Finite Group Databasebeta

← #150708

Subgroups — #150708

Order 512.

257 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 52 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#30089C211yesmaximal, characteristic
2#12803C211yesmaximal, characteristic
3#11535C211yesmaximal, characteristic
4#7436C212yesmaximal, complemented
5#7413C212yesmaximal, complemented
6#2647C411yescharacteristic
7#2326C2211yescharacteristic
8C2×#355—21no—
9#2095C2211yescharacteristic
10#2093C411yescharacteristic
11#2083—21no—
12#2074C2211yescharacteristic
13#2073C411yescharacteristic
14#2042—21no—
15#1863—21no—
16#1862—21no—
17#1840C411yescharacteristic
18#1801—21no—
19#1465—21no—
20#1394C2212yes—
21#1325—21no—
22#1319C2212yes—
23C2×#124D411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×D4D411yescharacteristic
26C2×#107—21no—
27C2×#107—21no—
28#423—41no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#102—21no—
32C2×#102—21no—
33C2×#101—22no—
34#407—41no—
35#400—41no—
36#398—41no—
37#391C2311yescharacteristic
38#390C2×C411yescharacteristic
39#389—21no—
40#389—21no—
41#388—41no—
42#385—41no—
43#384C2×C411yescharacteristic
44#384D411yescharacteristic
45#382C2×C411yescharacteristic
46#376C2×C411yescharacteristic
47#355—42no—
48#343—41no—
49#340—21no—
50#340—21no—
51#336—21no—
52#336—21no—
53#336C2×C411yescharacteristic
54#336C2×C411yescharacteristic
55#324—22no—
56#323—41no—
57C22×Q8C2×D411yescharacteristic
58C22×D4C22⋊C411yescharacteristic
59C23×C4—21no—
60C2×M16—42no—
61C2×M16—81no—
62C22×C8—21no—
63C22×C8—21no—
64#124—21no—
65#124—21no—
66#124—41no—
67C4×Q8—81no—
68C4×D4—81no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C42—21no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22×C411yescharacteristic
85#107—81no—
86#107—81no—
87C4⋊C8—82no—
88#103—41no—
89#103—41no—
90#103—81no—
91#102—42no—
92#102—81no—
93#101—42no—
94#101—42no—
95C22⋊C8—42no—
96C22⋊C8—81no—
97#95—21no—
98#95—21no—
99#95—41no—
100#95—41no—
101#95—41no—
102#95—41no—
103C24—21no—
104C2×Q8—21no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8AGL(1,ℤ/8)11yescharacteristic
108C2×Q8#13711yescharacteristic
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—41no—
112C2×D4—41no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—161no—
123C22×C4#9911yescharacteristic
124C22×C4#9911yescharacteristic
125C22×C4C2×C22⋊C411yescharacteristic
126M16—82no—
127M16—82no—
128M16—162no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—82no—
136C2×C8—82no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—81no—
144C4⋊C4—161no—
145C4⋊C4—161no—
146C4⋊C4—161no—
147C4⋊C4—161no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—81no—
151C22⋊C4—161no—
152C42—21no—
153C42—21no—
154C42—41no—
155C42—41no—
156C42—42no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—81no—
161C23—81no—
162C23C2×#9911yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—81no—
166D4—42no—
167D4—81no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—82no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—161no—
196C2×C4#41711yescharacteristic
197C2×C4#42111yescharacteristic
198C8—81no—
199C8—81no—
200C8—162no—
201C8—162no—
202C22—21no—
203C22—21no—
204C22—82no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—161no—
209C22#151311yescharacteristic
210C22#212311yescharacteristic
211C22#212711yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—81no—
220C4—81no—
221C4—81no—
222C4—161no—
223C4—161no—
224C2—21no—
225C2—21no—
226C2—161no—
227C2#1182711yescharacteristic
228C2#1360511yescharacteristic
229C2#1361211yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2326 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7436C2semidirect
5. #7413C2semidirect

Maximal subgroups

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

subgroupindex
#300892
#128032
#115352
#74362
#74132