Finite Group Databasebeta

← #355929

Subgroups — #355929

Order 512.

3626 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#407C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3C22×C4×C22⋊C4C211yesmaximal, characteristic
4#10733C214yesmaximal, complemented
5#10732C214yesmaximal, complemented
6#10719C214yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C23×M16C2212yes—
9C22×#117C2212yes—
10C23×C22⋊C4—21no—
11C23×C42C2212yes—
12C2×#407—24no—
13C2×#407—21no—
14C2×#407—21no—
15C2×#407C418yescomplemented
16C2×#407C2212yes—
17C2×#407C2212yes—
18C22×C22⋊C8—21no—
19C22×C22⋊C8—21no—
20#2280—21no—
21#2280—21no—
22#2280—21no—
23#2280—21no—
24#2280C412yescomplemented
25#2280C412yescomplemented
26#2280C2212yes—
27#2280C2212yes—
28C2×#381—21no—
29C2×#380C412yescomplemented
30C2×#380C412yescomplemented
31C2×#379—21no—
32C2×C4×C22⋊C4—24no—
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4—21no—
36C2×C4×C22⋊C4—21no—
37C2×C4×C22⋊C4C418yescomplemented
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95—21no—
41C2×#339C22116yescomplemented
42C26D411yescharacteristic
43C24×C4—22no—
44C24×C4—48no—
45C24×C4C2×C414yes—
46C24×C4C2×C414yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D414yes—
57C24×C4D411yescharacteristic
58C24×C4D411yescharacteristic
59C24×C4Q814yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16—24no—
64C22×M16—22no—
65C22×M16—48no—
66C22×M16C2×C418yescomplemented
67C22×M16C2314yes—
68C23×C8—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—48no—
72C2×#117C2×C414yescomplemented
73C2×#117C2314yes—
74C22×C4⋊C4C2×C414yescomplemented
75C22×C22⋊C4—216no—
76C22×C22⋊C4—28no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—22no—
83C22×C42—24no—
84C22×C42—22no—
85C22×C42—48no—
86C22×C42C2×C418yescomplemented
87C22×C42C2314yes—
88#407—28no—
89#407—28no—
90#407—24no—
91#407—24no—
92#407C2×C418yescomplemented
93#407C2×C418yescomplemented
94C2×C22⋊C8—28no—
95C2×C22⋊C8—28no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100#381—22no—
101#381—22no—
102#381—22no—
103#381—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380C2×C414yes—
109#380C2×C414yes—
110#380C2×C414yescomplemented
111#380C2×C414yescomplemented
112#379—22no—
113#379—22no—
114#379—22no—
115#379—22no—
116C4×C22⋊C4—28no—
117C4×C22⋊C4—28no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—24no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C418yes—
129C4×C22⋊C4C2×C418yescomplemented
130C2×#95—28no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—22no—
135#339—416no—
136C25—28no—
137C25—44no—
138C25—44no—
139C25C22⋊C411yescharacteristic
140C25C22⋊C411yescharacteristic
141C25C22⋊C414yes—
142C25C22⋊C411yescharacteristic
143C25C22⋊C411yescharacteristic
144C25C4⋊C414yes—
145C25C2×D411yescharacteristic
146C25C2×D411yescharacteristic
147C25C2×D411yescharacteristic
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—216no—
152C23×C4—28no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—416no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—416no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—416no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—416no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—48no—
171C23×C4—48no—
172C23×C4C4214yes—
173C23×C4C4214yes—
174C23×C4C4214yes—
175C23×C4C4214yes—
176C23×C4C22⋊C414yes—
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C414yes—
181C23×C4C22⋊C414yes—
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C414yes—
192C23×C4C22⋊C414yes—
193C23×C4C22⋊C414yes—
194C23×C4C22⋊C414yes—
195C23×C4C4⋊C414yes—
196C23×C4C4⋊C414yes—
197C23×C4C4⋊C414yes—
198C23×C4C4⋊C414yes—
199C23×C4C4⋊C414yes—
200C23×C4C4⋊C414yes—
201C23×C4C4⋊C414yes—
202C23×C4C4⋊C414yes—
203C23×C4C4⋊C414yes—
204C23×C4C4⋊C414yes—
205C23×C4C4⋊C414yes—
206C23×C4C22×C412yes—
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C412yes—
212C23×C4C22×C412yes—
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×Q812yes—
231C23×C4C2×Q812yes—
232C23×C4C2×Q812yes—
233C23×C4C2411yescharacteristic
234C2×M16—28no—
235C2×M16—24no—
236C2×M16—216no—
237C2×M16—416no—
238C2×M16—416no—
239C2×M16—48no—
240C2×M16C22×C418yes—
241C22×C8—28no—
242C22×C8—24no—
243C22×C8—48no—
244#117—216no—
245#117—48no—
246#117—48no—
247#117—48no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4C22×C418yes—
251C2×C22⋊C4—264no—
252C2×C22⋊C4—216no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C42—216no—
266C2×C42—24no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—416no—
271C2×C42—48no—
272C2×C42—48no—
273C2×C42—416no—
274C2×C42C22×C418yes—
275C22⋊C8—28no—
276C22⋊C8—28no—
277C22⋊C8—28no—
278C22⋊C8—28no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—216no—
291C24—216no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—42no—
300C24—42no—
301C24—48no—
302C24—42no—
303C24—42no—
304C24—48no—
305C24—42no—
306C24—42no—
307C24—48no—
308C24—48no—
309C24—42no—
310C24—42no—
311C24—464no—
312C24#9514yes—
313C24#9514yes—
314C24#9514yes—
315C24#9514yes—
316C24#10418yes—
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C412yes—
320C24C2×C22⋊C412yes—
321C24C2×C22⋊C412yes—
322C24C2×C22⋊C411yescharacteristic
323C24C2×C22⋊C411yescharacteristic
324C24C2×C22⋊C411yescharacteristic
325C24C2×C22⋊C411yescharacteristic
326C24C2×C4⋊C412yes—
327C24C2×C4⋊C412yes—
328C24C2×C4⋊C412yes—
329C24C22×D411yescharacteristic
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—232no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—416no—
396C22×C4—464no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4#9514yes—
404C22×C4#9514yes—
405C22×C4#9514yes—
406C22×C4#9514yes—
407C22×C4#9514yes—
408C22×C4#9514yes—
409C22×C4#9514yes—
410C22×C4#9514yes—
411C22×C4#9514yes—
412C22×C4#9514yes—
413C22×C4#9514yes—
414C22×C4#9514yes—
415C22×C4C2×C4212yes—
416C22×C4C2×C4212yes—
417C22×C4C2×C4212yes—
418C22×C4C2×C4212yes—
419C22×C4C2×C4212yes—
420C22×C4C2×C4212yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C412yes—
423C22×C4C2×C22⋊C412yes—
424C22×C4C2×C22⋊C412yes—
425C22×C4C2×C22⋊C412yes—
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C412yes—
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C4⋊C412yes—
437C22×C4C2×C4⋊C412yes—
438C22×C4C2×C4⋊C412yes—
439C22×C4C2×C4⋊C412yes—
440C22×C4C2×C4⋊C412yes—
441C22×C4C2×C4⋊C412yes—
442C22×C4C2×C4⋊C412yes—
443C22×C4C2×C4⋊C412yes—
444C22×C4C2×C4⋊C412yes—
445C22×C4C2×C4⋊C412yes—
446C22×C4C2×C4⋊C412yes—
447C22×C4C2×C4⋊C412yes—
448C22×C4C2×C4⋊C412yes—
449C22×C4C2×C4⋊C412yes—
450C22×C4C2×C4⋊C412yes—
451C22×C4C23×C411yescharacteristic
452C22×C4C23×C411yescharacteristic
453C22×C4C23×C411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×Q811yescharacteristic
457C22×C4H(5,𝔽2)18yes—
458M16—216no—
459M16—416no—
460M16—48no—
461C2×C8—28no—
462C2×C8—416no—
463C2×C8—48no—
464C4⋊C4—216no—
465C4⋊C4—48no—
466C22⋊C4—232no—
467C22⋊C4—232no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C42—216no—
475C42—416no—
476C42—48no—
477C42—48no—
478C42—48no—
479C23—232no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—416no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—416no—
527C23—416no—
528C23—416no—
529C23—416no—
530C23—464no—
531C23—464no—
532C23—416no—
533C23#339116yescomplemented
534C23C2×#9512yes—
535C23C2×#9512yes—
536C23C2×#9512yes—
537C23C2×#9512yes—
538C23C2×#9512yes—
539C23C2×#9512yes—
540C23C2×#10414yes—
541C23C2×#10414yes—
542C23C2×#10414yes—
543C23#42114yes—
544C23C22×C22⋊C411yescharacteristic
545C23C22×C22⋊C411yescharacteristic
546C23C22×C4⋊C411yescharacteristic
547C23#53414yes—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—216no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—464no—
598C2×C4—432no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—816no—
605C2×C4C2×#9512yes—
606C2×C4C2×#9512yes—
607C2×C4C2×#9512yes—
608C2×C4C2×#9512yes—
609C2×C4C2×#9512yes—
610C2×C4C2×#9512yes—
611C2×C4C22×C4211yescharacteristic
612C2×C4C22×C22⋊C411yescharacteristic
613C2×C4C22×C4⋊C411yescharacteristic
614C2×C4C22×C4⋊C411yescharacteristic
615C2×C4#51814yes—
616C2×C4#51814yes—
617C2×C4#51814yes—
618C2×C4#53414yes—
619C2×C4#53414yes—
620C2×C4#54314yes—
621C8—48no—
622C22—216no—
623C22—216no—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—28no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—416no—
648C22—48no—
649C22—48no—
650C22—48no—
651C22—48no—
652C22—416no—
653C22—416no—
654C22—416no—
655C22—44no—
656C22—44no—
657C22C2×#33914yes—
658C22C2×#33914yes—
659C22C2×#33914yes—
660C22#173314yes—
661C22C22×#9511yescharacteristic
662C22#230812yes—
663C22#230812yes—
664C22#231512yes—
665C22C22×#10412yes—
666C22C2×#42112yes—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—416no—
674C4—88no—
675C4#227312yes—
676C4#230812yes—
677C4#231512yes—
678C4#231512yes—
679C2—22no—
680C2—22no—
681C2—24no—
682C2—24no—
683C2—24no—
684C2—24no—
685C2—42no—
686C2—42no—
687C2#1379411yescharacteristic
688C2C22×#33911yescharacteristic
689C2C2×#173311yescharacteristic
690C2#2177314yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10733C2semidirect
5. #10732C2semidirect
6. #10719C2semidirect
15. C2×#407C4semidirect
24. #2280C4semidirect
25. #2280C4semidirect
29. C2×#380C4semidirect
30. C2×#380C4semidirect
37. C2×C4×C22⋊C4C4semidirect
41. C2×#339C22semidirect
66. C22×M16C2×C4semidirect
72. C2×#117C2×C4semidirect
74. C22×C4⋊C4C2×C4semidirect
86. C22×C42C2×C4semidirect
92. #407C2×C4semidirect
93. #407C2×C4semidirect
110. #380C2×C4semidirect
111. #380C2×C4semidirect
129. C4×C22⋊C4C2×C4semidirect
533. C23#339semidirect

Maximal subgroups

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

subgroupindex
C22×#4072
C2×#22802
C22×C4×C22⋊C42
#107332
#107322
#107192