Finite Group Databasebeta

← #382778

Subgroups — #382778

Order 512.

1514 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×#2293C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic, complemented
3C2×#1798C211yesmaximal, characteristic, complemented
4#11202C212yesmaximal, complemented
5#11147C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#10964C212yesmaximal
8#10795C211yesmaximal, characteristic, complemented
9#10795C211yesmaximal, characteristic, complemented
10#10713C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20#2293—24no—
21#2293—24no—
22C2×#390C412yes—
23C2×#389C412yes—
24C2×#388C412yes—
25C2×#388C412yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1818—24no—
31#1798—24no—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C23×D4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—42no—
57C2×#117—42no—
58C2×#117—42no—
59C2×#117—42no—
60C2×#117D414yes—
61C2×#117D414yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C42C2×C412yes—
73C22×C42C2×C412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—24no—
80C2×C4⋊C8—42no—
81C2×C4⋊C8—42no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—42no—
93C2×C22⋊C8—42no—
94#390—28no—
95#389—28no—
96#388—28no—
97#388—28no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C43—22no—
108C43—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#336—22no—
113#336—24no—
114#336—24no—
115#336—24no—
116#336—42no—
117#336—42no—
118#336—42no—
119#336—42no—
120C25C22×C411yescharacteristic
121C25C22×C411yescharacteristic
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—42no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C22×C412yes—
145C23×C4C22×C412yes—
146C23×C4C22×C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4#4114yes—
155C23×C4C2411yescharacteristic
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—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—
184C4×D4—216no—
185C4×D4—216no—
186#117—48no—
187#117—48no—
188#117—48no—
189#117—48no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22⋊C418yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4C22⋊C418yes—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4114yes—
248C4⋊C8—48no—
249C22⋊C8—48no—
250C22⋊C8—48no—
251C22⋊C8—48no—
252C22⋊C8—48no—
253#95—24no—
254#95—24no—
255#95—24no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24C2×C22⋊C412yes—
277C24#11712yes—
278C24C23×C411yescharacteristic
279C2×D4—28no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—44no—
283C2×D4—44no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
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—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
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×C4C2×C22⋊C414yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C414yes—
363C22×C4#11712yes—
364C22×C4#11712yes—
365C22×C4#11712yes—
366C22×C4#13114yes—
367C22×C4#13114yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+411yescharacteristic
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—
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—
406C4⋊C4—28no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—48no—
423C42—48no—
424C42—42no—
425C42—42no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C22×C22⋊C411yescharacteristic
477C23C2×#11711yescharacteristic
478C23C2×#11711yescharacteristic
479C23#51811yescharacteristic
480C23#51811yescharacteristic
481C23#52011yescharacteristic
482C23#52011yescharacteristic
483C23C2×#13112yes—
484C23C2×#13112yes—
485C23#56812yes—
486C23#56812yes—
487D4—416no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
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—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4#43314yes—
548C2×C4#52512yes—
549C2×C4#52512yes—
550C2×C4#53412yes—
551C2×C4#53412yes—
552C2×C4#53511yescharacteristic
553C2×C4#53611yescharacteristic
554C2×C4#53711yescharacteristic
555C2×C4#53711yescharacteristic
556C2×C4#53812yes—
557C2×C4#53812yes—
558C2×C4#53912yes—
559C2×C4#53912yes—
560C2×C4#54012yes—
561C2×C4#54012yes—
562C2×C4#54212yes—
563C2×C4#54212yes—
564C8—82no—
565C8—82no—
566C8—82no—
567C8—82no—
568C22—42no—
569C22—42no—
570C22—44no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22#228011yescharacteristic
591C22#230811yescharacteristic
592C22#231111yescharacteristic
593C22#231811yescharacteristic
594C22#231811yescharacteristic
595C22#231911yescharacteristic
596C22#231911yescharacteristic
597C22#288012yes—
598C22#288012yes—
599C22C2×#43311yescharacteristic
600C22#292111yescharacteristic
601C22#292211yescharacteristic
602C22#292311yescharacteristic
603C22#292412yes—
604C22#292512yes—
605C22#297912yes—
606C22#298012yes—
607C22#298112yes—
608C22#298112yes—
609C22#299212yes—
610C22#299312yes—
611C22#299412yes—
612C22#299512yes—
613C4—24no—
614C4—42no—
615C4—42no—
616C4—44no—
617C4—42no—
618C4—44no—
619C4—44no—
620C4—82no—
621C4#263012yes—
622C4#263012yes—
623C4#263512yes—
624C4#263812yes—
625C2—82no—
626C2—82no—
627C2#1434411yescharacteristic
628C2#2071611yescharacteristic
629C2#2072011yescharacteristic
630C2#2174512yes—
631C2#2185512yes—
632C2#2200512yes—
633C2#2200712yes—
634C2#2200911yescharacteristic
635C2#2201111yescharacteristic
636C2#2201311yescharacteristic
637C2#2201511yescharacteristic
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×#1818C2semidirect
3. C2×#1798C2semidirect
4. #11202C2semidirect
5. #11147C2semidirect
6. #11042C2semidirect
8. #10795C2semidirect
9. #10795C2semidirect
10. #10713C2semidirect
11. #10713C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
C2×#18182
C2×#17982
#112022
#111472
#110422
#109642
#107952
#107132