Finite Group Databasebeta

← #214133

Subgroups — #214133

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#21847C211yesmaximal, characteristic
3#21373C211yesmaximal, characteristic
4#9073C212yesmaximal, complemented
5#9071C212yesmaximal, complemented
6#8938C212yesmaximal, complemented
7#8937C212yesmaximal, complemented
8#8708C212yesmaximal, complemented
9#8311C212yesmaximal, complemented
10C2×#536C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2327—21no—
18#2326—21no—
19#2323C412yes—
20#2323C412yes—
21#2322—21no—
22#2322—21no—
23#2317C411yescharacteristic
24#2311C411yescharacteristic
25#2310C411yescharacteristic
26C4×C4⋊Q8—21no—
27C4×#124—21no—
28C4×C22⋊Q8C2212yes—
29C4×C22⋊Q8C2212yes—
30C4×#123—21no—
31C4×#123—21no—
32C4×#117C2211yescharacteristic
33#1984—21no—
34#1984—21no—
35#1954—21no—
36#1954—21no—
37#1869—21no—
38#1869—21no—
39#1869—21no—
40#1869—21no—
41#1797C2212yes—
42#1797C2212yes—
43C4×C22⋊C8C2212yes—
44C4×C22⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×#41C2×C411yescharacteristic
54C23×Q8C2×C411yescharacteristic
55C23×C8—21no—
56C23×C8—21no—
57#536—24no—
58#536C818yescomplemented
59C2×C4⋊Q8—21no—
60C2×#124—21no—
61C2×#123—21no—
62C2×#123—21no—
63C2×C22⋊Q8C2×C412yes—
64C2×C22⋊Q8C2×C412yes—
65C4×#41—41no—
66C4×#41—41no—
67C4×#41—41no—
68C4×#41—41no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—41no—
72C2×C4×Q8—41no—
73C2×C4×Q8—41no—
74C2×C4×Q8—41no—
75C2×C4×Q8C2×C412yes—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8C2312yes—
79C2×C4×D4D411yescharacteristic
80C2×C4×D4D411yescharacteristic
81C2×C4×D4D411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×#117C2×C411yescharacteristic
86C2×#117C2×C411yescharacteristic
87C2×#117C2×C411yescharacteristic
88C2×#117C2×C411yescharacteristic
89C2×#117C2×C411yescharacteristic
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2×C411yescharacteristic
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#431—41no—
97#431—41no—
98#431—41no—
99#431—41no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—41no—
103C2×C4⋊C8—41no—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—41no—
115C2×C4×C8—41no—
116C2×C4×C8C2312yes—
117C2×C4×C8C2312yes—
118#391—21no—
119#391—21no—
120#391—21no—
121#391C2×C412yes—
122#391C2×C412yes—
123#390—21no—
124#389—22no—
125#389—22no—
126#389C2×C412yes—
127#389C2×C412yes—
128#388—22no—
129#388—22no—
130#388—22no—
131#388—22no—
132#387—22no—
133#387—22no—
134#387—22no—
135#387—22no—
136#386—21no—
137#386—21no—
138#385—22no—
139#385—22no—
140#385—21no—
141#385—21no—
142#384—21no—
143#382—21no—
144#381—21no—
145#381—21no—
146#380C2×C412yes—
147#380C2×C412yes—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4C2×C412yes—
151C4×C4⋊C4C2312yes—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156#376—21no—
157C43—21no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162Q8⋊C8—42no—
163Q8⋊C8—42no—
164Q8⋊C8—42no—
165Q8⋊C8—42no—
166C2×#41—41no—
167C2×#41—41no—
168C2×#41—41no—
169C2×#41—41no—
170C2×#41C2×C814yes—
171C2×#41M1614yes—
172C22×Q8—21no—
173C22×Q8—22no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8C22⋊C411yescharacteristic
179C22×Q8C2×C814yescomplemented
180C22×Q8C22×C412yes—
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22⋊C411yescharacteristic
183C22×D4C22⋊C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—44no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—82no—
203C4⋊Q8—28no—
204#124—28no—
205#123—28no—
206#123—28no—
207C22⋊Q8—24no—
208C22⋊Q8—24no—
209C22⋊Q8C2×C818yescomplemented
210C22⋊Q8C2×C818yescomplemented
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—42no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237#117—22no—
238#117—41no—
239#117—41no—
240#117—41no—
241#117—41no—
242#117—41no—
243#117—41no—
244#117—41no—
245#117—41no—
246#117C2×C814yescomplemented
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C4⋊C4C22×C412yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
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×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2411yescharacteristic
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315C22⋊C8—44no—
316C22⋊C8—44no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—82no—
320C22⋊C8—82no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—42no—
326C4×C8—42no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340#41—44no—
341#41—44no—
342#41—44no—
343#41—44no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8C22⋊C814yes—
359C2×Q8C22×C814yes—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4C22⋊C814yes—
367C2×D4C22⋊C814yes—
368C2×D4C22⋊C814yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—82no—
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C22×C812yes—
421C22×C4C22×C812yes—
422C22×C4C22×C812yes—
423C22×C4C2×M1612yes—
424C22×C4C2×M1612yes—
425C22×C4C2×M1612yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C2×C8—24no—
430C2×C8—42no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—84no—
443C4⋊C4—22no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4C22×C814yes—
463C22⋊C4—24no—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—24no—
494C42—44no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—82no—
501C42H(5,𝔽2)11yescharacteristic
502C42H(5,𝔽2)11yescharacteristic
503C42H(5,𝔽2)11yescharacteristic
504C42H(5,𝔽2)11yescharacteristic
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23C2×C22⋊C812yes—
518C23C2×C22⋊C812yes—
519C23C2×C22⋊C812yes—
520C23C22×C22⋊C411yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—44no—
524Q8—48no—
525Q8—42no—
526Q8—42no—
527D4—44no—
528D4—44no—
529D4—44no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—84no—
590C2×C4C2×C22⋊C812yes—
591C2×C4C2×C22⋊C812yes—
592C2×C4C2×C22⋊C812yes—
593C2×C4#51811yescharacteristic
594C2×C4#51811yescharacteristic
595C2×C4#53411yescharacteristic
596C2×C4#53411yescharacteristic
597C2×C4#53411yescharacteristic
598C2×C4#53411yescharacteristic
599C2×C4C23×C811yescharacteristic
600C2×C4C22×M1611yescharacteristic
601C2×C4#56811yescharacteristic
602C2×C4#56811yescharacteristic
603C2×C4#57712yes—
604C2×C4#57812yes—
605C8—44no—
606C8—82no—
607C8—82no—
608C8—82no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22#230811yescharacteristic
625C22C22×C22⋊C811yescharacteristic
626C22#287911yescharacteristic
627C22#289812yes—
628C22#290112yes—
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—44no—
641C4—44no—
642C4—82no—
643C4—82no—
644C4#297211yescharacteristic
645C4#297211yescharacteristic
646C4#297311yescharacteristic
647C4#297311yescharacteristic
648C4#320011yescharacteristic
649C4#320112yes—
650C4#320211yescharacteristic
651C2—42no—
652C2—42no—
653C2—42no—
654C2#1743912yes—
655C2#1750612yes—
656C2#2173811yescharacteristic
657C2#2177111yescharacteristic
658C2#2177511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9073C2semidirect
5. #9071C2semidirect
6. #8938C2semidirect
7. #8937C2semidirect
8. #8708C2semidirect
9. #8311C2semidirect
58. #536C8semidirect
179. C22×Q8C2×C8semidirect
209. C22⋊Q8C2×C8semidirect
210. C22⋊Q8C2×C8semidirect
246. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#218472
#213732
#90732
#90712
#89382
#89372
#87082
#83112