Finite Group Databasebeta

← #214086

Subgroups — #214086

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2#21742C211yesmaximal, characteristic, complemented
3#21738C211yesmaximal, characteristic, complemented
4#9085C212yesmaximal, complemented
5#9083C212yesmaximal, complemented
6#9081C212yesmaximal, complemented
7#9079C212yesmaximal, complemented
8#8701C212yesmaximal, complemented
9#8300C212yesmaximal, complemented
10C22×#121C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C2×#407C2212yes—
14C22×C22⋊C8C2212yes—
15#2324—24no—
16#2324—24no—
17C2×#390C411yescharacteristic
18C2×#390C411yescharacteristic
19C2×#390C411yescharacteristic
20C2×#389C411yescharacteristic
21C2×#388C411yescharacteristic
22C2×#388C411yescharacteristic
23C2×#386C2211yescharacteristic
24C2×#386C2211yescharacteristic
25C2×#385C2211yescharacteristic
26C2×#385C2211yescharacteristic
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1870—24no—
30#1869—24no—
31#1817C2212yes—
32#1817C2212yes—
33#1779C2212yes—
34#1779C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C23×D4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16—22no—
46C23×C8—22no—
47C2×#121—24no—
48C2×#121—24no—
49C2×#121—24no—
50C2×C4×D4—44no—
51C2×C4×D4D414yes—
52C2×C4×D4D414yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—44no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
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⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2311yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2311yescharacteristic
79#407—44no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—44no—
87C2×C22⋊C8C2314yes—
88C2×C22⋊C8C2314yes—
89#390—24no—
90#390—24no—
91#390—24no—
92#389—24no—
93#388—24no—
94#388—24no—
95#386—22no—
96#386—22no—
97#386—22no—
98#386—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#380—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2311yescharacteristic
111C2×#95C2311yescharacteristic
112#336—24no—
113#336—24no—
114#324—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—24no—
127C22×D4—42no—
128C22×D4—81no—
129C22×D4—81no—
130C22×D4—81no—
131C22×D4—81no—
132C22×D4C22⋊C414yes—
133C22×D4M1614yescomplemented
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—42no—
141C23×C4—81no—
142C23×C4—81no—
143C23×C4—81no—
144C23×C4—81no—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
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×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2411yescharacteristic
161C2×M16—44no—
162C2×M16—82no—
163C2×M16—82no—
164C22×C8—24no—
165C22×C8—24no—
166C22×C8—44no—
167C22×C8—82no—
168C22×C8—82no—
169#121—232no—
170C4×D4—44no—
171C4×D4—44no—
172C4×D4—44no—
173C4×D4—44no—
174#117—48no—
175#117—48no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—81no—
190C2×C4⋊C4—81no—
191C2×C4⋊C4C22⋊C414yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—81no—
211C2×C22⋊C4—81no—
212C2×C22⋊C4—81no—
213C2×C22⋊C4—81no—
214C2×C22⋊C4—81no—
215C2×C22⋊C4—81no—
216C2×C22⋊C4C22⋊C414yes—
217C2×C22⋊C4C22⋊C414yes—
218C2×C22⋊C4M1614yescomplemented
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—42no—
224C2×C42—81no—
225C2×C42—81no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C22⋊C8—44no—
233C22⋊C8—44no—
234C22⋊C8—82no—
235C22⋊C8—82no—
236C22⋊C8—82no—
237C22⋊C8—82no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24—22no—
247C24—22no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24C2×C22⋊C412yes—
269C24C2×C22⋊C412yes—
270C24C2×M1612yes—
271C24C2×M1612yes—
272C24C2×M1612yes—
273C24C23×C411yescharacteristic
274C2×D4—28no—
275C2×D4—28no—
276C2×D4—28no—
277C2×D4—28no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—82no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—84no—
286C2×D4—84no—
287C2×D4—84no—
288C2×D4—84no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—28no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—48no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—48no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—88no—
351C22×C4—88no—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×M1612yes—
357C22×C4C2×M1612yes—
358C22×C4C2×M1612yes—
359C22×C4#13114yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4H(5,𝔽2)14yes—
363M16—88no—
364C2×C8—44no—
365C2×C8—44no—
366C2×C8—44no—
367C2×C8—44no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—88no—
371C4⋊C4—28no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42—24no—
393C42—24no—
394C42—44no—
395C42—82no—
396C42—82no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23#40714yes—
452C23C22×C22⋊C411yescharacteristic
453C23#51812yes—
454C23C22×M1611yescharacteristic
455C23C22×M1611yescharacteristic
456C23C2×#13112yes—
457D4—416no—
458D4—88no—
459D4—88no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—416no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#40814yes—
520C2×C4#53412yes—
521C2×C4#53412yes—
522C8—84no—
523C8—84no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—84no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#230811yescharacteristic
560C22C2×#40711yescharacteristic
561C22C2×#40811yescharacteristic
562C22#288311yescharacteristic
563C22#288311yescharacteristic
564C22#288811yescharacteristic
565C22#288911yescharacteristic
566C22#297412yes—
567C22#297412yes—
568C4—24no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—81no—
581C2—22no—
582C2—22no—
583C2—44no—
584C2—82no—
585C2—82no—
586C2#1736711yescharacteristic
587C2#1736811yescharacteristic
588C2#1738811yescharacteristic
589C2#1738911yescharacteristic
590C2#2174011yescharacteristic
591C2#2175011yescharacteristic
592C2#2175711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21742C2semidirect
3. #21738C2semidirect
4. #9085C2semidirect
5. #9083C2semidirect
6. #9081C2semidirect
7. #9079C2semidirect
8. #8701C2semidirect
9. #8300C2semidirect
133. C22×D4M16semidirect
218. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23242
#217422
#217382
#90852
#90832
#90812
#90792
#87012
#83002