Finite Group Databasebeta

← #137758

Subgroups — #137758

Order 512.

372 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 96 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#33419C211yesmaximal, characteristic
2#21024C211yesmaximal, characteristic
3#12847C211yesmaximal, characteristic
4#11440C212yesmaximal
5#7185C212yesmaximal
6C22×C4⋊C8C2211yescharacteristic
7C2×#405C2211yescharacteristic
8#2296C411yescharacteristic
9#2164—24no—
10#2150—21no—
11#2112C2212yes—
12#2112C2212yes—
13C4×M32—21no—
14#1917—21no—
15#1917—21no—
16#1916—21no—
17#1916—21no—
18#1729C411yescharacteristic
19#1729C411yescharacteristic
20#1729C2211yescharacteristic
21#1728C411yescharacteristic
22#1377—21no—
23#1377—21no—
24#1322—24no—
25C22×M16C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×#117C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C2×M32—22no—
33C2×M32—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—22no—
38#405—22no—
39#405—22no—
40C2×#97D411yescharacteristic
41C2×#97D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#389—21no—
45#389—21no—
46#385C811yescharacteristic
47#385C811yescharacteristic
48#385C811yescharacteristic
49#385C811yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4—21no—
52C4⋊C16—24no—
53C22⋊C16—22no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16D414yes—
57#346—22no—
58C4×C16—22no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67C23×C4C2×C811yescharacteristic
68C23×C4C2×C811yescharacteristic
69C23×C4C22×C411yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D412yes—
88C22×C8#4112yes—
89#117—22no—
90#117—22no—
91#117—41no—
92#117—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4M1611yescharacteristic
99C2×C4⋊C4M1611yescharacteristic
100C2×C22⋊C4C2×C811yescharacteristic
101C2×C22⋊C4C2×C811yescharacteristic
102C2×C22⋊C4C2×C811yescharacteristic
103C2×C22⋊C4C2×C811yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—21no—
108C2×C42—22no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C22⋊C411yescharacteristic
111M32—42no—
112M32—42no—
113M32—42no—
114M32—42no—
115C2×C16—24no—
116C2×C16—42no—
117C4⋊C8—41no—
118C4⋊C8—42no—
119C4⋊C8—41no—
120#97—22no—
121#97—22no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130C24C22×C811yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—81no—
143C22×C4C22⋊C811yescharacteristic
144C22×C4C22⋊C811yescharacteristic
145C22×C4C22⋊C811yescharacteristic
146C22×C4C22⋊C811yescharacteristic
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C4×D412yes—
149C22×C4C2×M1611yescharacteristic
150C22×C4#13112yes—
151M16—44no—
152M16—82no—
153C2×C8—21no—
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—42no—
163C2×C8—42no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—82no—
171C2×C8#10112yes—
172C2×C8#12011yescharacteristic
173C2×C8#12112yes—
174C2×C8#12311yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—41no—
191C42—41no—
192C42—41no—
193C42—41no—
194C42—42no—
195C16—44no—
196C16—82no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23C2×C22⋊C811yescharacteristic
202C23C8×D412yes—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
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—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—82no—
232C2×C4#38611yescharacteristic
233C2×C4#40811yescharacteristic
234C2×C4#41011yescharacteristic
235C2×C4#41011yescharacteristic
236C2×C4C2×#10111yescharacteristic
237C2×C4#41311yescharacteristic
238C2×C4#43512yes—
239C8—21no—
240C8—22no—
241C8—21no—
242C8—22no—
243C8—41no—
244C8—81no—
245C8—81no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22#145811yescharacteristic
254C22#146111yescharacteristic
255C22#186911yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4#147011yescharacteristic
266C4#147311yescharacteristic
267C4#189911yescharacteristic
268C4#190111yescharacteristic
269C2—21no—
270C2—21no—
271C2—42no—
272C2#908211yescharacteristic
273C2#1298111yescharacteristic
274C2#1298911yescharacteristic
subgroupG/H
centerC2×C4#386
commutator, prime radicalC24C22×C8
Frattini, cosocleC23×C8C23
socleC22#1869
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×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
#334192
#210242
#128472
#114402
#71852