122
#106932