Finite Group Databasebeta

← #403726

Subgroups — #403726

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2370C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4#15128C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11328C212yesmaximal
7#11228C211yesmaximal, characteristic
8#11202C211yesmaximal, characteristic
9#10907C212yesmaximal
10#10884C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C2×#395—21no—
17#2370—24no—
18#2370—24no—
19C2×#389—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#387—21no—
24C2×#387—21no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2057—24no—
31#1987—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1444—21no—
44#1441—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125Q814yescomplemented
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—42no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—42no—
85#395—44no—
86#389—28no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—24no—
91#389—42no—
92#389D414yes—
93#387—28no—
94#387—28no—
95C4×C4⋊C4—28no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D414yes—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340Q818yescomplemented
123#336—22no—
124#336—24no—
125#336—42no—
126C22×Q8—44no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—82no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
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—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C2×Q814yes—
158C4⋊Q8—28no—
159C4⋊Q8—48no—
160C4⋊Q8—44no—
161#125—28no—
162#125—28no—
163#125—28no—
164#125—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
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—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
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⋊C4C2×Q812yes—
214C2×C4⋊C4C2×Q812yes—
215C2×C4⋊C4C2×Q814yes—
216C2×C4⋊C4#4114yes—
217C2×C4⋊C4#4112yes—
218C2×C4⋊C4#4112yes—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—82no—
235C2×C42—82no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240#107—48no—
241C4⋊C8—44no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—22no—
248#95—22no—
249#95—44no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—82no—
254C24C22×D411yescharacteristic
255C2×Q8—48no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—88no—
259C2×Q8—88no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
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—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4#12012yes—
322C22×C4#12012yes—
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4#13512yes—
326C22×C4AGL(1,ℤ/8)12yes—
327C22×C4#13712yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C22×Q812yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C42−1+412yes—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—216no—
353C4⋊C4—216no—
354C4⋊C4—22no—
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—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—84no—
378C4⋊C4—88no—
379C4⋊C4C22⋊Q814yes—
380C4⋊C4C22⋊Q814yes—
381C4⋊C4C4⋊Q814yes—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—48no—
393C42—84no—
394C42—84no—
395C23C2×#12011yescharacteristic
396C23C2×C22⋊Q811yescharacteristic
397C23#52511yescharacteristic
398C23#54612yes—
399C23D4×Q812yes—
400C23C2×#13511yescharacteristic
401C23C2×AGL(1,ℤ/8)11yescharacteristic
402C23C2×#13711yescharacteristic
403C23#57511yescharacteristic
404C23#57612yes—
405C23#57812yes—
406Q8—816no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
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#45214yes—
475C2×C4C2×C22⋊Q811yescharacteristic
476C2×C4C2×C22⋊Q811yescharacteristic
477C2×C4#52511yescharacteristic
478C2×C4#52511yescharacteristic
479C2×C4#52511yescharacteristic
480C2×C4C2×C4⋊Q811yescharacteristic
481C2×C4#53312yes—
482C2×C4#53312yes—
483C2×C4#53511yescharacteristic
484C2×C4#53611yescharacteristic
485C2×C4#54212yes—
486C2×C4#54212yes—
487C2×C4#54612yes—
488C2×C4D4×Q812yes—
489C2×C4#55412yes—
490C2×C4#55612yes—
491C2×C4#55712yes—
492C2×C4Q8212yes—
493C8—44no—
494C8—82no—
495C22#240411yescharacteristic
496C22#243411yescharacteristic
497C22#245112yes—
498C22#246512yes—
499C22#284811yescharacteristic
500C22C2×#45211yescharacteristic
501C22#300611yescharacteristic
502C22#300811yescharacteristic
503C22#301111yescharacteristic
504C22#307611yescharacteristic
505C22#307911yescharacteristic
506C22#308312yes—
507C22#310611yescharacteristic
508C22#311611yescharacteristic
509C22#312412yes—
510C22#330012yes—
511C22#331212yes—
512C22#331812yes—
513C22#332212yes—
514C22#338012yes—
515C22#338512yes—
516C22#338612yes—
517C22#339012yes—
518C4—24no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—44no—
523C4—42no—
524C4—44no—
525C4—44no—
526C4—84no—
527C4—81no—
528C4—82no—
529C4—84no—
530C4#239911yescharacteristic
531C4#245112yes—
532C4#246512yes—
533C4#256711yescharacteristic
534C4#258111yescharacteristic
535C4#263211yescharacteristic
536C2#1488111yescharacteristic
537C2#2396711yescharacteristic
538C2#2397311yescharacteristic
539C2#2583911yescharacteristic
540C2#2584911yescharacteristic
541C2#2645112yes—
542C2#2645612yes—
543C2#2701112yes—
544C2#2702812yes—
545C2#3295811yescharacteristic
546C2#3296311yescharacteristic
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
59. C2×#125Q8semidirect
122. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23702
C2×#20572
C2×#19872
#151282
#113852
#113282
#112282
#112022
#109072
#108842
#108062