Finite Group Databasebeta

← #205097

Subgroups — #205097

Order 512.

1071 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic
2#21816C211yesmaximal, characteristic, complemented
3#21815C211yesmaximal, characteristic, complemented
4#21360C211yesmaximal, characteristic, complemented
5#17561C211yesmaximal, characteristic, complemented
6#17544C211yesmaximal, characteristic, complemented
7#17324C211yesmaximal, characteristic, complemented
8#9721C214yesmaximal, complemented
9#9719C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3245—22no—
13#3226C412yescomplemented
14#3226C412yescomplemented
15#3205C412yescomplemented
16#3205C412yescomplemented
17#3186—22no—
18#3169C2212yes—
19#3169C2212yes—
20#3100—22no—
21#3100—22no—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2283C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1980C2214yescomplemented
28#1980C2214yescomplemented
29#1963—22no—
30#1962—22no—
31#1949—22no—
32#1854—22no—
33#1789—22no—
34#1789—22no—
35C4×#107C2212yescomplemented
36C4×#106C2212yescomplemented
37#1514—22no—
38#1510—22no—
39#1507—22no—
40#1505—22no—
41#1502C2214yescomplemented
42#1502C2214yescomplemented
43#1454—22no—
44#1453—22no—
45#1451C2212yescomplemented
46C8×M16C2212yescomplemented
47C22×#41—21no—
48C22×#41—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532C2×C412yes—
62#532C2×C412yes—
63#532C2312yes—
64#532C2312yes—
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×#127C2×C411yescharacteristic
68C2×#127C2×C411yescharacteristic
69C2×#124—21no—
70C2×#124—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#500C2×C412yes—
82#500C2×C412yes—
83#493C2×C412yes—
84#493C2×C412yes—
85#486C2×C412yes—
86#486C2×C412yes—
87#486C2×C412yes—
88#486C2×C412yes—
89#480—24no—
90#480—24no—
91#474—24no—
92#474—24no—
93#466—24no—
94#466—24no—
95#459—24no—
96#459—24no—
97#447—24no—
98#447—24no—
99C4×D8—24no—
100C4×D8—24no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432D412yes—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2×C412yes—
109#431C2×C412yes—
110#431D412yes—
111#431D412yes—
112#431C2312yes—
113C2×#107—22no—
114C2×#106—22no—
115#423—21no—
116#423D412yes—
117C2×C4⋊C8D412yes—
118C2×#104—28no—
119C2×#104—28no—
120#407D412yes—
121#407D412yes—
122C2×C22⋊C8D412yes—
123C4×M16—21no—
124C4×M16C2312yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127#383—21no—
128#382—21no—
129#381—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138#376—21no—
139C43—21no—
140#340—22no—
141#340—22no—
142#327—28no—
143#327—28no—
144#322—22no—
145#322—24no—
146C82—22no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—41no—
152C2×#41—41no—
153C2×#41—41no—
154C2×#41—41no—
155C2×#41—84no—
156C2×#41—84no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—41no—
166C22×D4—41no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C2×D8—24no—
173C2×D8—24no—
174C2×M16—22no—
175C2×M16C2×D414yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C22×C412yes—
185C4⋊Q8C22×C412yes—
186#127—21no—
187#127—21no—
188#127—21no—
189#127—21no—
190#127—21no—
191#127—21no—
192#127C22×C412yes—
193#127C22×C412yes—
194#124—22no—
195#124—22no—
196#124—22no—
197#124—22no—
198#124—41no—
199#124—41no—
200#124—41no—
201#124—41no—
202#121—22no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209#121—22no—
210#121—41no—
211#121—41no—
212#121—41no—
213#121—41no—
214#121—41no—
215#121—41no—
216#121—41no—
217#121—41no—
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—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234#117—44no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42C2411yescharacteristic
263#107—24no—
264#107—24no—
265#107—44no—
266#106—44no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8C22⋊C414yes—
272C4⋊C8C22⋊C414yes—
273#104—88no—
274#104—88no—
275#102—24no—
276#102—24no—
277#102—24no—
278#102—24no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8C22⋊C414yes—
283C22⋊C8C22⋊C414yes—
284C22⋊C8C2×D414yes—
285#97—22no—
286#97—22no—
287C4×C8—22no—
288C4×C8—22no—
289C4×C8C22×C412yes—
290C4×C8C22×C412yes—
291C4×C8C2×D412yes—
292C4×C8C2×D412yes—
293#95—22no—
294#95—22no—
295#95—41no—
296#95—41no—
297#95—41no—
298#95—41no—
299C24—41no—
300C24—41no—
301C24#12711yescharacteristic
302#41—48no—
303#41—48no—
304#41—84no—
305#41—84no—
306#41—84no—
307#41—84no—
308#41—84no—
309#41—84no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4—82no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4—82no—
338C2×D4—84no—
339C2×D4—84no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—81no—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12411yescharacteristic
364C22×C4#12411yescharacteristic
365C22×C4#12711yescharacteristic
366C22×C4#12711yescharacteristic
367C22×C4#12711yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371D8—84no—
372D8—84no—
373M16—28no—
374M16—82no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8C2×C22⋊C414yes—
380C2×C8C2×C22⋊C414yes—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C22⋊C4—44no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—82no—
403C42—44no—
404C42—44no—
405C42—44no—
406C42—82no—
407C42C2×C22⋊C411yescharacteristic
408C42C2×C22⋊C411yescharacteristic
409C42C4×D411yescharacteristic
410C42C4×D411yescharacteristic
411C42C4×D411yescharacteristic
412C42C4×D411yescharacteristic
413C42#12111yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12111yescharacteristic
417C42#12411yescharacteristic
418C42#12411yescharacteristic
419C42C23×C411yescharacteristic
420C42C22×D411yescharacteristic
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C23—22no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23#39011yescharacteristic
435C23#39011yescharacteristic
436C23C2×#12711yescharacteristic
437Q8—44no—
438Q8—44no—
439Q8—82no—
440Q8—82no—
441D4—44no—
442D4—44no—
443D4—44no—
444D4—44no—
445D4—82no—
446D4—82no—
447D4—82no—
448D4—82no—
449D4—82no—
450D4—82no—
451D4—82no—
452D4—82no—
453D4—84no—
454D4—84no—
455D4—84no—
456D4—84no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4#39011yescharacteristic
489C2×C4#39011yescharacteristic
490C2×C4#39011yescharacteristic
491C2×C4#39011yescharacteristic
492C2×C4#39011yescharacteristic
493C2×C4#39011yescharacteristic
494C2×C4C22×C22⋊C411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12411yescharacteristic
500C2×C4#57614yes—
501C8—24no—
502C8—24no—
503C8—44no—
504C8—44no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—164no—
521C22—164no—
522C22C2×#39011yescharacteristic
523C22#315512yes—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—162no—
537C4#290112yes—
538C4#294812yes—
539C4#305812yes—
540C2—22no—
541C2—41no—
542C2—41no—
543C2—162no—
544C2—162no—
545C2#2123311yescharacteristic
546C2#2124411yescharacteristic
547C2#2125311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
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. #21816C2semidirect
3. #21815C2semidirect
4. #21360C2semidirect
5. #17561C2semidirect
6. #17544C2semidirect
7. #17324C2semidirect
8. #9721C2semidirect
9. #9719C2semidirect
13. #3226C4semidirect
14. #3226C4semidirect
15. #3205C4semidirect
16. #3205C4semidirect
27. #1980C22semidirect
28. #1980C22semidirect
35. C4×#107C22semidirect
36. C4×#106C22semidirect
41. #1502C22semidirect
42. #1502C22semidirect
45. #1451C22semidirect
46. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
#609402
#218162
#218152
#213602
#175612
#175442
#173242
#97212
#97192