Finite Group Databasebeta

← #455468

Subgroups — #455468

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2066C212yesmaximal, complemented
3C2×#2026C211yesmaximal, characteristic, complemented
4C2×#1798C211yesmaximal, characteristic
5#14258C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11228C212yesmaximal, complemented
9#11095C212yesmaximal
10#11050C211yesmaximal, characteristic, complemented
11#10970C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#106C2212yes—
16C22×C4⋊C8C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20#2629—28no—
21C2×#398—22no—
22C2×#396—22no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29#2066—28no—
30#2026—24no—
31#1798—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1441—22no—
40#1440—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#125—24no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—42no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—42no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8Q814yes—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—22no—
58C2×#117—42no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D412yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#106—24no—
76C2×#106—42no—
77C2×#106—42no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—42no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—42no—
90#398—216no—
91#396—216no—
92#391—24no—
93#391—22no—
94#391—22no—
95#389—24no—
96#389D418yes—
97#384—24no—
98#380—24no—
99#380—24no—
100#380—42no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#336—22no—
114#336D414yes—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8C2×Q812yes—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×Q812yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
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×C8—22no—
158C22×C8—22no—
159C22×C8C2×D414yes—
160#125—48no—
161#125—48no—
162C22⋊Q8—48no—
163C22⋊Q8—48no—
164#117—48no—
165#117—48no—
166C2×C4⋊C4—22no—
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—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×Q812yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C2×D414yes—
222C2×C42C2×D412yes—
223C2×C42#4112yes—
224#106—48no—
225C4⋊C8—48no—
226#103—48no—
227#103—48no—
228C22⋊C8—48no—
229#95—28no—
230#95—28no—
231#95—42no—
232#95—44no—
233#95—44no—
234#95—82no—
235#95—82no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24C22⋊Q812yes—
247C24C22×D411yescharacteristic
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
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—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4#12014yes—
335C22×C4#12114yes—
336C22×C4C22⋊Q812yes—
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C42−1+411yescharacteristic
346C22×C42−1+412yes—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—416no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—88no—
379C4⋊C4—88no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—88no—
383C22⋊C4—84no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—24no—
388C23—24no—
389C23—24no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23C2×#12011yescharacteristic
412C23C2×#12111yescharacteristic
413C23C2×C22⋊Q811yescharacteristic
414C23#53411yescharacteristic
415C23#53511yescharacteristic
416C23#53611yescharacteristic
417C23#53711yescharacteristic
418C23#57712yes—
419C23#57712yes—
420C23#57712yes—
421C23#57812yes—
422Q8—44no—
423Q8—44no—
424Q8—82no—
425Q8—82no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#53512yes—
488C2×C4#53712yes—
489C2×C4#53712yes—
490C2×C4#53812yes—
491C2×C4#54212yes—
492C2×C4#54312yes—
493C2×C4D4212yes—
494C2×C4#54611yescharacteristic
495C2×C4#54612yes—
496C2×C4#54812yes—
497C2×C4D4×Q811yescharacteristic
498C2×C4D4×Q812yes—
499C2×C4#55211yescharacteristic
500C2×C4#55411yescharacteristic
501C2×C4#56212yes—
502C2×C4#56312yes—
503C8—44no—
504C8—82no—
505C22—42no—
506C22—42no—
507C22—44no—
508C22—42no—
509C22—44no—
510C22—82no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22#240911yescharacteristic
517C22#241011yescharacteristic
518C22#241711yescharacteristic
519C22#261511yescharacteristic
520C22#262011yescharacteristic
521C22#262611yescharacteristic
522C22#281411yescharacteristic
523C22#300912yes—
524C22#301312yes—
525C22#306012yes—
526C22#306312yes—
527C22#308212yes—
528C22#308412yes—
529C22#320112yes—
530C22#320412yes—
531C22#322111yescharacteristic
532C22#322111yescharacteristic
533C22#322211yescharacteristic
534C22#322311yescharacteristic
535C22#324212yes—
536C22#324412yes—
537C22#325812yes—
538C22#326012yes—
539C4—44no—
540C4—44no—
541C4—42no—
542C4—82no—
543C4—84no—
544C4—82no—
545C4—84no—
546C4—161no—
547C4—161no—
548C4#267612yes—
549C4#268312yes—
550C4#269212yes—
551C4#270212yes—
552C2—82no—
553C2#1626511yescharacteristic
554C2#2431012yes—
555C2#2439512yes—
556C2#2485412yes—
557C2#2967311yescharacteristic
558C2#2967611yescharacteristic
559C2#2985011yescharacteristic
560C2#2985611yescharacteristic
561C2#2996311yescharacteristic
562C2#2996511yescharacteristic
563C2#3275512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2066C2semidirect
3. C2×#2026C2semidirect
6. #11385C2semidirect
7. #11323C2semidirect
8. #11228C2semidirect
10. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#20662
C2×#20262
C2×#17982
#142582
#113852
#113232
#112282
#110952
#110502
#109702