Finite Group Databasebeta

← #181744

Subgroups — #181744

Order 512.

3346 conjugacy classes of subgroups.

The subgroup lattice has 709 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#8020C214yesmaximal, complemented
6#8016C214yesmaximal, complemented
7C25×C4C2211yescharacteristic
8C23×M16—21no—
9C22×#117C412yescomplemented
10C23×C4⋊C4—21no—
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—21no—
21#2280—21no—
22#2280—21no—
23#2280—21no—
24#2280C412yescomplemented
25#2280C412yescomplemented
26#2280C2212yes—
27#2280C2212yes—
28C2×#381—21no—
29C2×#380—24no—
30C2×#380—21no—
31C2×#380—21no—
32C2×#380C418yescomplemented
33C2×#380C412yescomplemented
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#379—21no—
38C2×C4×C22⋊C4—21no—
39C2×C4×C22⋊C4—21no—
40C22×#95C2212yes—
41#1294C22116yescomplemented
42C26D411yescharacteristic
43C24×C4—22no—
44C24×C4—48no—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C414yes—
47C24×C4C2×C414yes—
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4D411yescharacteristic
57C24×C4D414yes—
58C24×C4D411yescharacteristic
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—22no—
69C2×#117—22no—
70C2×#117—48no—
71C2×#117C2×C414yes—
72C2×#117C2×C414yescomplemented
73C22×C4⋊C4—28no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—48no—
77C22×C4⋊C4C2×C414yescomplemented
78C22×C22⋊C4—28no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—48no—
85C22×C42—22no—
86#407—28no—
87#407—28no—
88#407—24no—
89#407—24no—
90#407C2×C418yescomplemented
91#407C2×C418yescomplemented
92C2×C22⋊C8—28no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8C2×C418yescomplemented
98C2×C22⋊C8C2314yes—
99#381—22no—
100#381—22no—
101#381—22no—
102#381—22no—
103#380—28no—
104#380—28no—
105#380—22no—
106#380—22no—
107#380—24no—
108#380—24no—
109#380—22no—
110#380—22no—
111#380C2×C414yes—
112#380C2×C418yes—
113#380C2×C414yescomplemented
114#380C2×C414yescomplemented
115#380C2×C418yescomplemented
116#380C2314yes—
117#379—22no—
118#379—22no—
119#379—22no—
120#379—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—24no—
132C2×#95—22no—
133C2×#95C2×C418yescomplemented
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—28no—
153C23×C4—24no—
154C23×C4—416no—
155C23×C4—416no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—416no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—416no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—48no—
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⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C414yes—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C414yes—
189C23×C4C22⋊C414yes—
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C414yes—
192C23×C4C22⋊C414yes—
193C23×C4C22⋊C414yes—
194C23×C4C22⋊C411yescharacteristic
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×C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C412yes—
210C23×C4C22×C412yes—
211C23×C4C22×C412yes—
212C23×C4C22×C412yes—
213C23×C4C22×C412yes—
214C23×C4C22×C411yescharacteristic
215C23×C4C22×C412yes—
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×Q812yes—
231C23×C4C2×Q812yes—
232C23×C4C2×Q812yes—
233C23×C4C2411yescharacteristic
234C2×M16—28no—
235C2×M16—28no—
236C2×M16—48no—
237C2×M16—432no—
238C2×M16—48no—
239C22×C8—28no—
240C22×C8—24no—
241C22×C8—48no—
242#117—48no—
243#117—48no—
244#117—48no—
245#117—48no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—432no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4C22×C418yes—
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—432no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—44no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—48no—
272C22⋊C8—28no—
273C22⋊C8—28no—
274C22⋊C8—28no—
275C22⋊C8—24no—
276C22⋊C8C22×C418yes—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95C22×C418yes—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—28no—
297C24—28no—
298C24—24no—
299C24—24no—
300C24—216no—
301C24—42no—
302C24—42no—
303C24—48no—
304C24—48no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—48no—
310C24—48no—
311C24—42no—
312C24—42no—
313C24—432no—
314C24—432no—
315C24#9514yes—
316C24#9514yes—
317C24#9514yes—
318C24#9514yes—
319C24C2×C22⋊C412yes—
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322C24C2×C22⋊C411yescharacteristic
323C24C2×C22⋊C411yescharacteristic
324C24C2×C22⋊C411yescharacteristic
325C24C2×C22⋊C411yescharacteristic
326C24C2×C22⋊C412yes—
327C24C2×C22⋊C412yes—
328C24C2×C4⋊C412yes—
329C24C2×C4⋊C412yes—
330C24C2×C4⋊C412yes—
331C24AGL(1,ℤ/8)14yes—
332C24#13714yes—
333C24C22×D411yescharacteristic
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—216no—
338C22×C4—216no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—216no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
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—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—48no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—416no—
402C22×C4—48no—
403C22×C4—416no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—832no—
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×C4#9514yes—
416C22×C4#9514yes—
417C22×C4#9514yes—
418C22×C4#9514yes—
419C22×C4#9918yes—
420C22×C4C2×C4212yes—
421C22×C4C2×C4212yes—
422C22×C4C2×C4212yes—
423C22×C4C2×C4212yes—
424C22×C4C2×C4212yes—
425C22×C4C2×C4212yes—
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
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×C4C2×C4⋊C412yes—
452C22×C4C2×C4⋊C412yes—
453C22×C4C2×C4⋊C412yes—
454C22×C4C2×C4⋊C412yes—
455C22×C4C2×C4⋊C412yes—
456C22×C4C23×C411yescharacteristic
457C22×C4C23×C411yescharacteristic
458C22×C4C23×C411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×Q811yescharacteristic
462M16—416no—
463M16—416no—
464C2×C8—28no—
465C2×C8—48no—
466C2×C8—416no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—48no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C42—48no—
481C42—48no—
482C23—216no—
483C23—216no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—48no—
531C23—48no—
532C23—416no—
533C23—416no—
534C23—416no—
535C23—416no—
536C23—432no—
537C23—432no—
538C23—464no—
539C23—48no—
540C23—48no—
541C23C2×#9512yes—
542C23C2×#9512yes—
543C23C2×#9512yes—
544C23C2×#9512yes—
545C23C2×#9512yes—
546C23C2×#9512yes—
547C23C2×#9914yes—
548C23#41714yes—
549C23#41714yes—
550C23#41814yes—
551C23#41914yes—
552C23#42114yes—
553C23#42818yes—
554C23C22×C22⋊C411yescharacteristic
555C23C22×C22⋊C411yescharacteristic
556C23C22×C4⋊C411yescharacteristic
557C23C2×AGL(1,ℤ/8)12yes—
558C23C2×#13712yes—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
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—48no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—816no—
617C2×C4—832no—
618C2×C4—816no—
619C2×C4#342116yes—
620C2×C4C2×#9512yes—
621C2×C4C2×#9512yes—
622C2×C4C2×#9512yes—
623C2×C4C2×#9512yes—
624C2×C4C2×#9512yes—
625C2×C4C2×#9512yes—
626C2×C4C2×#9914yes—
627C2×C4C2×#9914yes—
628C2×C4C22×C4211yescharacteristic
629C2×C4C22×C22⋊C411yescharacteristic
630C2×C4C22×C4⋊C411yescharacteristic
631C2×C4C22×C4⋊C411yescharacteristic
632C8—48no—
633C22—28no—
634C22—28no—
635C22—216no—
636C22—21no—
637C22—21no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—24no—
649C22—28no—
650C22—24no—
651C22—28no—
652C22—28no—
653C22—28no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—416no—
664C22—416no—
665C22—416no—
666C22—416no—
667C22—44no—
668C22—44no—
669C22—48no—
670C22—44no—
671C22—44no—
672C22#173314yes—
673C22#173614yes—
674C22#173614yes—
675C22C2×#34214yes—
676C22C22×#9511yescharacteristic
677C22C22×#9912yes—
678C22C2×#41811yescharacteristic
679C22C2×#41911yescharacteristic
680C22C2×#41712yes—
681C22C2×#42112yes—
682C22C2×#42812yes—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—21no—
691C4—88no—
692C4—88no—
693C4C2×#34214yes—
694C4C2×#34214yes—
695C2—21no—
696C2—21no—
697C2—22no—
698C2—24no—
699C2—24no—
700C2—24no—
701C2—24no—
702C2—42no—
703C2—42no—
704C2#1759214yes—
705C2C2×#173311yescharacteristic
706C2C2×#173611yescharacteristic
707C2C22×#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. #8020C2semidirect
6. #8016C2semidirect
9. C22×#117C4semidirect
15. C2×#407C4semidirect
24. #2280C4semidirect
25. #2280C4semidirect
32. C2×#380C4semidirect
33. C2×#380C4semidirect
41. #1294C22semidirect
72. C2×#117C2×C4semidirect
77. C22×C4⋊C4C2×C4semidirect
90. #407C2×C4semidirect
91. #407C2×C4semidirect
97. C2×C22⋊C8C2×C4semidirect
113. #380C2×C4semidirect
114. #380C2×C4semidirect
115. #380C2×C4semidirect
133. C2×#95C2×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
#80202
#80162