Finite Group Databasebeta

← #468116

Subgroups — #468116

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1864C211yesmaximal, characteristic
4#15200C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11268C212yesmaximal, complemented
9#11189C212yesmaximal
10#11123C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19#2563—24no—
20#2563—24no—
21C2×#393—22no—
22C2×#390C2212yes—
23C2×#389C2211yescharacteristic
24C2×#382C2212yes—
25C2×#380—21no—
26C2×#380—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2057—24no—
29#1864—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39#1445—21no—
40#1442—22no—
41#1442—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8D414yes—
55C2×C4×D4—22no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×#107—24no—
74C2×#107—42no—
75C2×#107—42no—
76C2×#106—22no—
77C2×#106—42no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—22no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102Q814yescomplemented
86#393—216no—
87#390—24no—
88#390—24no—
89#389—22no—
90#389—22no—
91#382—24no—
92#382—24no—
93#380—28no—
94#380—28no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—24no—
108#340—22no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—24no—
114#336—22no—
115#336—22no—
116C25—21no—
117C22×Q8—24no—
118C22×Q8—44no—
119C22×Q8—81no—
120C22×Q8—82no—
121C22×Q8—81no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4C2×D412yes—
130C22×D4C2×Q812yes—
131C22×D4#4112yes—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C2×Q812yes—
163C22×C8#4114yes—
164C4⋊Q8—48no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4×D4—44no—
168C4×D4—44no—
169C4×D4—48no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
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—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×Q812yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223#107—48no—
224#106—44no—
225#102—28no—
226#102—48no—
227#102—44no—
228#95—28no—
229#95—28no—
230#95—24no—
231#95—44no—
232#95—42no—
233#95—42no—
234#95—81no—
235#95—82no—
236#95—82no—
237#95—82no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—42no—
242C24—42no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—41no—
247C24—42no—
248C24—41no—
249C24C22×D411yescharacteristic
250C2×Q8—48no—
251C2×Q8—84no—
252C2×Q8—84no—
253C2×Q8—88no—
254C2×Q8—88no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4C22⋊Q814yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—48no—
313C22×C4—41no—
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—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4#12012yes—
328C22×C4#12012yes—
329C22×C4#13512yes—
330C22×C4#13712yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q811yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+411yescharacteristic
339C22×C42−1+412yes—
340C22×C42−1+411yescharacteristic
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—44no—
365C4⋊C4—48no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—44no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—84no—
380C22⋊C4—48no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C23—24no—
391C23—42no—
392C23—44no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—42no—
399C23—44no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23C2×#12011yescharacteristic
414C23#53512yes—
415C23#54712yes—
416C23#54811yescharacteristic
417C23D4×Q811yescharacteristic
418C23C2×#13511yescharacteristic
419C23C2×#13711yescharacteristic
420C23#57512yes—
421C23#57612yes—
422C23#57611yescharacteristic
423C23#57711yescharacteristic
424Q8—816no—
425D4—24no—
426D4—24no—
427D4—42no—
428D4—42no—
429D4—44no—
430D4—44no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—816no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4#45214yes—
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4#53211yescharacteristic
487C2×C4#53411yescharacteristic
488C2×C4#53411yescharacteristic
489C2×C4#53412yes—
490C2×C4#53611yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#53912yes—
493C2×C4#53912yes—
494C2×C4#54212yes—
495C2×C4#54412yes—
496C2×C4#54811yescharacteristic
497C2×C4#54812yes—
498C2×C4D4×Q811yescharacteristic
499C2×C4#55112yes—
500C2×C4#55412yes—
501C2×C4#56312yes—
502C2×C4#56312yes—
503C8—42no—
504C8—44no—
505C8—81no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—84no—
510C22—82no—
511C22—82no—
512C22—84no—
513C22—84no—
514C22—82no—
515C22—82no—
516C22—84no—
517C22#240911yescharacteristic
518C22#243211yescharacteristic
519C22#261311yescharacteristic
520C22#272012yes—
521C22#272312yes—
522C22C2×#45211yescharacteristic
523C22#300511yescharacteristic
524C22#300811yescharacteristic
525C22#301211yescharacteristic
526C22#321112yes—
527C22#321512yes—
528C22#322012yes—
529C22#322412yes—
530C22#334112yes—
531C22#334712yes—
532C22#334812yes—
533C22#335712yes—
534C22#336111yescharacteristic
535C22#337012yes—
536C22#337211yescharacteristic
537C22#338111yescharacteristic
538C22#338511yescharacteristic
539C22#338912yes—
540C4—24no—
541C4—42no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—84no—
547C4—162no—
548C4—161no—
549C4#256411yescharacteristic
550C4#262711yescharacteristic
551C4#266311yescharacteristic
552C4#272812yes—
553C4#272912yes—
554C4#281711yescharacteristic
555C2—84no—
556C2#1660111yescharacteristic
557C2#2433111yescharacteristic
558C2#2433911yescharacteristic
559C2#2570411yescharacteristic
560C2#2571011yescharacteristic
561C2#2962611yescharacteristic
562C2#2963011yescharacteristic
563C2#3235312yes—
564C2#3237312yes—
565C2#3243712yes—
566C2#3245512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2057C2semidirect
4. #15200C2semidirect
5. #11384C2semidirect
6. #11346C2semidirect
7. #11325C2semidirect
8. #11268C2semidirect
85. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#20572
C2×#18642
#152002
#113842
#113462
#113252
#112682
#111892
#111232
#107882