Finite Group Databasebeta

← #224834

Subgroups — #224834

Order 512.

1099 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36580C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3#13935C211yesmaximal, characteristic, complemented
4#9716C212yesmaximal, complemented
5#9712C211yesmaximal, characteristic
6#9682C212yesmaximal, complemented
7#9680C211yesmaximal, characteristic
8#8572C211yesmaximal, characteristic, complemented
9#8571C212yesmaximal, complemented
10#8216C212yesmaximal, complemented
11C4×#328C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic, complemented
13C22×C4×Q8C2211yescharacteristic
14C2×#446—21no—
15#2856C2211yescharacteristic
16#2850C2212yes—
17#2592C412yescomplemented
18#2582C411yescharacteristic, complemented
19#2580C412yescomplemented
20#2548C411yescharacteristic, complemented
21#2458—22no—
22#2448—21no—
23#2443—22no—
24#2436—21no—
25#2367—21no—
26#2361—21no—
27#2355C411yescharacteristic, complemented
28C4×C22⋊Q8C2211yescharacteristic
29C4×#121C2212yes—
30C2×#391—21no—
31C2×#389—21no—
32#1985—22no—
33#1984—21no—
34#1797C2212yes—
35#1797C2211yescharacteristic
36#1749C2212yes—
37C4×C22⋊C8C2211yescharacteristic
38C2×#328C2211yescharacteristic
39C2×#328C2211yescharacteristic
40C2×#328C2212yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45#1441—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2211yescharacteristic
48#1434C2212yes—
49C4×#95C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#559—44no—
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#124—21no—
58C2×#124C2×C412yes—
59C2×#123—22no—
60C2×C22⋊Q8C2311yescharacteristic
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2312yes—
64C2×C4×Q8—44no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D412yes—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4C2×C412yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#446—44no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8C2311yescharacteristic
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2311yescharacteristic
88C2×#97—22no—
89C2×C4×C8—21no—
90#399—22no—
91#398—22no—
92#397—22no—
93#394—22no—
94#393—22no—
95#393—22no—
96#391—21no—
97#391—21no—
98#391—44no—
99#391C2×C412yes—
100#391C2×C411yescharacteristic
101#390—21no—
102#390—21no—
103#390—21no—
104#390C2×C412yes—
105#390C2×C412yes—
106#390C2×C412yes—
107#389—21no—
108#389—44no—
109#389C2×C411yescharacteristic
110#389C2×C412yes—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#387—22no—
117#387—21no—
118#386—22no—
119#386—22no—
120#386—22no—
121#385—21no—
122#385—22no—
123#385—22no—
124#384C2×C411yescharacteristic
125#382C2×C412yes—
126#380C2311yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—21no—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141#336—22no—
142#336—21no—
143#336—22no—
144#336—21no—
145#328—22no—
146#328—22no—
147#328—42no—
148#327—24no—
149#327—24no—
150#327—42no—
151#327—42no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8C22⋊C411yescharacteristic
157C22×Q8C22⋊C411yescharacteristic
158C22×Q8C22⋊C412yes—
159C22×Q8C22×C411yescharacteristic
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22×C412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2411yescharacteristic
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176#127—42no—
177#124—44no—
178#124—44no—
179#124—44no—
180#123—48no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185#121—22no—
186#121—22no—
187#121—22no—
188#121—22no—
189#121—42no—
190#121—42no—
191#121—81no—
192#121—81no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—82no—
203C4×D4—82no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
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×C42—21no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42C22×C412yes—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C412yes—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D412yes—
257C4⋊C8—42no—
258C4⋊C8—44no—
259C22⋊C8—21no—
260C22⋊C8—22no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—22no—
264C22⋊C8#4112yes—
265#97—44no—
266C4×C8—42no—
267#95—21no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—22no—
280#95—21no—
281#95—21no—
282#95—22no—
283#95—42no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—81no—
288#95#4112yes—
289C24—22no—
290C24C22×D411yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
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—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
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—41no—
355C22×C4—41no—
356C22×C4—41no—
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⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4#13712yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C42−1+411yescharacteristic
383C2×C8—22no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—41no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—81no—
409C4⋊C4—82no—
410C4⋊C4—81no—
411C4⋊C4—84no—
412C4⋊C4—88no—
413C4⋊C4#9914yes—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C42—22no—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—41no—
434C42—42no—
435C42—41no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—82no—
440C42—84no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23C22×C22⋊C411yescharacteristic
453C23#54811yescharacteristic
454C23#55011yescharacteristic
455C23C2×AGL(1,ℤ/8)11yescharacteristic
456C23C2×#13711yescharacteristic
457C23#57611yescharacteristic
458C23#57711yescharacteristic
459Q8—48no—
460Q8—44no—
461Q8—44no—
462Q8—81no—
463Q8—81no—
464D4—82no—
465D4—82no—
466D4—82no—
467D4—82no—
468D4—82no—
469D4—82no—
470D4—84no—
471D4—162no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—164no—
544C2×C4C2×#9914yes—
545C2×C4C2×#9912yes—
546C2×C4#41712yes—
547C2×C4#41712yes—
548C2×C4#42112yes—
549C2×C4C4×#4111yescharacteristic
550C2×C4#52011yescharacteristic
551C2×C4#54811yescharacteristic
552C2×C4#55011yescharacteristic
553C8—42no—
554C8—81no—
555C8—82no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22#234111yescharacteristic
576C22C22×#9911yescharacteristic
577C22#288411yescharacteristic
578C22C2×#41711yescharacteristic
579C22#289711yescharacteristic
580C22C2×#42111yescharacteristic
581C22#290111yescharacteristic
582C22#337311yescharacteristic
583C22#337511yescharacteristic
584C22#339911yescharacteristic
585C22#340011yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—22no—
591C4—24no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—82no—
598C4—82no—
599C4—81no—
600C4—82no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—162no—
605C4#151314yes—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—162no—
611C2C2×#151311yescharacteristic
612C2#1760211yescharacteristic
613C2#1760311yescharacteristic
614C2#1762811yescharacteristic
615C2#2253411yescharacteristic
616C2#2254611yescharacteristic
617C2#2255111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
3. #13935C2semidirect
4. #9716C2semidirect
6. #9682C2semidirect
8. #8572C2semidirect
9. #8571C2semidirect
10. #8216C2semidirect
11. C4×#328C2semidirect
12. C2×#559C4semidirect
17. #2592C4semidirect
18. #2582C4semidirect
19. #2580C4semidirect
20. #2548C4semidirect
27. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#365802
#225102
#139352
#97162
#97122
#96822
#96802
#85722
#85712
#82162
C4×#3282