Finite Group Databasebeta

← #215137

Subgroups — #215137

Order 512.

1036 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2861C211yesmaximal, characteristic
2#20778C212yesmaximal, complemented
3#9652C218yesmaximal, complemented
4#8306C212yesmaximal, complemented
5#8305C212yesmaximal
6C22×#125C2211yescharacteristic
7C22×#124C2211yescharacteristic
8#2861—22no—
9#2861—22no—
10#2861—22no—
11#2861—22no—
12C2×#391C411yescharacteristic
13C2×#391C411yescharacteristic
14C2×#390C411yescharacteristic
15C2×#390C411yescharacteristic
16C2×#388C414yes—
17C2×#388C2214yes—
18C2×C43C2211yescharacteristic
19#1839—22no—
20#1831—22no—
21#1817C2214yes—
22#1797C2214yes—
23#1767—22no—
24#1749C2214yes—
25#1745—22no—
26C2×#324C2218yes—
27C2×#323C2218yes—
28C23×Q8C2×C411yescharacteristic
29C23×D4C2×C411yescharacteristic
30C2×#125—22no—
31C2×#125—22no—
32C2×#125—44no—
33C2×#124—22no—
34C2×#124—22no—
35C2×#124—44no—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2312yes—
40C22×C22⋊C4C2×C412yes—
41C22×C22⋊C4C2×C414yes—
42C22×C22⋊C4C2×C414yes—
43C22×C22⋊C4C2312yes—
44C22×C42C2×C412yes—
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42C2311yescharacteristic
48C22×C42C2312yes—
49C2×C4⋊C8—24no—
50C2×C22⋊C8C2318yes—
51C2×#97—24no—
52#391—22no—
53#391—22no—
54#391—21no—
55#391—21no—
56#391—21no—
57#391—21no—
58#390—22no—
59#390—22no—
60#390—21no—
61#390—21no—
62#390—21no—
63#390—21no—
64#388—24no—
65#388—24no—
66#388—24no—
67#388—24no—
68#388—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72C2×#95C2×C414yes—
73C2×#95C2×C414yes—
74C43D412yes—
75C43D412yes—
76C43D412yes—
77C43D412yes—
78#336—24no—
79#336—24no—
80#336—24no—
81#336—24no—
82#324—48no—
83#323—48no—
84C25C22×C412yes—
85C22×Q8—21no—
86C22×Q8—21no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—42no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4—42no—
93C22×D4—42no—
94C22×D4—42no—
95C23×C4C22×C412yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C412yes—
99C23×C4C22×C412yes—
100C23×C4C22×C412yes—
101C23×C4C22×C412yes—
102C23×C4C2411yescharacteristic
103C22×C8—28no—
104#125—48no—
105#125—48no—
106#124—48no—
107#124—48no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—48no—
126C2×C22⋊C4—44no—
127C2×C22⋊C4—48no—
128C2×C42—24no—
129C2×C42—42no—
130C2×C42—42no—
131C2×C42—44no—
132C2×C42—44no—
133C2×C42C22⋊C412yes—
134C2×C42C22⋊C412yes—
135C2×C42C22⋊C412yes—
136C2×C42C22⋊C412yes—
137C2×C42C22⋊C412yes—
138C2×C42C22⋊C412yes—
139C2×C42C22⋊C411yescharacteristic
140C2×C42C22⋊C411yescharacteristic
141C2×C42C22⋊C411yescharacteristic
142C2×C42C22⋊C411yescharacteristic
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C4⋊C8—48no—
150C22⋊C8—216no—
151#97—48no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—24no—
156C24—22no—
157C24—24no—
158C24—28no—
159C24—44no—
160C24—42no—
161C24—42no—
162C24C23×C411yescharacteristic
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—88no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—88no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C411yescharacteristic
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C22×D412yes—
207C22×C4H(5,𝔽2)14yes—
208C2×C8—48no—
209C2×C8—48no—
210C2×C8—48no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—48no—
216C4⋊C4—48no—
217C22⋊C4—48no—
218C22⋊C4—48no—
219C22⋊C4—48no—
220C22⋊C4—48no—
221C22⋊C4—48no—
222C22⋊C4—48no—
223C42—24no—
224C42—24no—
225C42—24no—
226C42—24no—
227C42—24no—
228C42—24no—
229C42—42no—
230C42—42no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—42no—
236C42—42no—
237C42—42no—
238C42—42no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—42no—
243C23—44no—
244C23—44no—
245C23—42no—
246C23—42no—
247C23—48no—
248C23—84no—
249C23—84no—
250C23—88no—
251C23—88no—
252C23C22×C22⋊C411yescharacteristic
253C23#51812yes—
254C23#56814yes—
255Q8—84no—
256Q8—84no—
257D4—84no—
258D4—84no—
259C2×C4—28no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—44no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—48no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4#41014yes—
295C2×C4#41312yes—
296C2×C4#41312yes—
297C2×C4#53514yes—
298C8—88no—
299C22—22no—
300C22—22no—
301C22—24no—
302C22—24no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—84no—
307C22—84no—
308C22—168no—
309C22#230911yescharacteristic
310C22#287912yes—
311C22C2×#41012yes—
312C22C2×#41311yescharacteristic
313C22C2×#41311yescharacteristic
314C22#298014yes—
315C4—24no—
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—42no—
321C4—42no—
322C4—42no—
323C4—82no—
324C4—82no—
325C4—82no—
326C2—24no—
327C2—162no—
328C2#1739512yes—
329C2#1740112yes—
330C2#2179111yescharacteristic
331C2#2179411yescharacteristic
332C2#2179811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #9652C2semidirect
4. #8306C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#207782
#96522
#83062
#83052