Finite Group Databasebeta

← #102468

Subgroups — #102468

Order 512.

496 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 98 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#33148C212yesmaximal
2#30291C211yesmaximal, characteristic
3#11346C211yesmaximal, characteristic
4#7667C212yesmaximal
5#7020C211yesmaximal, characteristic
6#2857C2212yes—
7C2×C4×C4⋊C4C2211yescharacteristic
8#2083—22no—
9#2057—22no—
10#1987—22no—
11#1948—22no—
12#1920—24no—
13#1853—22no—
14#1823—22no—
15#1808—22no—
16#1797C2212yes—
17C2×#336—22no—
18C2×#336C2211yescharacteristic
19C2×#336C2211yescharacteristic
20#1272—24no—
21C23×C8—21no—
22C23×C8D412yes—
23C2×C4⋊Q8D412yes—
24C2×C22⋊Q8—22no—
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C22×C42—21no—
28C22×C42C2311yescharacteristic
29C2×#107—22no—
30C2×#107—22no—
31C2×#106—24no—
32C2×C4⋊C8—22no—
33C2×C22⋊C8—22no—
34#391—22no—
35#389—22no—
36#389—22no—
37#384D412yes—
38#380—22no—
39#380—22no—
40C4×C4⋊C4—22no—
41C4×C4⋊C4—22no—
42C2×#95—21no—
43#340—24no—
44#340—24no—
45#340—24no—
46#340—24no—
47#336—22no—
48#336—22no—
49#336—44no—
50#336D414yes—
51C22×Q8—22no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4C2×D411yescharacteristic
55C23×C4C2×D411yescharacteristic
56C23×C4#4111yescharacteristic
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—24no—
61C22×C8—42no—
62C22×C8—42no—
63C22×C8—41no—
64C22×C8—42no—
65C22×C8—41no—
66C22×C8—42no—
67C22×C8SD1614yes—
68C4⋊Q8—44no—
69C22⋊Q8—84no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4Q1612yes—
82C2×C22⋊C4—42no—
83C2×C42—42no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42—42no—
87C2×C42—42no—
88C2×C42C2×D412yes—
89C2×C42C2×D412yes—
90#107—82no—
91#107—82no—
92#106—48no—
93C4⋊C8—44no—
94C22⋊C8—42no—
95C22⋊C8—42no—
96#95—42no—
97#95—42no—
98#95—42no—
99C24#12111yescharacteristic
100C2×Q8—44no—
101C2×Q8—82no—
102C2×Q8—82no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4#12012yes—
124C22×C4#12011yescharacteristic
125C22×C4#12112yes—
126C22×C4#12711yescharacteristic
127C22×C4C2×SD1612yes—
128C22×C4C2×Q1611yescharacteristic
129C22×C4#13511yescharacteristic
130C22×C4#13712yes—
131C22×C4#13712yes—
132C2×C8—24no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—44no—
136C2×C8—44no—
137C2×C8—44no—
138C2×C8—42no—
139C2×C8—44no—
140C2×C8—44no—
141C2×C8—82no—
142C2×C8—81no—
143C2×C8—82no—
144C2×C8—81no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—82no—
155C4⋊C4—84no—
156C22⋊C4—84no—
157C42—24no—
158C42—42no—
159C42—82no—
160C42—81no—
161C42—81no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23#39211yescharacteristic
167C23#46112yes—
168C23#46711yescharacteristic
169C23#47011yescharacteristic
170C23#47212yes—
171Q8—164no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4#44812yes—
204C2×C4#45112yes—
205C2×C4#45212yes—
206C2×C4#45511yescharacteristic
207C2×C4#45611yescharacteristic
208C2×C4#45812yes—
209C2×C4#49211yescharacteristic
210C2×C4#49711yescharacteristic
211C8—44no—
212C8—84no—
213C8—82no—
214C22—21no—
215C22—21no—
216C22—22no—
217C22—22no—
218C22—21no—
219C22—21no—
220C22—22no—
221C22—22no—
222C22#167512yes—
223C22#169512yes—
224C22#200111yescharacteristic
225C22#200711yescharacteristic
226C22#201011yescharacteristic
227C22#201311yescharacteristic
228C22#201711yescharacteristic
229C22#201811yescharacteristic
230C22#202211yescharacteristic
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—82no—
237C4—82no—
238C4—162no—
239C4—162no—
240C4#160812yes—
241C4#161112yes—
242C2—22no—
243C2—22no—
244C2#997311yescharacteristic
245C2#997711yescharacteristic
246C2#1002811yescharacteristic
247C2#1003611yescharacteristic
248C2#1025911yescharacteristic
249C2#1027811yescharacteristic
250C2#1032511yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#331482
#302912
#113462
#76672
#70202