Finite Group Databasebeta

← #150500

Subgroups — #150500

Order 512.

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

normal subgroups only

This group has 348 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#28718C211yesmaximal, characteristic
2C4×#353C211yesmaximal, characteristic
3#11529C211yesmaximal, characteristic
4#7412C214yesmaximal, complemented
5C2×#486—21no—
6#2862C411yescharacteristic, complemented
7C4×#127C2211yescharacteristic
8C2×#353—21no—
9#2073C412yescomplemented
10#2073C2212yes—
11#1990—21no—
12#1804—21no—
13#1804—21no—
14C4×C4⋊C8C411yescharacteristic, complemented
15C4×#99—21no—
16#1465—21no—
17#1393C2212yes—
18#1325—21no—
19#1318C2212yes—
20C2×C4⋊Q8—21no—
21C2×#127D411yescharacteristic
22C2×C4×D4—41no—
23C2×C4×D4D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25#486—44no—
26C2×C4⋊C8—21no—
27C2×#102—22no—
28C2×#102—22no—
29C2×#99—22no—
30#407—41no—
31C2×C4×C8—21no—
32#391—41no—
33#391—41no—
34#390—41no—
35#390C2×C411yescharacteristic
36#390C2311yescharacteristic
37#384D411yescharacteristic
38#382—21no—
39#382C2×C412yes—
40C4×C22⋊C4—41no—
41C43C2×C411yescharacteristic
42#353—24no—
43#342—41no—
44#340—22no—
45#340—22no—
46#336C2×C412yes—
47#324—22no—
48#323—41no—
49C22×Q8—41no—
50C22×D4—41no—
51C22×D4C22⋊C411yescharacteristic
52C22×D4C2×D411yescharacteristic
53C23×C4—21no—
54C23×C4—21no—
55C2×M16—81no—
56C22×C8—22no—
57C4⋊Q8—42no—
58C4⋊Q8—42no—
59#127—22no—
60#127—22no—
61C4×D4—41no—
62C4×D4—41no—
63C4×D4—41no—
64C4×D4—41no—
65C4×D4—84no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4C22⋊C411yescharacteristic
70C2×C4⋊C4C2×D411yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42—81no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82C4⋊C8—41no—
83C4⋊C8—41no—
84#102—44no—
85#102—44no—
86#99—24no—
87#99—42no—
88C22⋊C8—42no—
89C22⋊C8—81no—
90C4×C8—42no—
91#95—42no—
92#95—41no—
93#95—41no—
94#95—81no—
95C24—21no—
96C24—21no—
97C2×Q8—82no—
98C2×Q8—82no—
99C2×Q8—82no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—44no—
108C2×D4—82no—
109C2×D4—82no—
110C2×D4AGL(1,ℤ/8)12yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—82no—
121C22×C4—82no—
122C22×C4—82no—
123C22×C4#9911yescharacteristic
124C22×C4#9911yescharacteristic
125C22×C4C2×C22⋊C411yescharacteristic
126M16—162no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—82no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C4⋊C4—82no—
137C22⋊C4—44no—
138C22⋊C4—81no—
139C22⋊C4—81no—
140C22⋊C4—82no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—41no—
147C42—42no—
148C42—42no—
149C42—162no—
150C42#10412yes—
151C23—22no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—81no—
158C23—81no—
159C23—81no—
160C23—81no—
161C23C2×#9911yescharacteristic
162Q8—164no—
163D4—42no—
164D4—42no—
165D4—81no—
166D4—81no—
167D4—88no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—84no—
188C2×C4—82no—
189C2×C4—161no—
190C2×C4—161no—
191C2×C4—162no—
192C2×C4—162no—
193C2×C4#35411yescharacteristic
194C2×C4#35411yescharacteristic
195C2×C4#41811yescharacteristic
196C2×C4C2×#10411yescharacteristic
197C8—82no—
198C8—162no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—82no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—161no—
210C22—161no—
211C22#150711yescharacteristic
212C22#212311yescharacteristic
213C22C2×#35411yescharacteristic
214C4—24no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—41no—
220C4—81no—
221C4—81no—
222C4—81no—
223C4—162no—
224C4—164no—
225C2—21no—
226C2—21no—
227C2—82no—
228C2—161no—
229C2#1177711yescharacteristic
230C2#1359511yescharacteristic
231C2#1360211yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7412C2semidirect
6. #2862C4semidirect
9. #2073C4semidirect
14. C4×C4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#287182
C4×#3532
#115292
#74122