Finite Group Databasebeta

← #406717

Subgroups — #406717

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1825C212yesmaximal
3#13795C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11293C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11268C212yesmaximal, complemented
8#10730C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2628—28no—
16C2×#400—21no—
17C2×#399—21no—
18C2×#395—21no—
19C2×#393—21no—
20C2×#389C2212yes—
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23#1825—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C4⋊#95—21no—
36C4⋊#95—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C22⋊#95—21no—
40C22⋊#95—21no—
41C23×Q8—21no—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×#117—24no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—41no—
56C2×#117—41no—
57C2×#117Q814yescomplemented
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D411yescharacteristic
74C22×C22⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81#400—28no—
82#399—28no—
83#395—28no—
84#393—28no—
85#389—24no—
86#389—24no—
87#382—24no—
88#380—24no—
89#380D418yes—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#336—24no—
107#336—41no—
108#336—41no—
109C25C2×D411yescharacteristic
110C22×Q8—24no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—82no—
114C22×Q8—82no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—82no—
129C23×C4—82no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D414yes—
136C23×C4C2×Q812yes—
137C23×C4#4114yes—
138C23×C4#4112yes—
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
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—
154C4⋊Q8—48no—
155C4⋊Q8—48no—
156#117—216no—
157#117—48no—
158#117—48no—
159#117—48no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
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—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
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—28no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×Q814yes—
200C2×C4⋊C4#4114yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42#4114yes—
224C22⋊C8—48no—
225#95—24no—
226#95—24no—
227#95—28no—
228#95—28no—
229#95—82no—
230#95—82no—
231#95—82no—
232#95—82no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24#12012yes—
242C24C22⋊Q812yes—
243C24#12412yes—
244C24C22×D411yescharacteristic
245C2×Q8—48no—
246C2×Q8—88no—
247C2×Q8—88no—
248C2×Q8—84no—
249C2×Q8—84no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—48no—
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—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—88no—
316C22×C4—88no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4#12012yes—
320C22×C4C22⋊Q818yes—
321C22×C4C22⋊Q812yes—
322C22×C4#12312yes—
323C22×C4#12312yes—
324C22×C4#12418yes—
325C22×C4#12412yes—
326C22×C4C22×D412yes—
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4112yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C42−1+411yescharacteristic
335C22×C42−1+411yescharacteristic
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—88no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—48no—
374C22⋊C4—88no—
375C42—24no—
376C42—24no—
377C42—42no—
378C42—42no—
379C42—48no—
380C23—22no—
381C23—24no—
382C23—24no—
383C23—22no—
384C23—24no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—24no—
391C23—22no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23#39314yes—
403C23C2×#12011yescharacteristic
404C23C2×C22⋊Q811yescharacteristic
405C23C2×#12311yescharacteristic
406C23#52511yescharacteristic
407C23C2×#12411yescharacteristic
408C23#53411yescharacteristic
409C23#53611yescharacteristic
410C23#57612yes—
411C23#57712yes—
412C23#57712yes—
413C23#57812yes—
414Q8—816no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—816no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#39314yes—
475C2×C4C2×C22⋊Q812yes—
476C2×C4C2×#12412yes—
477C2×C4#52912yes—
478C2×C4#53312yes—
479C2×C4#53911yescharacteristic
480C2×C4#53911yescharacteristic
481C2×C4#54411yescharacteristic
482C2×C4#54411yescharacteristic
483C2×C4#54612yes—
484C2×C4#54612yes—
485C2×C4#54712yes—
486C2×C4#54712yes—
487C2×C4#55112yes—
488C2×C4#55112yes—
489C2×C4#55212yes—
490C2×C4#55212yes—
491C8—82no—
492C8—82no—
493C22—24no—
494C22—28no—
495C22—44no—
496C22—42no—
497C22—42no—
498C22—44no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22C2×#39311yescharacteristic
505C22#240511yescharacteristic
506C22#240611yescharacteristic
507C22#250311yescharacteristic
508C22#256111yescharacteristic
509C22#262711yescharacteristic
510C22#285511yescharacteristic
511C22#301212yes—
512C22#301312yes—
513C22#308312yes—
514C22#308412yes—
515C22#309612yes—
516C22#309712yes—
517C22#311812yes—
518C22#312112yes—
519C22#313612yes—
520C22#313712yes—
521C22#319911yescharacteristic
522C22#320011yescharacteristic
523C22#320011yescharacteristic
524C22#320211yescharacteristic
525C22#324312yes—
526C22#324412yes—
527C4—24no—
528C4—42no—
529C4—42no—
530C4—84no—
531C4—84no—
532C4—84no—
533C4—81no—
534C4—84no—
535C4—81no—
536C4#243812yes—
537C4#243812yes—
538C4#248812yes—
539C4#248812yes—
540C2—44no—
541C2#1493411yescharacteristic
542C2#2345812yes—
543C2#2401511yescharacteristic
544C2#2401611yescharacteristic
545C2#2407512yes—
546C2#2797112yes—
547C2#2911611yescharacteristic
548C2#2911811yescharacteristic
549C2#2997611yescharacteristic
550C2#2997711yescharacteristic
551C2#3315512yes—
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
4. #11384C2semidirect
5. #11293C2semidirect
6. #11292C2semidirect
7. #11268C2semidirect
57. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#18252
#137952
#113842
#112932
#112922
#112682
#107302
#107132