Finite Group Databasebeta

← #369001

Subgroups — #369001

Order 512.

1432 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic
4C2×#1989C211yesmaximal, characteristic
5C2×#1985C211yesmaximal, characteristic
6#11208C212yesmaximal
7#11208C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16#2584—24no—
17#2584C418yescomplemented
18C2×#398—21no—
19C2×#395—21no—
20#2355—24no—
21#2355—24no—
22C2×#391—21no—
23C2×#389—22no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387—22no—
27C2×#384C2211yescharacteristic
28C2×#382C2212yes—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1992—24no—
34#1989—24no—
35#1985—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C23×Q8D411yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125C2×C414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8C2×C414yes—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D412yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C42—21no—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×#97—22no—
87C2×#97—22no—
88C2×#97—22no—
89#398—28no—
90#395—28no—
91#391—28no—
92#389—216no—
93#389—24no—
94#389—24no—
95#389C2×C418yes—
96#389C2×C418yes—
97#387—216no—
98#384—22no—
99#384D414yes—
100#382—24no—
101#382—24no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C414yes—
108C4×C4⋊C4D414yes—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340Q818yescomplemented
126#336—24no—
127#336—24no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8C22⋊C414yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C2×Q814yes—
161#125—28no—
162#125—28no—
163#125—44no—
164#125—44no—
165C4×Q8—28no—
166C4×Q8—28no—
167C4×Q8—28no—
168C4×Q8—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
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—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C22⋊C418yes—
214C2×C4⋊C4C22⋊C414yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C4⋊C4C2×Q814yes—
223C2×C4⋊C4#4114yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—28no—
230C2×C42—24no—
231C2×C42—44no—
232C2×C42—81no—
233C2×C42—81no—
234C2×C42—82no—
235C2×C42—82no—
236C2×C42C22×C414yes—
237C2×C42C22×C412yes—
238C2×C42C2×D414yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C4⋊C8—44no—
245#97—44no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—24no—
250#95—28no—
251#95—28no—
252#95—24no—
253C24C22×D411yescharacteristic
254C2×Q8—28no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—44no—
258C2×Q8—44no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
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—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
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—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C2×C22⋊C414yes—
314C22×C4C2×C22⋊C414yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4#12012yes—
318C22×C4#12012yes—
319C22×C4#12312yes—
320C22×C4#12312yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×Q812yes—
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4112yes—
329C22×C42−1+412yes—
330C22×C42−1+412yes—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—216no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—48no—
364C4⋊C4—416no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—84no—
375C4⋊C4—88no—
376C4⋊C4C4×Q818yes—
377C4⋊C4C4⋊Q814yes—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42—48no—
383C42—84no—
384C42—88no—
385C42C4×D414yes—
386C42C4×D414yes—
387C42#12114yes—
388C42#12114yes—
389C23C22×C22⋊C411yescharacteristic
390C23C2×#12011yescharacteristic
391C23C2×#12311yescharacteristic
392C23#54812yes—
393C23D4×Q812yes—
394C23#57512yes—
395C23#57512yes—
396C23#57611yescharacteristic
397C23#57711yescharacteristic
398C23#57711yescharacteristic
399C23#57811yescharacteristic
400Q8—416no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—28no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4#38614yes—
462C2×C4#38614yes—
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×C4×Q812yes—
466C2×C4#51912yes—
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×C4⋊Q811yescharacteristic
470C2×C4#53211yescharacteristic
471C2×C4#53312yes—
472C2×C4#53611yescharacteristic
473C2×C4#53611yescharacteristic
474C2×C4#54112yes—
475C2×C4#54712yes—
476C2×C4#54812yes—
477C2×C4D4×Q812yes—
478C2×C4D4×Q812yes—
479C2×C4#55612yes—
480C2×C4#55712yes—
481C8—44no—
482C8—81no—
483C8—81no—
484C22C2×#38611yescharacteristic
485C22Q8×C22⋊C412yes—
486C22#243211yescharacteristic
487C22#243411yescharacteristic
488C22#244612yes—
489C22#289612yes—
490C22#289711yescharacteristic
491C22#289811yescharacteristic
492C22#300511yescharacteristic
493C22#300511yescharacteristic
494C22#301211yescharacteristic
495C22#301311yescharacteristic
496C22#309312yes—
497C22#309811yescharacteristic
498C22#309911yescharacteristic
499C22#337012yes—
500C22#337112yes—
501C22#337612yes—
502C22#337712yes—
503C22#338812yes—
504C22#338812yes—
505C22#338912yes—
506C22#339012yes—
507C4—24no—
508C4—24no—
509C4—44no—
510C4—44no—
511C4—44no—
512C4—48no—
513C4—82no—
514C4—84no—
515C4—82no—
516C4—82no—
517C4C4×C4⋊Q811yescharacteristic
518C4#232711yescharacteristic
519C4#236512yes—
520C4#242912yes—
521C4#254511yescharacteristic
522C4#254611yescharacteristic
523C2#1402411yescharacteristic
524C2#2091011yescharacteristic
525C2#2091511yescharacteristic
526C2#2252612yes—
527C2#2252712yes—
528C2#2565611yescharacteristic
529C2#2565811yescharacteristic
530C2#2584511yescharacteristic
531C2#2584611yescharacteristic
532C2#2628912yes—
533C2#2629112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
17. #2584C4semidirect
125. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
C2×#23552
C2×#19922
C2×#19892
C2×#19852
#112082
#108852
#108842
#108492
#107032