Finite Group Databasebeta

← #147721

Subgroups — #147721

Order 512.

1124 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2546C211yesmaximal, characteristic
2C22×#356C214yesmaximal
3#10763C212yesmaximal
4C22×C4⋊Q8C411yescharacteristic
5C22×C4⋊Q8C2212yes—
6C22×C4×Q8C2211yescharacteristic
7C22×#101C2214yes—
8#2546—28no—
9C2×#398—22no—
10C2×#391C412yes—
11C2×#389—22no—
12C2×C4×C4⋊C4C411yescharacteristic
13C2×#356—216no—
14C2×#356—28no—
15C2×#343—22no—
16#1728—22no—
17C23×Q8C2×C412yes—
18C23×Q8C2311yescharacteristic
19C22×M16—24no—
20C2×C4⋊Q8—24no—
21C2×C4⋊Q8—24no—
22C2×C4⋊Q8—28no—
23C2×C4⋊Q8—28no—
24C2×C4⋊Q8—42no—
25C2×C4⋊Q8—42no—
26C2×C4⋊Q8D418yes—
27C2×C4×Q8—24no—
28C2×C4×Q8—42no—
29C2×C4×Q8—42no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—24no—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4D412yes—
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C412yes—
38C22×C42D412yes—
39C2×#101—28no—
40C2×#101—28no—
41C2×#101—24no—
42C2×#101D418yes—
43#398—216no—
44#391—24no—
45#391—24no—
46#389—216no—
47C4×C4⋊C4—24no—
48C2×#95—21no—
49C2×#95—24no—
50#356—232no—
51#343—44no—
52#336—44no—
53C22×Q8—22no—
54C22×Q8—28no—
55C22×Q8—41no—
56C22×Q8—41no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8—42no—
60C22×Q8—44no—
61C22×Q8C22⋊C414yes—
62C22×Q8C2×D414yes—
63C22×Q8C2×D414yes—
64C22×Q8#4114yes—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22×C411yescharacteristic
71C23×C4C2×D412yes—
72C2×M16—28no—
73C2×M16—44no—
74C2×M16—44no—
75C2×M16—44no—
76C2×M16—44no—
77C2×M16—44no—
78C22×C8—44no—
79C4⋊Q8—28no—
80C4⋊Q8—216no—
81C4⋊Q8—28no—
82C4⋊Q8—416no—
83C4⋊Q8—44no—
84C4⋊Q8—48no—
85C4⋊Q8—44no—
86C4×Q8—416no—
87C2×C4⋊C4—28no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—216no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—48no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—44no—
105C2×C42—22no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—28no—
109C2×C42—24no—
110C2×C42—22no—
111C2×C42—24no—
112C2×C42—42no—
113C2×C42C22⋊C414yes—
114#101—216no—
115#101—48no—
116#95—24no—
117#95—216no—
118C24C2×C22⋊C411yescharacteristic
119C2×Q8—28no—
120C2×Q8—44no—
121C2×Q8—44no—
122C2×Q8—44no—
123C2×Q8—44no—
124C2×Q8—44no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—88no—
132C2×Q8—88no—
133C2×Q8#12014yes—
134C2×Q8#12118yes—
135C2×Q8#12314yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—28no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4#9912yes—
163C22×C4#9912yes—
164C22×C4C2×C22⋊C412yes—
165C22×C4C4×D414yes—
166C22×C4#12014yes—
167C22×C4#12114yes—
168C22×C4#12412yes—
169C22×C4#12712yes—
170M16—48no—
171M16—48no—
172M16—84no—
173M16—84no—
174C2×C8—48no—
175C2×C8—84no—
176C2×C8—84no—
177C4⋊C4—216no—
178C4⋊C4—48no—
179C4⋊C4—44no—
180C4⋊C4—48no—
181C4⋊C4—48no—
182C4⋊C4—416no—
183C4⋊C4—416no—
184C42—28no—
185C42—28no—
186C42—48no—
187C42—44no—
188C23—22no—
189C23—22no—
190C23#37911yescharacteristic
191C23#39012yes—
192C23C2×#9911yescharacteristic
193C23C2×#9911yescharacteristic
194C23#41012yes—
195Q8—44no—
196Q8—48no—
197Q8—44no—
198Q8—88no—
199Q8—88no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—28no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—416no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4—82no—
233C2×C4#35614yescomplemented
234C2×C4#38614yes—
235C2×C4#39214yes—
236C2×C4#39414yes—
237C8—88no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—24no—
243C22—24no—
244C22#143811yescharacteristic
245C22#178111yescharacteristic
246C22#178211yescharacteristic
247C22#196412yes—
248C22#196412yes—
249C22C2×#35612yes—
250C22#213512yes—
251C4—24no—
252C4—24no—
253C4—24no—
254C4—24no—
255C4—48no—
256C4—42no—
257C4—82no—
258C4—84no—
259C2—21no—
260C2—22no—
261C2—21no—
262C2#1098011yescharacteristic
263C2#1270011yescharacteristic
264C2#1270411yescharacteristic
265C2#1322714yes—
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
233. C2×C4#356semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C22×#3562
#107632