Finite Group Databasebeta

← #361118

Subgroups — #361118

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2C2×#1921C212yesmaximal, complemented
3#13920C211yesmaximal, characteristic, complemented
4#13820C211yesmaximal, characteristic, complemented
5#11066C212yesmaximal, complemented
6#11022C212yesmaximal
7#10931C212yesmaximal, complemented
8#10820C211yesmaximal, characteristic, complemented
9#10797C211yesmaximal, characteristic, complemented
10#10788C211yesmaximal, characteristic, complemented
11#10788C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C4×D8—28no—
16C2×C4×D8C418yescomplemented
17C22×#107C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#380—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4—21no—
29#1921—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1439—22no—
38#1437—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C22⋊#95—21no—
42C22⋊#95—21no—
43C4×#95C2211yescharacteristic
44C23×D4C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C24×C4—21no—
48C22×D8—22no—
49C22×D8—24no—
50C22×D8—22no—
51C22×D8—24no—
52C22×D8C2×C414yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4C2×C414yes—
63C2×C4×D4C2×C414yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—22no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C4×D8—232no—
79C2×#107C2×C414yes—
80C2×#107D418yes—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102C2×C414yes—
90C2×#102C2×C414yes—
91C2×C4×C8—24no—
92C2×C4×C8C2×C414yes—
93#390—24no—
94#390—24no—
95#390—24no—
96#390—42no—
97#389—24no—
98#389—42no—
99#380—44no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—42no—
102C4×C22⋊C4—44no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#336—24no—
117#336—22no—
118#336—42no—
119C25—21no—
120C25—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C22×C412yes—
137C22×D4C22×C412yes—
138C22×D4#4112yes—
139C22×D4#4114yes—
140C22×D4#4112yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C2×D8—216no—
161C2×D8—216no—
162C2×D8—28no—
163C2×D8—28no—
164C2×D8—28no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C22×C412yes—
180C22×C8C22×C412yes—
181C22×C8C2×D414yes—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—44no—
185C4×D4—44no—
186C4×D4—48no—
187C4×D4—48no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42—82no—
238C2×C42C22×C412yes—
239C2×C42#4112yes—
240#107C22⋊C4116yescomplemented
241#102—28no—
242#102—28no—
243#102—48no—
244#102—48no—
245C4×C8—28no—
246#95—24no—
247#95—22no—
248#95—22no—
249#95—44no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—81no—
254#95—81no—
255C24—22no—
256C24—22no—
257C24—24no—
258C24—22no—
259C24—22no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—41no—
264C24—41no—
265C24—42no—
266C24—42no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24—42no—
275C24—41no—
276C24—41no—
277C24C22×D411yescharacteristic
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—22no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4#11714yes—
313C2×D4#11714yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4#13514yes—
370C22×C4AGL(1,ℤ/8)12yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C2×#4112yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4111yescharacteristic
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C42−1+411yescharacteristic
382D8—232no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
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×C8C2×C22⋊C414yes—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—88no—
413C4⋊C4C2×C22⋊C414yes—
414C4⋊C4C2×C22⋊C414yes—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—416no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C42—22no—
422C42—22no—
423C42—24no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—88no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—81no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—81no—
462C23—81no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—81no—
475C23C2×C4×D411yescharacteristic
476C23#52512yes—
477C23D4212yes—
478C23#54611yescharacteristic
479C23#54811yescharacteristic
480C23C2×#13512yes—
481C23C2×AGL(1,ℤ/8)11yescharacteristic
482C23C2×AGL(1,ℤ/8)11yescharacteristic
483C23#57612yes—
484C23#57611yescharacteristic
485C23#57811yescharacteristic
486D4—24no—
487D4—24no—
488D4—24no—
489D4—24no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—44no—
495D4—44no—
496D4—44no—
497D4—44no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—816no—
535C2×C4—816no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#44214yes—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C2×#11711yescharacteristic
550C2×C4C2×#11711yescharacteristic
551C2×C4C4×#4111yescharacteristic
552C2×C4#51811yescharacteristic
553C2×C4#52011yescharacteristic
554C2×C4#52512yes—
555C2×C4#52512yes—
556C2×C4#52912yes—
557C2×C4#53812yes—
558C2×C4#53912yes—
559C2×C4#54612yes—
560C2×C4#54812yes—
561C2×C4#55012yes—
562C2×C4#55112yes—
563C2×C4#55211yescharacteristic
564C2×C4#55312yes—
565C2×C4#55311yescharacteristic
566C8—24no—
567C8—42no—
568C8—82no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22#229011yescharacteristic
592C22D4×C22⋊C411yescharacteristic
593C22#235211yescharacteristic
594C22#245612yes—
595C22#246312yes—
596C22C4×#13511yescharacteristic
597C22C2×#44211yescharacteristic
598C22#294811yescharacteristic
599C22#295111yescharacteristic
600C22#310512yes—
601C22#310612yes—
602C22#312212yes—
603C22#312412yes—
604C22#326412yes—
605C22#326412yes—
606C22#328512yes—
607C22#328712yes—
608C22#329712yes—
609C22#332511yescharacteristic
610C22#333011yescharacteristic
611C22#336112yes—
612C22#336611yescharacteristic
613C22#336711yescharacteristic
614C4—24no—
615C4—44no—
616C4—44no—
617C4—42no—
618C4—44no—
619C4—84no—
620C4—82no—
621C4—162no—
622C4#228011yescharacteristic
623C4#231811yescharacteristic
624C4#234111yescharacteristic
625C4#235311yescharacteristic
626C4#246012yes—
627C4#246712yes—
628C2—84no—
629C2—84no—
630C2#1389011yescharacteristic
631C2#2111811yescharacteristic
632C2#2112911yescharacteristic
633C2#2243611yescharacteristic
634C2#2245111yescharacteristic
635C2#2275811yescharacteristic
636C2#2277211yescharacteristic
637C2#2661712yes—
638C2#2661912yes—
639C2#2688812yes—
640C2#2689012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1921C2semidirect
3. #13920C2semidirect
4. #13820C2semidirect
5. #11066C2semidirect
7. #10931C2semidirect
8. #10820C2semidirect
9. #10797C2semidirect
10. #10788C2semidirect
11. #10788C2semidirect
16. C2×C4×D8C4semidirect
240. #107C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#19212
#139202
#138202
#110662
#110222
#109312
#108202
#107972
#107882