Finite Group Databasebeta

← #194359

Subgroups — #194359

Order 512.

1421 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#60144C211yesmaximal, characteristic, complemented
3#21386C211yesmaximal, characteristic, complemented
4C2×#1980C211yesmaximal, characteristic, complemented
5#17485C211yesmaximal, characteristic, complemented
6#8991C218yesmaximal, complemented
7C2×#1303C211yesmaximal, characteristic, complemented
8#7882C211yesmaximal, characteristic, complemented
9C2×#568—21no—
10C2×#532C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3245C414yescomplemented
13#3226—21no—
14#3226C412yescomplemented
15#3205—21no—
16#3205C412yescomplemented
17#3186—22no—
18#3169—22no—
19#3100—22no—
20#3100C2214yes—
21#2935—22no—
22#2935—22no—
23D4×M16—22no—
24D4×M16C2214yescomplemented
25#2932—22no—
26#2931—22no—
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C22×#104—21no—
31C22×#104—21no—
32#2879—21no—
33C2×C4×M16C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1988—21no—
36#1982—21no—
37#1980—22no—
38#1980C2214yes—
39#1874—22no—
40C4×#102C2214yescomplemented
41#1492C2214yes—
42#1486—22no—
43#1480—22no—
44#1303C2214yescomplemented
45#1278C2214yescomplemented
46C22×#41—21no—
47C22×#41—21no—
48#568—44no—
49#568—44no—
50C2×#131—24no—
51C2×#131—24no—
52C22×M16—21no—
53C22×M16—24no—
54C22×M16D411yescharacteristic
55C22×M16D411yescharacteristic
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532C2×C412yes—
67#532C2×C412yes—
68#532C2×C412yes—
69#532D412yes—
70#532D412yes—
71#532C2312yes—
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×C4⋊Q8D411yescharacteristic
74C2×#127C2×C411yescharacteristic
75C2×#127D411yescharacteristic
76C2×#124—21no—
77C2×#124—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×#121—21no—
81C2×#121D412yes—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2312yes—
86C22×C4⋊C4—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#500—22no—
90#493—22no—
91#486—22no—
92#486—22no—
93#480—24no—
94#480—24no—
95#474—24no—
96#474C2×C418yescomplemented
97#466—28no—
98#459—24no—
99#459C2×C418yescomplemented
100#447—24no—
101#447—24no—
102C4×D8—28no—
103#436—24no—
104#436—24no—
105#435—28no—
106#435—28no—
107C8×D4—24no—
108C8×D4—24no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431C2×C412yes—
116#431C2312yes—
117#423—21no—
118#423—21no—
119#423D412yes—
120#423C2312yes—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2×C412yes—
124C2×C4⋊C8D412yes—
125C2×#104—24no—
126C2×#104—24no—
127C2×#104—44no—
128C2×#104—44no—
129C2×#102—24no—
130#408—24no—
131#408—24no—
132#407—24no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C4×M16—21no—
136C4×M16—21no—
137C4×M16D412yes—
138C4×M16C2312yes—
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8C2312yes—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2312yes—
147C2×#95—21no—
148#340—24no—
149D4⋊C8—28no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—41no—
157C2×#41—41no—
158C2×#41—42no—
159C2×#41—42no—
160C2×#41—42no—
161C2×#41—42no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—41no—
175C22×D4C2×D412yes—
176C23×C4—22no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×D8—28no—
183#131—416no—
184#131—416no—
185C2×M16—21no—
186C2×M16—22no—
187C2×M16—28no—
188C2×M16—22no—
189C2×M16—21no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—48no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—48no—
197C2×M16D814yescomplemented
198C2×M16C2×D412yes—
199C2×M16C2×D412yes—
200C22×C8—21no—
201C22×C8—28no—
202C22×C8—22no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8C22⋊C412yes—
212C4⋊Q8C22⋊C412yes—
213C4⋊Q8C22×C412yes—
214#127—21no—
215#127—21no—
216#127—21no—
217#127—21no—
218#127—21no—
219#127C22⋊C412yes—
220#127C22⋊C412yes—
221#127C22×C412yes—
222#124—22no—
223#124—22no—
224#124—22no—
225#124—22no—
226#124—22no—
227#124—22no—
228#124—41no—
229#124—41no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—22no—
238#121—22no—
239#121—22no—
240#121—41no—
241#121—41no—
242#121—41no—
243#121—41no—
244#121C22⋊C414yes—
245#121C22⋊C414yes—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4C22×C414yes—
261C4×D4C2×D414yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4C2×D412yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C42—22no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—41no—
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42C2411yescharacteristic
289#107—28no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—41no—
295C4⋊C8—41no—
296C4⋊C8C22×C414yes—
297C4⋊C8C2×D414yes—
298#104—48no—
299#104—48no—
300#102—28no—
301#102—28no—
302#102—48no—
303C22⋊C8—28no—
304C22⋊C8—28no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309#97—22no—
310#97—22no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—22no—
315C4×C8—22no—
316C4×C8—22no—
317#95—22no—
318#95—41no—
319C24—22no—
320C24—41no—
321C24#12111yescharacteristic
322#41—44no—
323#41—44no—
324#41—44no—
325#41—44no—
326#41—44no—
327#41—44no—
328#41—44no—
329#41—44no—
330#41—42no—
331#41—42no—
332#41—42no—
333#41—42no—
334#41—44no—
335#41—44no—
336#41—82no—
337#41—82no—
338#41—82no—
339#41—82no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4C2×C22⋊C414yes—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—81no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4C4×D411yescharacteristic
399C22×C4C4×D411yescharacteristic
400C22×C4#12011yescharacteristic
401C22×C4#12011yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12311yescharacteristic
408C22×C4#12311yescharacteristic
409C22×C4C2×D812yes—
410C22×C4#13512yes—
411C22×C4AGL(1,ℤ/8)12yes—
412C22×C4AGL(1,ℤ/8)12yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4111yescharacteristic
416D8—88no—
417M16—22no—
418M16—24no—
419M16—22no—
420M16—22no—
421M16—416no—
422M16—42no—
423M16—42no—
424M16—44no—
425M16—42no—
426M16C2×D814yes—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—416no—
433C2×C8—44no—
434C2×C8—42no—
435C2×C8—44no—
436C2×C8—42no—
437C2×C8—82no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4C2×C22⋊C414yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C42—42no—
458C42—42no—
459C42—82no—
460C42C2×C22⋊C411yescharacteristic
461C42C2×C22⋊C411yescharacteristic
462C42C4×D411yescharacteristic
463C42C4×D411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42#12011yescharacteristic
467C42#12011yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42#12311yescharacteristic
471C42#12311yescharacteristic
472C42C23×C411yescharacteristic
473C42C22×D411yescharacteristic
474C42C22×D411yescharacteristic
475C42C2×#4111yescharacteristic
476C23—24no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23#38611yescharacteristic
488C23#38611yescharacteristic
489C23#45912yes—
490C23#46312yes—
491C23C2×#12111yescharacteristic
492Q8—44no—
493Q8—44no—
494Q8—42no—
495Q8—42no—
496Q8—82no—
497D4—28no—
498D4—44no—
499D4—44no—
500D4—48no—
501D4—48no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—42no—
506D4—42no—
507D4—82no—
508D4—82no—
509D4—82no—
510D4—82no—
511D4—84no—
512D4—82no—
513D4—82no—
514D4—82no—
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—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—82no—
537C2×C4—88no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#38611yescharacteristic
544C2×C4#38611yescharacteristic
545C2×C4#38611yescharacteristic
546C2×C4#38611yescharacteristic
547C2×C4#38611yescharacteristic
548C2×C4#38611yescharacteristic
549C2×C4C4×D812yes—
550C2×C4C4×D812yes—
551C2×C4#44212yes—
552C2×C4#44212yes—
553C2×C4#44712yes—
554C2×C4#44712yes—
555C2×C4#44912yes—
556C2×C4#44912yes—
557C2×C4#45912yes—
558C2×C4#46312yes—
559C2×C4C22×C22⋊C411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×#12011yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12311yescharacteristic
565C2×C4C22×D811yescharacteristic
566C2×C4C2×#13511yescharacteristic
567C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
568C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
569C8—24no—
570C8—44no—
571C8—44no—
572C8—42no—
573C8—82no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—162no—
589C22—162no—
590C22—164no—
591C22#187912yes—
592C22#187912yes—
593C22C2×#38611yescharacteristic
594C22C2×#45911yescharacteristic
595C22C2×#46311yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—44no—
604C4—84no—
605C4—82no—
606C4#187912yes—
607C4#187912yes—
608C4#290112yes—
609C4C2×C4×D811yescharacteristic
610C4C2×#44211yescharacteristic
611C4C2×#44711yescharacteristic
612C4C2×#44911yescharacteristic
613C2—21no—
614C2—22no—
615C2—21no—
616C2—22no—
617C2—84no—
618C2—162no—
619C2#2090311yescharacteristic
620C2C2×#187911yescharacteristic
621C2#2093911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#60940C2
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
2. #60144C2semidirect
3. #21386C2semidirect
4. C2×#1980C2semidirect
5. #17485C2semidirect
6. #8991C2semidirect
7. C2×#1303C2semidirect
8. #7882C2semidirect
12. #3245C4semidirect
14. #3226C4semidirect
16. #3205C4semidirect
24. D4×M16C22semidirect
40. C4×#102C22semidirect
44. #1303C22semidirect
45. #1278C22semidirect
96. #474C2×C4semidirect
99. #459C2×C4semidirect
197. C2×M16D8semidirect

Maximal subgroups

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

subgroupindex
#609402
#601442
#213862
C2×#19802
#174852
#89912
C2×#13032
#78822