Finite Group Databasebeta

← #134143

Subgroups — #134143

Order 512.

1260 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×#2858C211yesmaximal, characteristic
2#11505C214yesmaximal, complemented
3#8551C212yesmaximal, complemented
4C22×C4⋊Q8C2211yescharacteristic
5C22×#127C411yescharacteristic
6C22×#124—22no—
7#2858—28no—
8C2×#390—22no—
9C2×#390C412yes—
10C2×#390C2212yes—
11C2×C43C411yescharacteristic
12#1784—22no—
13#1463—22no—
14#1463C2214yes—
15#1318—28no—
16C23×Q8D412yes—
17C23×D4—21no—
18C23×D4—21no—
19C23×D4C2×C412yes—
20C2×C4⋊Q8—24no—
21C2×C4⋊Q8—42no—
22C2×C4⋊Q8—42no—
23C2×#127—24no—
24C2×#127—42no—
25C2×#127—42no—
26C2×#124—216no—
27C2×#124—416no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D412yes—
30C22×C4⋊C4C2311yescharacteristic
31C22×C22⋊C4—24no—
32C22×C22⋊C4—24no—
33C22×C42—21no—
34C22×C42—21no—
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C412yes—
37C2×#101—44no—
38#407—24no—
39#390—216no—
40#390—24no—
41#390—24no—
42#390—24no—
43#390D418yes—
44C43—24no—
45#324—28no—
46#324—44no—
47C25—22no—
48C22×Q8—24no—
49C22×Q8—24no—
50C22×Q8—42no—
51C22×Q8—48no—
52C22×D4—24no—
53C22×D4—24no—
54C22×D4—22no—
55C22×D4—44no—
56C22×D4—42no—
57C22×D4—42no—
58C22×D4—48no—
59C22×D4—42no—
60C22×D4—42no—
61C22×D4—48no—
62C22×D4—44no—
63C22×D4—42no—
64C22×D4—42no—
65C22×D4C22⋊C414yes—
66C23×C4—22no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C412yes—
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D412yes—
72C2×M16—44no—
73C4⋊Q8—416no—
74#127—416no—
75#124—464no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4C2×D414yes—
85C2×C22⋊C4—216no—
86C2×C22⋊C4—216no—
87C2×C22⋊C4—416no—
88C2×C22⋊C4—416no—
89C2×C42—24no—
90C2×C42—28no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—24no—
95C2×C42—22no—
96C2×C42—44no—
97C2×C42—44no—
98C2×C42C22⋊C414yes—
99#101—88no—
100C22⋊C8—44no—
101C22⋊C8—44no—
102C24—24no—
103C24—48no—
104C24—48no—
105C24—44no—
106C24—44no—
107C24—42no—
108C24—42no—
109C24C2×C22⋊C411yescharacteristic
110C2×Q8—48no—
111C2×Q8—416no—
112C2×Q8—48no—
113C2×Q8—84no—
114C2×Q8—88no—
115C2×D4—28no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—416no—
119C2×D4—48no—
120C2×D4—44no—
121C2×D4—48no—
122C2×D4—416no—
123C2×D4—44no—
124C2×D4—44no—
125C2×D4—88no—
126C2×D4—88no—
127C2×D4—88no—
128C2×D4—88no—
129C2×D4—84no—
130C2×D4—84no—
131C2×D4—84no—
132C2×D4—84no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—28no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4#9912yes—
154C22×C4#9912yes—
155C22×C4C2×C22⋊C412yes—
156C22×C4#12014yes—
157M16—88no—
158C2×C8—84no—
159C4⋊C4—44no—
160C4⋊C4—48no—
161C4⋊C4—44no—
162C22⋊C4—432no—
163C22⋊C4—432no—
164C42—216no—
165C42—28no—
166C42—44no—
167C42—48no—
168C42—44no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—48no—
173C23—48no—
174C23—44no—
175C23—44no—
176C23—44no—
177C23—84no—
178C23—84no—
179C23—84no—
180C23—84no—
181C23—816no—
182C23—816no—
183C23—84no—
184C23—84no—
185C23#37911yescharacteristic
186C23C2×#9911yescharacteristic
187C23C2×#9911yescharacteristic
188Q8—816no—
189D4—88no—
190D4—88no—
191D4—816no—
192D4—816no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—28no—
206C2×C4—24no—
207C2×C4—44no—
208C2×C4—416no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#35314yes—
218C2×C4#35514yes—
219C2×C4#41314yes—
220C8—164no—
221C22—28no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—88no—
229C22—88no—
230C22—84no—
231C22—84no—
232C22—84no—
233C22—168no—
234C22—168no—
235C22#178111yescharacteristic
236C22#178512yes—
237C22C2×#35312yes—
238C22C2×#35512yes—
239C4—24no—
240C4—24no—
241C4—24no—
242C4—48no—
243C4—44no—
244C4—82no—
245C2—22no—
246C2—22no—
247C2—22no—
248C2—164no—
249C2#855211yescharacteristic
250C2#1269311yescharacteristic
251C2#1269911yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC22×#127C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11505C2semidirect
3. #8551C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28582
#115052
#85512