Finite Group Databasebeta

← #126012

Subgroups — #126012

Order 512.

469 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 64 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#16248C211yesmaximal, characteristic
2#10343C212yesmaximal
3#9972C211yesmaximal, characteristic
4#7132C211yesmaximal, characteristic
5#7132C212yesmaximal
6C2×#484—21no—
7#2659C2211yescharacteristic
8#1824—22no—
9C2×#328—21no—
10#1446—22no—
11#1446—21no—
12#1445C2212yes—
13#1445C2211yescharacteristic
14#1436—21no—
15#1436—22no—
16#1294—22no—
17#1280C2212yes—
18#1280C2211yescharacteristic
19C23×Q8—21no—
20C2×C22⋊Q8—21no—
21C22×C4⋊C4D411yescharacteristic
22C22×C4⋊C4D412yes—
23C22×C42—21no—
24#484—42no—
25#484—42no—
26C2×#107—41no—
27C2×#103—41no—
28C2×C22⋊C8—21no—
29C2×C22⋊C8—22no—
30#398—41no—
31#395—41no—
32#380—22no—
33C2×#95—22no—
34C2×#95—22no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—21no—
38C2×#95—21no—
39C2×#95—22no—
40C2×#95D412yes—
41C2×#95D411yescharacteristic
42C2×#95C2311yescharacteristic
43#340—42no—
44#340—42no—
45#328—42no—
46C22×Q8—21no—
47C22×Q8—44no—
48C22×Q8—42no—
49C22×Q8—81no—
50C22×Q8—81no—
51C22×Q8—82no—
52C23×C4—21no—
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—22no—
56C23×C4C2×D411yescharacteristic
57C23×C4C2×D412yes—
58C22×C8—41no—
59C22×C8—42no—
60C22⋊Q8—41no—
61C22⋊Q8—41no—
62C22⋊Q8—41no—
63C22⋊Q8—41no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4SD1612yes—
75C2×C4⋊C4Q1614yes—
76C2×C22⋊C4—41no—
77C2×C42—82no—
78C2×C42—81no—
79#107—44no—
80#103—44no—
81C22⋊C8—22no—
82C22⋊C8—41no—
83C22⋊C8—42no—
84C22⋊C8—42no—
85#95—41no—
86#95—41no—
87#95—42no—
88#95—44no—
89#95—44no—
90#95—44no—
91#95—42no—
92#95—82no—
93#95—81no—
94#95—81no—
95#95—82no—
96C24#12011yescharacteristic
97C2×Q8—41no—
98C2×Q8—41no—
99C2×Q8—84no—
100C2×Q8—88no—
101C2×Q8—84no—
102C2×Q8—88no—
103C2×Q8—82no—
104C2×Q8—84no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—41no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4C2×SD1611yescharacteristic
139C22×C4C2×Q1612yes—
140C22×C4#13711yescharacteristic
141C22×C4#13712yes—
142C2×C8—42no—
143C2×C8—82no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—81no—
152C4⋊C4—82no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C22⋊C4—81no—
158C22⋊C4—81no—
159C42—84no—
160C23—22no—
161C23—21no—
162C23—21no—
163C23#44811yescharacteristic
164C23#45112yes—
165C23#45311yescharacteristic
166C23#45512yes—
167C23B(4,𝔽2)11yescharacteristic
168Q8—82no—
169Q8—816no—
170Q8—161no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—81no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—81no—
205C2×C4—82no—
206C2×C4—81no—
207C2×C4—82no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4—81no—
213C8—82no—
214C8—162no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22#159611yescharacteristic
224C22#160212yes—
225C22#163311yescharacteristic
226C22#164512yes—
227C22#165911yescharacteristic
228C22#219812yes—
229C22#219811yescharacteristic
230C22#220211yescharacteristic
231C4—41no—
232C4—41no—
233C4—42no—
234C4—42no—
235C4—84no—
236C4—82no—
237C4—82no—
238C4—84no—
239C4—161no—
240C4—161no—
241C2—21no—
242C2—21no—
243C2—22no—
244C2#775911yescharacteristic
245C2#1212211yescharacteristic
246C2#1212711yescharacteristic
247C2#1221612yes—
248C2#1222112yes—
subgroupG/H
center, socleC23B(4,𝔽2)
commutator, Frattini, cosocleC2×#95C23
prime radicalC24#120
LCSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#162482
#103432
#99722
#71322