Finite Group Databasebeta

← #357902

Subgroups — #357902

Order 512.

3490 conjugacy classes of subgroups.

The subgroup lattice has 647 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
4C2×#1736C212yesmaximal, complemented
5C2×#1736C212yesmaximal, complemented
6#11200C214yesmaximal, complemented
7#11199C214yesmaximal, complemented
8C25×C4C2211yescharacteristic
9C23×M16C2212yes—
10C22×#117C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C22⋊C4—21no—
14C2×#407—24no—
15C2×#407—28no—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18#2280—28no—
19C2×#381—21no—
20C2×#380—24no—
21C2×#380—21no—
22C2×#380—21no—
23C2×#380C418yescomplemented
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#379—21no—
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4—21no—
31C22×#95—21no—
32#1736—28no—
33#1736—28no—
34C2×#340C2218yescomplemented
35C2×#340C2218yescomplemented
36C26D411yescharacteristic
37C24×C4—24no—
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—24no—
41C24×C4—24no—
42C24×C4—24no—
43C24×C4—21no—
44C24×C4—21no—
45C24×C4C2×C414yes—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D414yes—
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C22×M16—24no—
54C22×M16—24no—
55C22×M16—24no—
56C22×M16—24no—
57C22×M16—24no—
58C22×M16—24no—
59C22×M16D418yes—
60C23×C8C2314yes—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—24no—
65C2×#117—24no—
66C2×#117D418yes—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2×C414yes—
76C22×C4⋊C4C2×C414yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C22⋊C4—28no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—48no—
87C22×C42—22no—
88#407—232no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—28no—
92C2×C22⋊C8—28no—
93#381—28no—
94#380—28no—
95#380—28no—
96#380—24no—
97#380—24no—
98#380—28no—
99#380—28no—
100#380C2×C418yes—
101#380C2×C418yes—
102#379—28no—
103C4×C22⋊C4—28no—
104C4×C22⋊C4—28no—
105C2×#95—28no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110#340—216no—
111#340—216no—
112C25—24no—
113C25—24no—
114C25—24no—
115C25—22no—
116C25—24no—
117C25—22no—
118C25—42no—
119C25—42no—
120C25C22⋊C414yes—
121C25C2×D411yescharacteristic
122C25C2×D411yescharacteristic
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—216no—
131C23×C4—216no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—28no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—216no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—216no—
143C23×C4—28no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—48no—
162C23×C4—48no—
163C23×C4—48no—
164C23×C4—48no—
165C23×C4—48no—
166C23×C4—48no—
167C23×C4—48no—
168C23×C4—48no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4C22⋊C414yes—
177C23×C4C22⋊C414yes—
178C23×C4C22⋊C414yes—
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C22×C412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4#4114yes—
191C23×C4#4114yes—
192C23×C4#4114yes—
193C23×C4#4114yes—
194C23×C4C2411yescharacteristic
195C2×M16—216no—
196C2×M16—216no—
197C2×M16—216no—
198C2×M16—48no—
199C2×M16—48no—
200C2×M16—48no—
201C2×M16—48no—
202C22×C8—24no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8—24no—
208C22×C8C2×D418yes—
209#117—216no—
210#117—216no—
211#117—48no—
212#117—48no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C2×D418yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—432no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—44no—
251C2×C42—28no—
252C2×C42—44no—
253C2×C42—44no—
254C22⋊C8—216no—
255C22⋊C8—216no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—232no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—28no—
280C24—28no—
281C24—28no—
282C24—28no—
283C24—28no—
284C24—44no—
285C24—44no—
286C24—48no—
287C24—48no—
288C24—48no—
289C24—48no—
290C24—416no—
291C24—416no—
292C24—416no—
293C24—48no—
294C24—48no—
295C24C2×C22⋊C412yes—
296C24C2×C22⋊C412yes—
297C24C2×C22⋊C412yes—
298C24#12012yes—
299C24#12114yes—
300C24#12314yes—
301C24#12312yes—
302C24AGL(1,ℤ/8)14yes—
303C24#13714yes—
304C24C22×D411yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—432no—
360C22×C4—432no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—432no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—432no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—816no—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4#11714yes—
393C22×C4#11714yes—
394C22×C4C4×D418yes—
395C22×C4C4×D418yes—
396C22×C4#12014yes—
397C22×C4#12012yes—
398C22×C4#12118yes—
399C22×C4#12118yes—
400C22×C4#12118yes—
401C22×C4#12114yes—
402C22×C4#12318yes—
403C22×C4#12314yes—
404C22×C4#12318yes—
405C22×C4#12318yes—
406C22×C4#12314yes—
407C22×C4#12312yes—
408C22×C4#12418yes—
409C22×C4#12418yes—
410C22×C4#12618yes—
411C22×C4#12618yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420M16—432no—
421C2×C8—28no—
422C2×C8—28no—
423C2×C8—28no—
424C2×C8—44no—
425C2×C8—44no—
426C2×C8—44no—
427C2×C8—44no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C42—416no—
445C23—28no—
446C23—28no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—232no—
458C23—28no—
459C23—28no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—28no—
477C23—28no—
478C23—416no—
479C23—416no—
480C23—416no—
481C23—416no—
482C23—464no—
483C23—464no—
484C23—416no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—416no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23#38114yes—
499C23#38614yes—
500C23#38614yes—
501C23#39418yes—
502C23#39718yes—
503C23#41814yes—
504C23#41914yes—
505C23C22×C22⋊C411yescharacteristic
506C23C2×#12011yescharacteristic
507C23C2×#12112yes—
508C23C2×#12312yes—
509C23C2×#12311yescharacteristic
510C23C2×AGL(1,ℤ/8)12yes—
511C23C2×#13712yes—
512C23#57512yes—
513C23#57512yes—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—416no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—416no—
530C2×C4—416no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—816no—
545C2×C4—816no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—88no—
552C2×C4—84no—
553C2×C4#38114yes—
554C2×C4#38614yes—
555C2×C4#38614yes—
556C2×C4#388116yes—
557C2×C4#388116yes—
558C2×C4#394116yes—
559C2×C4#39418yes—
560C2×C4#397116yes—
561C2×C4#39718yes—
562C2×C4C2×#11711yescharacteristic
563C2×C4C2×#11711yescharacteristic
564C2×C4C2×C4×D412yes—
565C2×C4C2×C4×D412yes—
566C2×C4C2×#12011yescharacteristic
567C2×C4C2×#12112yes—
568C2×C4C2×#12112yes—
569C2×C4C2×#12112yes—
570C2×C4C2×#12312yes—
571C2×C4C2×#12312yes—
572C2×C4C2×#12311yescharacteristic
573C2×C4C2×#12312yes—
574C2×C4C2×#12412yes—
575C2×C4C2×#12412yes—
576C2×C4C2×#12612yes—
577C2×C4C2×#12612yes—
578C8—48no—
579C22—216no—
580C22—24no—
581C22—24no—
582C22—216no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—464no—
594C22—42no—
595C22—42no—
596C22—48no—
597C22—48no—
598C22—42no—
599C22—42no—
600C22—48no—
601C22—48no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22#144018yes—
607C22C2×#38111yescharacteristic
608C22C2×#38611yescharacteristic
609C22C2×#38611yescharacteristic
610C22C2×#39412yes—
611C22C2×#39712yes—
612C22C2×#41811yescharacteristic
613C22C2×#41911yescharacteristic
614C22#289611yescharacteristic
615C22#289611yescharacteristic
616C22#300612yes—
617C22#300712yes—
618C22#305514yes—
619C22#305614yes—
620C22#309214yes—
621C22#309314yes—
622C22#309412yes—
623C22#309512yes—
624C4—42no—
625C4—42no—
626C4—84no—
627C4—88no—
628C4—84no—
629C4#144018yes—
630C4C2×#38812yes—
631C4C2×#38812yes—
632C4C2×#39412yes—
633C4C2×#39712yes—
634C2—28no—
635C2—44no—
636C2—44no—
637C2C2×#144011yescharacteristic
638C2#2070011yescharacteristic
639C2#2070111yescharacteristic
640C2#2091111yescharacteristic
641C2#2091211yescharacteristic
642C2#2097211yescharacteristic
643C2#2097311yescharacteristic
644C2#2348214yes—
645C2#2364214yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1736C2semidirect
5. C2×#1736C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
23. C2×#380C4semidirect
34. C2×#340C22semidirect
35. C2×#340C22semidirect

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
C2×#17362
#112002
#111992