Finite Group Databasebeta

← #387918

Subgroups — #387918

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2395C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4#11202C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#11147C212yesmaximal, complemented
7#10875C212yesmaximal, complemented
8#10715C212yesmaximal, complemented
9#10713C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2395—24no—
20#2395—24no—
21C2×#400C411yescharacteristic
22C2×#399C411yescharacteristic
23C2×#398C411yescharacteristic
24C2×#397C412yes—
25C2×#397C411yescharacteristic
26C2×#394C412yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1955—24no—
31#1870—24no—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—24no—
46C22×M16—41no—
47C22×M16—41no—
48C22×M16—42no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4—41no—
57C2×C4×D4D414yes—
58C2×C4×D4D414yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—41no—
62C2×#117—41no—
63C2×#117—41no—
64C2×#117—41no—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C42C2312yes—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—41no—
81C2×C4⋊C8—41no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×#97—22no—
95C2×#97—41no—
96C2×#97—41no—
97#400—24no—
98#399—24no—
99#398—24no—
100#397—28no—
101#397—24no—
102#394—28no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115#336—24no—
116#336—24no—
117#336—24no—
118#336—42no—
119#336—42no—
120#336—42no—
121#336—42no—
122#336—42no—
123#336—42no—
124C25C22×C411yescharacteristic
125C25C22×C411yescharacteristic
126C22×D4—41no—
127C22×D4—41no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4C22⋊C414yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2411yescharacteristic
160C2×M16—44no—
161C2×M16—48no—
162C2×M16—48no—
163C2×M16—44no—
164C2×M16—44no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
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#4114yes—
192C22×C8#4114yes—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4—44no—
196C4×D4—44no—
197#117—44no—
198#117—44no—
199#117—44no—
200#117—44no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C22⋊C414yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4C22⋊C414yes—
233C2×C22⋊C4C22⋊C414yes—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C4⋊C8—44no—
253C22⋊C8—48no—
254C22⋊C8—48no—
255C22⋊C8—48no—
256C22⋊C8—44no—
257#97—44no—
258#95—24no—
259#95—22no—
260#95—24no—
261#95—22no—
262#95—24no—
263#95—24no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24C2×C22⋊C412yes—
281C24C2×C22⋊C412yes—
282C24C23×C411yescharacteristic
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—48no—
287C2×D4—48no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
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—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
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—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—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4#13114yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4112yes—
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382M16—416no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
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—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
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—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C22×C22⋊C411yescharacteristic
477C23C4×#4112yes—
478C23#51811yescharacteristic
479C23#51811yescharacteristic
480C23#52012yes—
481C23C2×#13112yes—
482C23#56812yes—
483C23#56812yes—
484C23#56812yes—
485D4—416no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
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#53411yescharacteristic
547C2×C4#53411yescharacteristic
548C2×C4#53411yescharacteristic
549C2×C4#53411yescharacteristic
550C2×C4#53812yes—
551C2×C4#53912yes—
552C2×C4#54012yes—
553C2×C4#54212yes—
554C2×C4#54612yes—
555C2×C4#54612yes—
556C2×C4#54712yes—
557C2×C4#54712yes—
558C2×C4#55111yescharacteristic
559C2×C4#55312yes—
560C2×C4#55911yescharacteristic
561C2×C4#56011yescharacteristic
562C2×C4#56212yes—
563C2×C4#56311yescharacteristic
564C8—44no—
565C8—44no—
566C8—82no—
567C8—82no—
568C22—42no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22#230811yescharacteristic
587C22#233512yes—
588C22#233912yes—
589C22#237411yescharacteristic
590C22#237611yescharacteristic
591C22#287911yescharacteristic
592C22#287911yescharacteristic
593C22#288012yes—
594C22#296712yes—
595C22#296712yes—
596C22#296812yes—
597C22#296912yes—
598C22#297812yes—
599C22#297812yes—
600C22#297911yescharacteristic
601C22#298011yescharacteristic
602C22#298112yes—
603C22#299212yes—
604C22#299312yes—
605C22#299512yes—
606C22#299512yes—
607C4—42no—
608C4—42no—
609C4—44no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4#269711yescharacteristic
616C4#270112yes—
617C4#270812yes—
618C4#270911yescharacteristic
619C4#271711yescharacteristic
620C4#272411yescharacteristic
621C2—82no—
622C2—82no—
623C2#1450511yescharacteristic
624C2#2174311yescharacteristic
625C2#2174511yescharacteristic
626C2#2236412yes—
627C2#2237012yes—
628C2#2247212yes—
629C2#2247512yes—
630C2#2330112yes—
631C2#2332112yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
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
4. #11202C2semidirect
5. #11147C2semidirect
6. #11147C2semidirect
7. #10875C2semidirect
8. #10715C2semidirect
9. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23952
C2×#19552
C2×#18702
#112022
#111472
#108752
#107152
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