Finite Group Databasebeta

← #213550

Subgroups — #213550

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2325C211yesmaximal, characteristic
2#21790C211yesmaximal, characteristic, complemented
3#21742C211yesmaximal, characteristic, complemented
4#9086C212yesmaximal, complemented
5#9085C212yesmaximal, complemented
6#9084C212yesmaximal, complemented
7#9083C212yesmaximal, complemented
8#8698C212yesmaximal, complemented
9#8301C212yesmaximal, complemented
10C22×#121C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#432C2212yes—
14C2×#407C2212yes—
15#2325—24no—
16#2325—24no—
17C2×#388C411yescharacteristic
18C2×#388C2211yescharacteristic
19C2×#387C411yescharacteristic
20C2×#387C2211yescharacteristic
21C2×#386C411yescharacteristic
22C2×#386C411yescharacteristic
23C2×#386C411yescharacteristic
24C2×#386C2211yescharacteristic
25C2×#385C411yescharacteristic
26C2×#385C2211yescharacteristic
27C2×#380C411yescharacteristic
28C22×#95C2211yescharacteristic
29#1955—24no—
30#1870—24no—
31#1817C2212yes—
32#1779C2212yes—
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×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C2×#121—24no—
49C2×#121—24no—
50C2×#121—24no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—44no—
54C2×C4×D4—44no—
55C2×C4×D4D414yes—
56C2×C4×D4D414yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
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⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2311yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2311yescharacteristic
74#432—44no—
75C2×C4⋊C8—22no—
76#407—44no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8C2314yes—
85C2×C22⋊C8C2314yes—
86C2×#97—22no—
87#388—24no—
88#388—22no—
89#388—22no—
90#387—24no—
91#387—22no—
92#387—22no—
93#386—24no—
94#386—22no—
95#386—22no—
96#386—24no—
97#386—24no—
98#385—22no—
99#385—22no—
100#385—24no—
101#380—24no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2311yescharacteristic
113C2×#95C2311yescharacteristic
114#336—24no—
115#324—44no—
116#323—44no—
117C25C22×C411yescharacteristic
118C25C22×C411yescharacteristic
119C25C22×C411yescharacteristic
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—81no—
132C22×D4—81no—
133C22×D4C22⋊C414yes—
134C22×D4M1614yescomplemented
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—81no—
144C23×C4—81no—
145C23×C4—81no—
146C23×C4—81no—
147C23×C4—81no—
148C23×C4—81no—
149C23×C4—81no—
150C23×C4—81no—
151C23×C4—81no—
152C23×C4—81no—
153C23×C4M1614yescomplemented
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2411yescharacteristic
170C2×M16—44no—
171C2×M16—82no—
172C2×M16—82no—
173C22×C8—24no—
174C22×C8—24no—
175#121—232no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—48no—
181C4×D4—48no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—81no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4C22⋊C414yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—81no—
212C2×C22⋊C4—81no—
213C2×C22⋊C4—81no—
214C2×C22⋊C4—81no—
215C2×C22⋊C4C22⋊C414yes—
216C2×C22⋊C4C22⋊C414yes—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—42no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C4⋊C8—44no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227C22⋊C8—44no—
228C22⋊C8—82no—
229C22⋊C8—82no—
230C22⋊C8—82no—
231C22⋊C8—82no—
232C22⋊C8—82no—
233C22⋊C8—82no—
234#97—44no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24C2×C22⋊C412yes—
270C24C2×C22⋊C412yes—
271C24C2×M1612yes—
272C24C2×M1612yes—
273C24C2×M1612yes—
274C24C23×C411yescharacteristic
275C2×D4—28no—
276C2×D4—28no—
277C2×D4—28no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—48no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—84no—
289C2×D4—84no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—28no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
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—44no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—44no—
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—82no—
350C22×C4—82no—
351C22×C4—88no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—88no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×M1612yes—
366C22×C4C2×M1612yes—
367C22×C4C2×M1612yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4H(5,𝔽2)14yes—
371M16—88no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—44no—
378C4⋊C4—28no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—44no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23#40714yes—
454C23C22×C22⋊C411yescharacteristic
455C23#51812yes—
456C23C22×M1611yescharacteristic
457C23C22×M1611yescharacteristic
458C23#56812yes—
459D4—416no—
460D4—416no—
461D4—88no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
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—22no—
481C2×C4—24no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
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—84no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—88no—
517C2×C4—88no—
518C2×C4—88no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#53412yes—
524C2×C4#53412yes—
525C8—84no—
526C8—84no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—48no—
539C22—48no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—84no—
555C22—84no—
556C22—84no—
557C22—84no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22#230811yescharacteristic
563C22C2×#40711yescharacteristic
564C22#287911yescharacteristic
565C22#288311yescharacteristic
566C22#288311yescharacteristic
567C22#288811yescharacteristic
568C22#288911yescharacteristic
569C22#297312yes—
570C22#297512yes—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C2—22no—
585C2—22no—
586C2—44no—
587C2—82no—
588C2—82no—
589C2#1736711yescharacteristic
590C2#1736811yescharacteristic
591C2#1740511yescharacteristic
592C2#1740611yescharacteristic
593C2#2174211yescharacteristic
594C2#2174911yescharacteristic
595C2#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. #21790C2semidirect
3. #21742C2semidirect
4. #9086C2semidirect
5. #9085C2semidirect
6. #9084C2semidirect
7. #9083C2semidirect
8. #8698C2semidirect
9. #8301C2semidirect
134. C22×D4M16semidirect
153. C23×C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23252
#217902
#217422
#90862
#90852
#90842
#90832
#86982
#83012