Finite Group Databasebeta

← #137935

Subgroups — #137935

Order 512.

356 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 88 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#33487C211yesmaximal, characteristic
2#22064C211yesmaximal, characteristic
3#11466C211yesmaximal, characteristic
4#11444C212yesmaximal, complemented
5#7202C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C2×#424C2211yescharacteristic
8#2395C411yescharacteristic
9#2168—24no—
10#2163—21no—
11#2112—21no—
12#2112C2212yes—
13C2×C22⋊C16C2212yes—
14#1939—21no—
15#1923—21no—
16#1916—21no—
17#1870C411yescharacteristic
18#1851C411yescharacteristic
19#1850—21no—
20#1818C411yescharacteristic
21C8×C22⋊C4C2211yescharacteristic
22#1323—24no—
23#1322—21no—
24#1314—21no—
25C22×M16C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×C4×D4C2×C411yescharacteristic
29C2×#117C2×C411yescharacteristic
30C2×M32—22no—
31C2×M32—22no—
32C22×C16—22no—
33C8×D4—22no—
34C8×D4—22no—
35#424—22no—
36#424—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×C22⋊C8—21no—
42C2×C22⋊C8C2×C411yescharacteristic
43C2×C22⋊C8C2×C411yescharacteristic
44C2×C22⋊C8D411yescharacteristic
45C2×C4×C8—21no—
46C2×C4×C8D411yescharacteristic
47#400C811yescharacteristic
48#399C811yescharacteristic
49#398C811yescharacteristic
50#397—21no—
51#397C811yescharacteristic
52#394—21no—
53C4×C4⋊C4—21no—
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2×C411yescharacteristic
56C4⋊C16—22no—
57C22⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16—22no—
60C22⋊C16—22no—
61C22⋊C16D414yes—
62#346—22no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×D4M1611yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C2×M16—22no—
72C2×M16—81no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—81no—
78C22×C8—81no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C2×D412yes—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8#4112yes—
85C4×D4—22no—
86C4×D4—22no—
87#117—22no—
88#117—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4C2×C811yescharacteristic
92C2×C4⋊C4C2×C811yescharacteristic
93C2×C4⋊C4C2×C811yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4C2×C811yescharacteristic
99C2×C22⋊C4M1611yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C22⋊C411yescharacteristic
103M32—82no—
104M32—82no—
105C2×C16—24no—
106C2×C16—42no—
107C2×C16—82no—
108C4⋊C8—22no—
109C4⋊C8—41no—
110C4⋊C8—42no—
111C4⋊C8—41no—
112C4⋊C8—81no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C22⋊C8—22no—
116C22⋊C8—81no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—81no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95C2×C811yescharacteristic
125#95C2×C811yescharacteristic
126C24C22⋊C811yescharacteristic
127C24C2×M1611yescharacteristic
128C2×D4—22no—
129C2×D4—41no—
130C2×D4—41no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4C22⋊C811yescharacteristic
142C22×C4C22⋊C811yescharacteristic
143C22×C4C22⋊C811yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C4×D412yes—
146C22×C4C22×C811yescharacteristic
147C22×C4#13112yes—
148M16—82no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—41no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—41no—
155C2×C8—42no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—81no—
165C2×C8#12011yescharacteristic
166C2×C8#12112yes—
167C2×C8#12311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C42—21no—
180C42—21no—
181C42—41no—
182C42—41no—
183C42—41no—
184C42—41no—
185C16—44no—
186C16—82no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23C2×C22⋊C811yescharacteristic
197C23#43512yes—
198D4—44no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
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—81no—
227C2×C4#38611yescharacteristic
228C2×C4#40811yescharacteristic
229C2×C4C8×D412yes—
230C8—21no—
231C8—22no—
232C8—21no—
233C8—41no—
234C8—41no—
235C8—81no—
236C8—81no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—42no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22#186911yescharacteristic
250C22#211711yescharacteristic
251C22#211711yescharacteristic
252C4—21no—
253C4—22no—
254C4—42no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4—81no—
261C4#191311yescharacteristic
262C4#191511yescharacteristic
263C2—21no—
264C2—21no—
265C2—42no—
266C2—81no—
267C2#910811yescharacteristic
268C2#1298211yescharacteristic
269C2#1298411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1869
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11444C2semidirect
5. #7202C2semidirect

Maximal subgroups

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

subgroupindex
#334872
#220642
#114662
#114442
#72022