Finite Group Databasebeta

← #150591

Subgroups — #150591

Order 512.

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

normal subgroups only

This group has 356 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×#2074C211yesmaximal, characteristic
2#12860C211yesmaximal, characteristic
3#11506C211yesmaximal, characteristic
4#7413C214yesmaximal, complemented
5C22×#103—21no—
6C2×#391C2211yescharacteristic
7C2×#382C411yescharacteristic
8#2074—21no—
9#2074—21no—
10#2074C412yes—
11#2074C2212yes—
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—
22C22×C4⋊C4—21no—
23C22×C4⋊C4D411yescharacteristic
24C22×C42C2×C411yescharacteristic
25C2×#103—22no—
26C2×#103—22no—
27C2×#103—44no—
28C2×#101—22no—
29C2×#101—41no—
30#407—41no—
31#407—41no—
32#391—41no—
33#391C2×C411yescharacteristic
34#391D411yescharacteristic
35#391D411yescharacteristic
36#391C2311yescharacteristic
37#382—21no—
38#382—41no—
39#382C2×C412yes—
40C2×#95—41no—
41C2×#95—41no—
42#340—22no—
43#340—22no—
44#336—21no—
45#336C2×C412yes—
46#324—22no—
47#324—41no—
48C22×Q8—21no—
49C22×Q8—22no—
50C22×Q8—41no—
51C22×Q8—41no—
52C22×Q8—41no—
53C22×Q8—42no—
54C22×Q8C22⋊C411yescharacteristic
55C22×Q8C2×D411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—41no—
60C23×C4C22⋊C411yescharacteristic
61C2×M16—42no—
62C2×M16—81no—
63C22×C8—22no—
64C22×C8—41no—
65C22×C8—41no—
66C22×C8—42no—
67C22×C8—42no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4C22⋊C411yescharacteristic
76C2×C4⋊C4C2×D411yescharacteristic
77C2×C42—21no—
78C2×C42—42no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22×C411yescharacteristic
81#103—44no—
82#103—44no—
83#101—42no—
84#101—42no—
85#101—82no—
86C22⋊C8—42no—
87C22⋊C8—81no—
88C22⋊C8—81no—
89C22⋊C8—81no—
90#95—41no—
91#95—41no—
92#95—41no—
93#95—41no—
94#95—42no—
95#95—42no—
96C24#9911yescharacteristic
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—42no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×Q8—44no—
106C2×Q8—44no—
107C2×Q8—42no—
108C2×Q8—81no—
109C2×Q8—82no—
110C2×Q8—82no—
111C2×Q8AGL(1,ℤ/8)12yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
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—42no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—81no—
133C22×C4—82no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4#9911yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139M16—82no—
140M16—82no—
141M16—162no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8—82no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—81no—
154C4⋊C4—82no—
155C42—41no—
156C42—41no—
157C42—42no—
158C23—21no—
159C23—22no—
160C23—22no—
161C23—21no—
162C23#35411yescharacteristic
163C23#35411yescharacteristic
164C23C2×#9911yescharacteristic
165Q8—42no—
166Q8—42no—
167Q8—81no—
168Q8—84no—
169Q8—81no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
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—82no—
193C2×C4—82no—
194C2×C4—84no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—161no—
202C2×C4—164no—
203C2×C4#35212yes—
204C2×C4#41811yescharacteristic
205C2×C4#42111yescharacteristic
206C8—82no—
207C8—162no—
208C8—162no—
209C22—24no—
210C22—24no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22#141312yes—
218C22#141412yes—
219C22#151011yescharacteristic
220C22C2×#35211yescharacteristic
221C22C2×#35411yescharacteristic
222C4—21no—
223C4—22no—
224C4—21no—
225C4—42no—
226C4—81no—
227C4—81no—
228C4—81no—
229C4—81no—
230C4—81no—
231C4—162no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2#1179711yescharacteristic
237C2C2×#141311yescharacteristic
238C2C2×#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. #7413C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20742
#128602
#115062
#74132