Finite Group Databasebeta

← #405872

Subgroups — #405872

Order 512.

1376 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C211yesmaximal, characteristic
2C2×#1736C211yesmaximal, characteristic
3#13951C211yesmaximal, characteristic
4#13880C211yesmaximal, characteristic
5#11357C212yesmaximal, complemented
6#11257C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#385—22no—
20C2×#384C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1854—24no—
25#1736—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1444—21no—
37#1444—21no—
38#1439—22no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1434—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—42no—
48C22×M16—42no—
49C22×M16Q814yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×#117—22no—
54C2×#117—24no—
55C2×#117—42no—
56C2×#117D414yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2312yes—
77C2×#107—22no—
78C2×#107—24no—
79C2×#106—22no—
80C2×#106—42no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8Q814yes—
84#389—24no—
85#389—42no—
86#385—48no—
87#384—24no—
88#384—42no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104#340—22no—
105#340—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—42no—
113#336—42no—
114C25C2×D411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—44no—
126C23×C4—42no—
127C23×C4—44no—
128C23×C4—81no—
129C23×C4—81no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×Q812yes—
135C23×C4C2×Q812yes—
136C23×C4#4112yes—
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4C2411yescharacteristic
140C2×M16—28no—
141C2×M16—48no—
142C2×M16—44no—
143C2×M16—44no—
144C2×M16—48no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8C2×Q814yes—
160C22×C8C2×Q812yes—
161C22×C8C2×Q812yes—
162#117—28no—
163#117—44no—
164#117—48no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—28no—
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—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
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—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4#4114yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—81no—
216C2×C22⋊C4—81no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—44no—
225#107—44no—
226#106—44no—
227C22⋊C8—28no—
228#95—22no—
229#95—22no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—42no—
234#95—44no—
235#95—44no—
236#95—82no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24#12112yes—
246C24#12312yes—
247C24C4⋊Q812yes—
248C24C22×D411yescharacteristic
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
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—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—48no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
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—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—88no—
313C22×C4—88no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4#12112yes—
317C22×C4C22⋊Q814yes—
318C22×C4C22⋊Q814yes—
319C22×C4#12312yes—
320C22×C4#12514yes—
321C22×C4#12514yes—
322C22×C4C4⋊Q812yes—
323C22×C4AGL(1,ℤ/8)14yes—
324C22×C4#13714yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q811yescharacteristic
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4112yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C22×C4H(5,𝔽2)12yes—
335M16—416no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—88no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C4⋊C4#12314yes—
377C22⋊C4—216no—
378C22⋊C4—48no—
379C22⋊C4—84no—
380C22⋊C4—84no—
381C42—48no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C23—24no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—24no—
392C23—22no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23#39814yes—
409C23C2×#12111yescharacteristic
410C23C2×#12311yescharacteristic
411C23C2×C4⋊Q811yescharacteristic
412C23#54612yes—
413C23#54612yes—
414C23C2×AGL(1,ℤ/8)12yes—
415C23C2×#13712yes—
416C23#57611yescharacteristic
417C23#57711yescharacteristic
418C23#57711yescharacteristic
419C23#57811yescharacteristic
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—816no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4#39814yes—
480C2×C4C2×C22⋊Q811yescharacteristic
481C2×C4C2×C22⋊Q811yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4#52511yescharacteristic
484C2×C4C2×#12511yescharacteristic
485C2×C4C2×#12511yescharacteristic
486C2×C4#52912yes—
487C2×C4#53312yes—
488C2×C4#53411yescharacteristic
489C2×C4#53511yescharacteristic
490C2×C4#53812yes—
491C2×C4#54312yes—
492C2×C4#55112yes—
493C2×C4#55112yes—
494C2×C4#55312yes—
495C2×C4#55312yes—
496C2×C4#55412yes—
497C2×C4#55412yes—
498C8—42no—
499C8—44no—
500C8—42no—
501C22—24no—
502C22—24no—
503C22—24no—
504C22—44no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22C2×#39811yescharacteristic
513C22#241111yescharacteristic
514C22#241211yescharacteristic
515C22#244412yes—
516C22#254912yes—
517C22#303811yescharacteristic
518C22#303911yescharacteristic
519C22#304511yescharacteristic
520C22#304611yescharacteristic
521C22#309212yes—
522C22#309611yescharacteristic
523C22#309711yescharacteristic
524C22#316311yescharacteristic
525C22#316411yescharacteristic
526C22#316512yes—
527C22#330612yes—
528C22#331212yes—
529C22#331612yes—
530C22#331812yes—
531C22#332412yes—
532C22#332912yes—
533C22#333012yes—
534C22#333312yes—
535C4—24no—
536C4—44no—
537C4—44no—
538C4—44no—
539C4—84no—
540C4—82no—
541C4—162no—
542C4—161no—
543C4—161no—
544C4#248711yescharacteristic
545C4#250512yes—
546C4#256912yes—
547C4#257911yescharacteristic
548C4#262611yescharacteristic
549C4#285511yescharacteristic
550C2—44no—
551C2#1492011yescharacteristic
552C2#2365811yescharacteristic
553C2#2366211yescharacteristic
554C2#2449211yescharacteristic
555C2#2449811yescharacteristic
556C2#2454411yescharacteristic
557C2#2454611yescharacteristic
558C2#2621412yes—
559C2#2622012yes—
560C2#2878112yes—
561C2#2878212yes—
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. #11257C2semidirect
7. #11219C2semidirect
8. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
C2×#17362
#139512
#138802
#113572
#112572
#112192
#112072
#107302
#107222
#107152