Finite Group Databasebeta

← #149595

Subgroups — #149595

Order 512.

377 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 74 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#21872C211yesmaximal, characteristic
2#11450C211yesmaximal, characteristic, complemented
3#11448C211yesmaximal, characteristic, complemented
4#7358C212yesmaximal, complemented
5#7354C212yesmaximal, complemented
6C2×#540C411yescharacteristic
7C2×#433C411yescharacteristic
8#2317C2211yescharacteristic
9#1992C2211yescharacteristic
10#1991C411yescharacteristic
11#1961C411yescharacteristic
12#1957C2211yescharacteristic
13#1875—21no—
14#1874—21no—
15#1804—21no—
16#1802—21no—
17#1458—21no—
18C2×#324—21no—
19#1325—21no—
20#1325C2212yes—
21#1325C2212yes—
22#1324—21no—
23#540—22no—
24#540—42no—
25#540C814yescomplemented
26C2×#127C2×C411yescharacteristic
27C2×#125C2×C411yescharacteristic
28C2×#123—41no—
29C2×#121—21no—
30C2×#121—41no—
31C2×C4×D4—21no—
32C2×#117C2×C411yescharacteristic
33C2×#117D411yescharacteristic
34C22×C4⋊C4D411yescharacteristic
35#433—42no—
36#431—41no—
37C2×C4⋊C8—21no—
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×#102—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—41no—
44C2×#97C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46#389—41no—
47#385—41no—
48#384D411yescharacteristic
49#384C2311yescharacteristic
50#382C2×C411yescharacteristic
51#382D411yescharacteristic
52#340—21no—
53#340—21no—
54#340—21no—
55#340—21no—
56#324—22no—
57#324—22no—
58#324—41no—
59#323—41no—
60C22×D4—21no—
61C22×D4—41no—
62C22×D4—41no—
63C23×C4—21no—
64C23×C4—41no—
65C23×C4C22⋊C411yescharacteristic
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—81no—
69#127—21no—
70#127—41no—
71#127C2×C812yescomplemented
72#125—21no—
73#125—41no—
74#125C2×C812yescomplemented
75#123—44no—
76#123—44no—
77#121—24no—
78#121—44no—
79#121—44no—
80#121—42no—
81C4×D4—24no—
82C4×D4—42no—
83#117—22no—
84#117—81no—
85#117C2×C812yes—
86#117M1612yes—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4C22⋊C411yescharacteristic
95C2×C4⋊C4C22⋊C411yescharacteristic
96C2×C4⋊C4C2×D411yescharacteristic
97C2×C4⋊C4C2×D411yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C42—21no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22×C411yescharacteristic
105C4⋊C8—42no—
106C4⋊C8—81no—
107#102—81no—
108#102—81no—
109#102—81no—
110#102—81no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—81no—
115#97—41no—
116C4×C8—41no—
117C4×C8—81no—
118#95—41no—
119#95—41no—
120C24—41no—
121C24#9911yescharacteristic
122C2×D4—22no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4—41no—
131C2×D4—161no—
132C2×D4—161no—
133C2×D4—161no—
134C2×D4—161no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—44no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—161no—
148C22×C4—161no—
149C22×C4—161no—
150C22×C4C22⋊C812yes—
151C22×C4#9911yescharacteristic
152C22×C4#10012yes—
153C22×C4C2×C22⋊C411yescharacteristic
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—82no—
161C2×C8—82no—
162C2×C8—162no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—81no—
174C4⋊C4—161no—
175C4⋊C4—161no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—81no—
181C42—21no—
182C42—21no—
183C42—41no—
184C42—41no—
185C42C22⋊C812yes—
186C42C22×C811yescharacteristic
187C42C2×M1611yescharacteristic
188C23—21no—
189C23—21no—
190C23—41no—
191C23—41no—
192C23—42no—
193C23—81no—
194C23—81no—
195C23—81no—
196C23—81no—
197C23—81no—
198C23—81no—
199C23#32312yes—
200C23C2×#9911yescharacteristic
201D4—42no—
202D4—42no—
203D4—81no—
204D4—81no—
205D4—162no—
206D4—162no—
207D4—164no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—164no—
234C2×C4—161no—
235C2×C4#32312yes—
236C2×C4C2×C22⋊C811yescharacteristic
237C2×C4C2×#10011yescharacteristic
238C8—81no—
239C8—81no—
240C8—162no—
241C8—162no—
242C22—21no—
243C22—21no—
244C22—22no—
245C22—42no—
246C22—41no—
247C22—41no—
248C22—81no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22—161no—
253C22—161no—
254C22—161no—
255C22—161no—
256C22C2×#32311yescharacteristic
257C22#212711yescharacteristic
258C22#213311yescharacteristic
259C4—21no—
260C4—22no—
261C4—21no—
262C4—42no—
263C4—41no—
264C4—41no—
265C4—82no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4—81no—
270C2—21no—
271C2—21no—
272C2—42no—
273C2—162no—
274C2#1143211yescharacteristic
275C2#1360911yescharacteristic
276C2#1361111yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC42C22×C8
Frattini, cosocle#384C23
prime radicalC2×#540C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2317 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11450C2semidirect
3. #11448C2semidirect
4. #7358C2semidirect
5. #7354C2semidirect
25. #540C8semidirect
71. #127C2×C8semidirect
74. #125C2×C8semidirect

Maximal subgroups

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

subgroupindex
#218722
#114502
#114482
#73582
#73542