Finite Group Databasebeta

← #214358

Subgroups — #214358

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3#14684C211yesmaximal, characteristic, complemented
4#9158C214yesmaximal, complemented
5#9114C214yesmaximal, complemented
6#8717C212yesmaximal, complemented
7#8332C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#431C2212yes—
12#2857C411yescharacteristic, complemented
13#2857C411yescharacteristic, complemented
14#2856C411yescharacteristic, complemented
15#2856C411yescharacteristic, complemented
16#2854—21no—
17#2854C2212yes—
18#2850—21no—
19#2850—21no—
20#2847—21no—
21#2847C2212yes—
22C4×#121—21no—
23C4×#121—21no—
24C4×#121C412yescomplemented
25C4×#121C412yescomplemented
26#2297—21no—
27#2285C2211yescharacteristic
28C2×#381C2212yes—
29#1817C2212yes—
30#1797—21no—
31#1797—21no—
32#1779C2212yes—
33C4×C22⋊C8—21no—
34C4×C22⋊C8—21no—
35C2×#327C2218yes—
36#1434—21no—
37#1434—21no—
38#1434C2212yes—
39#1434C2212yes—
40C22⋊#95—21no—
41C22⋊#95—21no—
42#1294C2218yes—
43C22×#41—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47#532—44no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#127C2×C411yescharacteristic
50C2×#124—21no—
51#525—48no—
52C2×#123—22no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121C2×C412yes—
61C2×#121C2×C412yes—
62C2×#121C2312yes—
63C2×#120—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×#117—21no—
69C22×C22⋊C4—22no—
70C22×C42—44no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#431—44no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8C2314yes—
88C2×C4×C8—22no—
89#391—21no—
90#391—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390C2×C412yes—
96#390C2×C412yes—
97#389—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#382—21no—
114#381—44no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380C2×C412yes—
119#380C2×C412yes—
120#380C2312yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C412yes—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95C2314yes—
131#336—22no—
132#336—22no—
133#327—48no—
134C25C2×D411yescharacteristic
135C2×#41—44no—
136C2×#41—44no—
137C2×#41—44no—
138C22×Q8—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C22×C412yes—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
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×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8—24no—
175C4⋊Q8—42no—
176#127—42no—
177#124—44no—
178#123—48no—
179C22⋊Q8—44no—
180C22⋊Q8—44no—
181#121—28no—
182#121—44no—
183#121—44no—
184#121—44no—
185#121—44no—
186#121—44no—
187#121—44no—
188#121—44no—
189#120—48no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—48no—
193C4×D4—48no—
194#117—44no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C4⋊C8—44no—
249C22⋊C8—28no—
250C22⋊C8—44no—
251C22⋊C8—44no—
252C4×C8—44no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—28no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—41no—
272C24—41no—
273C24C2×C22⋊C411yescharacteristic
274C24C2×C22⋊C411yescharacteristic
275C24C22×D411yescharacteristic
276#41—816no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×D4—24no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—42no—
297C2×D4—42no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
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—41no—
351C22×C4—41no—
352C22×C4—416no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4#13514yes—
376C22×C4AGL(1,ℤ/8)14yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4H(5,𝔽2)14yes—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—44no—
383C4⋊C4—24no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—88no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—44no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—28no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23#41012yes—
453C23#41012yes—
454C23C22×C22⋊C411yescharacteristic
455C23#53412yes—
456C23C2×#13512yes—
457C23C2×AGL(1,ℤ/8)12yes—
458Q8—84no—
459D4—84no—
460D4—84no—
461D4—84no—
462D4—88no—
463D4—88no—
464D4—88no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—88no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4#41612yes—
529C2×C4#41612yes—
530C2×C4#41812yes—
531C2×C4#41812yes—
532C2×C4#42112yes—
533C2×C4#42112yes—
534C2×C4#51812yes—
535C2×C4#53412yes—
536C8—84no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—416no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22#230811yescharacteristic
566C22C2×#41011yescharacteristic
567C22C2×#41011yescharacteristic
568C22C2×#41611yescharacteristic
569C22C2×#41811yescharacteristic
570C22C2×#42111yescharacteristic
571C22C2×#42111yescharacteristic
572C22#318814yes—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—164no—
589C2—24no—
590C2—42no—
591C2—42no—
592C2—162no—
593C2#1756111yescharacteristic
594C2#1757111yescharacteristic
595C2#1757611yescharacteristic
596C2#1758911yescharacteristic
597C2#2174811yescharacteristic
598C2#2176111yescharacteristic
599C2#2177411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. #14684C2semidirect
4. #9158C2semidirect
5. #9114C2semidirect
6. #8717C2semidirect
7. #8332C2semidirect
12. #2857C4semidirect
13. #2857C4semidirect
14. #2856C4semidirect
15. #2856C4semidirect
24. C4×#121C4semidirect
25. C4×#121C4semidirect

Maximal subgroups

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

subgroupindex
#366242
#207142
#146842
#91582
#91142
#87172
#83322