Finite Group Databasebeta

← #163253

Subgroups — #163253

Order 512.

1730 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1730 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic
2#20715C212yesmaximal, complemented
3#7724C218yesmaximal, complemented
4#7626C214yesmaximal, complemented
5C22×#127C411yescharacteristic
6C22×#121—22no—
7C22×#121C412yes—
8C23×C42C2211yescharacteristic
9C2×#432—22no—
10#2856—22no—
11#2856—22no—
12#2856C414yes—
13#2856C2214yes—
14C2×#390—22no—
15C2×#390C2212yes—
16C2×#384C411yescharacteristic
17#1817C2214yescomplemented
18#1797—22no—
19#1797C2214yescomplemented
20#1779—22no—
21#1749—22no—
22#1749C2214yes—
23#1270C22116yescomplemented
24C23×D4—22no—
25C23×D4—22no—
26C23×D4C2×C412yes—
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29C2×#127—21no—
30C2×#127—21no—
31C2×#127—44no—
32C2×#127C2×C414yes—
33C2×#121—24no—
34C2×#121—24no—
35C2×#121—24no—
36C2×#121—24no—
37C2×#121—24no—
38C2×#121—416no—
39C2×#121—48no—
40C2×#121C2×C418yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C412yes—
43C22×C22⋊C4—22no—
44C22×C22⋊C4—24no—
45C22×C42—44no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C412yes—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42C2312yes—
52C22×C42C2311yescharacteristic
53#432—48no—
54C2×C4⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×#97—24no—
59#390—24no—
60#390—24no—
61#390—24no—
62#390—24no—
63#390—24no—
64#390C2×C414yes—
65#390C2314yes—
66#384—21no—
67#384—21no—
68#384C2×C414yes—
69#336—24no—
70#336C2318yes—
71C25—22no—
72C25C22⋊C411yescharacteristic
73C22×D4—28no—
74C22×D4—22no—
75C22×D4—24no—
76C22×D4—24no—
77C22×D4—44no—
78C22×D4—44no—
79C22×D4—48no—
80C22×D4—48no—
81C22×D4—48no—
82C22×D4—44no—
83C22×D4—416no—
84C22×D4—48no—
85C22×D4C22×C414yes—
86C23×C4—24no—
87C23×C4—28no—
88C23×C4—22no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—44no—
92C23×C4—44no—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C22×C8—28no—
104#127—48no—
105#127—44no—
106#127—44no—
107#121—416no—
108#121—416no—
109#121—416no—
110#121—416no—
111#121—416no—
112#121—416no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4C22×C414yes—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—416no—
123C2×C22⋊C4—48no—
124C2×C42—24no—
125C2×C42—28no—
126C2×C42—44no—
127C2×C42—44no—
128C2×C42—44no—
129C2×C42—42no—
130C2×C42—44no—
131C2×C42C22⋊C411yescharacteristic
132C2×C42C22⋊C412yes—
133C2×C42C22⋊C412yes—
134C2×C42C22⋊C411yescharacteristic
135C2×C42C22×C412yes—
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42C2411yescharacteristic
141C4⋊C8—48no—
142C22⋊C8—48no—
143C22⋊C8—48no—
144C22⋊C8—48no—
145#97—48no—
146C24—22no—
147C24—24no—
148C24—24no—
149C24—24no—
150C24—41no—
151C24—41no—
152C24—48no—
153C24—44no—
154C24—44no—
155C24—44no—
156C24—48no—
157C24C2×C22⋊C412yes—
158C24C2×C22⋊C411yescharacteristic
159C2×D4—28no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—48no—
164C2×D4—48no—
165C2×D4—44no—
166C2×D4—48no—
167C2×D4—48no—
168C2×D4—416no—
169C2×D4—416no—
170C2×D4—416no—
171C2×D4—44no—
172C2×D4—48no—
173C2×D4—48no—
174C2×D4—816no—
175C2×D4—832no—
176C2×D4—832no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—416no—
207C22×C4—432no—
208C22×C4—48no—
209C22×C4—44no—
210C22×C4#10014yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4AGL(1,ℤ/8)18yes—
217C22×C4C23×C411yescharacteristic
218C22×C4C22×D412yes—
219C2×C8—48no—
220C2×C8—48no—
221C2×C8—48no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—48no—
225C4⋊C4—48no—
226C22⋊C4—416no—
227C22⋊C4—416no—
228C22⋊C4—416no—
229C42—24no—
230C42—24no—
231C42—24no—
232C42—22no—
233C42—24no—
234C42—42no—
235C42—41no—
236C42—42no—
237C42—41no—
238C42#9914yes—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—28no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—48no—
250C23—48no—
251C23—44no—
252C23—44no—
253C23—44no—
254C23—44no—
255C23—44no—
256C23—48no—
257C23—48no—
258C23—88no—
259C23—816no—
260C23—88no—
261C23—816no—
262C23—88no—
263C23C2×#10012yes—
264C23#41814yes—
265C23#42114yes—
266C23C22×C22⋊C411yescharacteristic
267D4—88no—
268D4—816no—
269D4—816no—
270D4—88no—
271D4—832no—
272C2×C4—216no—
273C2×C4—28no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—88no—
300C2×C4C2×#9914yes—
301C2×C4C2×#9912yes—
302C2×C4C2×#10014yes—
303C2×C4#41814yes—
304C2×C4#56812yes—
305C2×C4C2×AGL(1,ℤ/8)14yes—
306C8—88no—
307C22—22no—
308C22—24no—
309C22—24no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22—24no—
314C22—416no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—84no—
320C22—88no—
321C22—88no—
322C22—84no—
323C22—84no—
324C22—1616no—
325C22#153614yes—
326C22#287911yescharacteristic
327C22C22×#9911yescharacteristic
328C22C22×#10011yescharacteristic
329C22C2×#41812yes—
330C22C2×#42112yes—
331C4—24no—
332C4—24no—
333C4—24no—
334C4—22no—
335C4—22no—
336C4—24no—
337C4—42no—
338C4—84no—
339C2—24no—
340C2—44no—
341C2—164no—
342C2#1739611yescharacteristic
343C2#1749512yes—
344C2#1757712yes—
345C2C2×#153611yescharacteristic
346C2#1774111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#2856C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #7724C2semidirect
4. #7626C2semidirect
17. #1817C22semidirect
19. #1797C22semidirect
23. #1270C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
#207152
#77242
#76262