Finite Group Databasebeta

← #482053

Subgroups — #482053

Order 512.

1114 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#35328C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic, complemented
3#17349C211yesmaximal, characteristic, complemented
4#11876C214yesmaximal, complemented
5#11875C214yesmaximal, complemented
6#11694C212yesmaximal, complemented
7#11693C212yesmaximal, complemented
8C2×#537—21no—
9C2×#532C411yescharacteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C2×#431—21no—
12C2×#410—21no—
13#2850—21no—
14#2850—21no—
15#2621C411yescharacteristic, complemented
16#2621C2211yescharacteristic
17#2619C411yescharacteristic, complemented
18#2619C2211yescharacteristic
19#2543—21no—
20#2543—21no—
21#2543C412yescomplemented
22#2543C2212yes—
23#2536—21no—
24#2536C412yescomplemented
25#2535—21no—
26#2535C2212yes—
27#2311—21no—
28#2285—21no—
29C2×#382C411yescharacteristic, complemented
30#1799—21no—
31#1799—21no—
32#1799C2212yes—
33#1799C2212yes—
34#1740C2212yes—
35#1461—21no—
36#1460C2212yes—
37#1459—21no—
38#1459C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1344C2218yes—
42#1343C2218yes—
43C22×#41—21no—
44C22×#41D411yescharacteristic
45#537—44no—
46#537—44no—
47#532—41no—
48#532—41no—
49#532—41no—
50#532—41no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#127C2×C411yescharacteristic
53C2×#124—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2×C412yes—
64C4×#41—41no—
65C4×#41—41no—
66C4×#41—41no—
67C4×#41—41no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C412yes—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—22no—
79C2×#117C2×C411yescharacteristic
80C2×#117D411yescharacteristic
81C2×#117C2311yescharacteristic
82C22×C4⋊C4—21no—
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86#431—22no—
87#431—22no—
88#431—41no—
89#431—41no—
90C2×C4⋊C8—41no—
91#410—44no—
92C2×#99—22no—
93C2×#99—22no—
94C2×C22⋊C8—22no—
95C2×C4×C8—41no—
96#398—21no—
97#398—21no—
98#398C2×C412yes—
99#398C2×C412yes—
100#397—21no—
101#397—21no—
102#397C2×C412yes—
103#397C2312yes—
104#394—21no—
105#394—21no—
106#394C2×C412yes—
107#394C2×C412yes—
108#392—21no—
109#392—21no—
110#392C2×C412yes—
111#392C2312yes—
112#391—21no—
113#391—21no—
114#390—21no—
115#390—21no—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#384—41no—
127#382—21no—
128#382—21no—
129#382—41no—
130#382C2×C411yescharacteristic
131#382C2×C411yescharacteristic
132#382C2×C412yes—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137C4×C4⋊C4—41no—
138C4×C4⋊C4—41no—
139C2×#95—21no—
140C2×#95C2×C412yes—
141#342—22no—
142#342C2314yes—
143#324—22no—
144#324—22no—
145#323—22no—
146#323C2314yes—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—44no—
153C2×#41—44no—
154C2×#41D814yes—
155C2×#41SD1614yes—
156C22×Q8—21no—
157C22×Q8C22⋊C411yescharacteristic
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22⋊C411yescharacteristic
164C22×D4C22×C412yes—
165C22×D4C2×D411yescharacteristic
166C22×D4C2×D411yescharacteristic
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C22×C8—42no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181#127—41no—
182#127—41no—
183#124—41no—
184#124—41no—
185#124—41no—
186#124—41no—
187C22⋊Q8—48no—
188C22⋊Q8—44no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191#121—42no—
192#121—42no—
193#121—48no—
194#121—44no—
195#121—44no—
196#121—44no—
197#121—44no—
198#121—44no—
199#121—42no—
200#121—42no—
201#121—42no—
202#121—42no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223#117—41no—
224#117—41no—
225#117—41no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—44no—
230#117—44no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22×C411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C22⋊C4C2411yescharacteristic
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—42no—
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C4⋊C8—42no—
289C4⋊C8—42no—
290#99—44no—
291#99—44no—
292C22⋊C8—24no—
293C22⋊C8—42no—
294C4×C8—42no—
295C4×C8—42no—
296#95—22no—
297#95—21no—
298#95—21no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—42no—
305#95—42no—
306#95C22×C412yes—
307#95C22×C412yes—
308C24—22no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24C22×D411yescharacteristic
312#41—44no—
313#41—44no—
314#41—44no—
315#41—44no—
316#41—88no—
317#41—88no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8#10214yes—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4#10214yes—
350C2×D4C2×D812yes—
351C2×D4C2×D812yes—
352C2×D4C2×SD1612yes—
353C2×D4C2×SD1612yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4#10214yes—
392C22×C4#10214yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×D812yes—
404C22×C4C2×SD1612yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C2×C8—44no—
408C2×C8—84no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—84no—
465C42#9911yescharacteristic
466C42#9911yescharacteristic
467C42#9911yescharacteristic
468C42#9911yescharacteristic
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C2×#10212yes—
489C23C2×#10212yes—
490C23C22×C22⋊C411yescharacteristic
491C23C22×D811yescharacteristic
492C23C22×SD1611yescharacteristic
493Q8—44no—
494Q8—82no—
495Q8—82no—
496D4—44no—
497D4—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—84no—
502D4—84no—
503D4—88no—
504D4—82no—
505D4—82no—
506D4—162no—
507D4—162no—
508D4—164no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
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—88no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—164no—
563C2×C4C2×#9911yescharacteristic
564C2×C4C2×#9911yescharacteristic
565C2×C4C2×#9911yescharacteristic
566C2×C4C2×#9911yescharacteristic
567C2×C4C2×#9911yescharacteristic
568C2×C4C2×#9911yescharacteristic
569C2×C4C2×#10212yes—
570C2×C4C2×#10212yes—
571C2×C4C2×#10212yes—
572C2×C4C2×#10212yes—
573C2×C4#42112yes—
574C8—84no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—22no—
579C22—21no—
580C22—22no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—81no—
593C22—81no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22C22×#9911yescharacteristic
602C22C22×#10211yescharacteristic
603C22C2×#42111yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—164no—
621C4#151912yes—
622C4#151912yes—
623C2—21no—
624C2—21no—
625C2—41no—
626C2—42no—
627C2—41no—
628C2—42no—
629C2—162no—
630C2C2×#151911yescharacteristic
631C2#3334711yescharacteristic
632C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35328C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20632C2semidirect
3. #17349C2semidirect
4. #11876C2semidirect
5. #11875C2semidirect
6. #11694C2semidirect
7. #11693C2semidirect
9. C2×#532C4semidirect
15. #2621C4semidirect
17. #2619C4semidirect
21. #2543C4semidirect
24. #2536C4semidirect
29. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
#353282
#206322
#173492
#118762
#118752
#116942
#116932