Finite Group Databasebeta

← #206408

Subgroups — #206408

Order 512.

1008 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
1#60941C211yesmaximal, characteristic
2#21844C211yesmaximal, characteristic, complemented
3#21815C211yesmaximal, characteristic
4#20472C211yesmaximal, characteristic, complemented
5#17815C211yesmaximal, characteristic, complemented
6#17501C211yesmaximal, characteristic, complemented
7#17500C211yesmaximal, 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×C4×M16C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1994C2212yescomplemented
30#1994C2212yescomplemented
31#1986—22no—
32#1980—22no—
33#1962C2212yes—
34#1962C2212yes—
35#1826—22no—
36#1789—22no—
37#1771—22no—
38#1753—22no—
39#1744C2211yescharacteristic, complemented
40#1744C2211yescharacteristic, complemented
41C4×#97C2211yescharacteristic, complemented
42C42×C8C2211yescharacteristic, complemented
43#1572—22no—
44#1572—22no—
45#1567C2212yescomplemented
46#1566C2212yescomplemented
47#1493C2212yescomplemented
48#1493C2212yescomplemented
49#1492C2212yes—
50#1492C2212yes—
51#1491—22no—
52#1490—22no—
53#1485—22no—
54#1484—22no—
55C22×#41—21no—
56C22×M16—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533C2×C412yes—
64#533C2312yes—
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532—21no—
71#532C2×C412yes—
72#532C2312yes—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77C2×#125—21no—
78C2×#124—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×#121—21no—
82C2×#121—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×#117—21no—
88C2×#117—21no—
89C22×C42C2311yescharacteristic
90#498C2×C412yes—
91#498C2×C412yes—
92#492C2×C412yes—
93#492C2×C412yes—
94#487C2×C412yes—
95#487C2×C412yes—
96#486C2×C412yes—
97#486C2×C412yes—
98#483—24no—
99#481—24no—
100#476—24no—
101#475—24no—
102#465—24no—
103#465—24no—
104#461—24no—
105#460—24no—
106#450—24no—
107#448—24no—
108C4×SD16—24no—
109C4×SD16—24no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431C2×C412yes—
119#431C2×C412yes—
120#431D412yes—
121#431D412yes—
122#423C2312yes—
123#423C2312yes—
124C2×C4⋊C8D412yes—
125C2×#104—24no—
126C2×#104—24no—
127#407—21no—
128#407—21no—
129C2×C22⋊C8D412yes—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16—21no—
133C4×M16—21no—
134C4×M16D412yes—
135C4×M16D412yes—
136C2×#97—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—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—44no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C22×C412yes—
197C4⋊Q8C22×C412yes—
198C4⋊Q8C22×C412yes—
199#127—21no—
200#127—21no—
201#127—21no—
202#127C22×C412yes—
203#125—22no—
204#125—22no—
205#125—41no—
206#125—41no—
207#124—22no—
208#124—22no—
209#124—41no—
210#124—41no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—41no—
216C22⋊Q8—41no—
217C22⋊Q8—41no—
218C22⋊Q8—41no—
219#121—22no—
220#121—22no—
221#121—22no—
222#121—22no—
223#121—41no—
224#121—41no—
225#121—41no—
226#121—41no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243#117—22no—
244#117—22no—
245#117—44no—
246#117—41no—
247#117—41no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—41no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42C2411yescharacteristic
280#106—24no—
281#106—24no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8C22⋊C414yes—
287C4⋊C8C22⋊C414yes—
288#104—84no—
289#104—84no—
290#103—24no—
291#103—24no—
292#102—24no—
293#102—24no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298C22⋊C8C22⋊C414yes—
299C22⋊C8C22⋊C414yes—
300#97—22no—
301#97—22no—
302#97—44no—
303#97—41no—
304#97—41no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—44no—
308C4×C8—41no—
309C4×C8—41no—
310C4×C8C22×C412yes—
311C4×C8C22×C412yes—
312C4×C8C2×D412yes—
313C4×C8C2×D412yes—
314C4×C8C2×D412yes—
315C4×C8C2×D412yes—
316#95—22no—
317#95—22no—
318C24—41no—
319C24#12111yescharacteristic
320#41—44no—
321#41—44no—
322#41—84no—
323#41—84no—
324#41—84no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—84no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—84no—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12411yescharacteristic
369C22×C4#12411yescharacteristic
370C22×C4#12711yescharacteristic
371C22×C4#12711yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375SD16—84no—
376SD16—84no—
377M16—44no—
378M16—44no—
379M16—44no—
380M16—44no—
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8C2×C22⋊C414yes—
390C2×C8C2×C22⋊C414yes—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C42—24no—
416C42—24no—
417C42—24no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42C2×C22⋊C411yescharacteristic
424C42C2×C22⋊C411yescharacteristic
425C42C4×D411yescharacteristic
426C42C4×D411yescharacteristic
427C42C4×D411yescharacteristic
428C42C4×D411yescharacteristic
429C42#12111yescharacteristic
430C42#12111yescharacteristic
431C42#12411yescharacteristic
432C42#12411yescharacteristic
433C42#12711yescharacteristic
434C42#12711yescharacteristic
435C42C23×C411yescharacteristic
436C42C22×D411yescharacteristic
437C42C22×D411yescharacteristic
438C42C2×#4111yescharacteristic
439C23—22no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23#39011yescharacteristic
447C23#39011yescharacteristic
448C23C2×#12111yescharacteristic
449Q8—44no—
450Q8—44no—
451Q8—44no—
452Q8—82no—
453Q8—82no—
454Q8—82no—
455D4—44no—
456D4—44no—
457D4—44no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—84no—
462D4—84no—
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—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4#39011yescharacteristic
496C2×C4#39011yescharacteristic
497C2×C4#39011yescharacteristic
498C2×C4#39011yescharacteristic
499C2×C4#39011yescharacteristic
500C2×C4#39011yescharacteristic
501C2×C4C22×C22⋊C411yescharacteristic
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×C4×D411yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12411yescharacteristic
506C2×C4C2×#12711yescharacteristic
507C2×C4#57714yes—
508C8—24no—
509C8—24no—
510C8—24no—
511C8—42no—
512C8—82no—
513C8—82no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—164no—
527C22C2×#39011yescharacteristic
528C22#305712yes—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—21no—
537C4—44no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—82no—
542C4#290112yes—
543C4#294912yes—
544C4#295612yes—
545C4#315412yes—
546C2—22no—
547C2—41no—
548C2—41no—
549C2—162no—
550C2#2121011yescharacteristic
551C2#2123111yescharacteristic
552C2#2124111yescharacteristic
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. #20472C2semidirect
5. #17815C2semidirect
6. #17501C2semidirect
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
47. #1493C22semidirect
48. #1493C22semidirect

Maximal subgroups

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

subgroupindex
#609412
#218442
#218152
#204722
#178152
#175012
#175002
#98542
#98512
#98462
#98432