Finite Group Databasebeta

← #136771

Subgroups — #136771

Order 512.

366 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 92 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#21635C211yesmaximal, characteristic
3#11639C212yesmaximal, complemented
4#11487C211yesmaximal, characteristic
5#7150C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2398C411yescharacteristic
9#2168—24no—
10#2163C2212yes—
11#2162C2212yes—
12#2112—21no—
13C2×C22⋊C16—21no—
14#1985C411yescharacteristic
15#1955C411yescharacteristic
16#1924—21no—
17#1924—21no—
18#1832C411yescharacteristic
19C8×C22⋊C4—21no—
20C8×C22⋊C4—21no—
21#1726C2211yescharacteristic
22Q8⋊C16—24no—
23#1314—21no—
24#1314—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×C4×D4C2×C411yescharacteristic
29C2×M32—22no—
30C22×C16—22no—
31C8×Q8—22no—
32C8×Q8—22no—
33C8×D4—22no—
34C8×D4—22no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8D411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42C2×#97—21no—
43C2×#97C2×C411yescharacteristic
44C2×C4×C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8C2311yescharacteristic
47#400—21no—
48#397C811yescharacteristic
49#397C811yescharacteristic
50#394—21no—
51#393C811yescharacteristic
52#393C811yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55#376C2×C411yescharacteristic
56C4⋊C16—22no—
57C4⋊C16D414yes—
58C22⋊C16—22no—
59C22⋊C16—22no—
60C22⋊C16—22no—
61C22⋊C16—22no—
62#346—22no—
63C4×C16—22no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×Q8C2×C811yescharacteristic
69C22×D4M1611yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—42no—
79C22×C8—42no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C4×Q8—22no—
85C4×Q8—22no—
86C4×D4—22no—
87C4×D4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2×C811yescharacteristic
90C2×C4⋊C4M1611yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4C2×C811yescharacteristic
96C2×C22⋊C4C2×C811yescharacteristic
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C22×C411yescharacteristic
100M32—82no—
101C2×C16—24no—
102C2×C16—42no—
103C2×C16—82no—
104C4⋊C8—22no—
105C4⋊C8—22no—
106C4⋊C8—42no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111#97—22no—
112#97—42no—
113C4×C8—42no—
114C4×C8—41no—
115C4×C8—41no—
116C4×C8C2×D412yes—
117C4×C8#4112yes—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95C2×C811yescharacteristic
124#95C2×C811yescharacteristic
125C24C22⋊C811yescharacteristic
126C24C22⋊C811yescharacteristic
127C2×Q8—22no—
128C2×Q8—42no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—41no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4C22⋊C811yescharacteristic
145C22×C4C22⋊C811yescharacteristic
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C22×C811yescharacteristic
148C22×C4C2×M1611yescharacteristic
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—81no—
164C2×C8—81no—
165C2×C8#12011yescharacteristic
166C2×C8#12112yes—
167C2×C8#12311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C42—41no—
178C42—41no—
179C42—41no—
180C42—41no—
181C42C4×D412yes—
182C42#13112yes—
183C16—44no—
184C16—82no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23#35012yes—
195C23C2×C22⋊C811yescharacteristic
196Q8—24no—
197Q8—42no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
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—81no—
225C2×C4—81no—
226C2×C4#38611yescharacteristic
227C2×C4#40811yescharacteristic
228C2×C4C8×D412yes—
229C2×C4#43512yes—
230C8—22no—
231C8—21no—
232C8—42no—
233C8—41no—
234C8—41no—
235C8—81no—
236C8#45512yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—81no—
248C22—81no—
249C22#186911yescharacteristic
250C22C2×#35011yescharacteristic
251C22#211711yescharacteristic
252C4—22no—
253C4—22no—
254C4—21no—
255C4—42no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4—81no—
261C4#188411yescharacteristic
262C4#188511yescharacteristic
263C2—21no—
264C2—21no—
265C2—42no—
266C2—81no—
267C2#895711yescharacteristic
268C2#1297811yescharacteristic
269C2#1298111yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11639C2semidirect
5. #7150C2semidirect

Maximal subgroups

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

subgroupindex
#334872
#216352
#116392
#114872
#71502