Finite Group Databasebeta

← #407038

Subgroups — #407038

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1872C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4C2×#1852C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6C2×C4⋊#95C211yesmaximal, characteristic
7#11379C212yesmaximal, complemented
8#11378C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C22×#95—22no—
26#1872—24no—
27#1872—24no—
28#1853—24no—
29#1852—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1436—22no—
37C4⋊#95—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C22⋊#95—22no—
43C23×Q8C2311yescharacteristic
44C24×C4—21no—
45C24×C4—21no—
46C24×C4D412yes—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—42no—
77C22×C42—42no—
78C22×C42D412yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—42no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106—42no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103—42no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8D414yes—
98C2×C22⋊C8D414yes—
99#391—42no—
100#391—42no—
101#380—22no—
102#380—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—24no—
108C2×#95—24no—
109C2×#95—21no—
110C2×#95—48no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#336—42no—
116#336—42no—
117C25C2×D411yescharacteristic
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8C2×Q812yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×Q812yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
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×C8C2×D414yes—
178C22×C8C2×D414yes—
179C22⋊Q8—28no—
180C22⋊Q8—28no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4#4114yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C42—24no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—48no—
239C2×C42—48no—
240#107—48no—
241#106—48no—
242#103—48no—
243#103—48no—
244C22⋊C8—28no—
245C22⋊C8—28no—
246#95—48no—
247#95—44no—
248#95—44no—
249#95—42no—
250#95—44no—
251#95—48no—
252#95—44no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24C22⋊Q812yes—
260C24#12312yes—
261C24#12712yes—
262C24#13514yes—
263C24#13714yes—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—216no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—416no—
323C22×C4—416no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—41no—
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—84no—
353C22×C4—84no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4C22⋊Q814yes—
359C22×C4C22⋊Q814yes—
360C22×C4C22⋊Q812yes—
361C22×C4#12314yes—
362C22×C4#12314yes—
363C22×C4#12312yes—
364C22×C4#12712yes—
365C22×C4AGL(1,ℤ/8)14yes—
366C22×C4#13714yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4112yes—
374C22×C4C2×#4112yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)12yes—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C22⋊C4—48no—
411C22⋊C4—84no—
412C42—48no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23#39914yes—
436C23#47918yes—
437C23#48518yes—
438C23#49514yes—
439C23#49714yes—
440C23C2×C22⋊Q811yescharacteristic
441C23C2×#12311yescharacteristic
442C23#52511yescharacteristic
443C23#52511yescharacteristic
444C23C2×#12711yescharacteristic
445C23#53411yescharacteristic
446C23#53411yescharacteristic
447C23C2×#13512yes—
448C23C2×AGL(1,ℤ/8)12yes—
449C23C2×#13712yes—
450C23C2×#13712yes—
451Q8—44no—
452Q8—44no—
453Q8—82no—
454Q8—82no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—432no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—816no—
490C2×C4—816no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4#39914yes—
515C2×C4C2×C22⋊Q811yescharacteristic
516C2×C4C2×C22⋊Q811yescharacteristic
517C2×C4C2×#12311yescharacteristic
518C2×C4C2×#12311yescharacteristic
519C2×C4#52912yes—
520C2×C4#52912yes—
521C2×C4#53812yes—
522C2×C4#53812yes—
523C2×C4#54612yes—
524C2×C4#54612yes—
525C2×C4#54612yes—
526C2×C4#54612yes—
527C2×C4#55312yes—
528C2×C4#55312yes—
529C2×C4#55412yes—
530C2×C4#55412yes—
531C8—44no—
532C8—44no—
533C22—28no—
534C22—28no—
535C22—28no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—28no—
543C22#209718yes—
544C22C2×#39911yescharacteristic
545C22#253911yescharacteristic
546C22#253911yescharacteristic
547C22#257911yescharacteristic
548C22#257911yescharacteristic
549C22#262211yescharacteristic
550C22#285311yescharacteristic
551C22C2×#47912yes—
552C22#307912yes—
553C22C2×#48512yes—
554C22#309212yes—
555C22#310512yes—
556C22#310812yes—
557C22#311012yes—
558C22#311312yes—
559C22C2×#49511yescharacteristic
560C22C2×#49711yescharacteristic
561C22#315111yescharacteristic
562C22#315211yescharacteristic
563C22#318812yes—
564C22#319012yes—
565C22#319312yes—
566C22#319512yes—
567C4—24no—
568C4—44no—
569C4—44no—
570C4—82no—
571C4—84no—
572C4—84no—
573C4—82no—
574C4—161no—
575C4—161no—
576C4#244412yes—
577C4#244412yes—
578C4#245012yes—
579C4#245012yes—
580C2—28no—
581C2#1494611yescharacteristic
582C2C2×#209711yescharacteristic
583C2#2373411yescharacteristic
584C2#2830311yescharacteristic
585C2#2830611yescharacteristic
586C2#2831111yescharacteristic
587C2#2831311yescharacteristic
588C2#2924312yes—
589C2#2924412yes—
590C2#2989412yes—
591C2#3310012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11379C2semidirect
8. #11378C2semidirect
9. #11124C2semidirect
10. #11122C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18722
C2×#18532
C2×#18522
#147022
C2×C4⋊#952
#113792
#113782
#111242
#111222
#107122