Finite Group Databasebeta

← #150499

Subgroups — #150499

Order 512.

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

normal subgroups only

This group has 307 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#28727C211yesmaximal, characteristic
2C4×#354C211yesmaximal, characteristic
3#11530C211yesmaximal, characteristic
4#7412C214yesmaximal
5C2×#489—21no—
6#2862C411yescharacteristic, complemented
7C4×#124C2211yescharacteristic
8C2×#354—21no—
9#2074C412yescomplemented
10#2073C2212yes—
11#1993—21no—
12#1805—21no—
13#1804—21no—
14#1767C411yescharacteristic, complemented
15C4×#99—21no—
16#1467—21no—
17#1393C2212yes—
18#1324—21no—
19#1318C2212yes—
20C2×C4⋊Q8—21no—
21C2×#124D411yescharacteristic
22C2×C4×Q8D411yescharacteristic
23C2×C4×D4D411yescharacteristic
24#489—44no—
25#432—41no—
26C2×C4⋊C8—21no—
27C2×#103—22no—
28C2×#102—22no—
29C2×#99—22no—
30C2×#97—21no—
31#391—41no—
32#391—41no—
33#391C2×C411yescharacteristic
34#390C2311yescharacteristic
35#388—41no—
36#382—21no—
37#382C2×C412yes—
38#382D411yescharacteristic
39C4×C22⋊C4—41no—
40C4×C22⋊C4—41no—
41C43C2×C411yescharacteristic
42#354—24no—
43#342—41no—
44#340—22no—
45#340—22no—
46#336C2×C412yes—
47#324—22no—
48#324—41no—
49C22×Q8—41no—
50C22×Q8C22⋊C411yescharacteristic
51C22×D4C2×D411yescharacteristic
52C23×C4—21no—
53C22×C8—22no—
54C4⋊Q8—42no—
55C4⋊Q8—42no—
56#124—22no—
57#124—22no—
58C4×Q8—41no—
59C4×Q8—41no—
60C4×D4—41no—
61C4×D4—41no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4C22⋊C411yescharacteristic
66C2×C4⋊C4C2×D411yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—41no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—41no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—21no—
74C2×C42—81no—
75C2×C42C22⋊C411yescharacteristic
76C2×C42C22⋊C411yescharacteristic
77C2×C42C22×C411yescharacteristic
78C4⋊C8—41no—
79C4⋊C8—41no—
80C4⋊C8—81no—
81#103—44no—
82#102—44no—
83#99—24no—
84#99—42no—
85C22⋊C8—42no—
86#97—42no—
87#97—81no—
88#95—41no—
89#95—41no—
90#95—42no—
91#95—41no—
92#95—41no—
93#95—81no—
94C24—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—41no—
98C2×Q8—41no—
99C2×Q8—82no—
100C2×Q8—82no—
101C2×Q8—82no—
102C2×D4—21no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4AGL(1,ℤ/8)12yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—82no—
115C22×C4—82no—
116C22×C4#9911yescharacteristic
117C22×C4#9911yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119C2×C8—42no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—82no—
123C4⋊C4—41no—
124C4⋊C4—41no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—81no—
128C4⋊C4—82no—
129C22⋊C4—44no—
130C22⋊C4—44no—
131C22⋊C4—81no—
132C22⋊C4—82no—
133C22⋊C4—82no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—41no—
138C42—41no—
139C42—41no—
140C42—81no—
141C42—81no—
142C42—162no—
143C42#10412yes—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—81no—
148C23—81no—
149C23C2×#9911yescharacteristic
150Q8—42no—
151Q8—81no—
152Q8—164no—
153D4—42no—
154D4—81no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—82no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—84no—
177C2×C4—82no—
178C2×C4—161no—
179C2×C4—162no—
180C2×C4—162no—
181C2×C4#35411yescharacteristic
182C2×C4#35411yescharacteristic
183C2×C4#41811yescharacteristic
184C2×C4C2×#10411yescharacteristic
185C8—82no—
186C8—162no—
187C22—21no—
188C22—21no—
189C22—42no—
190C22—82no—
191C22—81no—
192C22—81no—
193C22—161no—
194C22#150711yescharacteristic
195C22#212311yescharacteristic
196C22C2×#35411yescharacteristic
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—42no—
202C4—41no—
203C4—81no—
204C4—81no—
205C4—81no—
206C4—81no—
207C4—162no—
208C4—164no—
209C2—21no—
210C2—21no—
211C2—82no—
212C2#1177711yescharacteristic
213C2#1359411yescharacteristic
214C2#1360311yescharacteristic
subgroupG/H
center, socleC22C2×#354
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
6. #2862C4semidirect
9. #2074C4semidirect
14. #1767C4semidirect

Maximal subgroups

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

subgroupindex
#287272
C4×#3542
#115302
#74122