Finite Group Databasebeta

← #198028

Subgroups — #198028

Order 512.

1377 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60941C211yesmaximal, characteristic
2#34141C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic, complemented
4C2×#1503C211yesmaximal, characteristic, complemented
5#17485C211yesmaximal, characteristic
6#9235C214yesmaximal, complemented
7#9231C214yesmaximal, complemented
8C2×#1303C211yesmaximal, characteristic, complemented
9#7866C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12C2×C4×#41—21no—
13#3246C414yescomplemented
14#3229—21no—
15#3227C412yescomplemented
16#3208—21no—
17#3206C412yescomplemented
18#3187—22no—
19#3167—22no—
20#3103—22no—
21#3101C2214yes—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25#2367—22no—
26#2366—21no—
27#2365—21no—
28#2364—22no—
29#2361—22no—
30#2297—21no—
31#2293C2212yes—
32#2290—22no—
33C42×D4C2212yes—
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1995—22no—
37#1994C2214yes—
38#1733—21no—
39#1733—21no—
40#1503—22no—
41#1503C2212yes—
42#1503C2212yes—
43C2×#328—22no—
44#1492—22no—
45#1486C2214yes—
46#1480—22no—
47#1303C2212yes—
48#1303C2212yes—
49#1276C2214yes—
50C22×#41—21no—
51C22×M16—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533C2×C412yescomplemented
59#533C2312yes—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532C2×C412yes—
65#532C2×C412yes—
66#532D412yes—
67#532D412yes—
68C2×C4⋊Q8C2×C411yescharacteristic, complemented
69C2×C4⋊Q8C2×C411yescharacteristic, complemented
70C2×C4⋊Q8D411yescharacteristic
71C2×#127D411yescharacteristic
72C2×#125—21no—
73C2×#124—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×#121—21no—
77C2×#121D412yes—
78C4×#41—44no—
79C4×#41—44no—
80C2×C4×Q8—22no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×D4—24no—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4—21no—
87C2×C4×D4C2312yes—
88C2×#117—22no—
89C2×#117—21no—
90C2×#117D411yescharacteristic
91C2×#117D411yescharacteristic
92C22×C4⋊C4—21no—
93C22×C42—21no—
94C22×C42—22no—
95C22×C42C2311yescharacteristic
96#498—22no—
97#492—22no—
98#487—22no—
99#486—22no—
100#483—24no—
101#481—24no—
102#476C2×C418yescomplemented
103#475—24no—
104#465—28no—
105#461C2×C418yescomplemented
106#460—24no—
107#450—24no—
108#448—24no—
109C4×SD16—28no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431C2×C412yescomplemented
116#431C2312yes—
117#431C2312yes—
118#423—21no—
119#423—21no—
120#423—21no—
121#423C2312yes—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8C2×C412yescomplemented
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130#391—22no—
131#390—22no—
132#390—22no—
133#389—22no—
134#388—24no—
135#388—24no—
136#387—24no—
137#386—24no—
138#386—24no—
139#385—24no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4D412yes—
146C4×C4⋊C4C2311yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—21no—
149C4×C22⋊C4—21no—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—24no—
153#376D412yes—
154C2×#95D412yes—
155C43C2312yes—
156#339—24no—
157#339—24no—
158#328—24no—
159#328—24no—
160D4⋊C8—28no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41—42no—
168C2×#41—42no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×D4—22no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4C2×D412yes—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C2×SD16—28no—
183C2×M16—22no—
184C2×M16—41no—
185C2×M16—42no—
186C2×M16—42no—
187C22×C8—22no—
188C22×C8—41no—
189C22×C8—41no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8C22⋊C412yes—
199C4⋊Q8C22⋊C412yes—
200C4⋊Q8C22×C412yes—
201C4⋊Q8C22×C412yes—
202#127—21no—
203#127—21no—
204#127C22⋊C412yes—
205#127C22⋊C412yes—
206#125—22no—
207#125—22no—
208#125—41no—
209#125—41no—
210#124—22no—
211#124—22no—
212#124—22no—
213#124—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222#121—22no—
223#121—22no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—22no—
228#121C22⋊C414yes—
229#121C22⋊C414yes—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—48no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8—41no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—416no—
246C4×D4—48no—
247C4×D4—48no—
248C4×D4C22×C414yes—
249C4×D4C2×D414yes—
250#117—22no—
251#117—22no—
252#117—22no—
253#117—22no—
254#117—22no—
255#117—44no—
256#117—41no—
257#117—41no—
258#117—44no—
259#117SD1614yescomplemented
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4C2×D412yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—41no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42—41no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42#4111yescharacteristic
305C2×C42#4111yescharacteristic
306C2×C42#4111yescharacteristic
307C2×C42C2411yescharacteristic
308#106—28no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—41no—
314C4⋊C8—41no—
315C4⋊C8—41no—
316C4⋊C8C22×C414yes—
317#103—28no—
318#102—28no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322C22⋊C8—41no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—22no—
326C4×C8—41no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95C2×D412yes—
333#95C2×D412yes—
334C24—22no—
335C24#12011yescharacteristic
336#41—44no—
337#41—44no—
338#41—44no—
339#41—44no—
340#41—44no—
341#41—44no—
342#41—44no—
343#41—44no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—82no—
353C2×Q8—82no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—82no—
367C2×D4C2×C22⋊C414yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—44no—
386C22×C4—81no—
387C22×C4—84no—
388C22×C4—82no—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4#12011yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12311yescharacteristic
403C22×C4#12311yescharacteristic
404C22×C4C2×SD1612yes—
405C22×C4#13512yes—
406C22×C4AGL(1,ℤ/8)12yes—
407C22×C4#13712yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411SD16—88no—
412M16—42no—
413M16—42no—
414M16—82no—
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—82no—
419C2×C8—82no—
420C2×C8—82no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4C2×C22⋊C414yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—84no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—48no—
458C42—44no—
459C42—42no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—42no—
465C42—44no—
466C42—82no—
467C42C2×C22⋊C411yescharacteristic
468C42C2×C22⋊C411yescharacteristic
469C42C4×D411yescharacteristic
470C42C4×D411yescharacteristic
471C42C4×D411yescharacteristic
472C42C4×D411yescharacteristic
473C42#12111yescharacteristic
474C42#12111yescharacteristic
475C42#12111yescharacteristic
476C42#12111yescharacteristic
477C42#12311yescharacteristic
478C42#12311yescharacteristic
479C42C2×SD1614yes—
480C42C23×C411yescharacteristic
481C42C22×D411yescharacteristic
482C42C2×#4111yescharacteristic
483C42C2×#4111yescharacteristic
484C23—24no—
485C23—41no—
486C23—41no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23#38611yescharacteristic
492C23#38611yescharacteristic
493C23#45012yes—
494C23#45212yes—
495C23C2×#12011yescharacteristic
496Q8—44no—
497Q8—44no—
498Q8—44no—
499Q8—82no—
500Q8—82no—
501Q8—82no—
502D4—28no—
503D4—44no—
504D4—48no—
505D4—48no—
506D4—44no—
507D4—44no—
508D4—82no—
509D4—82no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—816no—
544C2×C4—88no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#38611yescharacteristic
548C2×C4#38611yescharacteristic
549C2×C4#38611yescharacteristic
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4#38611yescharacteristic
553C2×C4C4×SD1612yes—
554C2×C4C4×SD1612yes—
555C2×C4#44012yes—
556C2×C4#44012yes—
557C2×C4#45012yes—
558C2×C4#45212yes—
559C2×C4#46112yes—
560C2×C4#46112yes—
561C2×C4#46312yes—
562C2×C4#46312yes—
563C2×C4C22×C22⋊C411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12311yescharacteristic
569C2×C4C22×SD1611yescharacteristic
570C2×C4C2×#13511yescharacteristic
571C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
572C2×C4C2×#13711yescharacteristic
573C8—44no—
574C8—82no—
575C8—82no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—84no—
588C22—84no—
589C22#187712yes—
590C22#187712yes—
591C22C2×#38611yescharacteristic
592C22C2×#45011yescharacteristic
593C22C2×#45211yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—44no—
602C4—44no—
603C4—44no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4#187712yes—
609C4#187712yes—
610C4#290112yes—
611C4C2×C4×SD1611yescharacteristic
612C4C2×#44011yescharacteristic
613C4C2×#46111yescharacteristic
614C4C2×#46311yescharacteristic
615C2—21no—
616C2—21no—
617C2—22no—
618C2—22no—
619C2—84no—
620C2C2×#187711yescharacteristic
621C2#2098011yescharacteristic
622C2#2100011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21329C2semidirect
4. C2×#1503C2semidirect
6. #9235C2semidirect
7. #9231C2semidirect
8. C2×#1303C2semidirect
9. #7866C2semidirect
13. #3246C4semidirect
15. #3227C4semidirect
17. #3206C4semidirect
58. #533C2×C4semidirect
68. C2×C4⋊Q8C2×C4semidirect
69. C2×C4⋊Q8C2×C4semidirect
102. #476C2×C4semidirect
105. #461C2×C4semidirect
115. #431C2×C4semidirect
125. C2×C4⋊C8C2×C4semidirect
259. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#609412
#341412
#213292
C2×#15032
#174852
#92352
#92312
C2×#13032
#78662