Finite Group Databasebeta

← #216540

Subgroups — #216540

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3#14683C211yesmaximal, characteristic, complemented
4#9912C212yesmaximal, complemented
5#9715C212yesmaximal, complemented
6#9117C214yesmaximal, complemented
7#8575C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407C2212yes—
13#2857C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2856C411yescharacteristic, complemented
16#2856C2211yescharacteristic
17#2854—21no—
18#2854C412yescomplemented
19#2850—21no—
20#2850—21no—
21#2847—21no—
22#2847C2212yes—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121C412yescomplemented
26C4×#121C2212yes—
27#2297—21no—
28#2285C411yescharacteristic, complemented
29C2×#383—21no—
30#1797—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1728—21no—
35C2×#327C2218yes—
36#1434—21no—
37#1434C2212yes—
38#1434C2212yes—
39C22⋊#95—21no—
40C22⋊#95—21no—
41C22⋊#95C2212yes—
42#1294C2218yes—
43C22×#41—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48#532—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#124—21no—
52#525—48no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×#121C2×C412yes—
63C2×#121C2312yes—
64C2×#120—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117—21no—
70C22×C22⋊C4—22no—
71C22×C42—44no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#423—44no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87#407—44no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8C2314yes—
90#391—21no—
91#391—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2×C412yes—
98#389—21no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#384C2×C411yescharacteristic
113#384C2×C411yescharacteristic
114#383—44no—
115#382—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2312yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C412yes—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95C2314yes—
132#336—22no—
133#336—22no—
134#327—48no—
135C25C2×D411yescharacteristic
136C2×#41—44no—
137C2×#41—44no—
138C2×#41—44no—
139C22×Q8—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22×C412yes—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
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×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C2×M16—44no—
177C2×M16—44no—
178C2×M16—44no—
179C22×C8—24no—
180C4⋊Q8—42no—
181#127—42no—
182#124—44no—
183#123—48no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186#121—24no—
187#121—24no—
188#121—44no—
189#121—44no—
190#121—44no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—44no—
195#120—48no—
196C4×D4—44no—
197C4×D4—48no—
198C4×D4—48no—
199C4×D4—44no—
200#117—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
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×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C4⋊C8—44no—
255C4⋊C8—44no—
256C22⋊C8—28no—
257C22⋊C8—44no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—41no—
278C24—41no—
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C411yescharacteristic
281C24C22×D411yescharacteristic
282#41—816no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×D4—24no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
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—21no—
320C22×C4—21no—
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—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
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—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—416no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—416no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4AGL(1,ℤ/8)14yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+412yes—
390M16—88no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—44no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—88no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—44no—
433C42#9914yes—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—28no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23#41012yes—
465C23#42112yes—
466C23C22×C22⋊C411yescharacteristic
467C23#53712yes—
468C23C2×AGL(1,ℤ/8)12yes—
469C23#57512yes—
470Q8—84no—
471D4—84no—
472D4—84no—
473D4—84no—
474D4—88no—
475D4—88no—
476D4—88no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
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—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—88no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4C2×#9912yes—
544C2×C4C2×#9912yes—
545C2×C4C2×#9912yes—
546C2×C4#41812yes—
547C2×C4#41812yes—
548C2×C4#42112yes—
549C2×C4#52012yes—
550C2×C4#53511yescharacteristic
551C2×C4#53611yescharacteristic
552C8—84no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—416no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22#231111yescharacteristic
582C22C22×#9911yescharacteristic
583C22C2×#41011yescharacteristic
584C22C2×#41811yescharacteristic
585C22#289611yescharacteristic
586C22C2×#42111yescharacteristic
587C22C2×#42111yescharacteristic
588C22#325114yes—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—164no—
606C2—24no—
607C2—42no—
608C2—42no—
609C2—162no—
610C2#1755911yescharacteristic
611C2#1757711yescharacteristic
612C2#1758911yescharacteristic
613C2#1759311yescharacteristic
614C2#2185711yescharacteristic
615C2#2187411yescharacteristic
616C2#2189111yescharacteristic
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. #20626C2semidirect
3. #14683C2semidirect
4. #9912C2semidirect
5. #9715C2semidirect
6. #9117C2semidirect
7. #8575C2semidirect
8. C2×#532C4semidirect
13. #2857C4semidirect
15. #2856C4semidirect
18. #2854C4semidirect
25. C4×#121C4semidirect
28. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#366242
#206262
#146832
#99122
#97152
#91172
#85752