Finite Group Databasebeta

← #181741

Subgroups — #181741

Order 512.

3346 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#407C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3C22×#380C211yesmaximal, characteristic
4#8023C214yesmaximal, complemented
5#8019C214yesmaximal, complemented
6#8015C214yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C23×M16—21no—
9C22×#117—21no—
10C23×C4⋊C4C412yescomplemented
11C23×C22⋊C4—21no—
12C2×#407—24no—
13C2×#407—21no—
14C2×#407—21no—
15C2×#407C418yescomplemented
16C2×#407C2212yes—
17C2×#407C2212yes—
18C22×C22⋊C8—21no—
19C22×C22⋊C8C2212yes—
20#2280—24no—
21#2280C418yescomplemented
22C2×#381C2212yes—
23C2×#380—21no—
24C2×#380—21no—
25C2×#380—21no—
26C2×#380—21no—
27C2×#380—21no—
28C2×#380—21no—
29C2×#380—21no—
30C2×#380C412yescomplemented
31C2×#380C412yescomplemented
32C2×#380C412yescomplemented
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#379—21no—
38C2×C4×C22⋊C4—21no—
39C2×C4×C22⋊C4C2212yes—
40C22×#95—21no—
41#1294C22116yescomplemented
42C26D411yescharacteristic
43C24×C4—22no—
44C24×C4—48no—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C414yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C414yes—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4D411yescharacteristic
58C24×C4D414yes—
59C24×C4Q814yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16—28no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—48no—
67C23×C8—22no—
68C2×#117—28no—
69C2×#117—48no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—48no—
75C22×C4⋊C4C2×C414yes—
76C22×C4⋊C4C2×C414yes—
77C22×C4⋊C4C2×C414yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—48no—
88C22×C42—22no—
89#407—28no—
90#407—28no—
91#407—24no—
92#407—24no—
93#407C2×C418yescomplemented
94#407C2×C418yescomplemented
95C2×C22⋊C8—28no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8C2×C418yescomplemented
101C2×C22⋊C8C2314yes—
102#381—24no—
103#381C2×C418yescomplemented
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—28no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—24no—
118#380C2×C414yes—
119#380C2×C414yes—
120#380C2×C414yes—
121#380C2×C418yescomplemented
122#380C2314yes—
123#379—28no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4C2×C418yescomplemented
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95C2314yes—
135C25—24no—
136C25—24no—
137C25—44no—
138C25—44no—
139C25C22⋊C411yescharacteristic
140C25C22⋊C411yescharacteristic
141C25C22⋊C411yescharacteristic
142C25C22⋊C411yescharacteristic
143C25C22⋊C414yes—
144C25C4⋊C414yes—
145C25C2×D411yescharacteristic
146C25C2×D411yescharacteristic
147C25C2×D411yescharacteristic
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—416no—
156C23×C4—416no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—416no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—416no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—48no—
171C23×C4—48no—
172C23×C4—48no—
173C23×C4C4214yes—
174C23×C4C4214yes—
175C23×C4C4214yes—
176C23×C4C4214yes—
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C414yes—
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C414yes—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C414yes—
189C23×C4C22⋊C414yes—
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C414yes—
193C23×C4C22⋊C414yes—
194C23×C4C22⋊C414yes—
195C23×C4C22⋊C411yescharacteristic
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×C4C4⋊C414yes—
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C412yes—
211C23×C4C22×C412yes—
212C23×C4C22×C412yes—
213C23×C4C22×C412yes—
214C23×C4C22×C411yescharacteristic
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C412yes—
218C23×C4C22×C412yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×Q812yes—
232C23×C4C2×Q812yes—
233C23×C4C2×Q812yes—
234C23×C4C2411yescharacteristic
235C2×M16—28no—
236C2×M16—28no—
237C2×M16—48no—
238C2×M16—432no—
239C2×M16—48no—
240C22×C8—28no—
241C22×C8—24no—
242C22×C8—48no—
243#117—432no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4C22×C418yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—44no—
273C2×C42—28no—
274C2×C42—48no—
275C22⋊C8—28no—
276C22⋊C8—28no—
277C22⋊C8—28no—
278C22⋊C8—24no—
279C22⋊C8C22×C418yescomplemented
280#95—24no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95C22×C418yescomplemented
287C24—24no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—216no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—216no—
307C24—42no—
308C24—42no—
309C24—48no—
310C24—48no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—48no—
316C24—48no—
317C24—42no—
318C24—42no—
319C24—432no—
320C24—432no—
321C24#9514yes—
322C24#9514yes—
323C24#9514yes—
324C24#9514yes—
325C24C2×C22⋊C411yescharacteristic
326C24C2×C22⋊C411yescharacteristic
327C24C2×C22⋊C412yes—
328C24C2×C22⋊C412yes—
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C412yes—
334C24C2×C4⋊C412yes—
335C24C2×C4⋊C412yes—
336C24C2×C4⋊C412yes—
337C24AGL(1,ℤ/8)14yes—
338C24#13714yes—
339C24C22×D411yescharacteristic
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—216no—
344C22×C4—216no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—28no—
362C22×C4—28no—
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—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—832no—
417C22×C4#9514yes—
418C22×C4#9514yes—
419C22×C4#9514yes—
420C22×C4#9514yes—
421C22×C4#9514yes—
422C22×C4#9514yes—
423C22×C4#9514yes—
424C22×C4#9514yes—
425C22×C4#9514yes—
426C22×C4#9514yes—
427C22×C4#9514yes—
428C22×C4#9514yes—
429C22×C4#9918yes—
430C22×C4C2×C4212yes—
431C22×C4C2×C4212yes—
432C22×C4C2×C4212yes—
433C22×C4C2×C4212yes—
434C22×C4C2×C4212yes—
435C22×C4C2×C4212yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C22⋊C412yes—
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C412yes—
447C22×C4C2×C22⋊C412yes—
448C22×C4C2×C22⋊C412yes—
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C4⋊C412yes—
452C22×C4C2×C4⋊C412yes—
453C22×C4C2×C4⋊C412yes—
454C22×C4C2×C4⋊C412yes—
455C22×C4C2×C4⋊C412yes—
456C22×C4C2×C4⋊C412yes—
457C22×C4C2×C4⋊C412yes—
458C22×C4C2×C4⋊C412yes—
459C22×C4C2×C4⋊C412yes—
460C22×C4C2×C4⋊C412yes—
461C22×C4C2×C4⋊C412yes—
462C22×C4C2×C4⋊C412yes—
463C22×C4C2×C4⋊C412yes—
464C22×C4C2×C4⋊C412yes—
465C22×C4C2×C4⋊C412yes—
466C22×C4C23×C411yescharacteristic
467C22×C4C23×C411yescharacteristic
468C22×C4C23×C411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×Q811yescharacteristic
472M16—416no—
473M16—416no—
474C2×C8—28no—
475C2×C8—48no—
476C2×C8—416no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C42—416no—
490C23—216no—
491C23—216no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—24no—
526C23—28no—
527C23—24no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—464no—
551C23—464no—
552C23—48no—
553C23—48no—
554C23C2×#9512yes—
555C23C2×#9512yes—
556C23C2×#9512yes—
557C23C2×#9512yes—
558C23C2×#9512yes—
559C23C2×#9512yes—
560C23C2×#9914yes—
561C23#41718yes—
562C23#41812yes—
563C23#41812yes—
564C23#41912yes—
565C23#41912yes—
566C23#42114yes—
567C23#42818yes—
568C23C22×C22⋊C411yescharacteristic
569C23C22×C22⋊C411yescharacteristic
570C23C22×C4⋊C411yescharacteristic
571C23C2×AGL(1,ℤ/8)12yes—
572C23C2×#13712yes—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4—816no—
633C2×C4—816no—
634C2×C4—816no—
635C2×C4#342116yes—
636C2×C4C2×#9512yes—
637C2×C4C2×#9512yes—
638C2×C4C2×#9512yes—
639C2×C4C2×#9512yes—
640C2×C4C2×#9512yes—
641C2×C4C2×#9512yes—
642C2×C4C2×#9914yes—
643C2×C4C2×#9914yes—
644C2×C4C22×C4211yescharacteristic
645C2×C4C22×C22⋊C411yescharacteristic
646C2×C4C22×C4⋊C411yescharacteristic
647C2×C4C22×C4⋊C411yescharacteristic
648C8—48no—
649C22—232no—
650C22—21no—
651C22—21no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—21no—
658C22—21no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—48no—
679C22—48no—
680C22—48no—
681C22—48no—
682C22—48no—
683C22—416no—
684C22—48no—
685C22—416no—
686C22—416no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22#173314yes—
692C22#173614yes—
693C22#173614yes—
694C22C2×#34214yes—
695C22C22×#9511yescharacteristic
696C22C22×#9912yes—
697C22C2×#41811yescharacteristic
698C22C2×#41911yescharacteristic
699C22C2×#41712yes—
700C22C2×#42112yes—
701C22C2×#42812yes—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—21no—
710C4—88no—
711C4—88no—
712C4C2×#34214yes—
713C4C2×#34214yes—
714C2—22no—
715C2—24no—
716C2—24no—
717C2—24no—
718C2—22no—
719C2—24no—
720C2—42no—
721C2—42no—
722C2#1757712yes—
723C2#1757812yes—
724C2C2×#173311yescharacteristic
725C2C2×#173611yescharacteristic
726C2C22×#34211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8023C2semidirect
5. #8019C2semidirect
6. #8015C2semidirect
10. C23×C4⋊C4C4semidirect
15. C2×#407C4semidirect
21. #2280C4semidirect
30. C2×#380C4semidirect
31. C2×#380C4semidirect
32. C2×#380C4semidirect
41. #1294C22semidirect
93. #407C2×C4semidirect
94. #407C2×C4semidirect
100. C2×C22⋊C8C2×C4semidirect
103. #381C2×C4semidirect
121. #380C2×C4semidirect
126. C4×C22⋊C4C2×C4semidirect
279. C22⋊C8C22×C4semidirect
286. #95C22×C4semidirect

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×#3802
#80232
#80192
#80152