Finite Group Databasebeta

← #200860

Subgroups — #200860

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60618C211yesmaximal, characteristic
2#34388C211yesmaximal, characteristic
3#17754C211yesmaximal, characteristic
4#17596C211yesmaximal, characteristic, complemented
5C2×#1515C211yesmaximal, characteristic, complemented
6#9426C214yesmaximal, complemented
7#9425C214yesmaximal, complemented
8#7972C211yesmaximal, characteristic, complemented
9C2×#1288C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3142—22no—
13#3141C412yescomplemented
14#3141C412yescomplemented
15#3140—22no—
16#3139C2212yes—
17#3138C2212yes—
18#3135—22no—
19#3131C414yescomplemented
20C2×#432C2211yescharacteristic
21C2×#432C2211yescharacteristic
22#2896—22no—
23C2×C4×M16C2211yescharacteristic
24#2381C2212yes—
25#2380C2212yes—
26#2364—22no—
27#2353—22no—
28#2339—22no—
29#2337—22no—
30#2325—22no—
31#2318—22no—
32#2283C2211yescharacteristic
33C2×#383C2211yescharacteristic
34#1558C2212yes—
35#1555C2212yes—
36#1551—22no—
37#1551—22no—
38#1520—22no—
39#1515—22no—
40#1515C2214yes—
41#1515C2214yes—
42#1506—22no—
43C2×#328—22no—
44#1288C2214yes—
45#1288C2214yes—
46C22×#41D412yes—
47C22×M16—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533C2×C412yescomplemented
55#533C2312yes—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532C2×C412yes—
61#532C2×C412yes—
62#532D412yes—
63#532D412yes—
64C2×C4⋊Q8C2×C411yescharacteristic, complemented
65C2×C4⋊Q8C2×C411yescharacteristic, complemented
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#125—21no—
69C2×#124D412yes—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×D4—24no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#491—24no—
82#490—24no—
83#489—24no—
84#489—24no—
85#488—28no—
86#487C2×C414yescomplemented
87#487C2×C414yescomplemented
88#486C2×C414yescomplemented
89#486C2×C414yescomplemented
90#432—21no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432C2312yes—
97#432C2312yes—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100#419—24no—
101#418—24no—
102#417—24no—
103#417—24no—
104#416—28no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C4×M16—21no—
108C4×M16—21no—
109C4×M16C2×C412yescomplemented
110C4×M16C2312yes—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8C2×C412yescomplemented
115#391—24no—
116#388—24no—
117#388—24no—
118#387—24no—
119#387—24no—
120#386—24no—
121#386—24no—
122#386—24no—
123#386—24no—
124#385—24no—
125#383—21no—
126#383—21no—
127#383D412yes—
128#383C2312yes—
129#383C2312yes—
130#382—21no—
131#381—22no—
132#381—21no—
133#380—22no—
134#380D412yes—
135C4×C4⋊C4—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—21no—
139#376D412yes—
140C2×#95—22no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95D412yes—
144#333—24no—
145#331—24no—
146#328—28no—
147#328—28no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—48no—
151C2×#41C22⋊C414yes—
152C2×#41C22⋊C414yes—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8C2×D412yes—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C2×D412yes—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—44no—
169C22×C8—22no—
170C22×C8—41no—
171C22×C8—41no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C2×D412yes—
184#127—21no—
185#127—21no—
186#127C22×C412yes—
187#127C2×D412yes—
188#125—22no—
189#125—22no—
190#125—41no—
191#125—41no—
192#124—22no—
193#124—22no—
194#124C22⋊C414yes—
195#124C22⋊C414yes—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204#121—22no—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209#121—22no—
210#121—22no—
211#121—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
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—88no—
229#117—22no—
230#117—22no—
231#117—41no—
232#117—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—41no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—41no—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42C2411yescharacteristic
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269#103—28no—
270#103—28no—
271#102—28no—
272#102—28no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—41no—
276C22⋊C8—41no—
277#97—22no—
278#97—22no—
279#97—22no—
280#97—22no—
281#97—41no—
282#97—41no—
283C4×C8—22no—
284C4×C8C22×C414yes—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95C2×D414yes—
296#95C2×D414yes—
297C24—22no—
298C24#12011yescharacteristic
299#41—48no—
300#41—48no—
301#41—84no—
302#41—84no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—44no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8C2×C22⋊C414yes—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—44no—
320C2×D4—48no—
321C2×D4—81no—
322C2×D4—82no—
323C2×D4—81no—
324C2×D4C2×C22⋊C414yes—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—44no—
334C22×C4—84no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4#12311yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360M16—44no—
361M16—44no—
362C2×C8—24no—
363C2×C8—44no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—82no—
367C2×C8—82no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—82no—
385C4⋊C4—84no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C42—42no—
397C42—42no—
398C42—82no—
399C42—84no—
400C42C2×C22⋊C411yescharacteristic
401C42C2×C22⋊C411yescharacteristic
402C42C4×D411yescharacteristic
403C42C4×D411yescharacteristic
404C42C4×D411yescharacteristic
405C42C4×D411yescharacteristic
406C42#12111yescharacteristic
407C42#12111yescharacteristic
408C42#12111yescharacteristic
409C42#12111yescharacteristic
410C42#12311yescharacteristic
411C42#12311yescharacteristic
412C42C23×C411yescharacteristic
413C42C22×D411yescharacteristic
414C42C2×#4111yescharacteristic
415C42C2×#4111yescharacteristic
416C23—24no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—44no—
421C23—42no—
422C23—44no—
423C23#38611yescharacteristic
424C23#38611yescharacteristic
425C23#45514yes—
426C23C2×#12011yescharacteristic
427Q8—44no—
428Q8—82no—
429Q8—82no—
430Q8—82no—
431Q8—82no—
432Q8—82no—
433D4—44no—
434D4—48no—
435D4—48no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—82no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—164no—
474C2×C4—168no—
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#38611yescharacteristic
481C2×C4C22×C22⋊C411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C8—44no—
488C8—82no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—84no—
501C22—84no—
502C22#188512yes—
503C22#188512yes—
504C22C2×#38611yescharacteristic
505C22C2×#45512yes—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—44no—
519C4—84no—
520C4—84no—
521C4—82no—
522C4—82no—
523C2—22no—
524C2—22no—
525C2—22no—
526C2—84no—
527C2C2×#188511yescharacteristic
528C2#2099811yescharacteristic
529C2#2099811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
4. #17596C2semidirect
5. C2×#1515C2semidirect
6. #9426C2semidirect
7. #9425C2semidirect
8. #7972C2semidirect
9. C2×#1288C2semidirect
13. #3141C4semidirect
14. #3141C4semidirect
19. #3131C4semidirect
54. #533C2×C4semidirect
64. C2×C4⋊Q8C2×C4semidirect
65. C2×C4⋊Q8C2×C4semidirect
86. #487C2×C4semidirect
87. #487C2×C4semidirect
88. #486C2×C4semidirect
89. #486C2×C4semidirect
109. C4×M16C2×C4semidirect
114. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#606182
#343882
#177542
#175962
C2×#15152
#94262
#94252
#79722
C2×#12882