Finite Group Databasebeta

← #206355

Subgroups — #206355

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic
2#21415C212yesmaximal, complemented
3C4×#431C211yesmaximal, characteristic, complemented
4#17815C211yesmaximal, characteristic, complemented
5#17489C212yesmaximal, complemented
6#9848C211yesmaximal, characteristic, complemented
7#9847C211yesmaximal, characteristic, complemented
8#9846C212yesmaximal, complemented
9#9845C211yesmaximal, characteristic, complemented
10#9844C211yesmaximal, characteristic, complemented
11#9843C212yesmaximal, complemented
12#3444C2212yes—
13#3250—22no—
14#3231C412yescomplemented
15#3228C412yescomplemented
16#3210C412yescomplemented
17#3207C412yescomplemented
18#3191—22no—
19#3170C2211yescharacteristic
20#3169C2211yescharacteristic
21#3167C2212yes—
22#3102—24no—
23#2920C2212yes—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×C43C2211yescharacteristic
27#1994C2212yes—
28#1994C2212yes—
29#1962C2212yes—
30#1962C2212yes—
31#1896—24no—
32C4×C4⋊C8—22no—
33C4×#104—24no—
34C4×C22⋊C8—22no—
35#1744C2211yescharacteristic
36#1744C2211yescharacteristic
37C42×C8C2211yescharacteristic
38C42×C8C2211yescharacteristic
39#1572—24no—
40#1567C2211yescharacteristic
41#1567C2211yescharacteristic
42#1566C2212yes—
43#1493C2212yes—
44#1493C2212yes—
45#1492C2212yes—
46#1492C2212yes—
47#1488—24no—
48#1482—24no—
49#533C2×C412yes—
50#533C2×C412yes—
51#533C2312yes—
52#532C2×C412yes—
53#532C2×C412yes—
54#532C2312yes—
55#529—22no—
56#529—22no—
57#529—22no—
58#529—22no—
59#529—22no—
60#529—22no—
61#525—22no—
62#525—22no—
63#525—22no—
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—22no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71#500C2×C411yescharacteristic, complemented
72#500C2×C411yescharacteristic, complemented
73#498C2×C412yescomplemented
74#494C2×C411yescharacteristic, complemented
75#493C2×C411yescharacteristic, complemented
76#492C2×C412yescomplemented
77#487C2×C412yescomplemented
78#487C2×C412yescomplemented
79#486C2×C412yescomplemented
80#486C2×C412yescomplemented
81#485—24no—
82#482—24no—
83#479—24no—
84#478—24no—
85#467—22no—
86#466—22no—
87#465—24no—
88#464—24no—
89#463—24no—
90#452—24no—
91#449—24no—
92C4×Q16—22no—
93C4×SD16—24no—
94C4×D8—22no—
95#432—22no—
96#432—22no—
97#431—22no—
98#431—22no—
99#431—22no—
100#431C2×C412yes—
101#431C2×C412yes—
102#431D414yes—
103#431D414yes—
104#423C2312yes—
105#423C2312yes—
106C2×C4⋊C8—21no—
107C2×C4⋊C8D412yes—
108C2×#104—24no—
109C2×#104—24no—
110#407—22no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8D412yes—
113C4×M16—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C43—21no—
121C43—21no—
122C43D412yes—
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#339—24no—
126#339—24no—
127#336—22no—
128#336—22no—
129#335—22no—
130#335—22no—
131#334—24no—
132Q8⋊C8—24no—
133Q8⋊C8—24no—
134D4⋊C8—24no—
135D4⋊C8—24no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4111yescharacteristic
143C2×Q16—22no—
144C2×SD16—24no—
145C2×D8—22no—
146C2×M16—22no—
147C2×M16—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C4⋊Q8C22×C412yes—
153C4⋊Q8C22×C412yes—
154C4⋊Q8C22×C412yes—
155#127C22×C412yes—
156#126—42no—
157#126—42no—
158#125—22no—
159#125—22no—
160#125—22no—
161#124—22no—
162#124—22no—
163#124—22no—
164#123—42no—
165#123—42no—
166#123—42no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179#120—42no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—42no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—42no—
202#117—42no—
203#117—42no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42—42no—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42#4111yescharacteristic
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226#107—22no—
227#107—22no—
228#106—24no—
229C4⋊C8—24no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8C22⋊C414yes—
235C4⋊C8C22⋊C414yes—
236#104—48no—
237#104—48no—
238#103—24no—
239#103—24no—
240#102—24no—
241#102—24no—
242C22⋊C8—24no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245C22⋊C8C22⋊C414yes—
246C22⋊C8C22⋊C414yes—
247#97—42no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—44no—
251C4×C8—44no—
252C4×C8—42no—
253C4×C8C22×C411yescharacteristic
254C4×C8C22×C412yes—
255C4×C8C22×C411yescharacteristic
256C4×C8C2×D412yes—
257C4×C8C2×D412yes—
258C4×C8C2×D412yes—
259C4×C8C2×D412yes—
260C24#12111yescharacteristic
261#41—44no—
262#41—44no—
263#41—84no—
264#41—84no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C4×D412yes—
288C22×C4C4×D412yes—
289C22×C4#12111yescharacteristic
290C22×C4#12112yes—
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12412yes—
294C22×C4#12712yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298Q16—82no—
299SD16—84no—
300D8—82no—
301M16—44no—
302M16—44no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8C2×C22⋊C412yes—
312C2×C8C2×C22⋊C414yes—
313C2×C8C2×C22⋊C412yes—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C42—24no—
336C42—28no—
337C42—24no—
338C42—22no—
339C42—22no—
340C42—44no—
341C42—44no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42C2×C22⋊C412yes—
348C42C4×D412yes—
349C42C4×D412yes—
350C42#12112yes—
351C42#12412yes—
352C42#12712yes—
353C42#13514yes—
354C42#13514yes—
355C42C23×C411yescharacteristic
356C42C22×D411yescharacteristic
357C42C22×D411yescharacteristic
358C42C2×#4111yescharacteristic
359C23—22no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23#39012yes—
364C23C2×#12111yescharacteristic
365Q8—44no—
366Q8—44no—
367Q8—44no—
368D4—44no—
369D4—44no—
370D4—44no—
371D4—84no—
372D4—84no—
373D4—84no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—416no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#39012yes—
401C2×C4#39012yes—
402C2×C4#39012yes—
403C2×C4C22×C22⋊C411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×C4×D411yescharacteristic
406C2×C4C2×#12111yescharacteristic
407C2×C4C2×#12411yescharacteristic
408C2×C4C2×#12711yescharacteristic
409C2×C4C2×#13512yes—
410C2×C4C2×#13512yes—
411C8—22no—
412C8—24no—
413C8—22no—
414C8—42no—
415C8—42no—
416C8—82no—
417C8—82no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—44no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22C2×#39011yescharacteristic
428C22#305012yes—
429C4—28no—
430C4—21no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—21no—
435C4—44no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4#290112yes—
440C4C4×#13512yes—
441C4#295612yes—
442C4#314711yescharacteristic
443C4#314711yescharacteristic
444C2—22no—
445C2—42no—
446C2—162no—
447C2#2121011yescharacteristic
448C2#2122411yescharacteristic
449C2#2124111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60944C2
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. #21415C2semidirect
3. C4×#431C2semidirect
4. #17815C2semidirect
5. #17489C2semidirect
6. #9848C2semidirect
7. #9847C2semidirect
8. #9846C2semidirect
9. #9845C2semidirect
10. #9844C2semidirect
11. #9843C2semidirect
14. #3231C4semidirect
15. #3228C4semidirect
16. #3210C4semidirect
17. #3207C4semidirect
71. #500C2×C4semidirect
72. #500C2×C4semidirect
73. #498C2×C4semidirect
74. #494C2×C4semidirect
75. #493C2×C4semidirect
76. #492C2×C4semidirect
77. #487C2×C4semidirect
78. #487C2×C4semidirect
79. #486C2×C4semidirect
80. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#609442
#214152
C4×#4312
#178152
#174892
#98482
#98472
#98462
#98452
#98442
#98432