Finite Group Databasebeta

← #188270

Subgroups — #188270

Order 512.

1950 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C42×D4C211yesmaximal, characteristic
2C2×#1797C212yesmaximal, complemented
3C2×#1459C212yesmaximal
4#8560C212yesmaximal, complemented
5#8559C212yesmaximal, complemented
6#8558C212yesmaximal, complemented
7#8557C212yesmaximal, complemented
8#7907C212yesmaximal, complemented
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C42C411yescharacteristic
13C23×C42C2211yescharacteristic
14C22×C4⋊C8—22no—
15C22×C22⋊C8C2214yes—
16C42×D4—22no—
17C42×D4—22no—
18C42×D4—22no—
19C42×D4—22no—
20C2×C4×C4⋊C4C411yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C411yescharacteristic
24C2×C4×C22⋊C4C411yescharacteristic
25C2×C4×C22⋊C4C411yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C43C411yescharacteristic
30#1797—22no—
31#1797—22no—
32#1797C2214yes—
33#1797C2214yes—
34C2×#336—22no—
35#1459—28no—
36C2×#324C2214yes—
37C2×#323C2214yes—
38#1280C2214yes—
39#1280C2214yes—
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—24no—
48C2×C4×D4—24no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—24no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4C818yescomplemented
59C2×C4×D4D414yes—
60C2×C4×D4Q814yes—
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×C42—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—44no—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42D412yes—
90C22×C42D412yes—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—48no—
96C2×C22⋊C8—28no—
97C2×C22⋊C8—28no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8C2318yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122C2×#95C2311yescharacteristic
123C2×#95C2311yescharacteristic
124C43—21no—
125C43—21no—
126C43—21no—
127C43—21no—
128#336—24no—
129#336—24no—
130#324—28no—
131#323—28no—
132C25C22×C411yescharacteristic
133C25C22×C411yescharacteristic
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C4⋊C414yes—
139C22×D4C2×C814yes—
140C22×D4M1614yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C22⋊C412yes—
166C23×C4C22⋊C412yes—
167C23×C4C22⋊C412yes—
168C23×C4C22⋊C412yes—
169C23×C4C4⋊C414yes—
170C23×C4C2×C814yes—
171C23×C4C2×C814yescomplemented
172C23×C4M1614yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
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×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×Q812yes—
189C23×C4#4114yes—
190C23×C4C2411yescharacteristic
191C22×C8—28no—
192C22×C8—48no—
193C22×C8—48no—
194C22×C8—48no—
195C4×D4—216no—
196C4×D4—216no—
197C4×D4—216no—
198C4×D4—216no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4C2×C814yescomplemented
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×Q812yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4C4⋊C414yes—
227C2×C22⋊C4C2×C814yes—
228C2×C22⋊C4C2×C814yescomplemented
229C2×C22⋊C4M1614yes—
230C2×C22⋊C4C2×D412yes—
231C2×C22⋊C4C2×Q812yes—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—28no—
243C2×C42—24no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—42no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C412yes—
276C2×C42C22⋊C412yes—
277C2×C42C22⋊C411yescharacteristic
278C2×C42C4⋊C414yes—
279C2×C42C2×C814yescomplemented
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C4⋊C8—48no—
285C4⋊C8—48no—
286C22⋊C8—28no—
287C22⋊C8—28no—
288C22⋊C8—48no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24C2×C22⋊C412yes—
308C24C2×C4⋊C412yes—
309C24C2×C4⋊C412yes—
310C24#11712yes—
311C24C22×C812yes—
312C24C22×C812yes—
313C24C2×M1612yes—
314C24C2×M1612yes—
315C24C2×M1612yes—
316C24C2×M1612yes—
317C24C23×C411yescharacteristic
318C2×D4—28no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4C4⋊C8116yescomplemented
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4C22⋊C818yes—
415C22×C4C22⋊C818yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C4⋊C412yes—
425C22×C4C2×C4⋊C412yes—
426C22×C4C2×C4⋊C412yes—
427C22×C4C2×C4⋊C412yes—
428C22×C4#11712yes—
429C22×C4#12114yes—
430C22×C4#12114yes—
431C22×C4C22⋊Q814yes—
432C22×C4C22⋊Q814yes—
433C22×C4C22×C812yes—
434C22×C4C22×C812yes—
435C22×C4C22×C812yes—
436C22×C4C22×C812yes—
437C22×C4C22×C812yes—
438C22×C4C2×M1612yes—
439C22×C4C2×M1612yes—
440C22×C4C2×M1612yes—
441C22×C4C2×M1612yes—
442C22×C4C2×M1612yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×Q811yescharacteristic
446C22×C4C2×#4112yes—
447C2×C8—48no—
448C2×C8—48no—
449C2×C8—48no—
450C2×C8—816no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—28no—
462C42—24no—
463C42—24no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42#9914yes—
490C23—21no—
491C23—22no—
492C23—21no—
493C23—22no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23#38014yes—
533C23C2×C22⋊C814yes—
534C23#40714yes—
535C23C2×C4⋊C814yes—
536C23C2×C4⋊C814yescomplemented
537C23#42314yes—
538C23#43114yes—
539C23#43214yes—
540C23C22×C22⋊C411yescharacteristic
541C23C22×C4⋊C411yescharacteristic
542C23C2×#11711yescharacteristic
543C23C23×C811yescharacteristic
544C23C22×M1611yescharacteristic
545C23C22×M1611yescharacteristic
546C23C22×M1611yescharacteristic
547D4—232no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—216no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4#38012yes—
627C2×C4#38012yes—
628C2×C4C2×C22⋊C812yes—
629C2×C4C2×C22⋊C814yes—
630C2×C4C2×C22⋊C812yes—
631C2×C4C2×#9912yes—
632C2×C4C2×#9912yes—
633C2×C4C2×#9912yes—
634C2×C4#41012yes—
635C2×C4#41312yes—
636C2×C4#41312yes—
637C2×C4C2×C4⋊C814yescomplemented
638C2×C4C2×#12112yes—
639C2×C4C2×C22⋊Q812yes—
640C8—88no—
641C22—21no—
642C22—21no—
643C22—24no—
644C22—21no—
645C22—21no—
646C22—24no—
647C22—21no—
648C22—21no—
649C22—24no—
650C22—21no—
651C22—21no—
652C22—24no—
653C22—21no—
654C22—21no—
655C22—24no—
656C22—21no—
657C22—21no—
658C22—24no—
659C22—24no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22#179718yes—
675C22C2×#38011yescharacteristic
676C22C22×C22⋊C811yescharacteristic
677C22C2×#40711yescharacteristic
678C22C22×#9911yescharacteristic
679C22C2×#41011yescharacteristic
680C22C2×#41311yescharacteristic
681C22C2×#41311yescharacteristic
682C22C22×C4⋊C811yescharacteristic
683C22C2×#42311yescharacteristic
684C22C2×#43111yescharacteristic
685C22C2×#43211yescharacteristic
686C4—24no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—24no—
692C4—22no—
693C4—22no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4#145914yescomplemented
702C4#179914yes—
703C2—21no—
704C2—21no—
705C2—21no—
706C2—21no—
707C2—28no—
708C2—44no—
709C2C2×#145911yescharacteristic
710C2#1734911yescharacteristic
711C2#1736211yescharacteristic
712C2#1736311yescharacteristic
713C2C2×#179711yescharacteristic
714C2C2×#179911yescharacteristic
715C2#2063711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
4. #8560C2semidirect
5. #8559C2semidirect
6. #8558C2semidirect
7. #8557C2semidirect
8. #7907C2semidirect
58. C2×C4×D4C8semidirect
171. C23×C4C2×C8semidirect
210. C2×C4⋊C4C2×C8semidirect
228. C2×C22⋊C4C2×C8semidirect
279. C2×C42C2×C8semidirect
324. C2×D4C4⋊C8semidirect
536. C23C2×C4⋊C8semidirect
637. C2×C4C2×C4⋊C8semidirect
701. C4#1459semidirect

Maximal subgroups

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

subgroupindex
C2×C42×D42
C2×#17972
C2×#14592
#85602
#85592
#85582
#85572
#79072