Finite Group Databasebeta

← #98403

Subgroups — #98403

Order 512.

529 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 104 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#39641C211yesmaximal, characteristic
2#9903C212yesmaximal
3#8584C211yesmaximal, characteristic
4#7615C212yesmaximal
5#7602C211yesmaximal, characteristic
6#2868C2211yescharacteristic
7#2791—24no—
8#2647—21no—
9#2546—22no—
10#2546C412yes—
11#2507—22no—
12#2446—22no—
13#2365C412yes—
14#2330—21no—
15C4×#125C2211yescharacteristic
16#2295C2211yescharacteristic
17#1831C2212yes—
18#1767C2212yes—
19C2×#330—21no—
20#1296—24no—
21#1295—21no—
22#1272—22no—
23#1271—22no—
24#1271—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8C2×C412yes—
28C2×#125D411yescharacteristic
29C2×C4×Q8—22no—
30C2×C4×Q8C2×C411yescharacteristic
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×#97—22no—
34#400—24no—
35#398—24no—
36#398—24no—
37#398—24no—
38#398—24no—
39#391—22no—
40#391—22no—
41#391D412yes—
42#389—24no—
43#389—24no—
44#389—22no—
45#389—22no—
46#389—22no—
47#389D412yes—
48#389D412yes—
49#387—24no—
50#387—24no—
51#387—24no—
52#387—22no—
53#387—22no—
54#384—21no—
55#384—21no—
56#384D414yes—
57#382D411yescharacteristic
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4C2×C412yes—
62C4×C4⋊C4C2×C411yescharacteristic
63#376—21no—
64C43C2311yescharacteristic
65#336—22no—
66#336—22no—
67#336—22no—
68#330—42no—
69C22×Q8—22no—
70C22×Q8C22⋊C411yescharacteristic
71C22×C8—42no—
72C22×C8—42no—
73C4⋊Q8—44no—
74C4⋊Q8—84no—
75C4⋊Q8—84no—
76#125—22no—
77#125SD1612yes—
78#125Q1612yes—
79C4×Q8—42no—
80C4×Q8—84no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4C22⋊C411yescharacteristic
92C2×C4⋊C4C22⋊C412yes—
93C2×C4⋊C4C22⋊C412yes—
94C2×C4⋊C4C22⋊C414yes—
95C2×C4⋊C4C22⋊C412yes—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—24no—
100C2×C42C22×C411yescharacteristic
101C2×C42C2×D412yes—
102C2×C42C2×D412yes—
103C2×C42C2×D411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C4⋊C8—24no—
106C4⋊C8—42no—
107C4⋊C8—44no—
108#97—42no—
109#97—42no—
110#95—24no—
111#95—24no—
112#95—24no—
113#95—24no—
114#95—21no—
115#95—22no—
116#95—22no—
117#95—22no—
118#95—22no—
119#95—22no—
120#95—22no—
121C2×Q8—22no—
122C2×Q8—44no—
123C2×Q8—44no—
124C2×Q8—42no—
125C2×Q8—44no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—24no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4#12012yes—
136C22×C4#12012yes—
137C2×C8—44no—
138C2×C8—82no—
139C2×C8—82no—
140C2×C8—82no—
141C2×C8—82no—
142C2×C8—82no—
143C4⋊C4—22no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—44no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—84no—
157C4⋊C4—84no—
158C4⋊C4—84no—
159C4⋊C4—84no—
160C4⋊C4—84no—
161C4⋊C4#10314yes—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—41no—
166C42—42no—
167C42—42no—
168C42—42no—
169C42—44no—
170C42—84no—
171C42C2×SD1611yescharacteristic
172C42C2×Q1611yescharacteristic
173C42#13712yes—
174C23#37911yescharacteristic
175Q8—82no—
176Q8—88no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4#41312yes—
203C2×C4C2×#10311yescharacteristic
204C2×C4#41911yescharacteristic
205C2×C4#44812yes—
206C2×C4#45112yes—
207C2×C4#45412yes—
208C2×C4#45612yes—
209C8—84no—
210C8—162no—
211C22#178511yescharacteristic
212C22#178711yescharacteristic
213C22#179312yes—
214C22#179512yes—
215C22#179611yescharacteristic
216C4—22no—
217C4—22no—
218C4—22no—
219C4—21no—
220C4—21no—
221C4—22no—
222C4—22no—
223C4—41no—
224C4—44no—
225C4—81no—
226C4—82no—
227C4—82no—
228C4—81no—
229C4—84no—
230C2#839211yescharacteristic
231C2#842412yes—
232C2#847411yescharacteristic
233C2#849912yes—
234C2#852611yescharacteristic
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
#396412
#99032
#85842
#76152
#76022