Finite Group Databasebeta

← #416924

Subgroups — #416924

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic, complemented
3C2×#1991C211yesmaximal, characteristic, complemented
4C2×#1836C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6#14340C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#11200C212yesmaximal
10#11199C212yesmaximal
11#10841C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2629—24no—
19#2629—24no—
20C2×#398—22no—
21C2×#396—22no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28#1992—24no—
29#1991—24no—
30#1836—24no—
31#1750—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1441—22no—
43#1439—22no—
44C4⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125D414yes—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×#97—22no—
83C2×#97—22no—
84C2×#97D414yes—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88#398—216no—
89#396—216no—
90#389—24no—
91#389—24no—
92#384—22no—
93#384—22no—
94#384—22no—
95#384D414yes—
96#382—22no—
97#382—22no—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—24no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#336—22no—
117#336—22no—
118C25C2×D411yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8#4112yes—
168C22×C8#4112yes—
169#125—28no—
170#125—28no—
171#125—44no—
172#125—44no—
173#117—44no—
174#117—44no—
175#117—44no—
176#117—44no—
177C2×C4⋊C4—22no—
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—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C22⋊C8—28no—
238C22⋊C8—28no—
239#97—28no—
240C4×C8—44no—
241#95—28no—
242#95—28no—
243#95—44no—
244#95—44no—
245#95—82no—
246#95—82no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24#12412yes—
258C24C22×D411yescharacteristic
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
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—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—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4#12014yes—
333C22×C4#12114yes—
334C22×C4#12314yes—
335C22×C4#12412yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C42−1+411yescharacteristic
347C22×C42−1+411yescharacteristic
348C22×C42−1+411yescharacteristic
349C22×C42−1+411yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—48no—
387C4⋊C4—416no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—88no—
399C22⋊C4—48no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42#12114yes—
407C42#12114yes—
408C42#12314yes—
409C42#12414yes—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—42no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23C2×#12011yescharacteristic
434C23C2×#12111yescharacteristic
435C23C2×#12311yescharacteristic
436C23C2×#12411yescharacteristic
437C23#53211yescharacteristic
438C23#53611yescharacteristic
439C23#53611yescharacteristic
440C23#57612yes—
441C23#57712yes—
442C23#57712yes—
443C23#57812yes—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
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—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#39418yes—
501C2×C4C2×#12111yescharacteristic
502C2×C4C2×#12111yescharacteristic
503C2×C4C2×#12311yescharacteristic
504C2×C4#52511yescharacteristic
505C2×C4#52511yescharacteristic
506C2×C4C2×#12411yescharacteristic
507C2×C4#52911yescharacteristic
508C2×C4#52911yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#53611yescharacteristic
511C2×C4#54111yescharacteristic
512C2×C4#54111yescharacteristic
513C2×C4#54711yescharacteristic
514C2×C4#54711yescharacteristic
515C2×C4#54811yescharacteristic
516C2×C4#54811yescharacteristic
517C2×C4#54811yescharacteristic
518C2×C4#54811yescharacteristic
519C2×C4#54811yescharacteristic
520C2×C4#54811yescharacteristic
521C2×C4#55211yescharacteristic
522C2×C4#55211yescharacteristic
523C2×C4#55211yescharacteristic
524C2×C4#55211yescharacteristic
525C2×C4#55211yescharacteristic
526C2×C4#55211yescharacteristic
527C2×C4#55611yescharacteristic
528C2×C4#55611yescharacteristic
529C8—42no—
530C8—42no—
531C8—42no—
532C8—42no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—42no—
538C22—81no—
539C22—81no—
540C22—82no—
541C22—82no—
542C22—81no—
543C22—81no—
544C22C2×#39411yescharacteristic
545C22#243211yescharacteristic
546C22#243611yescharacteristic
547C22#249911yescharacteristic
548C22#250311yescharacteristic
549C22#262111yescharacteristic
550C22#285011yescharacteristic
551C22#301212yes—
552C22#301312yes—
553C22#306212yes—
554C22#306312yes—
555C22#309812yes—
556C22#309912yes—
557C22#313612yes—
558C22#313712yes—
559C22#317411yescharacteristic
560C22#317411yescharacteristic
561C22#317611yescharacteristic
562C22#317711yescharacteristic
563C22#324112yes—
564C22#324212yes—
565C22#324312yes—
566C22#324412yes—
567C4—28no—
568C4—42no—
569C4—84no—
570C4—82no—
571C4—84no—
572C4—162no—
573C4—162no—
574C4#241511yescharacteristic
575C4#241511yescharacteristic
576C4#242111yescharacteristic
577C4#242111yescharacteristic
578C4#248911yescharacteristic
579C4#248911yescharacteristic
580C4#253611yescharacteristic
581C4#253611yescharacteristic
582C2—82no—
583C2#1523611yescharacteristic
584C2#2348512yes—
585C2#2568211yescharacteristic
586C2#2568311yescharacteristic
587C2#2591612yes—
588C2#2788411yescharacteristic
589C2#2788511yescharacteristic
590C2#2797012yes—
591C2#2988512yes—
592C2#3305511yescharacteristic
593C2#3305611yescharacteristic
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
2. C2×#1992C2semidirect
3. C2×#1991C2semidirect
4. C2×#1836C2semidirect
7. #11293C2semidirect
8. #11292C2semidirect
11. #10841C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#19922
C2×#19912
C2×#18362
C2×#17502
#143402
#112932
#112922
#112002
#111992
#108412