Finite Group Databasebeta

← #224883

Subgroups — #224883

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36596C211yesmaximal, characteristic
2#22625C211yesmaximal, characteristic, complemented
3#13933C211yesmaximal, characteristic, complemented
4#9715C212yesmaximal, complemented
5#9712C211yesmaximal, characteristic, complemented
6#9680C211yesmaximal, characteristic, complemented
7#9680C212yesmaximal, complemented
8#8571C211yesmaximal, characteristic, complemented
9#8571C212yesmaximal, complemented
10#8220C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic, complemented
12C2×#553—21no—
13#2856C2211yescharacteristic
14#2856C2212yes—
15#2554—22no—
16#2552C411yescharacteristic, complemented
17#2543—22no—
18#2541C411yescharacteristic, complemented
19#2512C412yescomplemented
20#2490—21no—
21#2445C412yescomplemented
22#2416—21no—
23#2366C411yescharacteristic, complemented
24#2363—21no—
25#2354C411yescharacteristic, complemented
26C4×#121C2211yescharacteristic
27C4×#121C2212yes—
28C2×#389—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1939—21no—
31#1916—22no—
32#1797C2211yescharacteristic
33#1797C2212yes—
34C8×C4⋊C4—21no—
35C4×C22⋊C8C2211yescharacteristic
36C4×C22⋊C8C2212yes—
37C2×#327C2211yescharacteristic
38C2×#327C2211yescharacteristic
39C2×#327C2212yes—
40C2×#327C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44#1442—22no—
45C4⋊#95—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53#553—44no—
54#553—44no—
55C2×#127C2×C411yescharacteristic
56C2×#127C2×C412yes—
57C2×#126—22no—
58C2×#125—21no—
59C2×#123—22no—
60C2×#121—21no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121C2312yes—
67C2×#121C2311yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2312yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2311yescharacteristic
85C2×C22⋊C8C2311yescharacteristic
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88#398—22no—
89#397—22no—
90#396—22no—
91#394—22no—
92#394—22no—
93#392—22no—
94#390—21no—
95#390—21no—
96#390—22no—
97#390—22no—
98#390—21no—
99#390C2×C412yes—
100#390C2×C412yes—
101#390C2×C412yes—
102#390C2×C411yescharacteristic
103#390C2×C412yes—
104#389—21no—
105#389—44no—
106#389C2×C412yes—
107#389C2×C411yescharacteristic
108#388—22no—
109#388—21no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—21no—
114#386—22no—
115#386—22no—
116#384C2×C411yescharacteristic
117#384C2×C412yes—
118#380—21no—
119#380C2311yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95D412yes—
136C2×#95C2311yescharacteristic
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139#336—21no—
140#336—22no—
141#336—21no—
142#336—22no—
143#327—22no—
144#327—24no—
145#327—22no—
146#327—24no—
147#327—42no—
148#327—42no—
149#327—42no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4C22×C411yescharacteristic
157C22×D4C22×C412yes—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171#127—42no—
172#127—82no—
173#126—48no—
174#125—44no—
175#123—48no—
176#121—22no—
177#121—21no—
178#121—22no—
179#121—21no—
180#121—22no—
181#121—21no—
182#121—22no—
183#121—21no—
184#121—44no—
185#121—44no—
186#121—42no—
187#121—44no—
188#121—42no—
189#121—44no—
190#121—82no—
191#121—81no—
192#121—82no—
193#121—81no—
194C4×D4—42no—
195C4×D4—44no—
196C4×D4—42no—
197C4×D4—44no—
198C4×D4—42no—
199C4×D4—44no—
200C4×D4—82no—
201#117—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4D814yes—
222C2×C4⋊C4SD1614yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—82no—
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C412yes—
251C2×C42C22×C412yes—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C4⋊C8—42no—
258C4⋊C8—82no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—21no—
264C22⋊C8#4112yes—
265C4×C8—42no—
266C4×C8—82no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—21no—
276#95—22no—
277#95—42no—
278#95—44no—
279#95—82no—
280#95—81no—
281#95—82no—
282#95C22⋊C412yes—
283#95C22⋊C412yes—
284#95C22⋊C412yes—
285#95C22⋊C412yes—
286#95#4112yes—
287C24—21no—
288C24—22no—
289C24C22×D411yescharacteristic
290C2×D4—21no—
291C2×D4—22no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—44no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—82no—
317C2×D4—81no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—81no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
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—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×D812yes—
376C22×C4C2×D814yes—
377C22×C4C2×SD1612yes—
378C22×C4C2×SD1614yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4H(5,𝔽2)11yescharacteristic
384C22×C42−1+411yescharacteristic
385C2×C8—22no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—81no—
408C4⋊C4—82no—
409C4⋊C4—84no—
410C4⋊C4—88no—
411C4⋊C4#10214yes—
412C4⋊C4#10214yes—
413C4⋊C4#10218yes—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—42no—
420C22⋊C4—41no—
421C22⋊C4—42no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—81no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C42—22no—
433C42—22no—
434C42—42no—
435C42—42no—
436C42—41no—
437C42—41no—
438C42—42no—
439C42—42no—
440C42—44no—
441C42—42no—
442C42—84no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23C22×C22⋊C411yescharacteristic
462C23#54811yescharacteristic
463C23#55011yescharacteristic
464C23C22×D811yescharacteristic
465C23C22×SD1611yescharacteristic
466C23#57611yescharacteristic
467C23#57711yescharacteristic
468D4—81no—
469D4—82no—
470D4—81no—
471D4—82no—
472D4—82no—
473D4—81no—
474D4—82no—
475D4—81no—
476D4—82no—
477D4—81no—
478D4—82no—
479D4—81no—
480D4—84no—
481D4—164no—
482D4—164no—
483D4—162no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—84no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—164no—
553C2×C4#41012yes—
554C2×C4C2×#10214yes—
555C2×C4C2×#10212yes—
556C2×C4C2×#10214yes—
557C2×C4C2×#10212yes—
558C2×C4#42112yes—
559C2×C4C4×#4111yescharacteristic
560C2×C4#52011yescharacteristic
561C2×C4#54811yescharacteristic
562C2×C4#55011yescharacteristic
563C8—42no—
564C8—81no—
565C8—82no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—81no—
579C22—82no—
580C22—81no—
581C22—82no—
582C22—82no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—81no—
587C22—82no—
588C22—82no—
589C22—81no—
590C22—82no—
591C22—81no—
592C22#234111yescharacteristic
593C22C2×#41011yescharacteristic
594C22#288311yescharacteristic
595C22C22×#10211yescharacteristic
596C22#289711yescharacteristic
597C22C2×#42111yescharacteristic
598C22#290111yescharacteristic
599C22#335811yescharacteristic
600C22#336211yescharacteristic
601C22#339111yescharacteristic
602C22#339211yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—22no—
608C4—24no—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—41no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—162no—
621C4#151714yes—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—161no—
627C2—162no—
628C2C2×#151711yescharacteristic
629C2#1759911yescharacteristic
630C2#1760711yescharacteristic
631C2#1761311yescharacteristic
632C2#2253511yescharacteristic
633C2#2254011yescharacteristic
634C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36596C2
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. #22625C2semidirect
3. #13933C2semidirect
4. #9715C2semidirect
5. #9712C2semidirect
6. #9680C2semidirect
7. #9680C2semidirect
8. #8571C2semidirect
9. #8571C2semidirect
10. #8220C2semidirect
11. C4×#327C2semidirect
16. #2552C4semidirect
18. #2541C4semidirect
19. #2512C4semidirect
21. #2445C4semidirect
23. #2366C4semidirect
25. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#365962
#226252
#139332
#97152
#97122
#96802
#85712
#82202
C4×#3272