Finite Group Databasebeta

← #215509

Subgroups — #215509

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2#21916C211yesmaximal, characteristic, complemented
3#21912C211yesmaximal, characteristic, complemented
4#9653C212yesmaximal, complemented
5#9652C212yesmaximal, complemented
6#9651C212yesmaximal, complemented
7#9649C212yesmaximal, complemented
8#8302C212yesmaximal, complemented
9#8300C212yesmaximal, complemented
10C22×#121C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13#2324—24no—
14#2324—24no—
15C2×#390C411yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#389C411yescharacteristic
19C2×#388C2211yescharacteristic
20C2×#388C2211yescharacteristic
21C2×#386C411yescharacteristic
22C2×#386C411yescharacteristic
23C2×#385C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#380C411yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1917—24no—
28#1916—24no—
29#1817C2212yes—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37C2×#323C2214yes—
38C2×#323C2214yes—
39C23×D4C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C2×#121—24no—
46C2×#121—24no—
47C2×#121—24no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—44no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—44no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C22×C22⋊C4C2311yescharacteristic
72C22×C22⋊C4C2311yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C22⋊C8C2314yes—
80C2×C22⋊C8C2314yes—
81C2×#97—22no—
82C2×C4×C8—22no—
83#390—24no—
84#390—22no—
85#390—22no—
86#390—22no—
87#390—22no—
88#389—24no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#386—24no—
94#386—24no—
95#385—24no—
96#385—24no—
97#380—24no—
98C4×C4⋊C4D414yes—
99C4×C4⋊C4D414yes—
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106#336—22no—
107#336—22no—
108#336—24no—
109#336—24no—
110#336—22no—
111#336—22no—
112#324—44no—
113#323—44no—
114#323—44no—
115#323—44no—
116C25C22×C411yescharacteristic
117C25C22×C411yescharacteristic
118C25C22×C411yescharacteristic
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—24no—
128C22×D4—42no—
129C22×D4—81no—
130C22×D4—81no—
131C22×D4—81no—
132C22×D4—81no—
133C22×D4M1614yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—81no—
144C23×C4—81no—
145C23×C4—81no—
146C23×C4—81no—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—24no—
160C22×C8—24no—
161#121—232no—
162C4×D4—48no—
163C4×D4—48no—
164#117—48no—
165#117—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4—81no—
180C2×C4⋊C4C22⋊C414yes—
181C2×C4⋊C4C22⋊C414yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—81no—
205C2×C22⋊C4—81no—
206C2×C22⋊C4—81no—
207C2×C22⋊C4—81no—
208C2×C22⋊C4—81no—
209C2×C22⋊C4—81no—
210C2×C22⋊C4M1614yes—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—81no—
215C2×C42—81no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C4⋊C8—82no—
223C4⋊C8—82no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227C22⋊C8—24no—
228#97—82no—
229C4×C8—82no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95C22⋊C414yes—
235#95C22⋊C414yes—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24C2×M1612yes—
261C24C2×M1612yes—
262C24C2×M1612yes—
263C24C23×C411yescharacteristic
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—28no—
267C2×D4—48no—
268C2×D4—48no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4—84no—
276C2×D4—84no—
277C2×D4—84no—
278C2×D4—84no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—28no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—48no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—48no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—88no—
331C22×C4—88no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×M1612yes—
339C22×C4C2×M1612yes—
340C22×C4C2×M1612yes—
341C22×C4#13114yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4H(5,𝔽2)14yes—
345C2×C8—44no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—44no—
349C2×C8—44no—
350C2×C8—44no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C22⋊C4—28no—
360C22⋊C4—28no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C42—24no—
375C42—24no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—44no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23C22×C22⋊C411yescharacteristic
437C23#51812yes—
438C23C22×M1611yescharacteristic
439C23C22×M1611yescharacteristic
440C23C2×#13112yes—
441D4—416no—
442D4—88no—
443D4—88no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—416no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—88no—
493C2×C4—88no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4#40714yes—
499C2×C4#40814yes—
500C2×C4#53512yes—
501C2×C4#53512yes—
502C8—84no—
503C8—84no—
504C22—22no—
505C22—22no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—22no—
511C22—22no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—84no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22#230911yescharacteristic
540C22C2×#40711yescharacteristic
541C22C2×#40811yescharacteristic
542C22#288311yescharacteristic
543C22#288411yescharacteristic
544C22#288811yescharacteristic
545C22#288811yescharacteristic
546C22#297412yes—
547C22#297412yes—
548C4—24no—
549C4—22no—
550C4—22no—
551C4—24no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C2—22no—
562C2—22no—
563C2—82no—
564C2—82no—
565C2—82no—
566C2#1736711yescharacteristic
567C2#1737211yescharacteristic
568C2#1738911yescharacteristic
569C2#1739011yescharacteristic
570C2#2178811yescharacteristic
571C2#2179611yescharacteristic
572C2#2180111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21916C2semidirect
3. #21912C2semidirect
4. #9653C2semidirect
5. #9652C2semidirect
6. #9651C2semidirect
7. #9649C2semidirect
8. #8302C2semidirect
9. #8300C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23242
#219162
#219122
#96532
#96522
#96512
#96492
#83022
#83002