Finite Group Databasebeta

← #206403

Subgroups — #206403

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60941C211yesmaximal, characteristic
2#21844C211yesmaximal, characteristic, complemented
3#21815C211yesmaximal, characteristic
4#20462C211yesmaximal, characteristic, complemented
5#17815C211yesmaximal, characteristic, complemented
6#17479C211yesmaximal, characteristic, complemented
7#17478C211yesmaximal, characteristic
8#9854C212yesmaximal, complemented
9#9851C212yesmaximal, complemented
10#9846C212yesmaximal, complemented
11#9843C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3246—22no—
15#3229C412yescomplemented
16#3227C412yescomplemented
17#3208C412yescomplemented
18#3206C412yescomplemented
19#3187—22no—
20#3167C2212yes—
21#3167C2212yes—
22#3103—22no—
23#3101—22no—
24#2920C2211yescharacteristic
25#2920C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1994C2212yescomplemented
30#1994C2212yescomplemented
31#1986—22no—
32#1980—22no—
33#1962C2212yes—
34#1962C2212yes—
35#1826—22no—
36#1789—22no—
37#1768—22no—
38#1750—22no—
39#1744C2211yescharacteristic, complemented
40#1744C2211yescharacteristic, complemented
41C4×#97C2211yescharacteristic, complemented
42C42×C8C2211yescharacteristic, complemented
43#1572—22no—
44#1572—22no—
45#1567C2212yescomplemented
46#1566C2212yescomplemented
47#1497—22no—
48#1496—22no—
49#1493C2212yescomplemented
50#1493C2212yescomplemented
51#1492C2212yes—
52#1492C2212yes—
53#1479—22no—
54#1478—22no—
55C22×#41—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533C2×C412yes—
63#533C2312yes—
64#532—21no—
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532C2×C412yes—
71#532C2312yes—
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×#127C2×C411yescharacteristic
76C2×#125—21no—
77C2×#124—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×#121—21no—
81C2×#121—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×#117—21no—
87C2×#117—21no—
88C22×C42C2311yescharacteristic
89#498C2×C412yes—
90#498C2×C412yes—
91#492C2×C412yes—
92#492C2×C412yes—
93#487C2×C412yes—
94#487C2×C412yes—
95#486C2×C412yes—
96#486C2×C412yes—
97#483—24no—
98#481—24no—
99#476—24no—
100#475—24no—
101#465—24no—
102#465—24no—
103#461—24no—
104#460—24no—
105#450—24no—
106#448—24no—
107C4×SD16—24no—
108C4×SD16—24no—
109#432—21no—
110#432—21no—
111#432—21no—
112#432—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431—21no—
119#431C2×C412yes—
120#431C2×C412yes—
121#431D412yes—
122#431D412yes—
123#431D412yes—
124#431D412yes—
125#423C2312yes—
126#423C2312yes—
127C2×C4⋊C8—21no—
128C2×C4⋊C8D412yes—
129C2×#104—24no—
130C2×#104—24no—
131#407—21no—
132#407—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8D412yes—
135C4×M16—21no—
136C4×M16—21no—
137C2×#97—22no—
138C2×C4×C8—22no—
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C4×C4⋊C4—21no—
144C4×C4⋊C4D412yes—
145C4×C22⋊C4—21no—
146C4×C22⋊C4D412yes—
147#376—21no—
148C43C2311yescharacteristic
149C43C2311yescharacteristic
150#339—24no—
151#339—24no—
152#336—22no—
153#336—22no—
154#335—24no—
155#334—24no—
156Q8⋊C8—24no—
157Q8⋊C8—24no—
158D4⋊C8—24no—
159D4⋊C8—24no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—41no—
163C2×#41—41no—
164C2×#41—84no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—41no—
172C23×C4—41no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C2×SD16—24no—
177C2×SD16—24no—
178C2×M16—22no—
179C2×M16—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22×C412yes—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196#127—21no—
197#127—21no—
198#127—21no—
199#127C22×C412yes—
200#125—22no—
201#125—22no—
202#125—41no—
203#125—41no—
204#124—22no—
205#124—22no—
206#124—41no—
207#124—41no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—41no—
221#121—41no—
222#121—41no—
223#121—41no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240#117—22no—
241#117—22no—
242#117—44no—
243#117—41no—
244#117—41no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—41no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42C2411yescharacteristic
277#106—24no—
278#106—24no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8C22⋊C414yes—
286C4⋊C8C22⋊C414yes—
287#104—84no—
288#104—84no—
289#103—24no—
290#103—24no—
291#102—24no—
292#102—24no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299C22⋊C8C22⋊C414yes—
300C22⋊C8C22⋊C414yes—
301#97—44no—
302#97—41no—
303#97—41no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—44no—
307C4×C8—41no—
308C4×C8—41no—
309C4×C8C22×C412yes—
310C4×C8C22×C412yes—
311C4×C8C2×D412yes—
312C4×C8C2×D412yes—
313C4×C8C2×D412yes—
314C4×C8C2×D412yes—
315#95—22no—
316#95—22no—
317C24—41no—
318C24#12111yescharacteristic
319#41—44no—
320#41—44no—
321#41—84no—
322#41—84no—
323#41—84no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—44no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—84no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—84no—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12411yescharacteristic
368C22×C4#12411yescharacteristic
369C22×C4#12711yescharacteristic
370C22×C4#12711yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374SD16—84no—
375SD16—84no—
376M16—44no—
377M16—44no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8C2×C22⋊C414yes—
387C2×C8C2×C22⋊C414yes—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C42—24no—
413C42—24no—
414C42—24no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42C2×C22⋊C411yescharacteristic
421C42C2×C22⋊C411yescharacteristic
422C42C4×D411yescharacteristic
423C42C4×D411yescharacteristic
424C42C4×D411yescharacteristic
425C42C4×D411yescharacteristic
426C42#12111yescharacteristic
427C42#12111yescharacteristic
428C42#12411yescharacteristic
429C42#12411yescharacteristic
430C42#12711yescharacteristic
431C42#12711yescharacteristic
432C42C23×C411yescharacteristic
433C42C22×D411yescharacteristic
434C42C22×D411yescharacteristic
435C42C2×#4111yescharacteristic
436C23—22no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23#39011yescharacteristic
444C23#39011yescharacteristic
445C23C2×#12111yescharacteristic
446Q8—44no—
447Q8—44no—
448Q8—44no—
449Q8—82no—
450Q8—82no—
451Q8—82no—
452D4—44no—
453D4—44no—
454D4—44no—
455D4—82no—
456D4—82no—
457D4—82no—
458D4—84no—
459D4—84no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—84no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#39011yescharacteristic
493C2×C4#39011yescharacteristic
494C2×C4#39011yescharacteristic
495C2×C4#39011yescharacteristic
496C2×C4#39011yescharacteristic
497C2×C4#39011yescharacteristic
498C2×C4C22×C22⋊C411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×#12111yescharacteristic
502C2×C4C2×#12411yescharacteristic
503C2×C4C2×#12711yescharacteristic
504C2×C4#57714yes—
505C8—24no—
506C8—24no—
507C8—42no—
508C8—42no—
509C8—82no—
510C8—82no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—164no—
524C22C2×#39011yescharacteristic
525C22#305712yes—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4#290112yes—
540C4#294912yes—
541C4#295512yes—
542C4#315412yes—
543C2—22no—
544C2—41no—
545C2—41no—
546C2—162no—
547C2#2121011yescharacteristic
548C2#2123111yescharacteristic
549C2#2124211yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC2×#532C22
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. #21844C2semidirect
4. #20462C2semidirect
5. #17815C2semidirect
6. #17479C2semidirect
8. #9854C2semidirect
9. #9851C2semidirect
10. #9846C2semidirect
11. #9843C2semidirect
15. #3229C4semidirect
16. #3227C4semidirect
17. #3208C4semidirect
18. #3206C4semidirect
29. #1994C22semidirect
30. #1994C22semidirect
39. #1744C22semidirect
40. #1744C22semidirect
41. C4×#97C22semidirect
42. C42×C8C22semidirect
45. #1567C22semidirect
46. #1566C22semidirect
49. #1493C22semidirect
50. #1493C22semidirect

Maximal subgroups

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

subgroupindex
#609412
#218442
#218152
#204622
#178152
#174792
#174782
#98542
#98512
#98462
#98432