Finite Group Databasebeta

← #440870

Subgroups — #440870

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2073C211yesmaximal, characteristic
3C2×#1835C211yesmaximal, characteristic, complemented
4#15235C211yesmaximal, characteristic, complemented
5#11384C212yesmaximal, complemented
6#11372C211yesmaximal, characteristic, complemented
7#11268C212yesmaximal, complemented
8#11189C212yesmaximal
9#10925C211yesmaximal, characteristic
10#10874C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×C4×C8C2211yescharacteristic
17#2563—24no—
18#2563—24no—
19C2×#393—22no—
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380—22no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#2073—24no—
28#1835—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1443—22no—
39#1442—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—42no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×C4×D4—42no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—42no—
75C2×C4×C8—24no—
76C2×C4×C8—22no—
77#393—216no—
78#390—22no—
79#390—22no—
80#390—22no—
81#390D414yes—
82#382—22no—
83#382—22no—
84#382—22no—
85#382—22no—
86#382—22no—
87#382D414yes—
88#380—216no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340—42no—
106#336—24no—
107#336—22no—
108#336—24no—
109C25—21no—
110C22×Q8—24no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8—82no—
114C22×Q8—82no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4C2×D412yes—
122C22×D4#4114yes—
123C23×C4—24no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—42no—
130C23×C4—44no—
131C23×C4—42no—
132C23×C4—44no—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
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#4114yes—
152C4⋊Q8—48no—
153C4⋊Q8—48no—
154C4×D4—48no—
155C4×D4—48no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
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—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42#4114yes—
207#102—48no—
208#102—48no—
209C4×C8—28no—
210#95—28no—
211#95—28no—
212#95—24no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—82no—
217#95—82no—
218#95—82no—
219C24—22no—
220C24—24no—
221C24—41no—
222C24—41no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24C22×D411yescharacteristic
229C2×Q8—48no—
230C2×Q8—88no—
231C2×Q8—88no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4#12414yes—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—48no—
274C22×C4—416no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
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—84no—
302C22×C4—88no—
303C22×C4—84no—
304C22×C4#12014yes—
305C22×C4#12412yes—
306C22×C4#12412yes—
307C22×C4#13514yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4112yes—
313C22×C4H(5,𝔽2)12yes—
314C22×C42−1+412yes—
315C22×C42−1+412yes—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C22⋊C4—48no—
347C22⋊C4—88no—
348C22⋊C4—88no—
349C42—22no—
350C42—22no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—41no—
355C42—41no—
356C42—42no—
357C23—24no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—81no—
373C23—82no—
374C23—81no—
375C23—82no—
376C23—81no—
377C23—81no—
378C23C2×#12011yescharacteristic
379C23C2×#12411yescharacteristic
380C23#53511yescharacteristic
381C23#54712yes—
382C23#54812yes—
383C23C2×#13512yes—
384C23#57512yes—
385C23#57612yes—
386C23#57712yes—
387Q8—816no—
388D4—28no—
389D4—44no—
390D4—44no—
391D4—44no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—816no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#48814yes—
446C2×C4#52511yescharacteristic
447C2×C4C2×#12411yescharacteristic
448C2×C4#52912yes—
449C2×C4#53611yescharacteristic
450C2×C4#53611yescharacteristic
451C2×C4#53912yes—
452C2×C4#54112yes—
453C2×C4#54212yes—
454C2×C4#54612yes—
455C2×C4#54612yes—
456C2×C4#54712yes—
457C2×C4#54812yes—
458C2×C4#55112yes—
459C2×C4#55212yes—
460C2×C4#55212yes—
461C2×C4#56312yes—
462C8—44no—
463C8—82no—
464C22—44no—
465C22—48no—
466C22—82no—
467C22—82no—
468C22—84no—
469C22—84no—
470C22—84no—
471C22—84no—
472C22—84no—
473C22#240611yescharacteristic
474C22#243511yescharacteristic
475C22#243611yescharacteristic
476C22#248912yes—
477C22#272312yes—
478C22#300212yes—
479C22#301212yes—
480C22C2×#48811yescharacteristic
481C22#313612yes—
482C22#314011yescharacteristic
483C22#321112yes—
484C22#322011yescharacteristic
485C22#322211yescharacteristic
486C22#334012yes—
487C22#334112yes—
488C22#334712yes—
489C22#334812yes—
490C22#336012yes—
491C22#336112yes—
492C22#337012yes—
493C22#337612yes—
494C4—24no—
495C4—42no—
496C4—44no—
497C4—82no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—162no—
502C4—162no—
503C4#247712yes—
504C4#248812yes—
505C4#269012yes—
506C4#269812yes—
507C2—84no—
508C2#1567011yescharacteristic
509C2#2409312yes—
510C2#2585911yescharacteristic
511C2#2586711yescharacteristic
512C2#2593812yes—
513C2#2772812yes—
514C2#2774012yes—
515C2#3243512yes—
516C2#3243712yes—
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
3. C2×#1835C2semidirect
4. #15235C2semidirect
5. #11384C2semidirect
6. #11372C2semidirect
7. #11268C2semidirect
10. #10874C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#20732
C2×#18352
#152352
#113842
#113722
#112682
#111892
#109252
#108742
#107882