Finite Group Databasebeta

← #197235

Subgroups — #197235

Order 512.

1627 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3036C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#17574C211yesmaximal, characteristic, complemented
4#17574C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic, complemented
6#9179C218yesmaximal, complemented
7#8095C211yesmaximal, characteristic, complemented
8#7957C211yesmaximal, characteristic, complemented
9C22×AGL(1,ℤ/8)—21no—
10C2×#532C2211yescharacteristic
11C2×#518—21no—
12C22×C4×D4C2211yescharacteristic
13#3036—22no—
14#3036—22no—
15#3036C414yescomplemented
16#3036C2214yes—
17C2×#463—22no—
18C2×#461C412yescomplemented
19C2×#459C412yescomplemented
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417—21no—
24#2349—22no—
25#2349C2214yescomplemented
26D4×C4⋊C4—22no—
27D4×C4⋊C4—22no—
28#2316—21no—
29#2315—22no—
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32C2×#380—22no—
33#1519—22no—
34#1519—22no—
35#1517—22no—
36#1517C2214yescomplemented
37#1502—22no—
38#1502C2214yescomplemented
39#1478—22no—
40#1478C2214yescomplemented
41C2×D4⋊C8—22no—
42#1303C2214yescomplemented
43#1286C2214yescomplemented
44C22×#41—21no—
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C2×AGL(1,ℤ/8)—24no—
48C2×AGL(1,ℤ/8)—24no—
49C2×AGL(1,ℤ/8)—44no—
50C2×AGL(1,ℤ/8)—44no—
51C22×SD16—22no—
52C22×D8—22no—
53C22×M16—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532C2×C412yescomplemented
61#532C2312yes—
62C2×C4⋊Q8C2×C411yescharacteristic, complemented
63C2×#127C2×C411yescharacteristic, complemented
64C2×#124—21no—
65C2×#121—21no—
66C2×#121—21no—
67#518—44no—
68#518—44no—
69C2×C4×D4—24no—
70C2×C4×D4—24no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2×C412yes—
81C2×C4×D4D412yes—
82C2×C4×D4D412yes—
83C2×C4×D4D412yes—
84C2×C4×D4C2312yes—
85C2×#117—21no—
86C2×#117—21no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4D411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—24no—
91C22×C42C2311yescharacteristic
92#463—28no—
93#463—28no—
94#461—24no—
95#461C2×C418yescomplemented
96#459—24no—
97#459C2×C418yescomplemented
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104#431C2312yes—
105#431C2312yes—
106#423—21no—
107#423—21no—
108#423C2×C412yescomplemented
109#423C2312yes—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2×C412yescomplemented
114#417—24no—
115#417—24no—
116C2×#103—22no—
117C2×#102—22no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122#389—24no—
123#389—24no—
124#385—28no—
125#385—28no—
126#384—21no—
127#384—21no—
128#382D412yes—
129#382C2312yes—
130#380—24no—
131#380—24no—
132#380—21no—
133#380—21no—
134#380—24no—
135#380D412yes—
136#380D412yes—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4D412yes—
140C4×C4⋊C4C2312yes—
141C2×#95—24no—
142#327—28no—
143#327—28no—
144D4⋊C8—28no—
145C25—21no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—42no—
153C2×#41—42no—
154C22×Q8—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—41no—
164C22×D4C22⋊C412yes—
165C22×D4C22⋊C412yes—
166C22×D4D814yes—
167C22×D4SD1614yes—
168C22×D4C2×D412yes—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—28no—
177C23×C4—41no—
178C23×C4—84no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182AGL(1,ℤ/8)—416no—
183AGL(1,ℤ/8)—48no—
184AGL(1,ℤ/8)—48no—
185C2×SD16—28no—
186C2×SD16—48no—
187C2×D8—28no—
188C2×D8—48no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—42no—
192C2×M16—42no—
193C22×C8—22no—
194C22×C8—41no—
195C22×C8—41no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8C22×C412yes—
200#127—21no—
201#127—21no—
202#127—21no—
203#127C22×C412yes—
204#124—22no—
205#124—22no—
206#124—41no—
207#124—41no—
208#121—22no—
209#121—22no—
210#121—22no—
211#121—22no—
212#121—41no—
213#121—41no—
214#121—41no—
215#121—41no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—48no—
233C4×D4—416no—
234C4×D4—48no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4C22⋊C414yes—
240C4×D4C22⋊C414yes—
241C4×D4C22×C414yes—
242C4×D4C2×D414yes—
243#117—42no—
244#117—42no—
245#117—42no—
246#117—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—88no—
260C2×C4⋊C4C22⋊C412yes—
261C2×C4⋊C4C22⋊C412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—416no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42C2411yescharacteristic
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—41no—
295C4⋊C8—41no—
296C4⋊C8C22×C414yes—
297#103—28no—
298#102—28no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—41no—
302C22⋊C8—41no—
303C4×C8—22no—
304C4×C8—22no—
305C4×C8—41no—
306C4×C8—41no—
307#95—28no—
308#95—22no—
309#95—22no—
310#95C2×D414yes—
311#95C2×D414yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—41no—
318C24—82no—
319C24—82no—
320C24#12011yescharacteristic
321#41—42no—
322#41—42no—
323#41—42no—
324#41—42no—
325#41—44no—
326#41—44no—
327#41—82no—
328#41—82no—
329#41—82no—
330#41—82no—
331C2×Q8—22no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4#10218yes—
363C2×D4#10218yes—
364C2×D4C2×C22⋊C414yes—
365C2×D4C2×D814yes—
366C2×D4C2×SD1614yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4—48no—
378C22×C4—41no—
379C22×C4—416no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—81no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—82no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4C4×D411yescharacteristic
399C22×C4#12011yescharacteristic
400C22×C4#12011yescharacteristic
401C22×C4#12011yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12311yescharacteristic
407C22×C4#12311yescharacteristic
408C22×C4C2×D812yes—
409C22×C4C2×SD1612yes—
410C22×C4AGL(1,ℤ/8)12yes—
411C22×C4#13712yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415SD16—416no—
416D8—416no—
417M16—42no—
418M16—42no—
419M16—44no—
420C2×C8—24no—
421C2×C8—44no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—82no—
425C2×C8—82no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—44no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—88no—
441C4⋊C4C2×C22⋊C414yes—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—416no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C42—42no—
454C42—42no—
455C42—44no—
456C42C2×C22⋊C411yescharacteristic
457C42C2×C22⋊C411yescharacteristic
458C42C4×D411yescharacteristic
459C42C4×D411yescharacteristic
460C42C4×D411yescharacteristic
461C42C4×D411yescharacteristic
462C42#12111yescharacteristic
463C42#12111yescharacteristic
464C42#12111yescharacteristic
465C42#12111yescharacteristic
466C42#12311yescharacteristic
467C42#12311yescharacteristic
468C42C23×C411yescharacteristic
469C42C22×D411yescharacteristic
470C42C2×#4111yescharacteristic
471C42C2×#4111yescharacteristic
472C23—24no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23—41no—
478C23—41no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—88no—
489C23—88no—
490C23—82no—
491C23#38611yescharacteristic
492C23#38611yescharacteristic
493C23#44712yes—
494C23#44812yes—
495C23C2×#12011yescharacteristic
496Q8—42no—
497Q8—42no—
498Q8—82no—
499D4—24no—
500D4—24no—
501D4—24no—
502D4—28no—
503D4—28no—
504D4—42no—
505D4—42no—
506D4—82no—
507D4—82no—
508D4—82no—
509D4—84no—
510D4—82no—
511D4—82no—
512D4—82no—
513D4—82no—
514D4C2×#10218yes—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4—1616no—
547C2×C4#38611yescharacteristic
548C2×C4#38611yescharacteristic
549C2×C4#38611yescharacteristic
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4#38611yescharacteristic
553C2×C4C2×#10212yes—
554C2×C4C2×#10212yes—
555C2×C4#41712yes—
556C2×C4#41712yes—
557C2×C4#44712yes—
558C2×C4#44812yes—
559C2×C4#48012yes—
560C2×C4#48012yes—
561C2×C4#48112yes—
562C2×C4#48112yes—
563C2×C4C22×C22⋊C411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12311yescharacteristic
569C2×C4C22×D811yescharacteristic
570C2×C4C22×SD1611yescharacteristic
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—28no—
583C22—42no—
584C22—42no—
585C22—48no—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—88no—
590C22—88no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—164no—
595C22#187412yes—
596C22#187412yes—
597C22C2×#38611yescharacteristic
598C22C2×#44711yescharacteristic
599C22C2×#44811yescharacteristic
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—44no—
608C4—84no—
609C4—84no—
610C4—82no—
611C4#187412yes—
612C4#187412yes—
613C4#190412yes—
614C4C22×#10211yescharacteristic
615C4C2×#41711yescharacteristic
616C4C2×#48011yescharacteristic
617C4C2×#48111yescharacteristic
618C2—21no—
619C2—21no—
620C2—22no—
621C2—22no—
622C2—48no—
623C2—162no—
624C2C2×#187411yescharacteristic
625C2C2×#190411yescharacteristic
626C2#2100011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3036C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34181C2semidirect
3. #17574C2semidirect
4. #17574C2semidirect
5. C2×#1478C2semidirect
6. #9179C2semidirect
7. #8095C2semidirect
8. #7957C2semidirect
15. #3036C4semidirect
18. C2×#461C4semidirect
19. C2×#459C4semidirect
25. #2349C22semidirect
36. #1517C22semidirect
38. #1502C22semidirect
40. #1478C22semidirect
42. #1303C22semidirect
43. #1286C22semidirect
60. #532C2×C4semidirect
62. C2×C4⋊Q8C2×C4semidirect
63. C2×#127C2×C4semidirect
95. #461C2×C4semidirect
97. #459C2×C4semidirect
108. #423C2×C4semidirect
113. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30362
#341812
#175742
C2×#14782
#91792
#80952
#79572