Finite Group Databasebeta

← #106205

Subgroups — #106205

Order 512.

675 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 180 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×#2985C211yesmaximal, characteristic
2#17748C212yesmaximal
3#8006C211yesmaximal, characteristic
4#7995C212yesmaximal
5#7986C211yesmaximal, characteristic
6C2×#533C2211yescharacteristic
7#2985—24no—
8#2985—24no—
9C2×#446—22no—
10C2×#446C412yes—
11C2×C8×Q8C412yes—
12#2920C2212yes—
13C2×#433—22no—
14C2×#432C2212yes—
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#1558—22no—
18#1556—22no—
19#1554C418yes—
20#1550—24no—
21#1293—22no—
22#1291—24no—
23#1290—22no—
24#533—22no—
25#533—22no—
26#533C2×C414yes—
27#533D414yes—
28C2×C4⋊Q8D411yescharacteristic
29C2×C4⋊Q8D411yescharacteristic
30C2×#125—21no—
31C2×C22⋊Q8—21no—
32C2×C22⋊Q8Q812yes—
33C2×C4×Q8—21no—
34C2×C4×Q8C2×C412yes—
35C2×#117—21no—
36C22×C42C2311yescharacteristic
37#446—216no—
38#446—24no—
39#446—24no—
40C8×Q8—24no—
41C8×Q8—24no—
42#433—216no—
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#432—22no—
48#432—22no—
49#432C2×C414yes—
50#432D414yes—
51#431—22no—
52#431—22no—
53#431—22no—
54#431C2×C414yes—
55#423—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2×C412yes—
60C2×C4⋊C8D412yes—
61#407—22no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8D412yes—
65C4×M16—22no—
66C2×#97—22no—
67C2×#97—21no—
68C2×#97—22no—
69C2×C4×C8—22no—
70C2×C4×C8C2×C411yescharacteristic
71C2×C4×C8C2×C411yescharacteristic
72#333—28no—
73#332—28no—
74C22×Q8—21no—
75C22×Q8C4⋊C412yes—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×Q811yescharacteristic
78C23×C4#4111yescharacteristic
79C2×M16—42no—
80C22×C8—22no—
81C22×C8—42no—
82C22×C8C22⋊C412yes—
83C22×C8C22⋊C412yes—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86C4⋊Q8C22⋊C414yes—
87C4⋊Q8C22⋊C414yes—
88#125—24no—
89#125—24no—
90C22⋊Q8—24no—
91C22⋊Q8—24no—
92C22⋊Q8—24no—
93C22⋊Q8C4⋊C418yes—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—24no—
97C4×Q8—24no—
98C4×Q8C4214yes—
99C4×Q8C22⋊C414yes—
100#117—24no—
101#117—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4C4⋊C412yes—
108C2×C22⋊C4—22no—
109C2×C42—22no—
110C2×C42C22×C411yescharacteristic
111C2×C42C22×C412yes—
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42#4112yes—
117C2×C42#4111yescharacteristic
118C4⋊C8—28no—
119C4⋊C8—28no—
120C4⋊C8—22no—
121C4⋊C8—24no—
122C4⋊C8—24no—
123C4⋊C8—24no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126C4⋊C8C22⋊C414yes—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—24no—
130C22⋊C8—24no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#97—28no—
134#97—24no—
135#97—24no—
136#97—42no—
137#97—42no—
138C4×C8—28no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8—24no—
144C24C22⋊Q811yescharacteristic
145C2×Q8—22no—
146C2×Q8—24no—
147C2×Q8—44no—
148C2×Q8—82no—
149C2×Q8#9514yes—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—81no—
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×C4⋊C411yescharacteristic
157C22×C4C2×C4⋊C412yes—
158C22×C4#11711yescharacteristic
159C22×C4C4×D412yes—
160C22×C4C4×D412yes—
161C22×C4#12012yes—
162C22×C4#12111yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4C22⋊Q811yescharacteristic
165C22×C4C22⋊Q812yes—
166C22×C4#12312yes—
167M16—84no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—24no—
171C2×C8—44no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—44no—
175C2×C8—44no—
176C2×C8—84no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—28no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4#9514yes—
185C22⋊C4—28no—
186C42—24no—
187C42—24no—
188C42C2×C4211yescharacteristic
189C42C2×C22⋊C411yescharacteristic
190C42C2×C22⋊C412yes—
191C42C2×C22⋊C411yescharacteristic
192C42#11711yescharacteristic
193C42C4×D412yes—
194C42C4×D412yes—
195C42#12012yes—
196C42#12112yes—
197C42#12312yes—
198C23—22no—
199C23—41no—
200C23—41no—
201C23#38011yescharacteristic
202C23#38512yes—
203Q8—44no—
204Q8—44no—
205Q8—84no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4C2×#9511yescharacteristic
223C2×C4C4×C22⋊C411yescharacteristic
224C2×C4C4×C22⋊C411yescharacteristic
225C2×C4#37911yescharacteristic
226C2×C4#38011yescharacteristic
227C2×C4#38111yescharacteristic
228C2×C4#38512yes—
229C2×C4#38612yes—
230C2×C4#38612yes—
231C8—42no—
232C8—42no—
233C8—44no—
234C8—84no—
235C8—84no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—82no—
242C22C22⋊#9511yescharacteristic
243C22#181312yes—
244C4—21no—
245C4—21no—
246C4—22no—
247C4—22no—
248C4—21no—
249C4—22no—
250C4—21no—
251C4—21no—
252C4—44no—
253C4—44no—
254C4#179612yes—
255C2—22no—
256C2—42no—
257C2#1076011yescharacteristic
258C2#1077711yescharacteristic
259C2#1079311yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#29852
#177482
#80062
#79952
#79862