Finite Group Databasebeta

← #150588

Subgroups — #150588

Order 512.

347 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 56 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#23631C211yesmaximal, characteristic
2#12861C211yesmaximal, characteristic
3#11507C211yesmaximal, characteristic
4#7413C214yesmaximal, complemented
5C2×#417—21no—
6#2297C2211yescharacteristic
7C2×#382C411yescharacteristic
8#2074—21no—
9#2074C2212yes—
10#2073—21no—
11#2073C412yes—
12#1785—21no—
13#1736—21no—
14#1728C411yescharacteristic
15#1464—21no—
16#1394—21no—
17#1394C2212yes—
18#1319C2212yes—
19#1318—21no—
20C22×#41D411yescharacteristic
21C22×M16—21no—
22C2×#117D411yescharacteristic
23C22×C4⋊C4—21no—
24C22×C42C2×C411yescharacteristic
25#432—41no—
26#417—44no—
27C2×#103—22no—
28C2×#102—22no—
29C2×#101—22no—
30C2×#100—41no—
31#407—41no—
32#391D411yescharacteristic
33#391C2311yescharacteristic
34#390C2×C411yescharacteristic
35#390D411yescharacteristic
36#386—41no—
37#382—21no—
38#382—41no—
39#382C2×C412yes—
40C4×C22⋊C4—41no—
41C2×#95—41no—
42#340—22no—
43#340—22no—
44#336—21no—
45#336C2×C412yes—
46#324—22no—
47#323—41no—
48C2×#41—22no—
49C2×#41—41no—
50C2×#41—41no—
51C2×#41—41no—
52C2×#41—42no—
53C22×Q8C2×D411yescharacteristic
54C22×D4—21no—
55C22×D4C22⋊C411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—41no—
59C23×C4C22⋊C411yescharacteristic
60C2×M16—42no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—42no—
64C2×M16—42no—
65C22×C8—22no—
66#117—41no—
67#117—41no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4C22⋊C411yescharacteristic
73C2×C4⋊C4C2×D411yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—41no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—42no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82C4⋊C8—81no—
83#103—44no—
84#102—44no—
85#101—42no—
86#101—42no—
87#100—82no—
88C22⋊C8—42no—
89C22⋊C8—81no—
90C22⋊C8—81no—
91#97—81no—
92#95—41no—
93#95—41no—
94#95—42no—
95#95—42no—
96C24—21no—
97C24#9911yescharacteristic
98#41—44no—
99#41—44no—
100#41—82no—
101#41—82no—
102C2×Q8—21no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×Q8AGL(1,ℤ/8)12yes—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—81no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4—81no—
131C22×C4—82no—
132C22×C4—82no—
133C22×C4—82no—
134C22×C4#9911yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136M16—82no—
137M16—82no—
138M16—82no—
139M16—82no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—82no—
144C2×C8—82no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—82no—
150C22⋊C4—81no—
151C22⋊C4—82no—
152C22⋊C4—82no—
153C42—41no—
154C42—41no—
155C42—42no—
156C42—41no—
157C42—41no—
158C23—21no—
159C23—21no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—81no—
166C23—81no—
167C23#35212yes—
168C23C2×#9911yescharacteristic
169Q8—42no—
170Q8—81no—
171D4—42no—
172D4—81no—
173D4—84no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—84no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—164no—
202C2×C4#35411yescharacteristic
203C2×C4#35411yescharacteristic
204C2×C4#41811yescharacteristic
205C2×C4#42111yescharacteristic
206C8—82no—
207C8—162no—
208C8—162no—
209C22—21no—
210C22—21no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22—82no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22—161no—
221C22#151011yescharacteristic
222C22C2×#35211yescharacteristic
223C22C2×#35411yescharacteristic
224C4—21no—
225C4—21no—
226C4—21no—
227C4—21no—
228C4—42no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4—81no—
233C4—162no—
234C2—21no—
235C2—21no—
236C2—42no—
237C2—161no—
238C2#1179711yescharacteristic
239C2#1360011yescharacteristic
240C2#1360411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7413C2semidirect

Maximal subgroups

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

subgroupindex
#236312
#128612
#115072
#74132