Finite Group Databasebeta

← #403410

Subgroups — #403410

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×#2093C211yesmaximal, characteristic
3C2×#1926C211yesmaximal, characteristic
4#15157C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11051C212yesmaximal
9#10931C212yesmaximal
10#10907C212yesmaximal, complemented
11#10807C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C2×#393—21no—
19#2355—24no—
20#2355—24no—
21C2×#391—21no—
22C2×#390C2212yes—
23C2×#390C2211yescharacteristic
24C2×#387—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2093—24no—
31#1926—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41#1439—21no—
42#1439—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C42—21no—
68C22×C42—22no—
69C22×C42C2311yescharacteristic
70C22×C42C2312yes—
71C2×#107—24no—
72C2×#107—24no—
73C2×#107—42no—
74C2×#106—22no—
75C2×#106—42no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—22no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102Q814yescomplemented
84#393—44no—
85#391—28no—
86#390—22no—
87#390—22no—
88#390—24no—
89#390—24no—
90#387—216no—
91#384—22no—
92#384D414yes—
93#382—24no—
94#382—24no—
95C4×C4⋊C4—28no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108#340—24no—
109#340—22no—
110#340—24no—
111#340—42no—
112#340—42no—
113#336—22no—
114#336—22no—
115#336—24no—
116#336—24no—
117C25—21no—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—42no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C2×Q812yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×Q812yes—
159C22×C8#4114yes—
160C4×Q8—28no—
161C4×Q8—28no—
162C4×Q8—216no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
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—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×Q812yes—
198C2×C4⋊C4#4112yes—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—28no—
218C2×C42—44no—
219C2×C42—82no—
220C2×C42—82no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42#4114yes—
224C2×C42#4114yes—
225#107—48no—
226#106—44no—
227#102—28no—
228#102—48no—
229#102—44no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—24no—
234#95—44no—
235#95—42no—
236#95—82no—
237#95—81no—
238C24—22no—
239C24—41no—
240C24—42no—
241C24—41no—
242C24—42no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24—41no—
248C24—41no—
249C24C22×D411yescharacteristic
250C2×Q8—28no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×D4—22no—
256C2×D4—24no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
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—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—24no—
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—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4#12012yes—
321C22×C4#12012yes—
322C22×C4#12314yes—
323C22×C4#12314yes—
324C22×C4#13512yes—
325C22×C4AGL(1,ℤ/8)12yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4C2×#4112yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+411yescharacteristic
335C22×C42−1+411yescharacteristic
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—28no—
354C4⋊C4—216no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—48no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—84no—
374C4⋊C4C22⋊Q814yes—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C22⋊C4—84no—
378C22⋊C4—84no—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—48no—
387C42—88no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—42no—
392C23—44no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—81no—
406C23—82no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23C2×#12011yescharacteristic
412C23C2×#12312yes—
413C23#54811yescharacteristic
414C23D4×Q811yescharacteristic
415C23#55012yes—
416C23C2×#13511yescharacteristic
417C23C2×AGL(1,ℤ/8)11yescharacteristic
418C23#57512yes—
419C23#57612yes—
420C23#57711yescharacteristic
421C23#57811yescharacteristic
422Q8—416no—
423D4—42no—
424D4—42no—
425D4—44no—
426D4—81no—
427D4—81no—
428D4—82no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
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—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
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—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4#44914yes—
482C2×C4C2×C22⋊Q811yescharacteristic
483C2×C4#52511yescharacteristic
484C2×C4#52511yescharacteristic
485C2×C4#52512yes—
486C2×C4#52912yes—
487C2×C4#53211yescharacteristic
488C2×C4#53312yes—
489C2×C4#53511yescharacteristic
490C2×C4#53611yescharacteristic
491C2×C4#54212yes—
492C2×C4#54212yes—
493C2×C4#54612yes—
494C2×C4#54811yescharacteristic
495C2×C4D4×Q811yescharacteristic
496C2×C4#55212yes—
497C2×C4#55412yes—
498C2×C4#55612yes—
499C2×C4#55912yes—
500C8—42no—
501C8—44no—
502C8—81no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—84no—
507C22—84no—
508C22—161no—
509C22—161no—
510C22—161no—
511C22—162no—
512C22—161no—
513C22—162no—
514C22#240911yescharacteristic
515C22#242312yes—
516C22#243111yescharacteristic
517C22#244812yes—
518C22#284911yescharacteristic
519C22C2×#44911yescharacteristic
520C22#300511yescharacteristic
521C22#300811yescharacteristic
522C22#301311yescharacteristic
523C22#309112yes—
524C22#309412yes—
525C22#309612yes—
526C22#309812yes—
527C22#336411yescharacteristic
528C22#336811yescharacteristic
529C22#337012yes—
530C22#338111yescharacteristic
531C22#338611yescharacteristic
532C22#338912yes—
533C22#339612yes—
534C22#339912yes—
535C22#340112yes—
536C22#340212yes—
537C4—24no—
538C4—44no—
539C4—44no—
540C4—44no—
541C4—42no—
542C4—84no—
543C4—82no—
544C4—161no—
545C4—162no—
546C4#239611yescharacteristic
547C4#245812yes—
548C4#246512yes—
549C4#258011yescharacteristic
550C4#260411yescharacteristic
551C4#263211yescharacteristic
552C2—162no—
553C2#1487511yescharacteristic
554C2#2432811yescharacteristic
555C2#2433911yescharacteristic
556C2#2506412yes—
557C2#2507212yes—
558C2#2563311yescharacteristic
559C2#2563811yescharacteristic
560C2#2632612yes—
561C2#2633212yes—
562C2#3298011yescharacteristic
563C2#3298611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2355C2semidirect
5. #11380C2semidirect
6. #11357C2semidirect
7. #11325C2semidirect
10. #10907C2semidirect
11. #10807C2semidirect
83. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#20932
C2×#19262
#151572
#113802
#113572
#113252
#110512
#109312
#109072
#108072