Finite Group Databasebeta

← #214955

Subgroups — #214955

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35703C211yesmaximal, characteristic
2#20392C211yesmaximal, characteristic, complemented
3#17369C211yesmaximal, characteristic, complemented
4#9475C212yesmaximal, complemented
5#9407C212yesmaximal, complemented
6#9184C212yesmaximal, complemented
7#9010C212yesmaximal, complemented
8#8776C212yesmaximal, complemented
9#8366C212yesmaximal, complemented
10C2×#548C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2975C2212yes—
13C22×#99—21no—
14#2638C2211yescharacteristic
15#2631C411yescharacteristic, complemented
16#2617C411yescharacteristic, complemented
17#2610C411yescharacteristic, complemented
18#2599C411yescharacteristic, complemented
19#2561C2211yescharacteristic
20#2543C411yescharacteristic, complemented
21#2477—21no—
22#2472—21no—
23#2463—21no—
24#2459—21no—
25#2444—21no—
26#2410—21no—
27#2409C411yescharacteristic, complemented
28#2404C2211yescharacteristic
29#2398C411yescharacteristic, complemented
30#2388C2211yescharacteristic
31C2×#393C411yescharacteristic, complemented
32#2364—21no—
33#2358—21no—
34#2353C2211yescharacteristic
35#2273C2212yes—
36#1789C2214yescomplemented
37C4×#99—21no—
38C4×#99—21no—
39C4×#99C2212yes—
40C4×#99C2212yes—
41C2×#342—21no—
42#1508C2214yescomplemented
43#1465—21no—
44#1465C2212yes—
45#1462—21no—
46#1462C2212yes—
47#1460—21no—
48#1458—21no—
49#1297C2214yescomplemented
50#1284C2214yescomplemented
51C22×#41C2311yescharacteristic
52C23×Q8C2×C411yescharacteristic, complemented
53C23×D4D411yescharacteristic
54#548—44no—
55C2×#124—21no—
56C2×#124C2×C411yescharacteristic, complemented
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2311yescharacteristic
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C411yescharacteristic, complemented
63C2×#120—21no—
64C2×#120—21no—
65#518—44no—
66C2×C4×Q8C2×C411yescharacteristic, complemented
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4C2311yescharacteristic
85#436—44no—
86#431—22no—
87C2×#104—44no—
88C2×#99—28no—
89C2×#99—22no—
90C2×#99—22no—
91#407—22no—
92#407C2314yes—
93#400—21no—
94#398C2×C411yescharacteristic, complemented
95#397—21no—
96#397—21no—
97#397C2×C411yescharacteristic
98#397C2×C411yescharacteristic
99#397C2×C411yescharacteristic, complemented
100#395C2×C411yescharacteristic, complemented
101#395C2×C411yescharacteristic, complemented
102#394—21no—
103#394—21no—
104#394—21no—
105#394—21no—
106#394—21no—
107#394C2×C411yescharacteristic, complemented
108#393—21no—
109#393—21no—
110#393C2×C411yescharacteristic
111#393C2×C411yescharacteristic
112#393C2×C411yescharacteristic
113#393C2×C411yescharacteristic
114#393C2×C411yescharacteristic, complemented
115#393C2×C411yescharacteristic, complemented
116#392—21no—
117#392—21no—
118#392C2×C411yescharacteristic
119#392C2×C411yescharacteristic
120#392C2×C411yescharacteristic, complemented
121#391—21no—
122#390—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#386—21no—
145#385—21no—
146#385—21no—
147#385—21no—
148#383C2×C411yescharacteristic, complemented
149#381—21no—
150#381C2311yescharacteristic
151#380C2×C411yescharacteristic, complemented
152#379—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—22no—
158C4×C22⋊C4—44no—
159C4×C22⋊C4C2314yes—
160C4×C22⋊C4C2311yescharacteristic
161#376C2×C411yescharacteristic, complemented
162C2×#95—21no—
163C2×#95C2×C411yescharacteristic, complemented
164#342—22no—
165#342—22no—
166#328—44no—
167#324—22no—
168#323—22no—
169#323—22no—
170#323—22no—
171C25C22⋊C411yescharacteristic
172C25C2×D411yescharacteristic
173C2×#41—24no—
174C2×#41—82no—
175C22×Q8—21no—
176C22×Q8—42no—
177C22×Q8—82no—
178C22×Q8C22×C411yescharacteristic
179C22×Q8C22×C411yescharacteristic
180C22×D4—24no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4C22⋊C411yescharacteristic
189C22×D4C22⋊C411yescharacteristic
190C22×D4C22×C411yescharacteristic
191C22×D4C2×D411yescharacteristic
192C22×D4C2×D411yescharacteristic
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—42no—
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2411yescharacteristic
206C2×M16—24no—
207C2×M16—42no—
208#124—42no—
209#124—44no—
210C22⋊Q8—24no—
211C22⋊Q8—44no—
212C22⋊Q8—82no—
213#121—42no—
214#121—44no—
215#121—44no—
216#120—44no—
217#120—82no—
218C4×Q8—42no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—44no—
228C4×D4—82no—
229#117—42no—
230#117—42no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22×C411yescharacteristic
277C2×C22⋊C4C22×C411yescharacteristic
278C2×C22⋊C4C22×C411yescharacteristic
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C22⋊C4C2×D411yescharacteristic
281C2×C22⋊C4C2×D411yescharacteristic
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—24no—
288C2×C42—42no—
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C4⋊C8—42no—
294#104—88no—
295#99—28no—
296#99—28no—
297C22⋊C8—24no—
298C22⋊C8—24no—
299C22⋊C8—42no—
300C4×C8—42no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—24no—
319#95—42no—
320#95C22×C411yescharacteristic
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323#95C22×C411yescharacteristic
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—41no—
331C24—81no—
332C24—81no—
333C24—81no—
334C24—81no—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24C22×D411yescharacteristic
340#41—48no—
341#41—84no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—84no—
350C2×Q8—84no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—42no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4#10414yescomplemented
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—162no—
419C22×C4—162no—
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4H(5,𝔽2)14yes—
431M16—84no—
432M16—162no—
433C2×C8—44no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—28no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C42—22no—
482C42—22no—
483C42—41no—
484C42—41no—
485C42—42no—
486C42—81no—
487C42—81no—
488C42—84no—
489C42—162no—
490C23—24no—
491C23—24no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—41no—
504C23—41no—
505C23—44no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—82no—
522C23—82no—
523C23—84no—
524C23—161no—
525C23—161no—
526C23—162no—
527C23—162no—
528C23—162no—
529C23C2×#10412yes—
530C23C2×#10412yes—
531C23#42112yes—
532C23C22×C22⋊C411yescharacteristic
533C23#53412yes—
534Q8—24no—
535Q8—42no—
536Q8—82no—
537Q8—84no—
538D4—44no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—84no—
544D4—84no—
545D4—84no—
546D4—162no—
547D4—162no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—48no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—84no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—161no—
599C2×C4—161no—
600C2×C4—161no—
601C2×C4—161no—
602C2×C4—162no—
603C2×C4—168no—
604C2×C4C2×#10412yes—
605C2×C4#51812yes—
606C2×C4#53412yes—
607C8—84no—
608C22—21no—
609C22—21no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—84no—
640C22—84no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—164no—
645C22#230811yescharacteristic
646C22C22×#10411yescharacteristic
647C22C2×#42111yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—88no—
664C4—81no—
665C4—81no—
666C2—21no—
667C2—21no—
668C2—42no—
669C2—42no—
670C2—42no—
671C2—42no—
672C2—81no—
673C2—161no—
674C2#2177311yescharacteristic
675C2#2177611yescharacteristic
676C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35703C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20392C2semidirect
3. #17369C2semidirect
4. #9475C2semidirect
5. #9407C2semidirect
6. #9184C2semidirect
7. #9010C2semidirect
8. #8776C2semidirect
9. #8366C2semidirect
15. #2631C4semidirect
16. #2617C4semidirect
17. #2610C4semidirect
18. #2599C4semidirect
20. #2543C4semidirect
27. #2409C4semidirect
29. #2398C4semidirect
31. C2×#393C4semidirect
36. #1789C22semidirect
42. #1508C22semidirect
49. #1297C22semidirect
50. #1284C22semidirect
52. C23×Q8C2×C4semidirect
56. C2×#124C2×C4semidirect
62. C2×#121C2×C4semidirect
66. C2×C4×Q8C2×C4semidirect
94. #398C2×C4semidirect
99. #397C2×C4semidirect
100. #395C2×C4semidirect
101. #395C2×C4semidirect
107. #394C2×C4semidirect
114. #393C2×C4semidirect
115. #393C2×C4semidirect
120. #392C2×C4semidirect
148. #383C2×C4semidirect
151. #380C2×C4semidirect
161. #376C2×C4semidirect
163. C2×#95C2×C4semidirect
381. C2×D4#104semidirect

Maximal subgroups

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

subgroupindex
#357032
#203922
#173692
#94752
#94072
#91842
#90102
#87762
#83662