Finite Group Databasebeta

← #238516

Subgroups — #238516

Order 512.

1518 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1518 conjugacy classes of subgroups — too many to draw. Only its 202 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
1#35166C211yesmaximal, characteristic
2#21395C212yesmaximal, complemented
3C2×#1505C212yesmaximal, complemented
4#13856C211yesmaximal, characteristic, complemented
5#10402C218yesmaximal, complemented
6#7912C211yesmaximal, characteristic, complemented
7C2×#537—22no—
8C2×#532C2212yes—
9C2×#432C2212yes—
10#2856—22no—
11#2656C2214yes—
12#2542—22no—
13#2541—22no—
14#2540—24no—
15C2×C4×C4⋊C4—22no—
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1987—22no—
20#1982—22no—
21#1874C2218yes—
22#1762—24no—
23#1505—28no—
24C2×#327C2218yes—
25#1444—22no—
26#1444C2214yes—
27#1280C2214yes—
28C22×#41—22no—
29#537—416no—
30#532—24no—
31#532—24no—
32#532—24no—
33#532—24no—
34C2×C4⋊Q8D412yes—
35C2×#127—22no—
36C2×#127D412yes—
37C2×#124—22no—
38C2×C22⋊Q8—28no—
39C2×#121—28no—
40C2×#121—22no—
41C2×#121—28no—
42C2×#121C2314yes—
43C2×C4×D4—22no—
44C2×C4×D4D414yes—
45C22×C4⋊C4—24no—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2312yes—
48C22×C42—21no—
49C22×C42—21no—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52#432—24no—
53#432—24no—
54C2×C4⋊C8—22no—
55C2×#102—28no—
56C2×C22⋊C8C2314yes—
57C2×#97—22no—
58#399—28no—
59#392—28no—
60#390—28no—
61#390—24no—
62#384—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—48no—
68#376—22no—
69#376D414yescomplemented
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—24no—
73C2×#95C2312yes—
74C2×#95C2312yes—
75#340—28no—
76#327—216no—
77C2×#41—416no—
78C2×#41—44no—
79C2×#41—44no—
80C22×Q8—22no—
81C22×D4—24no—
82C22×D4—28no—
83C22×D4—22no—
84C22×D4—22no—
85C22×D4C2×D414yes—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—24no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D414yes—
92C23×C4C2411yescharacteristic
93C22×C8—24no—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96#127—24no—
97#127—24no—
98#127—84no—
99#124—44no—
100#124—44no—
101C22⋊Q8—432no—
102#121—24no—
103#121—24no—
104#121—24no—
105#121—24no—
106#121—432no—
107#121—416no—
108#121—44no—
109#121—44no—
110#121—416no—
111C4×D4—28no—
112C4×D4—28no—
113C4×D4—44no—
114C4×D4—44no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—48no—
124C2×C4⋊C4C2×D414yes—
125C2×C22⋊C4—216no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—24no—
128C2×C42—24no—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—44no—
134C2×C42—44no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42#4114yes—
141C2×C42#4112yes—
142C4⋊C8—44no—
143#102—416no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146#97—44no—
147#95—24no—
148#95—24no—
149#95—22no—
150#95—48no—
151#95—42no—
152#95C2×D414yes—
153C24—24no—
154C24C22×D411yescharacteristic
155#41—816no—
156#41—816no—
157C2×Q8—48no—
158C2×Q8—44no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—48no—
162C2×D4—48no—
163C2×D4—416no—
164C2×D4—44no—
165C2×D4—44no—
166C2×D4—416no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—88no—
172C2×D4—88no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—42no—
184C22×C4—416no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4#12012yes—
195C22×C4#12011yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12112yes—
198C22×C4#12112yes—
199C22×C4#12412yes—
200C22×C4#12412yes—
201C22×C4AGL(1,ℤ/8)18yes—
202C22×C4C22×D412yes—
203C22×C4C22×D411yescharacteristic
204C22×C4C2×#4112yes—
205C22×C4C2×#4111yescharacteristic
206C2×C8—48no—
207C2×C8—44no—
208C4⋊C4—216no—
209C4⋊C4—28no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—44no—
213C4⋊C4—416no—
214C4⋊C4—432no—
215C4⋊C4—44no—
216C22⋊C4—432no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—48no—
220C42—216no—
221C42—44no—
222C42—82no—
223C42—82no—
224C42—82no—
225C42#12114yes—
226C42#12114yes—
227C42#12314yes—
228C42#12314yes—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—44no—
233C23—48no—
234C23—44no—
235C23—44no—
236C23—48no—
237C23C2×#12011yescharacteristic
238C23C2×#12111yescharacteristic
239C23C2×#12411yescharacteristic
240C23C2×AGL(1,ℤ/8)14yes—
241Q8—88no—
242D4—48no—
243D4—84no—
244D4—88no—
245D4—88no—
246D4—88no—
247D4—88no—
248D4—1616no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—28no—
258C2×C4—28no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—48no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—44no—
271C2×C4—48no—
272C2×C4—44no—
273C2×C4—816no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#39412yes—
279C2×C4#39414yes—
280C2×C4#39412yes—
281C2×C4#45312yes—
282C2×C4#45612yes—
283C2×C4C2×#12112yes—
284C2×C4C2×#12312yes—
285C8—84no—
286C22—24no—
287C22—24no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—88no—
293C22—88no—
294C22—84no—
295C22—84no—
296C22—84no—
297C22C2×#39411yescharacteristic
298C22#300312yes—
299C22C2×#45311yescharacteristic
300C22C2×#45611yescharacteristic
301C22#302512yes—
302C22#303812yes—
303C22#313812yes—
304C4—22no—
305C4—24no—
306C4—22no—
307C4—22no—
308C4—48no—
309C4—48no—
310C4—84no—
311C4—84no—
312C4#203914yes—
313C2—22no—
314C2—22no—
315C2—164no—
316C2#1817812yes—
317C2#1818712yes—
318C2C2×#203911yescharacteristic
319C2#2350511yescharacteristic
320C2#2351411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radical#35166C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21395C2semidirect
3. C2×#1505C2semidirect
4. #13856C2semidirect
5. #10402C2semidirect
6. #7912C2semidirect
69. #376D4semidirect

Maximal subgroups

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

subgroupindex
#351662
#213952
C2×#15052
#138562
#104022
#79122