Finite Group Databasebeta

← #6528497

Subgroups — #6528497

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#29110C212yesmaximal
4#29103C212yesmaximal
5#27966C212yesmaximal, complemented
6#27960C212yesmaximal, complemented
7#27830C218yesmaximal, complemented
8#27744C218yesmaximal, complemented
9#21840C212yesmaximal, complemented
10#20739C212yesmaximal
11#20534C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—22no—
14C22×C4×D4C2211yescharacteristic
15C2×#432C2212yes—
16#2857C2212yes—
17#2851C2212yes—
18#2627—22no—
19#2606—22no—
20#2605—22no—
21#2570C2214yescomplemented
22#2550C2214yescomplemented
23#2542—22no—
24#2542—22no—
25#2542C2218yes—
26C2×#399—22no—
27#2329—22no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2283C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34#2100C2218yescomplemented
35#2099C2218yescomplemented
36#2073C2214yes—
37#2071C2214yes—
38#2025C2214yes—
39#2023C2214yes—
40#1993C2214yescomplemented
41#1993C2214yescomplemented
42#1992C2214yescomplemented
43#1990C2214yescomplemented
44#1987C2218yescomplemented
45#1936C2218yes—
46#1934C2218yes—
47#1836C2214yescomplemented
48#1834C2214yescomplemented
49#1823C2214yes—
50#1823C2214yes—
51#1823C2214yes—
52#1804C2214yes—
53#1804C2214yes—
54#1786C2214yes—
55#1771C2218yescomplemented
56#1762C2218yes—
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59C23×Q8—21no—
60C23×D4C2311yescharacteristic
61C24×C4—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—28no—
64C2×C4⋊Q8C2314yes—
65C2×C4⋊Q8C2312yes—
66C2×#125C2312yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—416no—
71C2×C4×D4—22no—
72C2×C4×D4—28no—
73C2×C4×D4C2314yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#432—28no—
88C2×C4⋊C8C2314yes—
89C2×#102C2318yes—
90C2×#102C2318yes—
91C2×C22⋊C8C2314yes—
92C2×#97C2314yes—
93C2×#97C2314yes—
94C2×C4×C8C2314yes—
95#400—24no—
96#399—28no—
97#399—28no—
98#398—24no—
99#398—24no—
100#395—24no—
101#391—22no—
102#391—24no—
103#390—22no—
104#390—21no—
105#390—21no—
106#390C2314yes—
107#390C2314yes—
108#389—22no—
109#389—24no—
110#389C2314yes—
111#389C2314yes—
112#387—24no—
113#387—24no—
114#387—24no—
115#385—24no—
116#384—22no—
117#384—21no—
118#384—21no—
119#384C2312yes—
120#384C2314yes—
121#384C2314yes—
122#383—22no—
123#382C2312yes—
124#382C2312yes—
125#382C2312yes—
126#381—22no—
127#380—22no—
128#380C2314yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2314yes—
133C4×C22⋊C4—22no—
134#376—22no—
135C2×#95D412yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#336C2314yes—
143#336C2314yes—
144C25—21no—
145C22×Q8—24no—
146C22×Q8—22no—
147C22×Q8—48no—
148C22×Q8—44no—
149C22×D4—24no—
150C22×D4—21no—
151C22×D4—24no—
152C22×D4—42no—
153C22×D4C2412yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—48no—
159C23×C4—44no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2411yescharacteristic
165C22×C8C2418yes—
166C4⋊Q8—28no—
167C4⋊Q8—28no—
168C4⋊Q8—44no—
169C4⋊Q8—48no—
170#125—44no—
171C22⋊Q8—432no—
172C22⋊Q8—432no—
173C22⋊Q8—48no—
174C4×D4—28no—
175C4×D4—28no—
176C4×D4—48no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2412yes—
200C2×C4⋊C4C2412yes—
201C2×C4⋊C4C2412yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2414yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—48no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—48no—
212C2×C42—24no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42#4118yes—
220C2×C42C2412yes—
221C2×C42C2412yes—
222C2×C42C2412yes—
223C4⋊C8—28no—
224#102—216no—
225#102—48no—
226C22⋊C8—28no—
227#97—28no—
228#97—44no—
229C4×C8—44no—
230#95—24no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95C2×D414yes—
240#95C2×D414yes—
241C24—24no—
242C24—22no—
243C24—42no—
244C24—44no—
245C24—42no—
246C24—44no—
247C24C22×D411yescharacteristic
248C2×Q8—416no—
249C2×Q8—48no—
250C2×Q8—44no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—816no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—416no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D42−1+414yes—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—416no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—48no—
284C22×C4—816no—
285C22×C4#12414yes—
286C22×C4#12414yes—
287C22×C4AGL(1,ℤ/8)18yes—
288C22×C4C22×D414yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C2×#4114yes—
295C22×C4C2511yescharacteristic
296C2×C8—28no—
297C2×C8—28no—
298C2×C8—28no—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—416no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4—82no—
319C22⋊C4—416no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C42—24no—
325C42—24no—
326C42—41no—
327C42—41no—
328C42—42no—
329C42—84no—
330C42#12418yes—
331C42C2×#4114yes—
332C42C2×#4114yes—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—24no—
337C23—48no—
338C23—416no—
339C23—44no—
340C23—42no—
341C23—42no—
342C23—44no—
343C23—88no—
344C23—84no—
345C23—84no—
346C23C2×#12412yes—
347C23C2×AGL(1,ℤ/8)14yes—
348C23C23×D411yescharacteristic
349Q8—816no—
350Q8—164no—
351D4—28no—
352D4—48no—
353D4—44no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—28no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—816no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4C2×#12412yes—
387C2×C4C2×#12414yes—
388C2×C4C2×#12414yes—
389C2×C4C2×AGL(1,ℤ/8)14yes—
390C2×C4C2×AGL(1,ℤ/8)14yes—
391C2×C4C22×#4112yes—
392C2×C4C2×2−1+412yes—
393C2×C4#58512yes—
394C8—48no—
395C22—24no—
396C22—22no—
397C22—24no—
398C22—22no—
399C22—44no—
400C22—48no—
401C22—816no—
402C22—84no—
403C22—82no—
404C22—82no—
405C22#313814yes—
406C22C22×#12411yescharacteristic
407C22#348511yescharacteristic
408C22#348611yescharacteristic
409C22C22×AGL(1,ℤ/8)12yes—
410C22#358612yes—
411C4—24no—
412C4—24no—
413C4—22no—
414C4—24no—
415C4—24no—
416C4—44no—
417C4—84no—
418C4—84no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4#349514yes—
423C2—24no—
424C2—84no—
425C2#3564411yescharacteristic
426C2C2×#313811yescharacteristic
427C2#6062011yescharacteristic
428C2#6170612yes—
429C2#6182712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #35104C2semidirect
5. #27966C2semidirect
6. #27960C2semidirect
7. #27830C2semidirect
8. #27744C2semidirect
9. #21840C2semidirect
11. #20534C2semidirect
21. #2570C22semidirect
22. #2550C22semidirect
34. #2100C22semidirect
35. #2099C22semidirect
40. #1993C22semidirect
41. #1993C22semidirect
42. #1992C22semidirect
43. #1990C22semidirect
44. #1987C22semidirect
47. #1836C22semidirect
48. #1834C22semidirect
55. #1771C22semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#25422
#291102
#291032
#279662
#279602
#278302
#277442
#218402
#207392
#205342