Finite Group Databasebeta

← #415886

Subgroups — #415886

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#487C211yesmaximal, characteristic, complemented
2C2×#2621C211yesmaximal, characteristic
3C2×#2628C211yesmaximal, characteristic
4C2×#1993C211yesmaximal, characteristic, complemented
5C2×#1960C211yesmaximal, characteristic, complemented
6C2×#1750C211yesmaximal, characteristic
7#11067C214yesmaximal, complemented
8#10970C214yesmaximal
9#10842C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C22⋊Q8C2212yes—
13C22×#117C2211yescharacteristic
14C2×#487—24no—
15C2×#487—24no—
16C2×#487—24no—
17C22×#103C2214yes—
18C22×#103C2214yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2628—24no—
23#2628—24no—
24#2621—24no—
25#2621—24no—
26C2×#400—21no—
27C2×#399—21no—
28C2×#398—21no—
29C2×#398—21no—
30C2×#397—21no—
31C2×#397—21no—
32C2×#395—21no—
33C2×#393—21no—
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#382C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39#1993—24no—
40#1960—24no—
41#1750—24no—
42C2×#340C2214yes—
43C2×#340C2214yes—
44C2×#336C2212yes—
45C23×Q8—21no—
46C23×Q8C2312yes—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8D414yes—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—44no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#487—232no—
89C2×#103—28no—
90C2×#103—28no—
91C2×#103—28no—
92C2×#103—28no—
93C2×#103—28no—
94C2×#103—44no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8D414yes—
104#400—28no—
105#399—28no—
106#398—28no—
107#398—28no—
108#397—28no—
109#397—28no—
110#395—28no—
111#393—28no—
112#391—24no—
113#391—24no—
114#389—24no—
115#389—24no—
116#382—22no—
117#382—22no—
118#382D414yes—
119#382D414yes—
120#380—24no—
121#380—24no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—21no—
130#340—28no—
131#340—44no—
132#336—24no—
133C25C2×D411yescharacteristic
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—24no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—84no—
147C22×Q8C2×D414yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4#4114yes—
167C23×C4C2411yescharacteristic
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×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D414yes—
181C22×C8#4114yes—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—28no—
185C4⋊Q8—28no—
186C4⋊Q8—44no—
187C4⋊Q8—44no—
188C4⋊Q8—44no—
189C4⋊Q8—44no—
190C22⋊Q8—416no—
191C22⋊Q8—416no—
192#117—44no—
193#117—44no—
194#117—44no—
195#117—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
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—28no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—84no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D414yes—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—84no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253#103—232no—
254#103—416no—
255C22⋊C8—216no—
256#97—44no—
257C4×C8—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—24no—
273C24—24no—
274C24#12412yes—
275C24C22×D411yescharacteristic
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—28no—
279C2×Q8—28no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—48no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—88no—
294C2×Q8—88no—
295C2×Q8—88no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4#12012yes—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#12412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C42−1+412yes—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—416no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—88no—
396C22⋊C4—416no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—88no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42#12014yes—
405C42#12114yes—
406C42#12114yes—
407C42#12714yes—
408C23—22no—
409C23—28no—
410C23—22no—
411C23—22no—
412C23—41no—
413C23—41no—
414C23—44no—
415C23—41no—
416C23—41no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23C2×#12011yescharacteristic
426C23C2×#12011yescharacteristic
427C23C2×#12111yescharacteristic
428C23C2×#12411yescharacteristic
429C23#53211yescharacteristic
430C23#53511yescharacteristic
431C23#53611yescharacteristic
432C23#57714yes—
433C23#57814yes—
434Q8—48no—
435Q8—48no—
436Q8—816no—
437Q8—84no—
438Q8—84no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4#39214yes—
487C2×C4#39214yes—
488C2×C4C2×#12011yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4#52512yes—
492C2×C4#52912yes—
493C2×C4C2×#12711yescharacteristic
494C2×C4#53712yes—
495C2×C4#54212yes—
496C2×C4#54612yes—
497C2×C4#54612yes—
498C2×C4#54612yes—
499C2×C4#54712yes—
500C2×C4#54812yes—
501C2×C4#54812yes—
502C2×C4#54812yes—
503C2×C4#55012yes—
504C8—44no—
505C8—44no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—42no—
510C22—82no—
511C22—82no—
512C22—81no—
513C22—81no—
514C22—82no—
515C22C2×#39211yescharacteristic
516C22#243111yescharacteristic
517C22#243211yescharacteristic
518C22#243511yescharacteristic
519C22#243611yescharacteristic
520C22#262011yescharacteristic
521C22#285011yescharacteristic
522C22#300912yes—
523C22#301012yes—
524C22#301112yes—
525C22#301112yes—
526C22#305712yes—
527C22#305912yes—
528C22#313714yes—
529C22#317814yes—
530C22#322514yes—
531C22#324214yes—
532C4—24no—
533C4—24no—
534C4—42no—
535C4—84no—
536C4—84no—
537C4—81no—
538C4—84no—
539C4—81no—
540C4—84no—
541C4#240712yes—
542C4#242612yes—
543C4#242712yes—
544C4#254012yes—
545C2—82no—
546C2#1521511yescharacteristic
547C2#2338811yescharacteristic
548C2#2338911yescharacteristic
549C2#2561312yes—
550C2#2568612yes—
551C2#2586612yes—
552C2#2591412yes—
553C2#2984812yes—
554C2#3304612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#487C2semidirect
4. C2×#1993C2semidirect
5. C2×#1960C2semidirect
7. #11067C2semidirect
9. #10842C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4872
C2×#26212
C2×#26282
C2×#19932
C2×#19602
C2×#17502
#110672
#109702
#108422