Finite Group Databasebeta

← #214959

Subgroups — #214959

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35840C211yesmaximal, characteristic
2#20392C211yesmaximal, characteristic, complemented
3#17369C211yesmaximal, characteristic, complemented
4#9477C212yesmaximal, complemented
5#9409C212yesmaximal, complemented
6#9186C212yesmaximal, complemented
7#9012C212yesmaximal, complemented
8#8776C212yesmaximal, complemented
9#8362C212yesmaximal, complemented
10C2×#553C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2975C2212yes—
13C22×#99—21no—
14#2635C2211yescharacteristic
15#2633C411yescharacteristic, complemented
16#2617C411yescharacteristic, complemented
17#2611C411yescharacteristic, complemented
18#2598C411yescharacteristic, complemented
19#2543C411yescharacteristic, complemented
20#2539C2211yescharacteristic
21#2487C411yescharacteristic, complemented
22#2482C2211yescharacteristic
23#2479—21no—
24#2467—21no—
25#2455—21no—
26#2443—21no—
27#2418—21no—
28#2410—21no—
29#2397C411yescharacteristic, complemented
30#2395C2211yescharacteristic
31C2×#399C411yescharacteristic, complemented
32#2361—21no—
33#2358—21no—
34#2353C2211yescharacteristic
35#2273C2212yes—
36#1826C2214yescomplemented
37C4×#99—21no—
38C4×#99—21no—
39C4×#99C2212yes—
40C4×#99C2212yes—
41C2×#342—21no—
42#1507C2214yescomplemented
43#1465—21no—
44#1465C2212yes—
45#1462—21no—
46#1462C2212yes—
47#1460—21no—
48#1458—21no—
49#1297C2214yescomplemented
50#1284C2214yescomplemented
51C23×D4D411yescharacteristic
52#553—44no—
53C2×#126—21no—
54C2×#125C2×C411yescharacteristic, complemented
55C2×#123—21no—
56C2×#123—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121C2×C411yescharacteristic, complemented
60C2×#121C2311yescharacteristic
61C2×#120—21no—
62C2×#120—21no—
63#518—44no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic, complemented
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic, complemented
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4C2311yescharacteristic
87#436—44no—
88#431—22no—
89C2×#99—28no—
90C2×#99—22no—
91C2×#99—22no—
92#407—22no—
93#407C2314yes—
94#400C2×C411yescharacteristic, complemented
95#400C2×C411yescharacteristic, complemented
96#399—21no—
97#399—21no—
98#399—21no—
99#399C2×C411yescharacteristic
100#399C2×C411yescharacteristic
101#399C2×C411yescharacteristic
102#399C2×C411yescharacteristic
103#399C2×C411yescharacteristic, complemented
104#399C2×C411yescharacteristic, complemented
105#398—21no—
106#398C2×C411yescharacteristic, complemented
107#397—21no—
108#397C2×C411yescharacteristic
109#397C2×C411yescharacteristic
110#397C2×C411yescharacteristic, complemented
111#394—21no—
112#394—21no—
113#394—21no—
114#394—21no—
115#394—21no—
116#394C2×C411yescharacteristic, complemented
117#392—21no—
118#392—21no—
119#392C2×C411yescharacteristic
120#392C2×C411yescharacteristic
121#392C2×C411yescharacteristic, complemented
122#390—21no—
123#390—21no—
124#389—21no—
125#389—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#387—21no—
134#387—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#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#383C2×C411yescharacteristic, complemented
150#381C2311yescharacteristic
151#380—21no—
152#380C2×C411yescharacteristic, complemented
153#379—21no—
154C4×C4⋊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×#95C2×C411yescharacteristic, complemented
163#342—22no—
164#342—22no—
165#339—44no—
166#327—44no—
167#324—22no—
168#323—22no—
169#323—22no—
170#323—22no—
171C25C22⋊C411yescharacteristic
172C25C2×D411yescharacteristic
173C22×D4—24no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22⋊C411yescharacteristic
183C22×D4C22×C411yescharacteristic
184C22×D4C2×D411yescharacteristic
185C22×D4C2×D411yescharacteristic
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—42no—
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
199C2×M16—24no—
200C2×M16—42no—
201#126—44no—
202#125—42no—
203#123—44no—
204#123—82no—
205#121—24no—
206#121—42no—
207#121—44no—
208#121—44no—
209#120—44no—
210#120—82no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—44no—
221C4×D4—82no—
222C4×D4—82no—
223#117—24no—
224#117—42no—
225#117—42no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
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⋊C4—42no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
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—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22×C411yescharacteristic
281C2×C22⋊C4C22×C411yescharacteristic
282C2×C22⋊C4C22×C411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2×D411yescharacteristic
285C2×C22⋊C4C2×D411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—24no—
292C2×C42—42no—
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C4⋊C8—42no—
298#99—28no—
299#99—28no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302C22⋊C8—42no—
303C4×C8—42no—
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—24no—
318#95—42no—
319#95C22×C411yescharacteristic
320#95C22×C411yescharacteristic
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—41no—
330C24—81no—
331C24—81no—
332C24—81no—
333C24—81no—
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C22×D411yescharacteristic
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—42no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—81no—
362C2×D4—81no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—81no—
366C2×D4—81no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4#10414yes—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—48no—
392C22×C4—42no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—162no—
407C22×C4—162no—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4H(5,𝔽2)14yes—
419M16—84no—
420M16—162no—
421C2×C8—44no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—28no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C42—22no—
477C42—22no—
478C42—41no—
479C42—41no—
480C42—42no—
481C42—42no—
482C42—81no—
483C42—84no—
484C42—162no—
485C23—24no—
486C23—24no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—41no—
499C23—41no—
500C23—44no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—82no—
517C23—82no—
518C23—84no—
519C23—161no—
520C23—161no—
521C23—162no—
522C23—162no—
523C23—162no—
524C23C2×#10412yes—
525C23C2×#10412yes—
526C23#42112yes—
527C23C22×C22⋊C411yescharacteristic
528C23#53412yes—
529D4—44no—
530D4—44no—
531D4—44no—
532D4—44no—
533D4—82no—
534D4—84no—
535D4—84no—
536D4—84no—
537D4—162no—
538D4—162no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—48no—
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—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—161no—
589C2×C4—161no—
590C2×C4—161no—
591C2×C4—161no—
592C2×C4—162no—
593C2×C4—162no—
594C2×C4—168no—
595C2×C4C2×#10412yes—
596C2×C4#51812yes—
597C2×C4#53412yes—
598C8—84no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—84no—
631C22—84no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—164no—
636C22#230811yescharacteristic
637C22C22×#10411yescharacteristic
638C22C2×#42111yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—88no—
654C4—81no—
655C4—81no—
656C4—81no—
657C2—21no—
658C2—21no—
659C2—42no—
660C2—42no—
661C2—42no—
662C2—42no—
663C2—81no—
664C2—161no—
665C2#2177311yescharacteristic
666C2#2177711yescharacteristic
667C2#2178211yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35840C2
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. #9477C2semidirect
5. #9409C2semidirect
6. #9186C2semidirect
7. #9012C2semidirect
8. #8776C2semidirect
9. #8362C2semidirect
15. #2633C4semidirect
16. #2617C4semidirect
17. #2611C4semidirect
18. #2598C4semidirect
19. #2543C4semidirect
21. #2487C4semidirect
29. #2397C4semidirect
31. C2×#399C4semidirect
36. #1826C22semidirect
42. #1507C22semidirect
49. #1297C22semidirect
50. #1284C22semidirect
54. C2×#125C2×C4semidirect
59. C2×#121C2×C4semidirect
69. C2×C4×D4C2×C4semidirect
80. C22×C4⋊C4C2×C4semidirect
94. #400C2×C4semidirect
95. #400C2×C4semidirect
103. #399C2×C4semidirect
104. #399C2×C4semidirect
106. #398C2×C4semidirect
110. #397C2×C4semidirect
116. #394C2×C4semidirect
121. #392C2×C4semidirect
149. #383C2×C4semidirect
152. #380C2×C4semidirect
161. #376C2×C4semidirect
162. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#358402
#203922
#173692
#94772
#94092
#91862
#90122
#87762
#83622