Finite Group Databasebeta

← #203130

Subgroups — #203130

Order 512.

810 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21370C211yesmaximal, characteristic, complemented
2C2×#1980C211yesmaximal, characteristic
3#17717C211yesmaximal, characteristic
4#17478C211yesmaximal, characteristic
5#17327C211yesmaximal, characteristic, complemented
6#9592C214yesmaximal, complemented
7#9590C214yesmaximal, complemented
8#8098C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×#104—22no—
16C2×C4×M16C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1980—22no—
19#1980C2212yes—
20#1980C2212yes—
21#1950C2212yescomplemented
22#1950C2212yescomplemented
23#1855—22no—
24C4×#108—22no—
25#1553C414yescomplemented
26#1549C411yescharacteristic, complemented
27#1549C411yescharacteristic, complemented
28#1545C411yescharacteristic, complemented
29#1545C411yescharacteristic, complemented
30#1540C2214yes—
31#1496—22no—
32#1492C2214yes—
33#1478—21no—
34#1478—21no—
35#1453—22no—
36#1451—21no—
37#1451C2212yescomplemented
38C8×M16—21no—
39C8×M16C2212yescomplemented
40#1303C2214yescomplemented
41#1279C2214yescomplemented
42C22×#41—21no—
43C22×M16D412yes—
44#532—21no—
45#532—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532C2×C412yes—
51#532C2312yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C22×C4⋊C4—21no—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#432—21no—
63#432—21no—
64#432—21no—
65#432C2×C412yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431C2×C411yescharacteristic
74#431C2×C411yescharacteristic
75#431C2×C412yes—
76#431D411yescharacteristic
77#431D411yescharacteristic
78#431D412yes—
79#431C2312yes—
80C2×#108—22no—
81C2×#108—22no—
82#423C2×C411yescharacteristic
83#423C2×C411yescharacteristic
84#423C2312yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×#104—24no—
88C2×#104—24no—
89C2×#104—48no—
90#407—21no—
91#407—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16D412yes—
99C4×M16C2311yescharacteristic
100C4×M16C2311yescharacteristic
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2312yes—
109C2×#95—21no—
110#341—22no—
111#341—22no—
112#332C2×C414yescomplemented
113#332C2×C414yescomplemented
114#331C2×C414yescomplemented
115#331C2×C414yescomplemented
116D4⋊C8—24no—
117D4⋊C8—24no—
118#322—22no—
119#322—22no—
120#322—22no—
121C82—22no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—44no—
125C2×#41—44no—
126C2×#41—41no—
127C2×#41—41no—
128C22×Q8—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—22no—
141C2×M16—24no—
142C2×M16—22no—
143C2×M16—42no—
144C2×M16C2×D412yes—
145C2×M16C2×D412yes—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8C22×C412yes—
153#127—21no—
154#127—21no—
155#127—21no—
156#127C22×C412yes—
157#124—22no—
158#124—22no—
159#124—41no—
160#124—41no—
161#121—22no—
162#121—22no—
163#121—22no—
164#121—22no—
165#121—41no—
166#121—41no—
167#121—41no—
168#121—41no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C42—22no—
186C2×C42—44no—
187C2×C42—41no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42C2411yescharacteristic
200#108—44no—
201#108—44no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8C22×C412yes—
211C4⋊C8C22×C412yes—
212C4⋊C8C22×C412yes—
213C4⋊C8C22×C412yes—
214#104—48no—
215#104—48no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226#97—22no—
227#97—22no—
228#97—21no—
229#97—21no—
230#97—41no—
231C4×C8—22no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—22no—
235C4×C8—21no—
236C4×C8—22no—
237C4×C8—21no—
238C4×C8—22no—
239C4×C8—41no—
240C4×C8C2×D412yes—
241C4×C8C2×D412yes—
242#95—22no—
243#95—41no—
244C24—41no—
245C24#12411yescharacteristic
246#41—48no—
247#41—44no—
248#41—44no—
249#41—82no—
250#41—82no—
251#41—84no—
252C2×Q8—22no—
253C2×Q8—44no—
254C2×Q8—41no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—82no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—84no—
279C22×C4—81no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4#11711yescharacteristic
283C22×C4#11711yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12411yescharacteristic
293C22×C4#12411yescharacteristic
294C22×C4#12411yescharacteristic
295C22×C4#12611yescharacteristic
296C22×C4#12611yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300M16—22no—
301M16—22no—
302M16—22no—
303M16—44no—
304M16—42no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16AGL(1,ℤ/8)14yes—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—44no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—82no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C42—44no—
334C42—82no—
335C42#11711yescharacteristic
336C42#11711yescharacteristic
337C42C4×D411yescharacteristic
338C42C4×D411yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42#12111yescharacteristic
342C42#12111yescharacteristic
343C42#12311yescharacteristic
344C42#12311yescharacteristic
345C42#12611yescharacteristic
346C42#12611yescharacteristic
347C42C23×C411yescharacteristic
348C42C22×D411yescharacteristic
349C42C2×#4111yescharacteristic
350C42C2×#4111yescharacteristic
351C23—21no—
352C23—21no—
353C23—22no—
354C23—41no—
355C23—42no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23#38811yescharacteristic
360C23#38811yescharacteristic
361C23C2×#12411yescharacteristic
362Q8—44no—
363Q8—82no—
364D4—44no—
365D4—42no—
366D4—42no—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—84no—
371D4—82no—
372D4—82no—
373D4—82no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—84no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—88no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4#38811yescharacteristic
402C2×C4#38811yescharacteristic
403C2×C4#38811yescharacteristic
404C2×C4#38811yescharacteristic
405C2×C4#38811yescharacteristic
406C2×C4#38811yescharacteristic
407C2×C4#44312yes—
408C2×C4#44412yes—
409C2×C4C2×#11711yescharacteristic
410C2×C4C2×C4×D411yescharacteristic
411C2×C4C2×C4×D411yescharacteristic
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12311yescharacteristic
414C2×C4C2×#12611yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)12yes—
416C2×C4#57512yes—
417C8—24no—
418C8—42no—
419C8—42no—
420C8—42no—
421C8—42no—
422C8—42no—
423C8—42no—
424C8—42no—
425C8—82no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—164no—
439C22#192912yes—
440C22C2×#38811yescharacteristic
441C22#313811yescharacteristic
442C22#314011yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—84no—
452C4—82no—
453C4#192912yes—
454C4C4×AGL(1,ℤ/8)12yes—
455C4C2×#44311yescharacteristic
456C4C2×#44411yescharacteristic
457C4#305512yes—
458C2—21no—
459C2—22no—
460C2—21no—
461C2—41no—
462C2—162no—
463C2C2×#192911yescharacteristic
464C2#2108011yescharacteristic
465C2#2109511yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
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
1. #21370C2semidirect
5. #17327C2semidirect
6. #9592C2semidirect
7. #9590C2semidirect
8. #8098C2semidirect
9. #7892C2semidirect
21. #1950C22semidirect
22. #1950C22semidirect
25. #1553C4semidirect
26. #1549C4semidirect
27. #1549C4semidirect
28. #1545C4semidirect
29. #1545C4semidirect
37. #1451C22semidirect
39. C8×M16C22semidirect
40. #1303C22semidirect
41. #1279C22semidirect
112. #332C2×C4semidirect
113. #332C2×C4semidirect
114. #331C2×C4semidirect
115. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#213702
C2×#19802
#177172
#174782
#173272
#95922
#95902
#80982
#78922