Finite Group Databasebeta

← #216179

Subgroups — #216179

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#126C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#9880C212yesmaximal, complemented
5#9651C212yesmaximal
6#9082C212yesmaximal, complemented
7#9080C212yesmaximal
8#8560C212yesmaximal, complemented
9#8559C212yesmaximal
10C22×#126C411yescharacteristic
11C2×#431C2212yes—
12C4×#126—21no—
13C4×#126—21no—
14C4×#126—21no—
15C4×#126—21no—
16C4×#126—21no—
17C4×#126—21no—
18C4×#126—21no—
19C4×#126—21no—
20C2×#388C411yescharacteristic
21C2×#388C2211yescharacteristic
22C2×#388C2211yescharacteristic
23C2×#387C411yescharacteristic
24C2×#387C2211yescharacteristic
25C2×#387C2211yescharacteristic
26C2×#383C411yescharacteristic
27C2×C4×C4⋊C4C411yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C411yescharacteristic
31C2×C4×C22⋊C4C411yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1842—21no—
35#1839—21no—
36#1831—21no—
37#1831—21no—
38#1817C2212yes—
39#1797C2212yes—
40#1797C2212yes—
41#1797C2212yes—
42C4×C4⋊C8—21no—
43C4×C4⋊C8—21no—
44C4×C4⋊C8—21no—
45C4×C22⋊C8C2212yes—
46#1744—21no—
47C2×#324C2214yes—
48C2×#323C2214yes—
49#1280C2214yes—
50#1280C2214yes—
51C24×C4C2×C411yescharacteristic
52C2×#126—21no—
53C2×#126—21no—
54C2×#126—21no—
55C2×#126—21no—
56C2×#126—24no—
57C2×#126C818yescomplemented
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#431—44no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8C2314yes—
86C2×C22⋊C8C2314yes—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#388—21no—
90#388—21no—
91#388—21no—
92#388—21no—
93#388—21no—
94#388—21no—
95#388—21no—
96#388—21no—
97#388—21no—
98#388—21no—
99#388—21no—
100#388—21no—
101#387—21no—
102#387—21no—
103#387—21no—
104#387—21no—
105#387—21no—
106#387—21no—
107#387—21no—
108#387—21no—
109#387—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#383—21no—
114#383—21no—
115#383—21no—
116#383—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C2×#95C2×C411yescharacteristic
142C2×#95C2×C411yescharacteristic
143C2×#95C2×C411yescharacteristic
144C2×#95C2×C411yescharacteristic
145C2×#95C2×C411yescharacteristic
146C2×#95C2×C411yescharacteristic
147C2×#95C2×C411yescharacteristic
148C2×#95C2×C411yescharacteristic
149C2×#95C2×C411yescharacteristic
150C2×#95C2×C411yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C43D411yescharacteristic
154C43D411yescharacteristic
155C43D411yescharacteristic
156C43D411yescharacteristic
157C43D411yescharacteristic
158C43D411yescharacteristic
159C43D411yescharacteristic
160C43D411yescharacteristic
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166#336—22no—
167#324—44no—
168#323—44no—
169C25C22×C411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2411yescharacteristic
189C22×C8—24no—
190C22×C8—24no—
191#126—28no—
192#126—28no—
193#126—28no—
194#126—28no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×C814yes—
215C2×C4⋊C4C2×C814yes—
216C2×C4⋊C4C2×C814yes—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4C2×C814yes—
237C2×C22⋊C4C2×C814yes—
238C2×C22⋊C4C2×C814yes—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C2×C814yes—
272C2×C42M1614yes—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C4⋊C8—44no—
286C4⋊C8—44no—
287C4⋊C8—44no—
288C4⋊C8—44no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293C4×C8—44no—
294C4×C8—44no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
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—21no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24—41no—
324C24—41no—
325C24—41no—
326C24—41no—
327C24C22×C812yes—
328C24C23×C411yescharacteristic
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—88no—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C22×C812yes—
394C22×C4C22×C812yes—
395C22×C4C22×C812yes—
396C22×C4C22×C812yes—
397C22×C4C22×C812yes—
398C22×C4C22×C812yes—
399C22×C4C2×M1612yes—
400C22×C4C2×M1612yes—
401C22×C4C2×M1612yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4H(5,𝔽2)12yes—
405C22×C42−1+412yes—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42C22⋊C814yes—
471C42C22⋊C814yes—
472C42C22⋊C814yes—
473C42C22⋊C814yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—24no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23C22×C22⋊C411yescharacteristic
496C23#52012yes—
497C23C23×C811yescharacteristic
498C23C22×M1611yescharacteristic
499C23#56812yes—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
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—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4C2×C22⋊C812yes—
575C2×C4C2×C22⋊C812yes—
576C2×C4C2×C22⋊C812yes—
577C2×C4C2×C22⋊C812yes—
578C2×C4C2×C22⋊C812yes—
579C2×C4C2×C22⋊C812yes—
580C2×C4#41012yes—
581C2×C4#41012yes—
582C2×C4#41312yes—
583C2×C4#41312yes—
584C2×C4#53511yescharacteristic
585C2×C4#53611yescharacteristic
586C2×C4#53712yes—
587C8—84no—
588C8—84no—
589C22—22no—
590C22—22no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22—84no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22#231111yescharacteristic
607C22C22×C22⋊C811yescharacteristic
608C22#287911yescharacteristic
609C22C2×#41011yescharacteristic
610C22C2×#41011yescharacteristic
611C22C2×#41311yescharacteristic
612C22C2×#41311yescharacteristic
613C22#298812yes—
614C22#298912yes—
615C4—22no—
616C4—22no—
617C4—21no—
618C4—22no—
619C4—21no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—81no—
637C2—22no—
638C2—22no—
639C2—82no—
640C2#1734911yescharacteristic
641C2#1734911yescharacteristic
642C2#1739511yescharacteristic
643C2#1739811yescharacteristic
644C2#2184711yescharacteristic
645C2#2187511yescharacteristic
646C2#2188011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
3. #20777C2semidirect
4. #9880C2semidirect
6. #9082C2semidirect
8. #8560C2semidirect
57. C2×#126C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1262
#208082
#207772
#98802
#96512
#90822
#90802
#85602
#85592