Finite Group Databasebeta

← #440952

Subgroups — #440952

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2453C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4#14155C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11346C211yesmaximal, characteristic
7#11219C212yesmaximal
8#11124C212yesmaximal
9#10932C211yesmaximal, characteristic
10#10884C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18#2453—24no—
19#2453—24no—
20C2×#400—21no—
21C2×#398—21no—
22C2×#391—21no—
23C2×#391—21no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389—21no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#387—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2057—24no—
33#2024—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—24no—
58C2×#125—42no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—42no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103Q818yescomplemented
90#400—28no—
91#398—28no—
92#391—22no—
93#391—22no—
94#391—22no—
95#391—44no—
96#391—44no—
97#391D414yes—
98#389—28no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389D414yes—
103#387—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—42no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120#340—24no—
121#340—24no—
122#340—24no—
123#340—42no—
124#340—42no—
125#336—22no—
126#336—22no—
127#336—42no—
128#336—42no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8C2×D412yes—
143C22×Q8C2×Q814yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8C2×Q814yes—
171C4⋊Q8—48no—
172C4⋊Q8—48no—
173#125—48no—
174#125—44no—
175#125—44no—
176C4×Q8—48no—
177C4×Q8—44no—
178C4×Q8—44no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
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—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C4⋊C4#4114yes—
223C2×C4⋊C4#4114yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241#107—48no—
242#103—48no—
243#103—48no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—42no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257C24C22×D411yescharacteristic
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—48no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8C4⋊Q814yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4#12012yes—
345C22×C4#12012yes—
346C22×C4#13512yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4AGL(1,ℤ/8)12yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×Q812yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4112yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—44no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4#12414yes—
395C42—24no—
396C42—24no—
397C42—48no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—42no—
403C42—42no—
404C42—84no—
405C23C2×#12011yescharacteristic
406C23#52511yescharacteristic
407C23#53511yescharacteristic
408C23#54612yes—
409C23D4×Q812yes—
410C23C2×#13511yescharacteristic
411C23C2×AGL(1,ℤ/8)11yescharacteristic
412C23C2×AGL(1,ℤ/8)11yescharacteristic
413C23#57511yescharacteristic
414C23#57812yes—
415C23#57812yes—
416Q8—28no—
417Q8—44no—
418Q8—44no—
419Q8—44no—
420Q8—816no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4#44914yes—
489C2×C4C2×#12411yescharacteristic
490C2×C4C2×C4⋊Q811yescharacteristic
491C2×C4#53312yes—
492C2×C4#53511yescharacteristic
493C2×C4#53611yescharacteristic
494C2×C4#54212yes—
495C2×C4#54212yes—
496C2×C4#54312yes—
497C2×C4#54612yes—
498C2×C4#54612yes—
499C2×C4#54612yes—
500C2×C4D4×Q812yes—
501C2×C4#55412yes—
502C2×C4#55412yes—
503C2×C4#55612yes—
504C2×C4#56212yes—
505C8—44no—
506C8—82no—
507C22#240411yescharacteristic
508C22#243411yescharacteristic
509C22#243511yescharacteristic
510C22#246512yes—
511C22#269212yes—
512C22C2×#44911yescharacteristic
513C22#300611yescharacteristic
514C22#301111yescharacteristic
515C22#301111yescharacteristic
516C22#310511yescharacteristic
517C22#311411yescharacteristic
518C22#312712yes—
519C22#321111yescharacteristic
520C22#321711yescharacteristic
521C22#322312yes—
522C22#330212yes—
523C22#330812yes—
524C22#331312yes—
525C22#333012yes—
526C22#338312yes—
527C22#338612yes—
528C22#338612yes—
529C22#339012yes—
530C4—24no—
531C4—44no—
532C4—42no—
533C4—44no—
534C4—42no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—82no—
539C4—84no—
540C4—82no—
541C4—84no—
542C4#249212yes—
543C4#254912yes—
544C4#269212yes—
545C4#270012yes—
546C2#1567111yescharacteristic
547C2#2393611yescharacteristic
548C2#2394311yescharacteristic
549C2#2584011yescharacteristic
550C2#2584911yescharacteristic
551C2#2588211yescharacteristic
552C2#2589611yescharacteristic
553C2#2700112yes—
554C2#2701612yes—
555C2#3153312yes—
556C2#3154012yes—
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
89. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24532
C2×#20572
C2×#20242
#141552
#113852
#113462
#112192
#111242
#109322
#108842
#107892