Finite Group Databasebeta

← #150544

Subgroups — #150544

Order 512.

343 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 58 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#28519C211yesmaximal, characteristic
2C4×#354C211yesmaximal, characteristic
3#11530C211yesmaximal, characteristic
4#7434C214yesmaximal
5C2×#477—21no—
6C2×#476—21no—
7#2868C411yescharacteristic, complemented
8C4×#124C2211yescharacteristic
9C2×#354—21no—
10#2096C412yescomplemented
11#2093C2212yes—
12#1959—21no—
13#1847—21no—
14#1767C411yescharacteristic, complemented
15C4×#99—21no—
16#1467—21no—
17#1393C2212yes—
18#1324—21no—
19#1318C2212yes—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8—41no—
22C2×#124D411yescharacteristic
23C2×C4×Q8D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25#477—42no—
26#477—42no—
27#476—42no—
28#476—42no—
29#432—41no—
30C2×#107—22no—
31C2×#106—22no—
32C2×C4⋊C8—21no—
33C2×#103—22no—
34C2×#102—22no—
35C2×#99—22no—
36C2×#97—21no—
37#391C2×C411yescharacteristic
38#390C2311yescharacteristic
39#388—41no—
40#384—21no—
41#384—41no—
42#384C2×C412yes—
43#382D411yescharacteristic
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C43C2×C411yescharacteristic
47#354—24no—
48#342—41no—
49#336C2×C412yes—
50#324—22no—
51#324—41no—
52C22×Q8—41no—
53C22×Q8C22⋊C411yescharacteristic
54C22×D4C2×D411yescharacteristic
55C23×C4—21no—
56C22×C8—22no—
57C4⋊Q8—41no—
58C4⋊Q8—41no—
59C4⋊Q8—41no—
60C4⋊Q8—41no—
61C4⋊Q8—84no—
62#124—22no—
63#124—22no—
64C4×Q8—41no—
65C4×Q8—41no—
66C4×D4—41no—
67C4×D4—41no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4C22⋊C411yescharacteristic
74C2×C4⋊C4C2×D411yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—41no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—81no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22×C411yescharacteristic
86#107—44no—
87#106—44no—
88C4⋊C8—22no—
89C4⋊C8—22no—
90C4⋊C8—81no—
91#103—44no—
92#102—44no—
93#99—24no—
94#99—42no—
95C22⋊C8—42no—
96#97—42no—
97#97—81no—
98#95—41no—
99#95—41no—
100#95—81no—
101C24—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8—81no—
107C2×Q8—81no—
108C2×Q8—84no—
109C2×D4—21no—
110C2×D4—41no—
111C2×D4—41no—
112C2×D4#13712yes—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—82no—
122C22×C4—82no—
123C22×C4#9911yescharacteristic
124C22×C4#9911yescharacteristic
125C22×C4C2×C22⋊C411yescharacteristic
126C2×C8—24no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—82no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—81no—
135C4⋊C4—84no—
136C4⋊C4—84no—
137C4⋊C4—81no—
138C4⋊C4—81no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C22⋊C4—81no—
142C22⋊C4—82no—
143C22⋊C4—82no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—41no—
148C42—42no—
149C42—42no—
150C42—41no—
151C42—41no—
152C42—162no—
153C42#10412yes—
154C23—22no—
155C23—41no—
156C23—41no—
157C23—81no—
158C23—81no—
159C23C2×#9911yescharacteristic
160Q8—42no—
161Q8—81no—
162Q8—162no—
163Q8—162no—
164D4—42no—
165D4—81no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—84no—
186C2×C4—84no—
187C2×C4—82no—
188C2×C4—161no—
189C2×C4—162no—
190C2×C4—162no—
191C2×C4#35311yescharacteristic
192C2×C4#35311yescharacteristic
193C2×C4#41911yescharacteristic
194C2×C4C2×#10411yescharacteristic
195C8—44no—
196C8—162no—
197C22—21no—
198C22—21no—
199C22—42no—
200C22—82no—
201C22—81no—
202C22—81no—
203C22—161no—
204C22#150811yescharacteristic
205C22#212311yescharacteristic
206C22C2×#35311yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—41no—
213C4—81no—
214C4—81no—
215C4—81no—
216C4—164no—
217C4—162no—
218C4#140712yes—
219C2—21no—
220C2—21no—
221C2—82no—
222C2#1178811yescharacteristic
223C2C2×#140711yescharacteristic
224C2#1358511yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #2868C4semidirect
10. #2096C4semidirect
14. #1767C4semidirect

Maximal subgroups

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

subgroupindex
#285192
C4×#3542
#115302
#74342