Finite Group Databasebeta

← #405506

Subgroups — #405506

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3#13795C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic, complemented
5#11385C211yesmaximal, characteristic, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11220C212yesmaximal, complemented
9#11219C212yesmaximal, complemented
10#10730C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2212yes—
18C22×C22⋊C8C2212yes—
19#2629—28no—
20C2×#398—22no—
21C2×#396—22no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#380C2212yes—
25#1854—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C4⋊#95—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C22⋊#95—22no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—24no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—42no—
49C2×#117—24no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—24no—
55C2×#117—42no—
56C2×#117—42no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—42no—
63C22×C4⋊C4—42no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4Q814yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D412yes—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—42no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8Q818yes—
87#398—216no—
88#396—216no—
89#389—24no—
90#389—24no—
91#384—24no—
92#380—24no—
93#380D418yes—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#336—22no—
107#336—22no—
108#336—42no—
109C25C2×D411yescharacteristic
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—24no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D414yes—
131C23×C4C2×Q814yes—
132C23×C4C2×Q812yes—
133C23×C4#4112yes—
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8C2×Q814yes—
148C22×C8C2×Q814yes—
149#125—48no—
150#125—48no—
151#117—48no—
152#117—48no—
153#117—48no—
154#117—48no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×Q812yes—
197C2×C4⋊C4C2×Q812yes—
198C2×C4⋊C4#4112yes—
199C2×C4⋊C4#4112yes—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42#4112yes—
219C2×C42#4112yes—
220#107—48no—
221#106—48no—
222C22⋊C8—216no—
223#95—28no—
224#95—28no—
225#95—82no—
226#95—82no—
227#95—82no—
228#95—82no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24#12012yes—
238C24C22⋊Q812yes—
239C24C4⋊Q812yes—
240C24C22×D411yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
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—48no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
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—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—88no—
308C22×C4—88no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4#12012yes—
312C22×C4C22⋊Q818yes—
313C22×C4C22⋊Q812yes—
314C22×C4#12314yes—
315C22×C4C4⋊Q818yes—
316C22×C4C4⋊Q812yes—
317C22×C4C22×D412yes—
318C22×C4C22×Q812yes—
319C22×C4C22×Q811yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)12yes—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—416no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—88no—
365C42—24no—
366C42—24no—
367C42—48no—
368C42—42no—
369C42—42no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—24no—
381C23—22no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23#39514yes—
393C23C2×#12011yescharacteristic
394C23C2×C22⋊Q811yescharacteristic
395C23C2×#12311yescharacteristic
396C23#52511yescharacteristic
397C23C2×C4⋊Q811yescharacteristic
398C23#53411yescharacteristic
399C23#53511yescharacteristic
400C23#57612yes—
401C23#57712yes—
402C23#57712yes—
403C23#57812yes—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—816no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#39514yes—
464C2×C4C2×C22⋊Q812yes—
465C2×C4#52911yescharacteristic
466C2×C4#52911yescharacteristic
467C2×C4C2×C4⋊Q812yes—
468C2×C4#53311yescharacteristic
469C2×C4#53311yescharacteristic
470C2×C4#53812yes—
471C2×C4#54312yes—
472C2×C4#54612yes—
473C2×C4#54612yes—
474C2×C4#54612yes—
475C2×C4#54612yes—
476C2×C4#55412yes—
477C2×C4#55412yes—
478C2×C4#55412yes—
479C2×C4#55412yes—
480C8—44no—
481C8—44no—
482C22—28no—
483C22—24no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—42no—
490C22—44no—
491C22—42no—
492C22—44no—
493C22C2×#39511yescharacteristic
494C22#240411yescharacteristic
495C22#240511yescharacteristic
496C22#250211yescharacteristic
497C22#258311yescharacteristic
498C22#262611yescharacteristic
499C22#285511yescharacteristic
500C22#301212yes—
501C22#301312yes—
502C22#308112yes—
503C22#308212yes—
504C22#309812yes—
505C22#309912yes—
506C22#311912yes—
507C22#312012yes—
508C22#316312yes—
509C22#316412yes—
510C22#319812yes—
511C22#320112yes—
512C22#322011yescharacteristic
513C22#322111yescharacteristic
514C22#322211yescharacteristic
515C22#322311yescharacteristic
516C4—24no—
517C4—44no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4—84no—
522C4—82no—
523C4—82no—
524C4—84no—
525C4#245012yes—
526C4#245012yes—
527C4#254912yes—
528C4#254912yes—
529C2—44no—
530C2#1491411yescharacteristic
531C2#2355712yes—
532C2#2394111yescharacteristic
533C2#2394411yescharacteristic
534C2#2401712yes—
535C2#2795711yescharacteristic
536C2#2795811yescharacteristic
537C2#2930912yes—
538C2#2996211yescharacteristic
539C2#2996311yescharacteristic
540C2#3315812yes—
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
4. #11385C2semidirect
5. #11385C2semidirect
6. #11324C2semidirect
7. #11323C2semidirect
8. #11220C2semidirect
9. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#18542
#137952
#113852
#113242
#113232
#112202
#112192
#107302
#107132