Finite Group Databasebeta

← #400368

Subgroups — #400368

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#1918C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal, complemented
5#15129C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10#10806C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C2×#394—21no—
19#2354—24no—
20#2354—24no—
21C2×#390—21no—
22C2×#389C2211yescharacteristic
23C2×#388—22no—
24C2×#384C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2084—24no—
31#1918—24no—
32#1854—28no—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1441—21no—
44#1440—22no—
45C4⋊#95C2211yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—42no—
60C2×#117—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42—22no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C2×#107—22no—
80C2×#107—22no—
81C2×#107D414yescomplemented
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—42no—
91#394—44no—
92#390—28no—
93#389—22no—
94#389—22no—
95#388—216no—
96#384—22no—
97#384D414yes—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—24no—
102#380—24no—
103#380—42no—
104C4×C4⋊C4—28no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—42no—
122#336—22no—
123#336—22no—
124C25C2×D412yes—
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—42no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4#4114yes—
152C23×C4#4114yes—
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C2×D412yes—
170C22×C8#4114yes—
171C4×D4—28no—
172C4×D4—28no—
173C4×D4—216no—
174#117—48no—
175#117—48no—
176#117—48no—
177#117—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4#4112yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—28no—
237C2×C42—24no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#107—28no—
245#106—48no—
246C22⋊C8—216no—
247C22⋊C8—48no—
248#95—28no—
249#95—24no—
250#95—44no—
251#95—42no—
252#95—44no—
253#95—42no—
254#95—81no—
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#12314yes—
270C24C22×D411yescharacteristic
271C2×D4—28no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—44no—
275C2×D4—44no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4#12012yes—
355C22×C4#12012yes—
356C22×C4#12314yes—
357C22×C4#13512yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+412yes—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—28no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—48no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4#12114yes—
412C22⋊C4—216no—
413C22⋊C4—416no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42—416no—
421C42—48no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
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—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23C2×#12011yescharacteristic
458C23C2×#12312yes—
459C23D4211yescharacteristic
460C23#54611yescharacteristic
461C23#54712yes—
462C23C2×#13511yescharacteristic
463C23C2×AGL(1,ℤ/8)11yescharacteristic
464C23#57512yes—
465C23#57712yes—
466C23#57711yescharacteristic
467C23#57811yescharacteristic
468D4—416no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4#44914yes—
532C2×C4C2×#12111yescharacteristic
533C2×C4#52511yescharacteristic
534C2×C4#52511yescharacteristic
535C2×C4#52512yes—
536C2×C4#52512yes—
537C2×C4#52912yes—
538C2×C4#53211yescharacteristic
539C2×C4#53511yescharacteristic
540C2×C4#53611yescharacteristic
541C2×C4#53912yes—
542C2×C4#53912yes—
543C2×C4D4211yescharacteristic
544C2×C4#54611yescharacteristic
545C2×C4#54812yes—
546C2×C4#54812yes—
547C2×C4#55112yes—
548C2×C4#55212yes—
549C2×C4#55312yes—
550C8—42no—
551C8—44no—
552C8—81no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22#240811yescharacteristic
569C22#241912yes—
570C22#243211yescharacteristic
571C22#244312yes—
572C22#284911yescharacteristic
573C22C2×#44911yescharacteristic
574C22#300511yescharacteristic
575C22#300911yescharacteristic
576C22#301311yescharacteristic
577C22#309112yes—
578C22#309412yes—
579C22#309712yes—
580C22#309912yes—
581C22#327112yes—
582C22#327211yescharacteristic
583C22#328311yescharacteristic
584C22#330311yescharacteristic
585C22#331712yes—
586C22#333011yescharacteristic
587C22#334012yes—
588C22#334912yes—
589C22#335012yes—
590C22#335412yes—
591C4—24no—
592C4—44no—
593C4—44no—
594C4—42no—
595C4—82no—
596C4—82no—
597C4—84no—
598C4—162no—
599C4—161no—
600C4#239211yescharacteristic
601C4#246012yes—
602C4#246312yes—
603C4#253611yescharacteristic
604C4#260311yescharacteristic
605C4#263311yescharacteristic
606C2—84no—
607C2#1480111yescharacteristic
608C2#2423711yescharacteristic
609C2#2425411yescharacteristic
610C2#2494612yes—
611C2#2495812yes—
612C2#2570011yescharacteristic
613C2#2571111yescharacteristic
614C2#2613912yes—
615C2#2615312yes—
616C2#3298011yescharacteristic
617C2#3298811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
3. C2×#1918C2semidirect
4. C2×#1854C2semidirect
6. #11371C2semidirect
7. #11200C2semidirect
8. #11095C2semidirect
10. #10806C2semidirect
11. #10731C2semidirect
81. C2×#107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#20842
C2×#19182
C2×#18542
#151292
#113712
#112002
#110952
#109642
#108062
#107312