Finite Group Databasebeta

← #440849

Subgroups — #440849

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1834C211yesmaximal, characteristic, complemented
4#15195C211yesmaximal, characteristic
5#11385C212yesmaximal, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11292C212yesmaximal, complemented
8#11199C212yesmaximal
9#10921C211yesmaximal, characteristic
10#10880C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16#2441—24no—
17#2441—24no—
18C2×#399—21no—
19C2×#396—21no—
20C2×#389—21no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#2100—24no—
30#1834—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—21no—
42#1442—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—24no—
53C2×#125—42no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—42no—
83C2×C4×C8—24no—
84C2×C4×C8—22no—
85#399—28no—
86#396—28no—
87#389—28no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389D414yes—
92#387—28no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384D414yes—
97#380—22no—
98#380—22no—
99#380—24no—
100#380—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#336—22no—
122#336—42no—
123C25C2×D411yescharacteristic
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4114yes—
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8#4114yes—
158#125—48no—
159#125—44no—
160#125—44no—
161#117—48no—
162#117—44no—
163#117—44no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4#4114yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—44no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C22⋊C8—48no—
234C4×C8—28no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—44no—
241#95—42no—
242#95—42no—
243#95—82no—
244#95—82no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24#12412yes—
256C24C22×D411yescharacteristic
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
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#12012yes—
326C22×C4#12012yes—
327C22×C4#12412yes—
328C22×C4#13512yes—
329C22×C4#13512yes—
330C22×C4#13712yes—
331C22×C4#13712yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+412yes—
339C22×C42−1+412yes—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—48no—
360C4⋊C4—48no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4#12414yes—
382C22⋊C4—48no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—84no—
386C22⋊C4—88no—
387C42—22no—
388C42—22no—
389C42—24no—
390C42—48no—
391C42—41no—
392C42—41no—
393C42—42no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23—41no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23C2×#12011yescharacteristic
418C23C2×#12411yescharacteristic
419C23#53611yescharacteristic
420C23#54612yes—
421C23#54712yes—
422C23C2×#13511yescharacteristic
423C23C2×#13511yescharacteristic
424C23C2×#13711yescharacteristic
425C23C2×#13711yescharacteristic
426C23#57612yes—
427C23#57712yes—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4#45214yes—
490C2×C4#45214yes—
491C2×C4#52511yescharacteristic
492C2×C4C2×#12411yescharacteristic
493C2×C4#52912yes—
494C2×C4#53511yescharacteristic
495C2×C4#53611yescharacteristic
496C2×C4#53912yes—
497C2×C4#54112yes—
498C2×C4#54212yes—
499C2×C4#54612yes—
500C2×C4#54712yes—
501C2×C4#54812yes—
502C2×C4#54812yes—
503C2×C4#55112yes—
504C2×C4#55212yes—
505C2×C4#55212yes—
506C2×C4#56312yes—
507C8—44no—
508C8—82no—
509C22—42no—
510C22—42no—
511C22—44no—
512C22—42no—
513C22—44no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—82no—
519C22—82no—
520C22#240611yescharacteristic
521C22#243511yescharacteristic
522C22#243611yescharacteristic
523C22#248812yes—
524C22#269812yes—
525C22C2×#45211yescharacteristic
526C22C2×#45211yescharacteristic
527C22#300811yescharacteristic
528C22#301011yescharacteristic
529C22#313111yescharacteristic
530C22#313612yes—
531C22#313911yescharacteristic
532C22#323111yescharacteristic
533C22#323111yescharacteristic
534C22#324112yes—
535C22#330312yes—
536C22#330512yes—
537C22#333112yes—
538C22#333312yes—
539C22#333712yes—
540C22#333912yes—
541C22#334412yes—
542C22#334612yes—
543C4—24no—
544C4—24no—
545C4—44no—
546C4—42no—
547C4—81no—
548C4—82no—
549C4—82no—
550C4—82no—
551C4—81no—
552C4—162no—
553C4—162no—
554C4#247712yes—
555C4#248912yes—
556C4#269012yes—
557C4#272312yes—
558C2—82no—
559C2#1567011yescharacteristic
560C2#2409211yescharacteristic
561C2#2409511yescharacteristic
562C2#2588311yescharacteristic
563C2#2588611yescharacteristic
564C2#2590411yescharacteristic
565C2#2591511yescharacteristic
566C2#2770212yes—
567C2#2770712yes—
568C2#3168712yes—
569C2#3168912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2100C2semidirect
3. C2×#1834C2semidirect
5. #11385C2semidirect
6. #11357C2semidirect
7. #11292C2semidirect
10. #10880C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#21002
C2×#18342
#151952
#113852
#113572
#112922
#111992
#109212
#108802
#107312