Finite Group Databasebeta

← #416751

Subgroups — #416751

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C211yesmaximal, characteristic
2C2×#1947C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4#13911C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11384C212yesmaximal
7#11219C212yesmaximal
8#11219C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2550—24no—
18#2550—24no—
19C2×#398—21no—
20C2×#395—21no—
21C2×#391—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382—21no—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1947—24no—
32#1836—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1442—21no—
43#1442—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—42no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—24no—
57C2×#125—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—24no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8Q814yescomplemented
84C2×#97—22no—
85C2×#97—22no—
86C2×#97Q814yescomplemented
87#398—28no—
88#395—28no—
89#391—28no—
90#389—28no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#384—22no—
96#384—22no—
97#382—22no—
98#382—44no—
99#382D414yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C4×C4⋊C4Q814yescomplemented
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—42no—
126#336—42no—
127C22×Q8—24no—
128C22×Q8—44no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×Q814yes—
158C22×C8C2×Q814yes—
159C4⋊Q8—48no—
160C4⋊Q8—44no—
161C4⋊Q8—44no—
162#125—48no—
163#125—44no—
164#125—44no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
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—24no—
178C2×C4⋊C4—24no—
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—21no—
186C2×C4⋊C4—21no—
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—24no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q814yes—
213C2×C4⋊C4#4114yes—
214C2×C4⋊C4#4114yes—
215C2×C42—24no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—82no—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×Q812yes—
231C2×C42C2×Q812yes—
232C2×C42C2×Q812yes—
233C2×C42#4112yes—
234#107—48no—
235C4⋊C8—28no—
236#97—28no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95—22no—
244#95—24no—
245#95—44no—
246#95—44no—
247#95—82no—
248#95—82no—
249C24C22×D411yescharacteristic
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8—88no—
253C2×Q8—84no—
254C2×Q8—88no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4#12012yes—
314C22×C4#12012yes—
315C22×C4C22⋊Q812yes—
316C22×C4C22⋊Q812yes—
317C22×C4C4⋊Q812yes—
318C22×C4C4⋊Q812yes—
319C22×C4AGL(1,ℤ/8)12yes—
320C22×C4#13712yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×Q811yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C22×Q811yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4112yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4#12414yes—
373C4⋊C4#12414yes—
374C42—48no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—42no—
380C42—42no—
381C42—84no—
382C42C22⋊Q814yes—
383C42C22⋊Q814yes—
384C42C4⋊Q814yes—
385C42C4⋊Q814yes—
386C23C2×#12011yescharacteristic
387C23C2×C22⋊Q811yescharacteristic
388C23C2×C4⋊Q811yescharacteristic
389C23#54612yes—
390C23#54812yes—
391C23C2×AGL(1,ℤ/8)11yescharacteristic
392C23C2×#13711yescharacteristic
393C23#57511yescharacteristic
394C23#57511yescharacteristic
395C23#57612yes—
396C23#57812yes—
397Q8—816no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#39514yes—
463C2×C4#39514yes—
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4C2×#12411yescharacteristic
467C2×C4C2×#12411yescharacteristic
468C2×C4C2×C4⋊Q811yescharacteristic
469C2×C4C2×C4⋊Q811yescharacteristic
470C2×C4#53312yes—
471C2×C4#53312yes—
472C2×C4#53511yescharacteristic
473C2×C4#53611yescharacteristic
474C2×C4#54412yes—
475C2×C4#54412yes—
476C2×C4#54612yes—
477C2×C4#54812yes—
478C2×C4#55412yes—
479C2×C4#55512yes—
480C2×C4#55612yes—
481C2×C4#55712yes—
482C8—44no—
483C8—44no—
484C22C2×#39511yescharacteristic
485C22#243511yescharacteristic
486C22#243611yescharacteristic
487C22#243912yes—
488C22#255012yes—
489C22#300611yescharacteristic
490C22#300711yescharacteristic
491C22#300811yescharacteristic
492C22#301111yescharacteristic
493C22#307711yescharacteristic
494C22#307811yescharacteristic
495C22#308312yes—
496C22#316412yes—
497C22#316511yescharacteristic
498C22#316611yescharacteristic
499C22#330812yes—
500C22#331312yes—
501C22#331412yes—
502C22#331912yes—
503C22#336612yes—
504C22#337012yes—
505C22#337412yes—
506C22#337712yes—
507C4—24no—
508C4—24no—
509C4—42no—
510C4—42no—
511C4—44no—
512C4—44no—
513C4—81no—
514C4—82no—
515C4—81no—
516C4—82no—
517C4—84no—
518C4—162no—
519C4#243912yes—
520C4#255012yes—
521C4#256311yescharacteristic
522C4#256811yescharacteristic
523C4#266011yescharacteristic
524C4#286211yescharacteristic
525C2#1523011yescharacteristic
526C2#2355411yescharacteristic
527C2#2355511yescharacteristic
528C2#2589611yescharacteristic
529C2#2589711yescharacteristic
530C2#2594311yescharacteristic
531C2#2594611yescharacteristic
532C2#2603312yes—
533C2#2603412yes—
534C2#2880412yes—
535C2#2880612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
83. C2×C4⋊C8Q8semidirect
86. C2×#97Q8semidirect
105. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
C2×#19472
C2×#18362
#139112
#113852
#113842
#112192
#108502
#108352
#107032