Finite Group Databasebeta

← #213685

Subgroups — #213685

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#9139C212yesmaximal, complemented
5#9137C212yesmaximal, complemented
6#9135C212yesmaximal, complemented
7#9135C212yesmaximal, complemented
8#8726C212yesmaximal, complemented
9#8343C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423C2212yes—
13C2×#423C2212yes—
14C2×#407—21no—
15C2×#407—21no—
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2855—21no—
21#2855C412yes—
22#2851—21no—
23#2851—21no—
24#2848—21no—
25#2848C412yes—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28C4×C22⋊Q8C412yes—
29C4×C22⋊Q8C412yes—
30#2296—21no—
31#2285C2211yescharacteristic
32#2280—21no—
33#1797—21no—
34#1797—21no—
35#1797—21no—
36#1797—21no—
37#1797C2212yes—
38#1797C2212yes—
39#1797C2212yes—
40#1797C2212yes—
41#1728—21no—
42#1728—21no—
43C2×#333C2214yescomplemented
44C2×#332C2214yescomplemented
45#1296C2214yescomplemented
46#1296C2214yescomplemented
47C24×C4C2×C411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52#533—44no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2×C412yes—
61C2×C22⋊Q8C2×C412yes—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C412yes—
64C2×#117—21no—
65C2×#117—21no—
66C22×C42—44no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#423—44no—
80#423—44no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8C2314yes—
84C2×C4⋊C8C2314yes—
85#407—44no—
86#407—44no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91#391—21no—
92#391—21no—
93#391C2×C412yes—
94#391C2×C412yes—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389C2×C412yes—
102#389C2×C412yes—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#381—22no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380C2×C412yes—
125#380C2×C412yes—
126#380C2×C412yes—
127#379—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C412yes—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#333—44no—
139#332—44no—
140C25C22⋊C411yescharacteristic
141C22×Q8—21no—
142C22×Q8C22×C412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
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×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C2×M16—44no—
173C2×M16—44no—
174C2×M16—44no—
175C2×M16—44no—
176C2×M16—44no—
177C2×M16—44no—
178C22×C8—24no—
179C22×C8—24no—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182#125—44no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C4×Q8—44no—
191C4×Q8—44no—
192#117—44no—
193#117—42no—
194#117—42no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—22no—
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×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2411yescharacteristic
247C4⋊C8—24no—
248C4⋊C8—24no—
249C4⋊C8—24no—
250C4⋊C8—24no—
251C4⋊C8—44no—
252C4⋊C8—44no—
253C22⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C2×Q8—24no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
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—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
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—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—416no—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4AGL(1,ℤ/8)12yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4#13712yes—
360C22×C4#13712yes—
361C22×C4#13714yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365M16—88no—
366M16—88no—
367C2×C8—44no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—44no—
373C4⋊C4—24no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—44no—
414C42—44no—
415C42H(5,𝔽2)14yes—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23#41712yes—
442C23#41712yes—
443C23#41912yes—
444C23C22×C22⋊C411yescharacteristic
445Q8—84no—
446Q8—82no—
447Q8—82no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
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—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4#41312yes—
509C2×C4#41312yes—
510C2×C4#41712yes—
511C2×C4#41712yes—
512C2×C4#41912yes—
513C2×C4#51812yes—
514C2×C4#53412yes—
515C2×C4#53412yes—
516C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
517C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
518C2×C4C2×#13712yes—
519C2×C4C2×#13711yescharacteristic
520C2×C4C2×#13711yescharacteristic
521C2×C4#57512yes—
522C8—84no—
523C8—84no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—416no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22#230811yescharacteristic
547C22C2×#41311yescharacteristic
548C22C2×#41311yescharacteristic
549C22C2×#41911yescharacteristic
550C22C2×#41711yescharacteristic
551C22C2×#41711yescharacteristic
552C22#289611yescharacteristic
553C4—22no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4#319312yes—
568C4#319412yes—
569C4#319512yes—
570C4#319612yes—
571C2—42no—
572C2—42no—
573C2—42no—
574C2#1773711yescharacteristic
575C2#1774711yescharacteristic
576C2#1774911yescharacteristic
577C2#1775211yescharacteristic
578C2#2175411yescharacteristic
579C2#2176511yescharacteristic
580C2#2176911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9139C2semidirect
5. #9137C2semidirect
6. #9135C2semidirect
7. #9135C2semidirect
8. #8726C2semidirect
9. #8343C2semidirect
43. C2×#333C22semidirect
44. C2×#332C22semidirect
45. #1296C22semidirect
46. #1296C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#206262
#91392
#91372
#91352
#87262
#83432