Finite Group Databasebeta

← #6524825

Subgroups — #6524825

Order 512.

3094 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3501C211yesmaximal, characteristic
2C2×#3236C212yesmaximal, complemented
3C2×#2562C211yesmaximal, characteristic
4#25894C218yesmaximal, complemented
5#25712C218yesmaximal, complemented
6#24006C214yesmaximal, complemented
7#24003C214yesmaximal, complemented
8#21790C211yesmaximal, characteristic
9#21394C212yesmaximal, complemented
10#3501—216no—
11C2×#554—24no—
12C2×#554C2214yes—
13C2×D4×Q8C2214yes—
14C2×#544—21no—
15C2×#543—22no—
16C2×#533C2212yes—
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2214yes—
19C2×#518—21no—
20C22×C4×D4C2211yescharacteristic
21#3236—216no—
22C2×#501C2214yescomplemented
23C2×#491C2214yescomplemented
24C2×#484C2214yes—
25C2×#483C2214yes—
26C2×#479C2218yescomplemented
27C2×#432C2212yes—
28#2562—24no—
29#2562C2218yes—
30C2×#400—21no—
31C2×#394—21no—
32C2×#388—22no—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#2036C22116yescomplemented
37#2035C22116yescomplemented
38#1988C2214yescomplemented
39#1981C2214yescomplemented
40#1955C2218yes—
41#1872C2218yes—
42#1853C2218yescomplemented
43#1852C2218yescomplemented
44#1787C2218yescomplemented
45#1787C2218yescomplemented
46#1779C2214yes—
47#1763C2218yescomplemented
48C23×Q8C2314yes—
49C23×D4D411yescharacteristic
50C24×C4C2312yes—
51#554—264no—
52#554—216no—
53#554—216no—
54D4×Q8—216no—
55D4×Q8—216no—
56#544—216no—
57#543—232no—
58#533—28no—
59#533D4116yes—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2312yes—
63C2×#125—24no—
64C2×#125—22no—
65C2×#125—22no—
66C2×#125—28no—
67C2×#125C2314yes—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—232no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8C2318yes—
75C2×C22⋊Q8C2314yes—
76#518—216no—
77C2×C4×Q8—22no—
78C2×C4×Q8C2314yes—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—28no—
82C2×C4×D4D414yes—
83C2×C4×D4C2314yes—
84C2×#117—22no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4D412yes—
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99#501—216no—
100#491—216no—
101#484—216no—
102#483—216no—
103#479—232no—
104#432—28no—
105C2×#107C2318yes—
106C2×#106C2318yes—
107C2×C4⋊C8C2314yes—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×#103C23116yes—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2318yes—
113C2×C22⋊C8C2318yes—
114C2×#97C2314yes—
115#400—28no—
116#394—28no—
117#388—216no—
118#380—24no—
119#380—24no—
120#380C2318yes—
121#380C2318yes—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2314yes—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127#340C23116yes—
128C25C2×D412yes—
129C22×Q8—216no—
130C22×Q8—22no—
131C22×Q8—24no—
132C22×Q8—48no—
133C22×Q8—48no—
134C22×Q8C2414yes—
135C22×D4—28no—
136C22×D4—24no—
137C22×D4C2×D414yes—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—28no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D414yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2414yes—
151C23×C4C2411yescharacteristic
152C22×C8C2418yes—
153C4⋊Q8—232no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8C2×D418yes—
157C4⋊Q8C2×D418yes—
158#125—232no—
159#125—216no—
160#125—28no—
161#125—28no—
162#125—48no—
163C22⋊Q8—232no—
164C22⋊Q8—232no—
165C22⋊Q8—264no—
166C22⋊Q8—264no—
167C22⋊Q8—216no—
168C22⋊Q8—216no—
169C22⋊Q8—28no—
170C22⋊Q8—216no—
171C22⋊Q8C2×D4116yes—
172C4×Q8—216no—
173C4×Q8—28no—
174C4×Q8C2×D4116yes—
175C4×D4—232no—
176C4×D4—232no—
177C4×D4—216no—
178C4×D4—28no—
179C4×D4—28no—
180#117—216no—
181C2×C4⋊C4—232no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—216no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2412yes—
203C2×C4⋊C4C2412yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C22⋊C4—232no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—84no—
214C2×C22⋊C4C2×D418yes—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—82no—
218C2×C42C2×D412yes—
219C2×C42C2×D418yes—
220C2×C42C2412yes—
221#107—216no—
222#106—216no—
223C4⋊C8—28no—
224#103—216no—
225#103—216no—
226#103—416no—
227C22⋊C8—28no—
228C22⋊C8—48no—
229C22⋊C8—48no—
230#97—28no—
231#95—24no—
232#95—28no—
233#95—28no—
234C24—24no—
235C24—24no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24#13718yes—
241C24C22×D414yes—
242C24C22×D411yescharacteristic
243C2×Q8—216no—
244C2×Q8—28no—
245C2×Q8—24no—
246C2×Q8—216no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—84no—
250C2×Q8—84no—
251C2×Q8—832no—
252C2×Q8C22×D418yes—
253C2×D4—232no—
254C2×D4—216no—
255C2×D4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—216no—
261C22×C4—216no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—816no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4#12014yes—
290C22×C4#12014yes—
291C22×C4C22×D414yes—
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D414yes—
296C22×C4C2511yescharacteristic
297C2×C8—28no—
298C2×C8—216no—
299C4⋊C4—216no—
300C4⋊C4—232no—
301C4⋊C4—216no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—216no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—28no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4C22×D418yes—
317C22⋊C4—232no—
318C22⋊C4—216no—
319C22⋊C4—216no—
320C22⋊C4—416no—
321C22⋊C4—88no—
322C42—216no—
323C42—84no—
324C42#12018yes—
325C42C22×D414yes—
326C42C22×D414yes—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—24no—
331C23—24no—
332C23—24no—
333C23—24no—
334C23—24no—
335C23—24no—
336C23—216no—
337C23—48no—
338C23—48no—
339C23—84no—
340C23—84no—
341C23—84no—
342C23—84no—
343C23C2×#12012yes—
344C23C2×#13718yes—
345C23C2×#13714yes—
346C23C23×D411yescharacteristic
347Q8—216no—
348Q8—48no—
349Q8—88no—
350Q8—816no—
351D4—232no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—28no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—416no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—88no—
377C2×C4—84no—
378C2×C4—816no—
379C2×C4—816no—
380C2×C4—88no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4C2×#12014yes—
384C2×C4C2×#12012yes—
385C2×C4C2×#12014yes—
386C2×C4C23×D412yes—
387C2×C4#58514yes—
388C8—48no—
389C22—24no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22—28no—
394C22—28no—
395C22—28no—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—44no—
400C22—816no—
401C22#300414yes—
402C22C22×#12011yescharacteristic
403C22#348512yes—
404C22C22×#13712yes—
405C22#358511yescharacteristic
406C22#358611yescharacteristic
407C4—22no—
408C4—24no—
409C4—24no—
410C4—44no—
411C4—48no—
412C4—48no—
413C4—84no—
414C4—82no—
415C4—82no—
416C4#350014yes—
417C2—24no—
418C2—48no—
419C2#3561611yescharacteristic
420C2C2×#300411yescharacteristic
421C2#6037111yescharacteristic
422C2#6170512yes—
423C2#6170812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3236C2semidirect
4. #25894C2semidirect
5. #25712C2semidirect
6. #24006C2semidirect
7. #24003C2semidirect
9. #21394C2semidirect
22. C2×#501C22semidirect
23. C2×#491C22semidirect
26. C2×#479C22semidirect
36. #2036C22semidirect
37. #2035C22semidirect
38. #1988C22semidirect
39. #1981C22semidirect
42. #1853C22semidirect
43. #1852C22semidirect
44. #1787C22semidirect
45. #1787C22semidirect
47. #1763C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35012
C2×#32362
C2×#25622
#258942
#257122
#240062
#240032
#217902
#213942