Finite Group Databasebeta

← #375656

Subgroups — #375656

Order 512.

1462 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2395C211yesmaximal, characteristic
2C2×#1750C212yesmaximal, complemented
3#11147C212yesmaximal, complemented
4#10964C211yesmaximal, characteristic
5#10964C211yesmaximal, characteristic
6#10922C212yesmaximal, complemented
7#10921C212yesmaximal, complemented
8#10875C212yesmaximal, complemented
9#10874C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C22×#97C2212yes—
18C22×C4×C8C2212yes—
19#2395—28no—
20C2×#400C411yescharacteristic
21C2×#399C411yescharacteristic
22C2×#398C411yescharacteristic
23C2×#397C411yescharacteristic
24C2×#397C412yes—
25C2×#394C412yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29#1750—28no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C23×D4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4×D4—24no—
47C2×C4×D4—42no—
48C2×C4×D4—41no—
49C2×C4×D4—41no—
50C2×#117—41no—
51C2×#117—42no—
52C2×#117—41no—
53C2×#117D418yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2312yes—
67C22×C42C2312yes—
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—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—41no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—41no—
81C2×#97—24no—
82C2×#97—42no—
83C2×#97—42no—
84C2×C4×C8—24no—
85C2×C4×C8—42no—
86C2×C4×C8—42no—
87#400—24no—
88#399—24no—
89#398—24no—
90#397—24no—
91#397—28no—
92#394—28no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104#336—24no—
105#336—24no—
106#336—24no—
107#336—24no—
108#336—24no—
109#336—24no—
110#336—41no—
111#336—42no—
112#336—41no—
113C25C22×C411yescharacteristic
114C25C22×C411yescharacteristic
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—41no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D414yes—
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8#4114yes—
172C22×C8#4114yes—
173C22×C8#4114yes—
174C22×C8#4114yes—
175C4×D4—48no—
176C4×D4—48no—
177#117—48no—
178#117—48no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4C22⋊C414yes—
193C2×C4⋊C4C22⋊C414yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4C22⋊C418yes—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42C2×D414yes—
227C2×C42C2×D414yes—
228C22⋊C8—216no—
229C22⋊C8—216no—
230C22⋊C8—44no—
231C22⋊C8—48no—
232C22⋊C8—44no—
233#97—48no—
234C4×C8—48no—
235#95—24no—
236#95—22no—
237#95—24no—
238#95—22no—
239#95—24no—
240#95—24no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24C2×C22⋊C412yes—
262C24#13112yes—
263C24C23×C411yescharacteristic
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4C2×C22⋊C414yes—
349C22×C4C2×C22⋊C414yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4#13114yes—
352C22×C4#13112yes—
353C22×C4C22×D412yes—
354C22×C4C2×#4112yes—
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4112yes—
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)12yes—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—44no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—24no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—41no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C22×C22⋊C411yescharacteristic
457C23C4×#4112yes—
458C23C4×#4112yes—
459C23#51811yescharacteristic
460C23#51811yescharacteristic
461C23C2×#13111yescharacteristic
462C23C2×#13112yes—
463C23C2×#13111yescharacteristic
464C23#56811yescharacteristic
465C23#56812yes—
466C23#56811yescharacteristic
467D4—416no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#52512yes—
527C2×C4#52512yes—
528C2×C4#53212yes—
529C2×C4#53212yes—
530C2×C4#53412yes—
531C2×C4#53412yes—
532C2×C4#55111yescharacteristic
533C2×C4#55112yes—
534C2×C4#55111yescharacteristic
535C2×C4#55312yes—
536C2×C4#55312yes—
537C2×C4#55312yes—
538C2×C4#55312yes—
539C2×C4#55911yescharacteristic
540C2×C4#55912yes—
541C2×C4#55911yescharacteristic
542C8—44no—
543C8—44no—
544C8—44no—
545C8—44no—
546C22—42no—
547C22—44no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—82no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22#229712yes—
569C22#230811yescharacteristic
570C22#237411yescharacteristic
571C22#237412yes—
572C22#237411yescharacteristic
573C22#288011yescharacteristic
574C22#288011yescharacteristic
575C22#288012yes—
576C22#296612yes—
577C22#296612yes—
578C22#296712yes—
579C22#296712yes—
580C22#296812yes—
581C22#296812yes—
582C22#296912yes—
583C22#296912yes—
584C22#297611yescharacteristic
585C22#297611yescharacteristic
586C22#297711yescharacteristic
587C22#297811yescharacteristic
588C22#297911yescharacteristic
589C22#298011yescharacteristic
590C22#298112yes—
591C4—42no—
592C4—44no—
593C4—44no—
594C4—42no—
595C4—44no—
596C4—81no—
597C4—82no—
598C4—81no—
599C4#260212yes—
600C4#260212yes—
601C4#260212yes—
602C4#260212yes—
603C2—82no—
604C2—82no—
605C2#1416711yescharacteristic
606C2#2138112yes—
607C2#2138112yes—
608C2#2174411yescharacteristic
609C2#2174511yescharacteristic
610C2#2329511yescharacteristic
611C2#2329711yescharacteristic
612C2#2329912yes—
613C2#2330112yes—
614C2#2330311yescharacteristic
615C2#2330511yescharacteristic
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
2. C2×#1750C2semidirect
3. #11147C2semidirect
6. #10922C2semidirect
7. #10921C2semidirect
8. #10875C2semidirect
9. #10874C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23952
C2×#17502
#111472
#109642
#109222
#109212
#108752
#108742