Finite Group Databasebeta

← #127814

Subgroups — #127814

Order 512.

576 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 128 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#21853C211yesmaximal, characteristic
2#20728C212yesmaximal
3#7284C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5#2920—22no—
6C2×#432C411yescharacteristic
7C2×#423C411yescharacteristic
8#1985C2212yes—
9#1955C412yes—
10#1870—22no—
11#1822C414yes—
12#1801—22no—
13#1800C414yes—
14#1797—22no—
15#1785—22no—
16C4×#101C2214yes—
17C4×#100—22no—
18C22×#41D411yescharacteristic
19C22×M16—21no—
20C4×#41—42no—
21C4×#41—42no—
22C2×C4×Q8D411yescharacteristic
23C2×C4×Q8C2311yescharacteristic
24C2×C4×D4C2×C412yes—
25C2×C4×D4C2×C411yescharacteristic
26C2×C4×D4D412yes—
27C2×C4×D4Q811yescharacteristic
28C2×#117C2×C412yes—
29C2×#117Q811yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32#432—42no—
33#432—42no—
34#432—42no—
35#431—42no—
36#431—42no—
37#423—24no—
38#423—42no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8C2×C412yes—
42C2×#101—24no—
43C2×#100—24no—
44#407—24no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—24no—
47C4×M16—24no—
48C2×#97C2×C412yes—
49#343—24no—
50#343C2×C418yes—
51#336—24no—
52C2×#41—42no—
53C2×#41—44no—
54C22×Q8C2×D411yescharacteristic
55C22×D4C22⋊C412yes—
56C22×D4C4⋊C411yescharacteristic
57C23×C4C4211yescharacteristic
58C23×C4C22⋊C412yes—
59C23×C4C22⋊C412yes—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C4⋊C411yescharacteristic
62C23×C4C4⋊C412yes—
63C2×M16—28no—
64C2×M16—42no—
65C2×M16—82no—
66C22×C8—22no—
67C22×C8—22no—
68C4×Q8—42no—
69C4×Q8—41no—
70C4×Q8—41no—
71C4×D4—42no—
72C4×D4—42no—
73C4×D4—41no—
74C4×D4—41no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78#117—42no—
79#117—42no—
80#117—42no—
81C2×C4⋊C4C22⋊C412yes—
82C2×C4⋊C4C4⋊C411yescharacteristic
83C2×C4⋊C4C22×C412yes—
84C2×C4⋊C4C2×Q811yescharacteristic
85C2×C22⋊C4C4212yes—
86C2×C22⋊C4C4⋊C412yes—
87C2×C22⋊C4C4⋊C412yes—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42C4211yescharacteristic
95C2×C42C22⋊C412yes—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C4⋊C412yes—
98C2×C42C4⋊C411yescharacteristic
99C2×C42C22×C411yescharacteristic
100C2×C42C2×D412yes—
101C4⋊C8—42no—
102C4⋊C8—44no—
103C4⋊C8—44no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106#101—48no—
107#100—48no—
108C22⋊C8—42no—
109C22⋊C8—44no—
110C22⋊C8—44no—
111C22⋊C8—84no—
112#97—42no—
113#97—44no—
114C4×C8—44no—
115C24#9512yes—
116C24#9511yescharacteristic
117#41—416no—
118C2×Q8—24no—
119C2×Q8—41no—
120C2×D4—24no—
121C2×D4—42no—
122C2×D4—41no—
123C2×D4—42no—
124C2×D4—42no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—82no—
143C22×C4—81no—
144C22×C4#9511yescharacteristic
145C22×C4#9511yescharacteristic
146C22×C4#9512yes—
147C22×C4#9511yescharacteristic
148C22×C4C2×C4211yescharacteristic
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C2×C22⋊C412yes—
151C22×C4C2×C4⋊C411yescharacteristic
152C22×C4C2×C4⋊C412yes—
153M16—48no—
154M16—48no—
155M16—84no—
156C2×C8—48no—
157C2×C8—42no—
158C2×C8—44no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—22no—
174C42—22no—
175C42—21no—
176C42—22no—
177C42—22no—
178C42—21no—
179C42#9912yes—
180C42#9912yes—
181C23—22no—
182C23—22no—
183C23—21no—
184C23—22no—
185C23—22no—
186C23—42no—
187C23—42no—
188C23—41no—
189C23—41no—
190C23C2×#9511yescharacteristic
191Q8—44no—
192D4—48no—
193D4—44no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4#34212yes—
223C2×C4#34212yes—
224C2×C4C2×#9911yescharacteristic
225C2×C4C2×#9911yescharacteristic
226C2×C4#42114yes—
227C8—82no—
228C8—88no—
229C8—82no—
230C22—21no—
231C22—21no—
232C22—44no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—82no—
240C22—82no—
241C22#173312yes—
242C22C2×#34211yescharacteristic
243C4—22no—
244C4—21no—
245C4—24no—
246C4—22no—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—42no—
251C4—41no—
252C4#213411yescharacteristic
253C4#213511yescharacteristic
254C2—22no—
255C2—82no—
256C2—81no—
257C2#804911yescharacteristic
258C2#1257911yescharacteristic
259C2#1258711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7284C2semidirect

Maximal subgroups

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

subgroupindex
#218532
#207282
#72842