Finite Group Databasebeta

← #405831

Subgroups — #405831

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1825C211yesmaximal, characteristic
2C2×#1736C211yesmaximal, characteristic
3#13939C211yesmaximal, characteristic
4#13880C211yesmaximal, characteristic
5#11357C212yesmaximal, complemented
6#11316C212yesmaximal, complemented
7#11315C212yesmaximal, complemented
8#11267C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C2×#385—22no—
17C2×#384C2212yes—
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21#1825—24no—
22#1736—24no—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1444—21no—
36#1444—21no—
37#1440—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—21no—
42#1434—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—24no—
49C22×M16—42no—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117D414yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D412yes—
76C22×C42—21no—
77C22×C42C2312yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81#385—48no—
82#384—24no—
83#384—42no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100#340—24no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—42no—
109#340—42no—
110#336—24no—
111#336—42no—
112C25C2×D411yescharacteristic
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—44no—
124C23×C4—42no—
125C23×C4—44no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4#4112yes—
137C23×C4C2411yescharacteristic
138C2×M16—28no—
139C2×M16—48no—
140C2×M16—48no—
141C2×M16—44no—
142C2×M16—44no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8#4114yes—
160#117—28no—
161#117—28no—
162#117—44no—
163#117—44no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—81no—
217C2×C22⋊C4—81no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—42no—
226C2×C42—42no—
227C22⋊C8—44no—
228#95—22no—
229#95—22no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—82no—
237#95—82no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24#12112yes—
247C24#12312yes—
248C24#12412yes—
249C24C22×D411yescharacteristic
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
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—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—88no—
316C22×C4—88no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4#12112yes—
320C22×C4#12314yes—
321C22×C4#12314yes—
322C22×C4#12312yes—
323C22×C4#12412yes—
324C22×C4#12614yes—
325C22×C4#12614yes—
326C22×C4AGL(1,ℤ/8)14yes—
327C22×C4#13714yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4C2×#4111yescharacteristic
336C22×C4H(5,𝔽2)12yes—
337C22×C42−1+412yes—
338M16—416no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—48no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—88no—
380C4⋊C4—88no—
381C22⋊C4—216no—
382C22⋊C4—48no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—48no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23#39714yes—
413C23C2×#12111yescharacteristic
414C23C2×#12311yescharacteristic
415C23C2×#12411yescharacteristic
416C23#54612yes—
417C23#54712yes—
418C23C2×AGL(1,ℤ/8)12yes—
419C23C2×#13712yes—
420C23#57611yescharacteristic
421C23#57711yescharacteristic
422C23#57711yescharacteristic
423C23#57811yescharacteristic
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—816no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
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—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#39714yes—
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4C2×#12611yescharacteristic
491C2×C4C2×#12611yescharacteristic
492C2×C4#52912yes—
493C2×C4#52912yes—
494C2×C4#53511yescharacteristic
495C2×C4#53611yescharacteristic
496C2×C4#53912yes—
497C2×C4#53912yes—
498C2×C4#55112yes—
499C2×C4#55112yes—
500C2×C4#55112yes—
501C2×C4#55212yes—
502C2×C4#55212yes—
503C2×C4#55212yes—
504C8—44no—
505C8—81no—
506C8—81no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—42no—
511C22—42no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22C2×#39711yescharacteristic
519C22#241111yescharacteristic
520C22#241311yescharacteristic
521C22#241912yes—
522C22#248812yes—
523C22#303811yescharacteristic
524C22#303911yescharacteristic
525C22#304511yescharacteristic
526C22#304611yescharacteristic
527C22#309212yes—
528C22#309811yescharacteristic
529C22#309911yescharacteristic
530C22#313611yescharacteristic
531C22#313711yescharacteristic
532C22#314112yes—
533C22#331812yes—
534C22#332012yes—
535C22#332412yes—
536C22#332812yes—
537C22#334312yes—
538C22#334412yes—
539C22#335412yes—
540C22#335712yes—
541C4—24no—
542C4—44no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—84no—
547C4—161no—
548C4—162no—
549C4—161no—
550C4#248211yescharacteristic
551C4#248411yescharacteristic
552C4#249312yes—
553C4#249312yes—
554C4#262311yescharacteristic
555C4#285311yescharacteristic
556C2—44no—
557C2#1491911yescharacteristic
558C2#2363011yescharacteristic
559C2#2363411yescharacteristic
560C2#2449211yescharacteristic
561C2#2449611yescharacteristic
562C2#2459411yescharacteristic
563C2#2459811yescharacteristic
564C2#2495112yes—
565C2#2495512yes—
566C2#2770612yes—
567C2#2772012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #11357C2semidirect
6. #11316C2semidirect
7. #11315C2semidirect
8. #11267C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18252
C2×#17362
#139392
#138802
#113572
#113162
#113152
#112672
#107302
#107222
#107152