Finite Group Databasebeta

← #404697

Subgroups — #404697

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1983C212yesmaximal
2C2×#1981C211yesmaximal, characteristic, complemented
3C2×#1928C212yesmaximal, complemented
4C2×#1925C211yesmaximal, characteristic
5#15243C211yesmaximal, characteristic
6#13810C211yesmaximal, characteristic, complemented
7#11322C211yesmaximal, characteristic, complemented
8#11077C212yesmaximal, complemented
9#11076C211yesmaximal, characteristic
10#11010C211yesmaximal, characteristic, complemented
11#11009C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×#376—21no—
29#1983—28no—
30#1981—24no—
31#1928—28no—
32#1925—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1439—21no—
38#1439—21no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43#1434—22no—
44C4×#95C2211yescharacteristic
45C23×Q8C2312yes—
46C23×D4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×#124—24no—
55C2×#124—24no—
56C2×#124—24no—
57C2×#124—24no—
58C2×#124—42no—
59C2×#124—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—22no—
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#107—22no—
75C2×#107—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8—42no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—42no—
100C2×#102—42no—
101#391—24no—
102#391—24no—
103#390—22no—
104#390—22no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4D414yes—
108C4×C4⋊C4D418yes—
109#376—44no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120C2×#95D412yes—
121#340—22no—
122#340—24no—
123#336—22no—
124#336—24no—
125C25—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8#4114yes—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4#4112yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8#4112yes—
177C4⋊Q8—28no—
178C4⋊Q8—48no—
179C4⋊Q8—44no—
180#124—216no—
181#124—48no—
182#124—48no—
183#124—48no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D414yes—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—28no—
226C2×C42—24no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42C2×D414yes—
232C2×C42C2×D412yes—
233C2×C42#4114yes—
234C2×C42#4112yes—
235#107—44no—
236C4⋊C8—48no—
237C4⋊C8—44no—
238#103—216no—
239#103—48no—
240#103—48no—
241#103—48no—
242#102—28no—
243#102—48no—
244#102—44no—
245#95—28no—
246#95—24no—
247#95—22no—
248#95—22no—
249#95—42no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—42no—
254#95—82no—
255#95—81no—
256C24—22no—
257C24—42no—
258C24—41no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—41no—
265C24—41no—
266C24—42no—
267C24C22×D411yescharacteristic
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—22no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—41no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4#12114yes—
364C22×C4#12112yes—
365C22×C4#12114yes—
366C22×C4#12112yes—
367C22×C4#13512yes—
368C22×C4AGL(1,ℤ/8)12yes—
369C22×C4C22×D412yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—42no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4#12714yes—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C42—24no—
417C42—22no—
418C42—24no—
419C42—22no—
420C42—24no—
421C42—24no—
422C42—24no—
423C42—416no—
424C42—48no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—44no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—82no—
434C23—81no—
435C23—82no—
436C23—81no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—82no—
442C23—82no—
443C23—81no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23C2×#12111yescharacteristic
449C23C2×#12112yes—
450C23#54611yescharacteristic
451C23#54612yes—
452C23#55011yescharacteristic
453C23C2×#13511yescharacteristic
454C23C2×AGL(1,ℤ/8)11yescharacteristic
455C23#57512yes—
456C23#57711yescharacteristic
457C23#57811yescharacteristic
458C23#57812yes—
459Q8—44no—
460Q8—44no—
461Q8—44no—
462Q8—44no—
463Q8—82no—
464Q8—82no—
465Q8—82no—
466Q8—82no—
467D4—44no—
468D4—42no—
469D4—42no—
470D4—81no—
471D4—82no—
472D4—81no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4#46314yes—
528C2×C4#52512yes—
529C2×C4#52511yescharacteristic
530C2×C4#52511yescharacteristic
531C2×C4#52512yes—
532C2×C4C2×#12711yescharacteristic
533C2×C4#53212yes—
534C2×C4#53211yescharacteristic
535C2×C4#53512yes—
536C2×C4#53511yescharacteristic
537C2×C4#53511yescharacteristic
538C2×C4#53512yes—
539C2×C4#55112yes—
540C2×C4#55112yes—
541C2×C4#55112yes—
542C2×C4#55311yescharacteristic
543C2×C4#55312yes—
544C2×C4#55512yes—
545C2×C4#55511yescharacteristic
546C8—42no—
547C8—81no—
548C8—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—84no—
553C22—84no—
554C22—162no—
555C22—162no—
556C22—161no—
557C22—161no—
558C22—161no—
559C22—161no—
560C22#241111yescharacteristic
561C22#241112yes—
562C22#241411yescharacteristic
563C22#241412yes—
564C22#285611yescharacteristic
565C22C2×#46311yescharacteristic
566C22#303512yes—
567C22#303612yes—
568C22#304011yescharacteristic
569C22#304412yes—
570C22#304411yescharacteristic
571C22#304611yescharacteristic
572C22#304612yes—
573C22#330012yes—
574C22#330311yescharacteristic
575C22#330812yes—
576C22#331512yes—
577C22#332112yes—
578C22#332712yes—
579C22#333011yescharacteristic
580C22#339311yescharacteristic
581C22#339711yescharacteristic
582C22#340212yes—
583C4—24no—
584C4—44no—
585C4—44no—
586C4—44no—
587C4—42no—
588C4—82no—
589C4—84no—
590C4—84no—
591C4—161no—
592C4#239711yescharacteristic
593C4#260312yes—
594C4#260311yescharacteristic
595C4#261011yescharacteristic
596C4#261012yes—
597C4#265711yescharacteristic
598C2—162no—
599C2#1490011yescharacteristic
600C2#2448111yescharacteristic
601C2#2448712yes—
602C2#2448912yes—
603C2#2449911yescharacteristic
604C2#2462711yescharacteristic
605C2#2463712yes—
606C2#2464012yes—
607C2#2464511yescharacteristic
608C2#3316411yescharacteristic
609C2#3316911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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×#1981C2semidirect
3. C2×#1928C2semidirect
6. #13810C2semidirect
7. #11322C2semidirect
8. #11077C2semidirect
10. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19832
C2×#19812
C2×#19282
C2×#19252
#152432
#138102
#113222
#110772
#110762
#110102
#110092