Finite Group Databasebeta

← #383857

Subgroups — #383857

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1924C212yesmaximal, complemented
2C2×#1780C212yesmaximal
3#13851C211yesmaximal, characteristic
4#11147C212yesmaximal, complemented
5#11042C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10922C211yesmaximal, characteristic, complemented
8#10922C211yesmaximal, characteristic, complemented
9#10922C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×C4⋊C8C2212yes—
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C2×C4×C22⋊C4C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1924—28no—
25#1780—28no—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1446C412yes—
34#1438C412yes—
35#1438C412yes—
36#1438C412yes—
37C22×#41—24no—
38C22×#41—24no—
39C22×#41—24no—
40C22×#41—41no—
41C22×#41—41no—
42C22×#41—42no—
43C22×#41—42no—
44C22×#41—42no—
45C23×Q8C2×C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C23×D4C2×C412yes—
48C24×C4C2312yes—
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2312yes—
64C2×C4⋊C8—24no—
65C2×C4⋊C8—42no—
66C2×C4⋊C8—42no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—42no—
81C2×C22⋊C8—42no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8D418yes—
85C2×#97—24no—
86C2×#97—41no—
87C2×#97—42no—
88C2×#97—41no—
89C4×C22⋊C4—24no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—42no—
92C4×C22⋊C4—41no—
93C4×C22⋊C4—41no—
94C4×C22⋊C4—42no—
95C4×C22⋊C4—42no—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102#336—24no—
103#336—24no—
104#336—42no—
105#336—42no—
106#336—42no—
107#336—41no—
108#336—41no—
109C25C22×C411yescharacteristic
110C25C22×C412yes—
111C2×#41—44no—
112C2×#41—48no—
113C2×#41—48no—
114C2×#41—48no—
115C2×#41—48no—
116C2×#41—48no—
117C2×#41—44no—
118C2×#41—48no—
119C2×#41—48no—
120C2×#41—48no—
121C2×#41—48no—
122C2×#41—48no—
123C2×#41—48no—
124C2×#41—48no—
125C2×#41—44no—
126C2×#41—44no—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×Q8—41no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—24no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—42no—
138C22×D4—41no—
139C22×D4—41no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C2×D414yes—
170C23×C4#4114yes—
171C23×C4#4114yes—
172C23×C4C2411yescharacteristic
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—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C2×D414yes—
200C22×C8C2×D414yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C4⋊C8—48no—
235C22⋊C8—48no—
236C22⋊C8—48no—
237C22⋊C8—48no—
238C22⋊C8—48no—
239C22⋊C8—48no—
240C22⋊C8—48no—
241#97—44no—
242#97—44no—
243#95—24no—
244#95—24no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—42no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24#11712yes—
272C24C4×D414yes—
273C24C23×C411yescharacteristic
274#41—432no—
275#41—432no—
276#41—432no—
277#41—432no—
278C2×Q8—48no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—48no—
282C2×Q8—44no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—44no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—48no—
294C2×D4—44no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
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—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
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—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4#11712yes—
377C22×C4C4×D414yes—
378C22×C4#13114yes—
379C22×C4C22×D412yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C4H(5,𝔽2)12yes—
385C22×C4H(5,𝔽2)11yescharacteristic
386C22×C4H(5,𝔽2)11yescharacteristic
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
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—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C22⋊C4—44no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—44no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—41no—
427C42—42no—
428C42—41no—
429C42—42no—
430C42—41no—
431C42—41no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C2×#11711yescharacteristic
489C23C2×C4×D412yes—
490C23#51811yescharacteristic
491C23#51811yescharacteristic
492C23#51812yes—
493C23C2×#13112yes—
494C23#56812yes—
495C23#56812yes—
496C23#56812yes—
497Q8—416no—
498D4—416no—
499D4—432no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
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—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
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#53412yes—
553C2×C4#53512yes—
554C2×C4#53812yes—
555C2×C4#53812yes—
556C2×C4#53812yes—
557C2×C4#53812yes—
558C2×C4D4211yescharacteristic
559C2×C4#54612yes—
560C2×C4#54612yes—
561C2×C4#54612yes—
562C2×C4#55112yes—
563C2×C4#55112yes—
564C2×C4#55111yescharacteristic
565C2×C4#56012yes—
566C2×C4#56011yescharacteristic
567C2×C4H(3,𝔽4)11yescharacteristic
568C8—44no—
569C8—44no—
570C8—82no—
571C8—82no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22#231211yescharacteristic
597C22#232112yes—
598C22#234212yes—
599C22#235811yescharacteristic
600C22#237511yescharacteristic
601C22#292311yescharacteristic
602C22#292311yescharacteristic
603C22#292512yes—
604C22#293012yes—
605C22#293212yes—
606C22#293312yes—
607C22#293512yes—
608C22#297612yes—
609C22#297711yescharacteristic
610C22#297812yes—
611C22#297812yes—
612C22#297811yescharacteristic
613C22#298012yes—
614C22#298012yes—
615C22#298112yes—
616C22#298112yes—
617C4—44no—
618C4—42no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#267812yes—
624C4#268412yes—
625C4#268512yes—
626C4#273512yes—
627C2—84no—
628C2—82no—
629C2#1436511yescharacteristic
630C2#2190711yescharacteristic
631C2#2191111yescharacteristic
632C2#2203512yes—
633C2#2203712yes—
634C2#2255712yes—
635C2#2256112yes—
636C2#2284912yes—
637C2#2331312yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1924C2semidirect
4. #11147C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10922C2semidirect
8. #10922C2semidirect
9. #10922C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19242
C2×#17802
#138512
#111472
#110422
#109222