Finite Group Databasebeta

← #205021

Subgroups — #205021

Order 512.

1168 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35103C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3C2×#1512C212yesmaximal, complemented
4C2×#1509C212yesmaximal, complemented
5#13804C211yesmaximal, characteristic, complemented
6#9716C212yesmaximal, complemented
7#9715C211yesmaximal, characteristic
8#9714C211yesmaximal, characteristic, complemented
9#9713C212yesmaximal, complemented
10#9712C211yesmaximal, characteristic
11#9711C211yesmaximal, characteristic, complemented
12C2×#529C2212yes—
13C2×#432C2212yes—
14#2857C2211yescharacteristic
15#2856C2211yescharacteristic
16#2850C2212yes—
17#2618—21no—
18#2618—21no—
19#2601—22no—
20#2601—22no—
21#2566C412yescomplemented
22#2565C412yescomplemented
23#2545C412yescomplemented
24#2544C412yescomplemented
25#2325—21no—
26#2325—21no—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1940—22no—
32#1940—22no—
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35#1797C2212yes—
36#1512—28no—
37#1509—28no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#328C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45C2×#327C2212yes—
46#1436—21no—
47#1436—21no—
48C4⋊#95—21no—
49#1434C2211yescharacteristic
50#1434C2211yescharacteristic
51C4×#95C2211yescharacteristic
52C4×#95C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55#529—24no—
56#529—24no—
57#529—24no—
58#529—24no—
59C2×#126—22no—
60C2×#125C2×C412yes—
61C2×#124C2×C412yes—
62C2×#124C2×C412yes—
63C2×#123—22no—
64C2×C22⋊Q8C2312yes—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121C2312yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×D4—22no—
70C2×C4×D4C2×C412yes—
71C2×#117—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#432—24no—
83#432—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90C2×#97—22no—
91#398—22no—
92#398—22no—
93#396—22no—
94#396—22no—
95#396—22no—
96#396—22no—
97#394—22no—
98#394—22no—
99#391C2×C412yes—
100#391C2×C412yes—
101#390—21no—
102#390—21no—
103#390C2×C412yes—
104#390C2×C412yes—
105#389—21no—
106#389—21no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#385—22no—
118#385—22no—
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#382—22no—
122#382C2×C412yes—
123#382D414yescomplemented
124#380—21no—
125#380—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—44no—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4D414yescomplemented
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95D411yescharacteristic
141C2×#95D411yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146#336—22no—
147#336—21no—
148#336—21no—
149#328—24no—
150#328—24no—
151#328—24no—
152#328—24no—
153#327—24no—
154#327—24no—
155#327—24no—
156#327—24no—
157C22×Q8C22×C412yes—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4C22×C412yes—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C22×C8—22no—
175C22×C8—22no—
176C4⋊Q8—81no—
177#127—81no—
178#126—44no—
179#126—44no—
180#125—24no—
181#125—24no—
182#124—24no—
183#124—24no—
184#124—82no—
185#123—44no—
186#123—44no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191#121—22no—
192#121—22no—
193#121—22no—
194#121—22no—
195#121—42no—
196#121—42no—
197#121—42no—
198#121—42no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—44no—
204C4×D4—44no—
205#117—44no—
206#117—44no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C412yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—81no—
253C2×C42—81no—
254C2×C42—81no—
255C2×C42—81no—
256C2×C42C22×C412yes—
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268C4⋊C8—44no—
269C4⋊C8—81no—
270C4⋊C8—81no—
271C4⋊C8—82no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276#97—44no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—24no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—81no—
293#95—81no—
294#95C22⋊C412yes—
295#95C22⋊C412yes—
296#95C22⋊C412yes—
297#95C22⋊C412yes—
298#95C2×D412yes—
299#95C2×D412yes—
300#95C2×D412yes—
301#95C2×D412yes—
302C24—22no—
303C24C22×D411yescharacteristic
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—81no—
321C2×D4—81no—
322C2×D4—81no—
323C2×D4—81no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
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—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—44no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4#12112yes—
389C22×C4#12112yes—
390C22×C4#12112yes—
391C22×C4#12112yes—
392C22×C4AGL(1,ℤ/8)12yes—
393C22×C4#13712yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—42no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42—42no—
444C42—41no—
445C42—41no—
446C42—48no—
447C42—44no—
448C42—84no—
449C42—84no—
450C42C4×D414yes—
451C42C4×D414yes—
452C42#12414yes—
453C42#12714yes—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C22×C22⋊C411yescharacteristic
466C23C2×#12111yescharacteristic
467C23C2×#12111yescharacteristic
468C23C2×AGL(1,ℤ/8)11yescharacteristic
469C23C2×#13711yescharacteristic
470C23#57511yescharacteristic
471C23#57511yescharacteristic
472Q8—82no—
473Q8—82no—
474D4—82no—
475D4—82no—
476D4—82no—
477D4—82no—
478D4—82no—
479D4—82no—
480D4—162no—
481D4—162no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
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—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—88no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4#39014yes—
563C2×C4#39014yes—
564C2×C4#41712yes—
565C2×C4#41712yes—
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×#12411yescharacteristic
569C2×C4C2×#12711yescharacteristic
570C8—82no—
571C8—82no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22C2×#39011yescharacteristic
592C22#288311yescharacteristic
593C22#288411yescharacteristic
594C22C2×#41711yescharacteristic
595C22#289611yescharacteristic
596C22#290111yescharacteristic
597C22#290111yescharacteristic
598C22#303611yescharacteristic
599C22#303611yescharacteristic
600C22#303711yescharacteristic
601C22#303711yescharacteristic
602C4—24no—
603C4—22no—
604C4—22no—
605C4—24no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—44no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—162no—
626C2#1762211yescharacteristic
627C2#1762411yescharacteristic
628C2#1762511yescharacteristic
629C2#1762811yescharacteristic
630C2#2121011yescharacteristic
631C2#2121411yescharacteristic
632C2#2125811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21955C2semidirect
3. C2×#1512C2semidirect
4. C2×#1509C2semidirect
5. #13804C2semidirect
6. #9716C2semidirect
8. #9714C2semidirect
9. #9713C2semidirect
11. #9711C2semidirect
21. #2566C4semidirect
22. #2565C4semidirect
23. #2545C4semidirect
24. #2544C4semidirect
123. #382D4semidirect
132. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#351032
#219552
C2×#15122
C2×#15092
#138042
#97162
#97152
#97142
#97132
#97122
#97112