Finite Group Databasebeta

← #406770

Subgroups — #406770

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1855C211yesmaximal, characteristic
2#20846C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C2×#1821C211yesmaximal, characteristic
5#20628C211yesmaximal, characteristic
6C2×#1736C211yesmaximal, characteristic
7#11303C212yesmaximal, complemented
8#11249C212yesmaximal, complemented
9#11242C212yesmaximal, complemented
10#11216C212yesmaximal, complemented
11#10739C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2212yes—
16C2×#428C2211yescharacteristic
17C2×#428C2211yescharacteristic
18C2×#424C2212yes—
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1855—24no—
24#1854—21no—
25#1854—21no—
26#1853—21no—
27#1852—21no—
28#1823—22no—
29#1823—22no—
30#1821—24no—
31#1798—22no—
32#1797—22no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#341C2212yes—
38#1736—21no—
39#1736—21no—
40#1736C2211yescharacteristic
41#1736C2212yes—
42#1736C2211yescharacteristic
43#1736C2212yes—
44C2×#340—21no—
45C2×#340—21no—
46C2×#340C2212yes—
47C2×#340C2212yes—
48#1729C2212yes—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C22×M16—21no—
52C22×M16—24no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—42no—
56C22×M16C2312yes—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2312yes—
60C23×C8—21no—
61C23×C8C2312yes—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—24no—
65C2×#117—42no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4Q814yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C2×#108—24no—
88C2×#108—24no—
89C2×#108—42no—
90#428—24no—
91#428—42no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96#424—24no—
97#424—42no—
98C2×C4⋊C8—24no—
99#407—22no—
100#407—22no—
101#407—22no—
102#407D414yes—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8D412yes—
110C2×C22⋊C8D412yes—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115C2×#95—21no—
116#343—24no—
117#343—24no—
118#343—24no—
119#343—42no—
120#341—24no—
121#340—24no—
122#340—24no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#336—24no—
132C25#4111yescharacteristic
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—42no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q812yes—
154C23×C4#4111yescharacteristic
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4C2411yescharacteristic
158C2×M16—24no—
159C2×M16—22no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—48no—
177C2×M16—48no—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16C2×D412yes—
181C2×M16C2×D412yes—
182C2×M16#4114yes—
183C22×C8—24no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8C2×D414yes—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202#117—44no—
203#117—44no—
204#117—41no—
205#117—41no—
206#117#4114yes—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×Q814yes—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—82no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—44no—
256#108—48no—
257#107—42no—
258#107—42no—
259#106—42no—
260#106—42no—
261C4⋊C8—48no—
262C22⋊C8—24no—
263C22⋊C8—24no—
264C22⋊C8—24no—
265C22⋊C8—24no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268#95—41no—
269#95—41no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24#12111yescharacteristic
277C24#12111yescharacteristic
278C24C22⋊Q812yes—
279C24#12612yes—
280C24C2×#4111yescharacteristic
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—88no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—88no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4C22⋊Q812yes—
341C22×C4C22⋊Q812yes—
342C22×C4C22⋊Q812yes—
343C22×C4C22⋊Q812yes—
344C22×C4C22⋊Q812yes—
345C22×C4#12612yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×Q811yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4112yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C42−1+412yes—
356M16—24no—
357M16—24no—
358M16—24no—
359M16—24no—
360M16—48no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16—42no—
365M16—42no—
366M16—42no—
367M16—42no—
368M16—42no—
369C2×C8—24no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—48no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4#12312yes—
410C4⋊C4#12312yes—
411C22⋊C4—24no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C42—24no—
418C42—42no—
419C42—42no—
420C42—42no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—24no—
430C23—24no—
431C23—21no—
432C23—21no—
433C23—24no—
434C23—24no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23#39612yes—
446C23#39612yes—
447C23#47112yes—
448C23#47212yes—
449C23C2×#12111yescharacteristic
450C23C2×C22⋊Q811yescharacteristic
451C23C2×#12611yescharacteristic
452C23#55112yes—
453C23#55212yes—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—88no—
489C2×C4—88no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—84no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—162no—
500C2×C4—162no—
501C2×C4#39612yes—
502C2×C4#39612yes—
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4C2×C22⋊Q811yescharacteristic
507C2×C4C2×#12311yescharacteristic
508C2×C4#52511yescharacteristic
509C2×C4#52912yes—
510C2×C4#53312yes—
511C2×C4#53411yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53912yes—
514C2×C4#54412yes—
515C2×C4#54612yes—
516C2×C4#54612yes—
517C2×C4#54712yes—
518C2×C4#54712yes—
519C2×C4#55112yes—
520C2×C4#55212yes—
521C8—44no—
522C8—42no—
523C8—44no—
524C8—42no—
525C8—82no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—48no—
544C22—48no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22#206314yes—
550C22C2×#39611yescharacteristic
551C22#241211yescharacteristic
552C22#241311yescharacteristic
553C22#243812yes—
554C22#249312yes—
555C22C2×#47111yescharacteristic
556C22C2×#47211yescharacteristic
557C22#306911yescharacteristic
558C22#306911yescharacteristic
559C4—22no—
560C4—22no—
561C4—42no—
562C4—42no—
563C4—84no—
564C4—81no—
565C4—81no—
566C4—161no—
567C4—161no—
568C4#242512yes—
569C4#243812yes—
570C4#248611yescharacteristic
571C4#253411yescharacteristic
572C4#262711yescharacteristic
573C4#285511yescharacteristic
574C2—24no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2#1493511yescharacteristic
579C2C2×#206311yescharacteristic
580C2#2360311yescharacteristic
581C2#2457011yescharacteristic
582C2#2457111yescharacteristic
583C2#2462011yescharacteristic
584C2#2462111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11303C2semidirect
8. #11249C2semidirect
9. #11242C2semidirect
10. #11216C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18552
#208462
C2×#18232
C2×#18212
#206282
C2×#17362
#113032
#112492
#112422
#112162
#107392