Finite Group Databasebeta

← #205937

Subgroups — #205937

Order 512.

1191 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic
2#21425C211yesmaximal, characteristic, complemented
3C2×#1965C211yesmaximal, characteristic
4#17817C211yesmaximal, characteristic, complemented
5#17749C211yesmaximal, characteristic, complemented
6#17741C211yesmaximal, characteristic
7#17327C211yesmaximal, characteristic, complemented
8#9805C212yesmaximal, complemented
9#9804C212yesmaximal, complemented
10#9796C212yesmaximal, complemented
11#9793C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C22×#127C2211yescharacteristic
14#3208—22no—
15#3208—22no—
16#3208—21no—
17#3208—21no—
18#3208—22no—
19C2×#492C2212yes—
20C2×#486C2212yes—
21C2×#460C414yes—
22C2×#450C414yes—
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C22×#100—22no—
28C2×C4×M16C2211yescharacteristic
29#1997C2211yescharacteristic, complemented
30#1997C2211yescharacteristic, complemented
31#1966C2211yescharacteristic, complemented
32#1966C2211yescharacteristic, complemented
33#1965—22no—
34#1965C2212yes—
35#1965C2212yes—
36#1901—22no—
37C4×#101—22no—
38#1574—22no—
39#1573—22no—
40#1567C2212yescomplemented
41#1566C2212yescomplemented
42#1559—22no—
43#1555—22no—
44#1550—22no—
45#1548C2212yescomplemented
46#1546C2212yescomplemented
47#1545—22no—
48#1545C2212yes—
49#1545C2212yes—
50#1453—22no—
51#1451—21no—
52#1451C2212yescomplemented
53C8×M16—21no—
54C8×M16C2212yescomplemented
55C23×D4—21no—
56C23×D4—21no—
57C23×D4C2×C412yes—
58C22×SD16C2×C418yes—
59C22×M16—21no—
60C22×M16D412yes—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533—21no—
67#533—21no—
68#533—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127—21no—
72C2×#127—21no—
73C2×#127—21no—
74C2×#127—21no—
75C2×#127—21no—
76C2×#127—21no—
77C2×#127—21no—
78C2×#127—21no—
79C2×#127—21no—
80C2×#127—21no—
81C2×#127C2×C412yes—
82C2×#127C2312yes—
83C2×#125—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8C2×C412yes—
86C2×C4×Q8—21no—
87C2×C4×Q8C2×C412yes—
88C2×#117—21no—
89C22×C42C2311yescharacteristic
90#492—22no—
91#492—22no—
92#492—24no—
93#486—22no—
94#486—22no—
95#486—24no—
96#460—28no—
97#460—24no—
98#460—24no—
99#450—24no—
100#450—24no—
101#450—28no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432D412yes—
106#431—21no—
107#431—21no—
108#431—21no—
109#431D412yes—
110#431D412yes—
111#431D411yescharacteristic
112#431D411yescharacteristic
113#431D412yes—
114#431C2312yes—
115#423—21no—
116#423—21no—
117#423—21no—
118#423D412yes—
119#423C2311yescharacteristic
120#423C2311yescharacteristic
121#423C2312yes—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8C2×C412yes—
125C2×#102C2×C418yes—
126C2×#101—22no—
127C2×#101—22no—
128C2×#100—24no—
129C2×#100—24no—
130C2×#100—48no—
131#407—21no—
132#407—21no—
133C2×C22⋊C8C2×C412yes—
134C4×M16—21no—
135C4×M16—21no—
136C4×M16—21no—
137C4×M16—21no—
138C4×M16D412yes—
139C4×M16C2311yescharacteristic
140C4×M16C2311yescharacteristic
141C2×#97—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2312yes—
144#343—22no—
145#343—22no—
146#335—24no—
147#334—24no—
148#333—24no—
149#332—24no—
150#331—24no—
151#331—24no—
152#322—22no—
153#322—22no—
154#322—22no—
155C82—22no—
156C25—22no—
157C22×Q8—21no—
158C22×Q8C22×C412yes—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—84no—
179C22×D4—84no—
180C22×D4C22×C414yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4111yescharacteristic
184C2×SD16—28no—
185C2×SD16—28no—
186C2×SD16—88no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—24no—
190C2×M16—22no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—42no—
194C2×M16—41no—
195C2×M16—44no—
196C2×M16C2×D412yes—
197C2×M16C2×D412yes—
198C22×C8—22no—
199C22×C8—41no—
200C22×C8C22×C414yes—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209#127—22no—
210#127—22no—
211#127—22no—
212#127—22no—
213#127—22no—
214#127—22no—
215#127—22no—
216#127—22no—
217#127—41no—
218#127—41no—
219#127—41no—
220#127—41no—
221#127—41no—
222#127—41no—
223#127—41no—
224#127—41no—
225#125—41no—
226#125—41no—
227#125—41no—
228#125—41no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—41no—
234C22⋊Q8—41no—
235C22⋊Q8—41no—
236C22⋊Q8—41no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—41no—
242C4×Q8—41no—
243C4×Q8—41no—
244C4×Q8—41no—
245#117—41no—
246#117—41no—
247#117—41no—
248#117—41no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C22×C412yes—
255C2×C22⋊C4—22no—
256C2×C42—22no—
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42C2411yescharacteristic
269C4⋊C8—22no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—22no—
273C4⋊C8—21no—
274C4⋊C8—22no—
275C4⋊C8—21no—
276C4⋊C8—41no—
277C4⋊C8—41no—
278C4⋊C8C22⋊C414yes—
279C4⋊C8C22⋊C412yes—
280C4⋊C8C22⋊C412yes—
281C4⋊C8C2×D414yes—
282#102—28no—
283#102—28no—
284#101—44no—
285#101—44no—
286#100—48no—
287#100—48no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292C22⋊C8C22⋊C412yes—
293C22⋊C8C22⋊C412yes—
294C22⋊C8C22⋊C414yes—
295#97—22no—
296#97—22no—
297#97—21no—
298#97—21no—
299#97—41no—
300C4×C8—22no—
301C4×C8—21no—
302C4×C8—22no—
303C4×C8—21no—
304C4×C8—41no—
305C4×C8C2×D412yes—
306C4×C8C2×D412yes—
307C4×C8C2×D414yes—
308C24—24no—
309C24—48no—
310C24—42no—
311C24—42no—
312C24—84no—
313C24—84no—
314C24C4×D411yescharacteristic
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—44no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—84no—
339C2×D4—84no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12411yescharacteristic
373C22×C4#12411yescharacteristic
374C22×C4#12711yescharacteristic
375C22×C4#12711yescharacteristic
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379SD16—88no—
380SD16—88no—
381M16—22no—
382M16—22no—
383M16—22no—
384M16—44no—
385M16—44no—
386M16—44no—
387M16—42no—
388M16—42no—
389M16—82no—
390M16AGL(1,ℤ/8)14yes—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—44no—
395C2×C8—44no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—82no—
400C2×C8C2×C22⋊C414yes—
401C2×C8C2×C22⋊C414yes—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C42—42no—
417C42—42no—
418C42C2×C22⋊C411yescharacteristic
419C42C2×C22⋊C411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12411yescharacteristic
427C42#12411yescharacteristic
428C42#12711yescharacteristic
429C42#12711yescharacteristic
430C42C22×D411yescharacteristic
431C42C22×D411yescharacteristic
432C42C22×D411yescharacteristic
433C42C2×#4111yescharacteristic
434C23—22no—
435C23—22no—
436C23—41no—
437C23—44no—
438C23—48no—
439C23—84no—
440C23—84no—
441C23—84no—
442C23—84no—
443C23—84no—
444C23—84no—
445C23—84no—
446C23—84no—
447C23—1616no—
448C23#39011yescharacteristic
449C23#39011yescharacteristic
450C23C2×C4×D411yescharacteristic
451Q8—44no—
452Q8—82no—
453Q8—81no—
454Q8—81no—
455Q8—82no—
456D4—48no—
457D4—84no—
458D4—84no—
459D4—84no—
460D4—84no—
461D4—84no—
462D4—84no—
463D4—84no—
464D4—82no—
465D4—82no—
466D4—84no—
467D4—84no—
468D4—84no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4#39011yescharacteristic
492C2×C4#39011yescharacteristic
493C2×C4#39011yescharacteristic
494C2×C4#39011yescharacteristic
495C2×C4#39011yescharacteristic
496C2×C4#39011yescharacteristic
497C2×C4#47114yes—
498C2×C4C22×C22⋊C411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4C2×#12111yescharacteristic
502C2×C4C2×#12411yescharacteristic
503C2×C4C2×#12711yescharacteristic
504C2×C4C2×AGL(1,ℤ/8)12yes—
505C2×C4#57512yes—
506C8—24no—
507C8—24no—
508C8—24no—
509C8—42no—
510C8—44no—
511C8—82no—
512C8—81no—
513C8—81no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—44no—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—168no—
526C22—168no—
527C22—168no—
528C22#197712yes—
529C22C2×#39011yescharacteristic
530C22C4×AGL(1,ℤ/8)12yes—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—82no—
539C4—82no—
540C4#197712yes—
541C4#288812yes—
542C4#305512yes—
543C4C2×#47112yes—
544C4#315111yescharacteristic
545C4#315311yescharacteristic
546C2—22no—
547C2—22no—
548C2—41no—
549C2—164no—
550C2#2121911yescharacteristic
551C2#2123611yescharacteristic
552C2C2×#197711yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ 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. #21425C2semidirect
4. #17817C2semidirect
5. #17749C2semidirect
7. #17327C2semidirect
8. #9805C2semidirect
9. #9804C2semidirect
10. #9796C2semidirect
11. #9793C2semidirect
29. #1997C22semidirect
30. #1997C22semidirect
31. #1966C22semidirect
32. #1966C22semidirect
40. #1567C22semidirect
41. #1566C22semidirect
45. #1548C22semidirect
46. #1546C22semidirect
52. #1451C22semidirect
54. C8×M16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
#214252
C2×#19652
#178172
#177492
#177412
#173272
#98052
#98042
#97962
#97932