Finite Group Databasebeta

← #196691

Subgroups — #196691

Order 512.

1546 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1553C212yesmaximal, complemented
5#9142C212yesmaximal, complemented
6#9141C212yesmaximal, complemented
7#9140C212yesmaximal, complemented
8#9139C212yesmaximal, complemented
9#8046C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#525—21no—
12C23×C42C2211yescharacteristic
13#2920—28no—
14C2×#432—21no—
15C2×#432C2212yes—
16C2×#432C2212yes—
17C2×#431—21no—
18C2×#431C2212yes—
19C2×#423C2212yes—
20C2×#407—21no—
21C2×C4×M16—21no—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2856C2211yescharacteristic
26#2854—21no—
27#2854C412yes—
28#2850—21no—
29#2850—21no—
30#2847—21no—
31#2847C412yes—
32C4×#121—21no—
33C4×#121—21no—
34C4×#121C412yes—
35C4×#121C412yes—
36#2297—21no—
37#2285C2211yescharacteristic
38#1817—21no—
39#1797C2212yes—
40#1797C2212yes—
41#1779—21no—
42#1749—21no—
43#1749—21no—
44#1553—28no—
45C2×#332C2214yescomplemented
46C2×#331C2214yescomplemented
47#1296C2214yescomplemented
48#1296C2214yescomplemented
49C22×#41—21no—
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C22×M16—22no—
54#532—24no—
55#532—24no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×#127C2311yescharacteristic
58C2×#124—21no—
59#525—48no—
60C2×#123—22no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121C2×C412yes—
69C2×#121C2×C412yes—
70C2×#121C2×C412yes—
71C2×#120—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C2×#117—21no—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#432—28no—
92#432—28no—
93#432—24no—
94#432—24no—
95#431—28no—
96#431—24no—
97#431D418yescomplemented
98#423—28no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8C2314yes—
101C2×C4⋊C8C2314yes—
102#407—28no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C4×M16—28no—
109C2×#97—22no—
110C2×#97—22no—
111C2×C4×C8—22no—
112#391—21no—
113#391—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#390C2×C412yes—
119#390C2×C412yes—
120#389—21no—
121#389—21no—
122#389C2×C412yes—
123#389C2×C412yes—
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#382—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380C2×C412yes—
141#380C2×C412yes—
142#380C2×C412yes—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2×C412yes—
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147#336—22no—
148#336—22no—
149#332—28no—
150#331—28no—
151C25C22⋊C411yescharacteristic
152C2×#41—44no—
153C2×#41—44no—
154C2×#41—42no—
155C2×#41—42no—
156C22×Q8—21no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4C22×C412yes—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
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×C4C22×C411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C2×M16—28no—
193C2×M16—44no—
194C2×M16—44no—
195C22×C8—24no—
196C22×C8—24no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199#127—22no—
200#127—22no—
201#124—42no—
202#124—42no—
203#123—48no—
204C22⋊Q8—44no—
205C22⋊Q8—44no—
206#121—28no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211#121—44no—
212#121—44no—
213#121—44no—
214#120—48no—
215C4×D4—28no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—44no—
219#117—44no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D414yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4114yes—
275C2×C42C2411yescharacteristic
276C4⋊C8—28no—
277C4⋊C8—24no—
278C4⋊C8—24no—
279C4⋊C8—24no—
280C4⋊C8C2×D418yes—
281C22⋊C8—28no—
282C22⋊C8—28no—
283C22⋊C8—44no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286#97—28no—
287#97—44no—
288C4×C8—28no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305C24C2×C22⋊C411yescharacteristic
306#41—88no—
307#41—88no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×D4—24no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—41no—
325C2×D4—42no—
326C2×D4—41no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C4×D414yes—
403C22×C4#12012yes—
404C22×C4#12114yes—
405C22×C4#12312yes—
406C22×C4AGL(1,ℤ/8)14yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410M16—416no—
411C2×C8—28no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—44no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—22no—
443C42—22no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—44no—
453C42C4×D414yes—
454C42#12012yes—
455C42#12114yes—
456C42#12312yes—
457C42C22×D412yes—
458C42C2×#4112yes—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23#38612yes—
492C23#41312yes—
493C23#41812yes—
494C23C22×C22⋊C411yescharacteristic
495Q8—82no—
496Q8—82no—
497D4—84no—
498D4—88no—
499D4—88no—
500D4—88no—
501D4—82no—
502D4—82no—
503D4—84no—
504C2×C4—28no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—88no—
559C2×C4#38612yes—
560C2×C4#38612yes—
561C2×C4#38612yes—
562C2×C4#41312yes—
563C2×C4#41812yes—
564C2×C4C2×C4×D412yes—
565C2×C4C2×#12011yescharacteristic
566C2×C4C2×#12112yes—
567C2×C4C2×#12311yescharacteristic
568C2×C4C2×AGL(1,ℤ/8)12yes—
569C2×C4#57512yes—
570C2×C4#57512yes—
571C2×C4#57512yes—
572C8—48no—
573C8—84no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—416no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22C2×#38611yescharacteristic
604C22C2×#41311yescharacteristic
605C22C2×#41311yescharacteristic
606C22C2×#41811yescharacteristic
607C22#289611yescharacteristic
608C22#289611yescharacteristic
609C22#289611yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#300612yes—
624C4#305414yes—
625C4#309412yes—
626C2—42no—
627C2—42no—
628C2—42no—
629C2—162no—
630C2#1774311yescharacteristic
631C2#1774411yescharacteristic
632C2#1775611yescharacteristic
633C2#1775711yescharacteristic
634C2#2091211yescharacteristic
635C2#2097011yescharacteristic
636C2#2098911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. #20715C2semidirect
4. C2×#1553C2semidirect
5. #9142C2semidirect
6. #9141C2semidirect
7. #9140C2semidirect
8. #9139C2semidirect
9. #8046C2semidirect
45. C2×#332C22semidirect
46. C2×#331C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect
97. #431D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366242
#207152
C2×#15532
#91422
#91412
#91402
#91392
#80462