Finite Group Databasebeta

← #374553

Subgroups — #374553

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic, complemented
4C2×#1991C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6#11208C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#10843C211yesmaximal, characteristic, complemented
9#10841C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2629—24no—
20#2629C418yescomplemented
21C2×#398—22no—
22C2×#396—22no—
23#2317—24no—
24#2317—24no—
25C2×#389—22no—
26C2×#389C2212yes—
27C2×#385—22no—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34#1992—24no—
35#1991—24no—
36#1750—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125C2×C414yescomplemented
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—24no—
63C2×#117—42no—
64C2×#117C2×C414yes—
65C2×#117D414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D412yes—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8Q818yes—
92C2×#97—22no—
93C2×#97—22no—
94C2×#97—22no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98#398—216no—
99#396—216no—
100#389—216no—
101#389—24no—
102#389C2×C418yescomplemented
103#385—216no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384C2×C414yescomplemented
109#384D414yes—
110#382—22no—
111#382—22no—
112#382—22no—
113#382D414yes—
114#380—24no—
115#380C2×C418yes—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#340—42no—
127#340—42no—
128#336—24no—
129C25C2×D411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×Q814yes—
153C23×C4#4114yes—
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C2×Q814yes—
168C22×C8C2×Q814yes—
169#125—28no—
170#125—28no—
171#125—44no—
172#125—44no—
173#117—28no—
174#117—28no—
175#117—28no—
176#117—28no—
177#117—48no—
178#117—48no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C22⋊C418yes—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4C22⋊C418yes—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—24no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C22⋊C8—216no—
255#97—44no—
256C4×C8—44no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24C2×C22⋊C412yes—
270C24#12412yes—
271C24C4⋊Q812yes—
272C24C22×D411yescharacteristic
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—48no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
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—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4C2×C22⋊C414yes—
341C22×C4C2×C22⋊C414yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C4×Q818yes—
344C22×C4#12014yes—
345C22×C4C22⋊Q818yes—
346C22×C4#12314yes—
347C22×C4#12412yes—
348C22×C4C4⋊Q812yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q812yes—
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4112yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C42−1+412yes—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C4⋊C4—216no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—48no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—416no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C22⋊C4—216no—
397C22⋊C4—48no—
398C22⋊C4—88no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—48no—
404C42C4×D414yes—
405C42C4×D414yes—
406C42#12114yes—
407C42#12114yes—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23#39114yes—
431C23C22×C22⋊C411yescharacteristic
432C23C2×#12011yescharacteristic
433C23C2×#12311yescharacteristic
434C23C2×#12411yescharacteristic
435C23C2×C4⋊Q811yescharacteristic
436C23#53511yescharacteristic
437C23#53611yescharacteristic
438C23#57612yes—
439C23#57712yes—
440C23#57712yes—
441C23#57812yes—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—816no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#38618yes—
501C2×C4#39114yes—
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4C2×C4×Q812yes—
505C2×C4#52012yes—
506C2×C4C2×#12111yescharacteristic
507C2×C4C2×#12111yescharacteristic
508C2×C4C2×C22⋊Q812yes—
509C2×C4#53712yes—
510C2×C4#54612yes—
511C2×C4#54612yes—
512C2×C4D4×Q812yes—
513C2×C4D4×Q812yes—
514C2×C4#55212yes—
515C2×C4#55212yes—
516C2×C4#55412yes—
517C2×C4#55412yes—
518C8—44no—
519C8—44no—
520C22—24no—
521C22—28no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—44no—
527C22—42no—
528C22—42no—
529C22—44no—
530C22—44no—
531C22C2×#38611yescharacteristic
532C22C2×#39111yescharacteristic
533C22#231111yescharacteristic
534C22#243411yescharacteristic
535C22#243511yescharacteristic
536C22#250211yescharacteristic
537C22#250311yescharacteristic
538C22#289712yes—
539C22#289812yes—
540C22#301212yes—
541C22#301312yes—
542C22#309812yes—
543C22#309912yes—
544C22#313612yes—
545C22#313712yes—
546C22#316312yes—
547C22#316412yes—
548C22#322011yescharacteristic
549C22#322111yescharacteristic
550C22#322211yescharacteristic
551C22#322311yescharacteristic
552C22#324112yes—
553C22#324212yes—
554C4—24no—
555C4—28no—
556C4—44no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—84no—
562C4—84no—
563C4#236012yes—
564C4#236012yes—
565C4#241712yes—
566C4#241712yes—
567C2—44no—
568C2#1413911yescharacteristic
569C2#2091512yes—
570C2#2132612yes—
571C2#2187211yescharacteristic
572C2#2187311yescharacteristic
573C2#2582212yes—
574C2#2586711yescharacteristic
575C2#2586811yescharacteristic
576C2#2795711yescharacteristic
577C2#2795811yescharacteristic
578C2#2797012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
3. C2×#1992C2semidirect
4. C2×#1991C2semidirect
6. #11208C2semidirect
7. #11207C2semidirect
8. #10843C2semidirect
9. #10841C2semidirect
20. #2629C4semidirect
55. C2×#125C2×C4semidirect
102. #389C2×C4semidirect
108. #384C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#23172
C2×#19922
C2×#19912
C2×#17502
#112082
#112072
#108432
#108412
#107312
#107302