Finite Group Databasebeta

← #136806

Subgroups — #136806

Order 512.

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

normal subgroups only

This group has 349 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#33497C211yesmaximal, characteristic, complemented
2#21634C211yesmaximal, characteristic
3#11636C212yesmaximal
4#11494C211yesmaximal, characteristic, complemented
5#7149C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×#436C2211yescharacteristic
8C4×#124C411yescharacteristic
9#2183—24no—
10#2163—21no—
11#2163C2212yes—
12#2163C2212yes—
13#2162—21no—
14#1984C411yescharacteristic
15#1954C411yescharacteristic
16#1924—21no—
17#1924—21no—
18#1831C411yescharacteristic
19#1753—21no—
20#1753—21no—
21C4×#97C2211yescharacteristic
22#1330—24no—
23#1322—21no—
24#1322—21no—
25C22×M16—21no—
26C2×#124C811yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×C4×D4C2×C411yescharacteristic
29C8×Q8—22no—
30C8×Q8—22no—
31#436—22no—
32#436—22no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8D411yescharacteristic
36C2×C4⋊C8D411yescharacteristic
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×#97—21no—
41C2×#97—21no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#391C811yescharacteristic
47#390C811yescharacteristic
48#388—21no—
49#388—21no—
50#382C811yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53C43C2×C411yescharacteristic
54C4⋊C16—22no—
55C4⋊C16—22no—
56C4⋊C16—22no—
57C4⋊C16D414yes—
58C22⋊C16—22no—
59C22⋊C16—22no—
60#346—22no—
61#346—22no—
62#346—22no—
63C4×C16—22no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×Q8C2×C811yescharacteristic
69C22×D4C2×C811yescharacteristic
70C23×C4—21no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#124—42no—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×D4—22no—
85C4×D4—22no—
86C2×C4⋊C4C2×C811yescharacteristic
87C2×C4⋊C4C2×C811yescharacteristic
88C2×C4⋊C4M1611yescharacteristic
89C2×C4⋊C4M1611yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42C2×C811yescharacteristic
100C2×C42C2×C811yescharacteristic
101C2×C42C22×C411yescharacteristic
102C2×C16—24no—
103C2×C16—42no—
104C4⋊C8—22no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111#97—42no—
112#97—42no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—22no—
118C4×C8C2×D412yes—
119C4×C8#4112yes—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124C24—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—41no—
128C2×D4—22no—
129C2×D4—41no—
130C2×D4—41no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4C22⋊C811yescharacteristic
141C22×C4C22⋊C811yescharacteristic
142C22×C4C22⋊C811yescharacteristic
143C22×C4C22⋊C811yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C22×C811yescharacteristic
146C22×C4C2×M1611yescharacteristic
147M16—82no—
148M16—82no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8#12011yescharacteristic
162C2×C8#12112yes—
163C2×C8#12311yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—81no—
172C22⋊C4—81no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—41no—
178C42—41no—
179C42—41no—
180C42—41no—
181C42—81no—
182C42C4×D412yes—
183C42#13112yes—
184C16—44no—
185C16—82no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23C2×C22⋊C811yescharacteristic
194Q8—44no—
195D4—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
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—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4#34912yes—
224C2×C4#38611yescharacteristic
225C2×C4#40811yescharacteristic
226C2×C4C8×D412yes—
227C2×C4#43512yes—
228C8—22no—
229C8—21no—
230C8—42no—
231C8—41no—
232C8—41no—
233C8—81no—
234C8#45412yes—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22#186911yescharacteristic
245C22C2×#34911yescharacteristic
246C22#211511yescharacteristic
247C4—21no—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4#188411yescharacteristic
258C4#188811yescharacteristic
259C2—21no—
260C2—21no—
261C2—161no—
262C2#896011yescharacteristic
263C2#1298511yescharacteristic
264C2#1298811yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
1. #33497C2semidirect
4. #11494C2semidirect
5. #7149C2semidirect

Maximal subgroups

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

subgroupindex
#334972
#216342
#116362
#114942
#71492