Finite Group Databasebeta

← #195560

Subgroups — #195560

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3205C211yesmaximal, characteristic
2C2×#2366C211yesmaximal, characteristic, complemented
3#21386C211yesmaximal, characteristic, complemented
4#21258C211yesmaximal, characteristic, complemented
5C2×#1545C211yesmaximal, characteristic, complemented
6#9061C214yesmaximal, complemented
7#9057C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×#127C2211yescharacteristic
12C22×C4×D4—21no—
13C22×C4×D4—21no—
14#3205—24no—
15#3205C418yescomplemented
16C2×#493C2212yes—
17C2×#486C2212yes—
18C2×#459—22no—
19C2×#447—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417—21no—
24C2×#417—21no—
25#2366—22no—
26#2366—22no—
27#2366—22no—
28#2366C2214yes—
29C2×#390—21no—
30C2×#390—21no—
31C2×#386—22no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1988C2212yescomplemented
35#1983—22no—
36#1982C2212yescomplemented
37#1982C2212yescomplemented
38#1981C2212yescomplemented
39#1874—21no—
40#1874—21no—
41C4×#102—22no—
42#1545—24no—
43C2×#331C2214yes—
44#1279C2214yescomplemented
45#1279C2214yescomplemented
46C22×#41—21no—
47C23×D4—21no—
48C23×D4—21no—
49C23×D4D412yes—
50C24×C4—22no—
51C22×D8—24no—
52C22×M16—21no—
53#532—22no—
54#532—22no—
55#532—22no—
56#532C2×C414yescomplemented
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127—21no—
59C2×#127—21no—
60C2×#127—22no—
61C2×#127—22no—
62C2×#127C2×C414yes—
63C2×#127D412yes—
64C2×#127D414yes—
65C2×#127C2311yescharacteristic
66C2×#127C2312yes—
67C2×#124—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—48no—
77C2×C4×D4—44no—
78C2×C4×D4—44no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#493—24no—
90#493C2×C418yescomplemented
91#486—24no—
92#486C2×C418yescomplemented
93#459—216no—
94#447—216no—
95#431—22no—
96#431—22no—
97#431—22no—
98#431C2×C414yescomplemented
99#423—22no—
100#423—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105#417—44no—
106#417—44no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#102—24no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—22no—
113C2×#102—22no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8C2312yes—
118#390—24no—
119#390—24no—
120#390—24no—
121#390—24no—
122#386—28no—
123#386—28no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4C2312yes—
131C4×C4⋊C4C2312yes—
132C2×#95—21no—
133C2×#95—21no—
134#340—22no—
135#340—22no—
136#331—28no—
137C25—22no—
138C2×#41—24no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C22×Q8—21no—
145C22×D4—24no—
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—24no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—41no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—82no—
166C22×D4—82no—
167C22×D4C22⋊C418yes—
168C22×D4C2×D414yes—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—28no—
174C23×C4—48no—
175C23×C4—48no—
176C23×C4—41no—
177C23×C4—84no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C2×D8—216no—
182C2×D8—84no—
183C2×D8—84no—
184C2×M16—42no—
185C2×M16—42no—
186C2×M16—42no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22×C412yescomplemented
194#127—24no—
195#127—24no—
196#127—24no—
197#127—24no—
198#127—24no—
199#127—21no—
200#127—22no—
201#127—22no—
202#127—21no—
203#127—21no—
204#127C22⋊C418yes—
205#127C22×C414yes—
206#127C22×C412yescomplemented
207#127C2×D414yes—
208#124—24no—
209#124—42no—
210#121—24no—
211#121—24no—
212#121—42no—
213#121—42no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—416no—
217C4×D4—48no—
218C4×D4—48no—
219C4×D4—416no—
220C4×D4—48no—
221C4×D4—48no—
222C4×D4—42no—
223C4×D4—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4D814yescomplemented
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—84no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—44no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C22×C412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4112yes—
261C2×C42#4112yes—
262C2×C42C2411yescharacteristic
263C4⋊C8—24no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—42no—
269#103—44no—
270#103—44no—
271#102—216no—
272#102—44no—
273#102—44no—
274#102—44no—
275#102—44no—
276C22⋊C8—24no—
277C22⋊C8—42no—
278C4×C8—22no—
279C4×C8—42no—
280C4×C8C22×C414yescomplemented
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285C24—24no—
286C24—28no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—41no—
294C24#12011yescharacteristic
295#41—48no—
296#41—48no—
297#41—84no—
298#41—84no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—416no—
319C2×D4—416no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—42no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—41no—
338C2×D4—44no—
339C2×D4—42no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4—88no—
349C2×D4—88no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4C2×C22⋊C418yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—416no—
364C22×C4—416no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—88no—
378C22×C4—832no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4C2×C22⋊C412yes—
382C22×C4C4×D412yes—
383C22×C4C4×D412yes—
384C22×C4#12011yescharacteristic
385C22×C4#12011yescharacteristic
386C22×C4#12011yescharacteristic
387C22×C4#12112yes—
388C22×C4#12112yes—
389C22×C4#12312yes—
390C22×C4C2×D812yes—
391C22×C4#13512yes—
392C22×C4AGL(1,ℤ/8)12yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397D8—816no—
398M16—84no—
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4C2×D814yes—
423C4⋊C4AGL(1,ℤ/8)14yes—
424C22⋊C4—416no—
425C22⋊C4—416no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—88no—
429C42—44no—
430C42—42no—
431C42—42no—
432C42—44no—
433C42—44no—
434C42C2×C22⋊C412yes—
435C42C4×D412yes—
436C42C4×D412yes—
437C42#12112yes—
438C42#12112yes—
439C42#12312yes—
440C42C23×C411yescharacteristic
441C42C22×D411yescharacteristic
442C42C2×#4111yescharacteristic
443C42C2×#4111yescharacteristic
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—28no—
449C23—416no—
450C23—416no—
451C23—416no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—88no—
460C23—88no—
461C23—88no—
462C23—88no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23#38612yes—
470C23#44712yes—
471C23#44912yes—
472C23C2×#12011yescharacteristic
473Q8—42no—
474Q8—42no—
475Q8—81no—
476Q8—81no—
477D4—48no—
478D4—416no—
479D4—44no—
480D4—416no—
481D4—44no—
482D4—42no—
483D4—44no—
484D4—42no—
485D4—48no—
486D4—44no—
487D4—44no—
488D4—44no—
489D4—48no—
490D4—84no—
491D4—88no—
492D4—82no—
493D4—88no—
494D4—81no—
495D4—81no—
496D4—82no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—832no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—88no—
528C2×C4#38612yes—
529C2×C4#38612yes—
530C2×C4#38612yes—
531C2×C4C4×D814yes—
532C2×C4#44214yes—
533C2×C4#44712yes—
534C2×C4#44912yes—
535C2×C4#45914yes—
536C2×C4#46314yes—
537C2×C4C22×C22⋊C411yescharacteristic
538C2×C4C2×C4×D411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×#12111yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12311yescharacteristic
543C2×C4C22×D811yescharacteristic
544C2×C4C2×#13511yescharacteristic
545C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
546C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
547C2×C4C2×AGL(1,ℤ/8)12yes—
548C2×C4#57512yes—
549C8—44no—
550C8—82no—
551C8—82no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—48no—
559C22—48no—
560C22—48no—
561C22—832no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—161no—
570C22—161no—
571C22—161no—
572C22—161no—
573C22#187914yes—
574C22C2×#38611yescharacteristic
575C22C2×#44711yescharacteristic
576C22C2×#44911yescharacteristic
577C22#300311yescharacteristic
578C22#300511yescharacteristic
579C22#301911yescharacteristic
580C22#301911yescharacteristic
581C4—22no—
582C4—21no—
583C4—21no—
584C4—22no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4#187914yes—
591C4C2×C4×D811yescharacteristic
592C4C2×#44211yescharacteristic
593C4C4×AGL(1,ℤ/8)12yes—
594C4C2×#45911yescharacteristic
595C4C2×#46311yescharacteristic
596C4#303612yes—
597C2—24no—
598C2—88no—
599C2—162no—
600C2#1874611yescharacteristic
601C2#1875011yescharacteristic
602C2#1876611yescharacteristic
603C2#1877011yescharacteristic
604C2C2×#187911yescharacteristic
605C2#2092511yescharacteristic
606C2#2095311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3205C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2366C2semidirect
3. #21386C2semidirect
4. #21258C2semidirect
5. C2×#1545C2semidirect
6. #9061C2semidirect
7. #9057C2semidirect
8. #7901C2semidirect
9. #7892C2semidirect
15. #3205C4semidirect
34. #1988C22semidirect
36. #1982C22semidirect
37. #1982C22semidirect
38. #1981C22semidirect
44. #1279C22semidirect
45. #1279C22semidirect
56. #532C2×C4semidirect
90. #493C2×C4semidirect
92. #486C2×C4semidirect
98. #431C2×C4semidirect
193. C4⋊Q8C22×C4semidirect
206. #127C22×C4semidirect
235. C2×C4⋊C4D8semidirect
280. C4×C8C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32052
C2×#23662
#213862
#212582
C2×#15452
#90612
#90572
#79012
#78922