Finite Group Databasebeta

← #150623

Subgroups — #150623

Order 512.

416 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#2026C211yesmaximal, characteristic
2#12860C211yesmaximal, characteristic
3#11506C211yesmaximal, characteristic
4#7429C214yesmaximal, complemented
5C22×#125C411yescharacteristic
6C22×#103—21no—
7C2×#391C2211yescharacteristic
8#2026—21no—
9#2026—21no—
10#2026C412yes—
11#2026C2212yes—
12#1784—21no—
13C2×#340—21no—
14C2×#336C411yescharacteristic
15#1463—21no—
16#1394—21no—
17#1394C2212yes—
18#1319—21no—
19#1319C2212yes—
20C23×Q8D411yescharacteristic
21C23×C8—21no—
22C2×#125—21no—
23C2×#125—41no—
24C2×#125—44no—
25C2×#125C2×C412yes—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C2×#103—22no—
31C2×#103—22no—
32C2×#103—44no—
33C2×#101—22no—
34C2×#101—41no—
35#407—41no—
36#407—41no—
37#391—41no—
38#391C2×C411yescharacteristic
39#391D411yescharacteristic
40#391D411yescharacteristic
41#391C2311yescharacteristic
42C2×#95—41no—
43#340—22no—
44#340—22no—
45#336—21no—
46#336C2×C412yes—
47#324—22no—
48#324—41no—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8—41no—
52C22×Q8—41no—
53C22×Q8—41no—
54C22×Q8—42no—
55C22×Q8C22⋊C411yescharacteristic
56C22×Q8C2×D411yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—41no—
61C23×C4C22⋊C411yescharacteristic
62C2×M16—42no—
63C2×M16—81no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—42no—
68C22×C8—42no—
69#125—44no—
70#125—44no—
71#125—44no—
72#125—44no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—48no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4C22⋊C411yescharacteristic
84C2×C4⋊C4C2×D411yescharacteristic
85C2×C42—21no—
86C2×C42—42no—
87C2×C42C22⋊C411yescharacteristic
88C2×C42C22×C411yescharacteristic
89#103—44no—
90#103—44no—
91#101—42no—
92#101—42no—
93#101—82no—
94C22⋊C8—42no—
95C22⋊C8—81no—
96C22⋊C8—81no—
97C22⋊C8—81no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102C24#9911yescharacteristic
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—42no—
107C2×Q8—41no—
108C2×Q8—41no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8—44no—
112C2×Q8—44no—
113C2×Q8—42no—
114C2×Q8—81no—
115C2×Q8—82no—
116C2×Q8—82no—
117C2×Q8#13712yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—81no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4#9911yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145M16—82no—
146M16—82no—
147M16—162no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—82no—
152C2×C8—82no—
153C2×C8—82no—
154C2×C8—82no—
155C4⋊C4—24no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—48no—
162C4⋊C4—44no—
163C4⋊C4—81no—
164C42—41no—
165C42—41no—
166C42—42no—
167C23—21no—
168C23—22no—
169C23—22no—
170C23—21no—
171C23#35411yescharacteristic
172C23#35411yescharacteristic
173C23C2×#9911yescharacteristic
174Q8—42no—
175Q8—42no—
176Q8—81no—
177Q8—84no—
178Q8—81no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—44no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—84no—
205C2×C4—88no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—161no—
212C2×C4AΣL(1,𝔽16)12yes—
213C2×C4#41911yescharacteristic
214C2×C4#42111yescharacteristic
215C8—82no—
216C8—162no—
217C8—162no—
218C22—24no—
219C22—24no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22#141312yes—
227C22#141412yes—
228C22#151111yescharacteristic
229C22C2×AΣL(1,𝔽16)11yescharacteristic
230C22C2×#35411yescharacteristic
231C4—21no—
232C4—22no—
233C4—21no—
234C4—42no—
235C4—81no—
236C4—81no—
237C4—84no—
238C4—81no—
239C4—81no—
240C4—81no—
241C2—21no—
242C2—22no—
243C2—22no—
244C2—21no—
245C2#1180811yescharacteristic
246C2C2×#141311yescharacteristic
247C2C2×#141411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7429C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20262
#128602
#115062
#74292