Finite Group Databasebeta

← #416917

Subgroups — #416917

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic, complemented
3C2×#1991C211yesmaximal, characteristic, complemented
4C2×#1835C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic
6#14682C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#11200C212yesmaximal
10#11199C212yesmaximal
11#10842C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2851—24no—
18#2851—24no—
19C2×#389C2212yes—
20C2×#387—22no—
21C2×#385—22no—
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#1991—24no—
29#1991—24no—
30#1835—24no—
31C4×C22⋊C8—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434—22no—
44C22⋊#95—22no—
45C24×C4D412yes—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125D414yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42D412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8D414yes—
84#389—24no—
85#389—24no—
86#387—216no—
87#385—216no—
88#382—22no—
89#382—22no—
90#382—22no—
91#382—22no—
92#382—22no—
93#382D414yes—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#336—24no—
116C25C2×D411yescharacteristic
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
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×C8C2×D414yes—
157C22×C8#4114yes—
158#125—28no—
159#125—28no—
160#125—44no—
161#125—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
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—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C22⋊C8—216no—
223C4×C8—28no—
224C4×C8—44no—
225#95—28no—
226#95—28no—
227#95—28no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—44no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24#12412yes—
244C24#13514yes—
245C24C22×D411yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—216no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—48no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
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#12014yes—
320C22×C4#12114yes—
321C22×C4#12314yes—
322C22×C4#12412yes—
323C22×C4#13514yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C42−1+412yes—
333C22×C42−1+412yes—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—416no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—416no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C22⋊C4—416no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—48no—
374C42#12114yes—
375C42#12114yes—
376C42#12314yes—
377C42#12414yes—
378C23—24no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—24no—
384C23—22no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—22no—
389C23—24no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—41no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23#48814yes—
401C23C2×#12011yescharacteristic
402C23C2×#12111yescharacteristic
403C23C2×#12311yescharacteristic
404C23C2×#12411yescharacteristic
405C23#53211yescharacteristic
406C23#53611yescharacteristic
407C23#53611yescharacteristic
408C23C2×#13512yes—
409C23C2×#13512yes—
410C23#57512yes—
411C23#57512yes—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—432no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#39418yes—
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×#12111yescharacteristic
474C2×C4C2×#12311yescharacteristic
475C2×C4#52512yes—
476C2×C4C2×#12411yescharacteristic
477C2×C4#52912yes—
478C2×C4#53612yes—
479C2×C4#54112yes—
480C2×C4#54712yes—
481C2×C4#54812yes—
482C2×C4#54812yes—
483C2×C4#54812yes—
484C2×C4#55212yes—
485C2×C4#55212yes—
486C2×C4#55212yes—
487C2×C4#55612yes—
488C8—44no—
489C8—44no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—42no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22C2×#39411yescharacteristic
502C22#243211yescharacteristic
503C22#243611yescharacteristic
504C22#249911yescharacteristic
505C22#250311yescharacteristic
506C22#262111yescharacteristic
507C22#285011yescharacteristic
508C22#300212yes—
509C22#300212yes—
510C22#305012yes—
511C22#305412yes—
512C22#309112yes—
513C22#309112yes—
514C22C2×#48811yescharacteristic
515C22#313111yescharacteristic
516C22#314011yescharacteristic
517C22#314211yescharacteristic
518C22#316812yes—
519C22#318312yes—
520C22#323212yes—
521C22#323212yes—
522C22#323312yes—
523C22#323312yes—
524C4—28no—
525C4—42no—
526C4—44no—
527C4—84no—
528C4—84no—
529C4—161no—
530C4—161no—
531C4—161no—
532C4—161no—
533C4#241512yes—
534C4#242112yes—
535C4#248912yes—
536C4#253612yes—
537C2—44no—
538C2#1523611yescharacteristic
539C2#2347912yes—
540C2#2567512yes—
541C2#2590611yescharacteristic
542C2#2590711yescharacteristic
543C2#2787912yes—
544C2#2796311yescharacteristic
545C2#2796411yescharacteristic
546C2#2987912yes—
547C2#3304411yescharacteristic
548C2#3305011yescharacteristic
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
2. C2×#1991C2semidirect
3. C2×#1991C2semidirect
4. C2×#1835C2semidirect
7. #11293C2semidirect
8. #11292C2semidirect
11. #10842C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#19912
C2×#18352
C2×C4×C22⋊C82
#146822
#112932
#112922
#112002
#111992
#108422