Finite Group Databasebeta

← #137754

Subgroups — #137754

Order 512.

396 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2162C211yesmaximal, characteristic
2#21020C211yesmaximal, characteristic
3#12844C211yesmaximal, characteristic
4C2×#1323C212yesmaximal
5#7186C212yesmaximal
6C2×#424C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2296C411yescharacteristic
9#2162—24no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12C2×C22⋊C16C2212yes—
13C2×C4×C16—21no—
14#1916—21no—
15#1916—21no—
16#1916—21no—
17#1916—21no—
18#1729C411yescharacteristic
19#1729C411yescharacteristic
20#1729C2211yescharacteristic
21#1728C411yescharacteristic
22#1374—21no—
23#1374—21no—
24#1323—24no—
25C22×M16C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×#117C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C16—22no—
33C22×C16—22no—
34#424—22no—
35#424—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4×C8—22no—
39C2×C4×C8—22no—
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#389C811yescharacteristic
45#389C811yescharacteristic
46#389C811yescharacteristic
47#389C811yescharacteristic
48#385—21no—
49#385—21no—
50C4×C4⋊C4—21no—
51C4×C4⋊C4—21no—
52C4⋊C16—24no—
53C22⋊C16—22no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16D414yes—
57C4×C16—24no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66C23×C4M1611yescharacteristic
67C23×C4M1611yescharacteristic
68C23×C4C22×C411yescharacteristic
69C2×M16—22no—
70C2×M16—22no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D412yes—
87C22×C8#4112yes—
88#117—22no—
89#117—22no—
90#117—41no—
91#117—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2×C811yescharacteristic
97C2×C4⋊C4C2×C811yescharacteristic
98C2×C4⋊C4C2×C811yescharacteristic
99C2×C4⋊C4C2×C811yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C22⋊C411yescharacteristic
108C2×C42C2×C811yescharacteristic
109C2×C42C2×C811yescharacteristic
110C2×C16—24no—
111C2×C16—42no—
112C2×C16—42no—
113C2×C16—42no—
114C2×C16—42no—
115C2×C16—42no—
116C4⋊C8—41no—
117C4⋊C8—42no—
118C4⋊C8—41no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—22no—
122C4×C8—22no—
123C4×C8—21no—
124C4×C8—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129C24C2×M1611yescharacteristic
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4C22⋊C811yescharacteristic
145C22×C4C22⋊C811yescharacteristic
146C22×C4C22⋊C811yescharacteristic
147C22×C4C22⋊C811yescharacteristic
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C4×D412yes—
150C22×C4C22×C811yescharacteristic
151C22×C4#13112yes—
152M16—82no—
153M16—82no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—44no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8#10112yes—
173C2×C8#12011yescharacteristic
174C2×C8#12112yes—
175C2×C8#12311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—22no—
191C42—22no—
192C42—41no—
193C42—41no—
194C42—41no—
195C42—41no—
196C16—44no—
197C16—82no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23C2×C22⋊C811yescharacteristic
205C23#43512yes—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—44no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4#35012yes—
236C2×C4#35012yes—
237C2×C4#35012yes—
238C2×C4#38611yescharacteristic
239C2×C4#40811yescharacteristic
240C2×C4#41011yescharacteristic
241C2×C4#41011yescharacteristic
242C2×C4C2×#10111yescharacteristic
243C2×C4#41311yescharacteristic
244C2×C4C8×D412yes—
245C8—21no—
246C8—22no—
247C8—21no—
248C8—22no—
249C8—41no—
250C8—81no—
251C8—81no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22#145811yescharacteristic
260C22#146111yescharacteristic
261C22#186911yescharacteristic
262C22C2×#35011yescharacteristic
263C22C2×#35011yescharacteristic
264C22C2×#35011yescharacteristic
265C22#211711yescharacteristic
266C4—22no—
267C4—22no—
268C4—22no—
269C4—22no—
270C4—41no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4#147011yescharacteristic
276C4#147311yescharacteristic
277C4#189911yescharacteristic
278C4#190111yescharacteristic
279C2—42no—
280C2#908211yescharacteristic
281C2#1146211yescharacteristic
282C2#1146411yescharacteristic
283C2#1149611yescharacteristic
284C2#1149811yescharacteristic
285C2#1297811yescharacteristic
286C2#1298311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#21622
#210202
#128442
C2×#13232
#71862