Finite Group Databasebeta

← #98707

Subgroups — #98707

Order 512.

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

normal subgroups only

This group has 596 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#59924C211yesmaximal, characteristic
2#9643C211yesmaximal, characteristic
3#8591C212yesmaximal
4#7653C211yesmaximal, characteristic
5#7591C212yesmaximal
6#2811—22no—
7#2567—24no—
8#2567—22no—
9#2551—22no—
10#2551C412yes—
11#2369—22no—
12#2369C412yes—
13C4×C4⋊Q8C2212yes—
14C4×C4⋊Q8C2211yescharacteristic
15#1842C2211yescharacteristic
16#1831C2212yes—
17#1744C2211yescharacteristic
18C2×#332—22no—
19C2×Q8⋊C8—22no—
20#1296—22no—
21#1272—22no—
22#1271—24no—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—24no—
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×C4⋊Q8D412yes—
27C2×C4×Q8—22no—
28C2×C4×Q8D412yes—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—22no—
31C2×C4⋊C8—21no—
32C2×C4×C8—21no—
33#395—24no—
34#395—28no—
35#395—24no—
36#391—28no—
37#391—24no—
38#391—22no—
39#391D412yes—
40#389—24no—
41#389—24no—
42#389—24no—
43#389—22no—
44#389—22no—
45#389D414yes—
46#384—24no—
47#384C2×C411yescharacteristic
48#384D412yes—
49C4×C4⋊C4—22no—
50C4×C4⋊C4—24no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4C2×C414yes—
54C43C2311yescharacteristic
55#336—22no—
56#336—22no—
57#336—22no—
58#332—44no—
59Q8⋊C8—44no—
60C22×Q8—24no—
61C22×Q8—22no—
62C22×Q8C22⋊C412yes—
63C22×C8—41no—
64C22×C8—42no—
65C22×C8—41no—
66C4⋊Q8—24no—
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69C4⋊Q8—88no—
70C4⋊Q8—82no—
71C4⋊Q8—82no—
72C4⋊Q8SD1614yes—
73C4⋊Q8Q1614yes—
74C4×Q8—42no—
75C4×Q8—42no—
76C4×Q8—84no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—28no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4C22⋊C414yes—
87C2×C4⋊C4C22⋊C412yes—
88C2×C4⋊C4C22⋊C414yes—
89C2×C42—24no—
90C2×C42—22no—
91C2×C42—24no—
92C2×C42C22×C411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D412yes—
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D412yes—
97C4⋊C8—42no—
98C4⋊C8—42no—
99C4⋊C8—42no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4×C8—22no—
103C4×C8—41no—
104#95—28no—
105#95—24no—
106#95—24no—
107#95—24no—
108#95—24no—
109C2×Q8—22no—
110C2×Q8—24no—
111C2×Q8—24no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—82no—
115C2×Q8—82no—
116C2×Q8—84no—
117C2×Q8—88no—
118C2×Q8#10114yes—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C2×C22⋊C412yes—
125C22×C4#12011yescharacteristic
126C22×C4#12012yes—
127C22×C4#12011yescharacteristic
128C2×C8—42no—
129C2×C8—82no—
130C2×C8—82no—
131C2×C8—82no—
132C2×C8—82no—
133C2×C8—82no—
134C2×C8—81no—
135C4⋊C4—28no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—42no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—82no—
145C4⋊C4—82no—
146C4⋊C4—84no—
147C4⋊C4—84no—
148C4⋊C4—88no—
149C4⋊C4#10318yes—
150C42—22no—
151C42—21no—
152C42—21no—
153C42—41no—
154C42—42no—
155C42—48no—
156C42—42no—
157C42—42no—
158C42—84no—
159C42C2×SD1612yes—
160C42C2×Q1612yes—
161C42#13712yes—
162C42#13712yes—
163C23#37911yescharacteristic
164Q8—44no—
165Q8—82no—
166Q8—84no—
167Q8—84no—
168Q8—168no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—28no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4C2×#10112yes—
189C2×C4C2×#10312yes—
190C2×C4#41912yes—
191C2×C4#44812yes—
192C2×C4#44812yes—
193C2×C4#45112yes—
194C2×C4#45112yes—
195C2×C4#45311yescharacteristic
196C2×C4#45511yescharacteristic
197C8—82no—
198C8—162no—
199C8—161no—
200C22#178411yescharacteristic
201C22#178712yes—
202C22#179211yescharacteristic
203C22#179411yescharacteristic
204C22#179512yes—
205C4—22no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—22no—
210C4—44no—
211C4—41no—
212C4—48no—
213C4—82no—
214C4—84no—
215C4#162311yescharacteristic
216C4#163311yescharacteristic
217C4#164512yes—
218C2#839612yes—
219C2#843212yes—
220C2#845111yescharacteristic
221C2#845711yescharacteristic
222C2#849111yescharacteristic
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
#599242
#96432
#85912
#76532
#75912