Finite Group Databasebeta

← #97508

Subgroups — #97508

Order 512.

588 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 136 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#59869C211yesmaximal, characteristic
2#9870C212yesmaximal
3#9637C211yesmaximal, characteristic
4#8587C211yesmaximal, characteristic
5#8586C212yesmaximal
6#2809C414yes—
7#2571—21no—
8#2568—22no—
9#2546—22no—
10#2508—24no—
11#2369—21no—
12#2355—22no—
13C4×C4⋊Q8C2212yes—
14C4×C4⋊Q8C2211yescharacteristic
15#1842C2211yescharacteristic
16#1842C2212yes—
17C42×C8C2211yescharacteristic
18C2×#333—21no—
19C2×#332—22no—
20C2×Q8⋊C8—22no—
21#1296—22no—
22#1296—21no—
23#1279—24no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8D411yescharacteristic
26C2×C4⋊Q8D412yes—
27C2×C4×Q8—22no—
28C2×C4×Q8D412yes—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—22no—
34C2×C4×C8—21no—
35C2×C4×C8—22no—
36#400—24no—
37#400—24no—
38#398—28no—
39#391—24no—
40#391—24no—
41#391—22no—
42#391C2×C412yes—
43#389—24no—
44#389—24no—
45#389—22no—
46#389—22no—
47#389C2×C414yes—
48#387—28no—
49#384D411yescharacteristic
50#384D412yes—
51#382—24no—
52#382—24no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—24no—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4D414yes—
58C43C2311yescharacteristic
59#333—42no—
60#332—44no—
61Q8⋊C8—44no—
62C22×Q8—22no—
63C22×Q8C22⋊C412yes—
64C22×C8—41no—
65C22×C8—42no—
66C22×C8—41no—
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72C4⋊Q8—82no—
73C4⋊Q8—82no—
74C4⋊Q8SD1612yes—
75C4⋊Q8Q1612yes—
76C4×Q8—42no—
77C4×Q8—42no—
78C4×Q8—42no—
79C4×Q8—42no—
80C4×Q8—42no—
81C4×Q8—42no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4C22⋊C414yes—
92C2×C4⋊C4C22⋊C412yes—
93C2×C4⋊C4C22⋊C414yes—
94C2×C42—24no—
95C2×C42—22no—
96C2×C42—24no—
97C2×C42C22×C411yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C42C2×D412yes—
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D412yes—
102C4⋊C8—42no—
103C4⋊C8—44no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4×C8—22no—
111C4×C8—24no—
112C4×C8—42no—
113C4×C8—41no—
114#95—24no—
115#95—28no—
116#95—28no—
117#95—24no—
118#95—24no—
119#95—24no—
120C2×Q8—22no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8#10314yes—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4C2×C22⋊C412yes—
136C22×C4#12011yescharacteristic
137C22×C4#12012yes—
138C22×C4#12011yescharacteristic
139C2×C8—42no—
140C2×C8—44no—
141C2×C8—82no—
142C2×C8—82no—
143C2×C8—82no—
144C2×C8—81no—
145C2×C8—82no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—44no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C42—21no—
166C42—22no—
167C42—21no—
168C42—22no—
169C42—41no—
170C42—44no—
171C42—44no—
172C42—42no—
173C42—42no—
174C42—84no—
175C42C2×SD1611yescharacteristic
176C42C2×Q1611yescharacteristic
177C42#13512yes—
178C42#13512yes—
179C42AGL(1,ℤ/8)12yes—
180C42#13712yes—
181C42#13712yes—
182C23#37911yescharacteristic
183Q8—44no—
184Q8—82no—
185Q8—82no—
186Q8—82no—
187Q8—82no—
188Q8—82no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4C2×#10311yescharacteristic
207C2×C4#41612yes—
208C2×C4#41712yes—
209C2×C4#41911yescharacteristic
210C2×C4#44812yes—
211C2×C4#44912yes—
212C2×C4#44912yes—
213C2×C4#45112yes—
214C2×C4#45212yes—
215C2×C4#45212yes—
216C2×C4#45311yescharacteristic
217C2×C4#45412yes—
218C2×C4#45511yescharacteristic
219C2×C4#45612yes—
220C8—44no—
221C8—161no—
222C8—162no—
223C22#178711yescharacteristic
224C22#178812yes—
225C22#179111yescharacteristic
226C22#179211yescharacteristic
227C22#179312yes—
228C4—21no—
229C4—21no—
230C4—22no—
231C4—22no—
232C4—41no—
233C4—44no—
234C4—84no—
235C4—82no—
236C4—84no—
237C4#162411yescharacteristic
238C4#162612yes—
239C4#163411yescharacteristic
240C4#163612yes—
241C4#164612yes—
242C4#164812yes—
243C4#165312yes—
244C2#835511yescharacteristic
245C2#835712yes—
246C2#844812yes—
247C2#845211yescharacteristic
248C2#853111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#598692
#98702
#96372
#85872
#85862