Finite Group Databasebeta

← #150905

Subgroups — #150905

Order 512.

326 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 64 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#29149C211yesmaximal, characteristic
2#12797C211yesmaximal, characteristic
3#11475C211yesmaximal, characteristic
4#7428C214yesmaximal, complemented
5#2859C411yescharacteristic, complemented
6C4×#127C2211yescharacteristic
7C2×#353—21no—
8#2025C412yescomplemented
9#2025C2212yes—
10#1961—21no—
11#1836—21no—
12#1802—21no—
13#1802—21no—
14#1799—21no—
15#1767C411yescharacteristic, complemented
16#1459—21no—
17#1393C2212yes—
18#1325—21no—
19#1318C2212yes—
20C2×#127D411yescharacteristic
21C2×#125C2×C412yes—
22C2×C4×D4—41no—
23C2×C4×D4D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25C2×C4⋊C8—21no—
26C2×#102—22no—
27C2×#102—22no—
28C2×#99—22no—
29C2×C22⋊C8—41no—
30C2×#97—21no—
31#390—41no—
32#390C2×C411yescharacteristic
33#390C2311yescharacteristic
34#389—41no—
35#389—41no—
36#384—21no—
37#384D411yescharacteristic
38#382—21no—
39#380—41no—
40C43C2×C411yescharacteristic
41#353—42no—
42#342—41no—
43#340—22no—
44#340—22no—
45#336C2×C412yes—
46#324—22no—
47#323—41no—
48C22×D4—41no—
49C22×D4C22⋊C411yescharacteristic
50C22×D4C2×D411yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C22×C8—22no—
54C22×C8—81no—
55#127—41no—
56#127SD1612yes—
57#127Q1612yes—
58#125—82no—
59C4×D4—41no—
60C4×D4—41no—
61C4×D4—41no—
62C4×D4—41no—
63C4×D4—84no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—81no—
70C2×C4⋊C4C22⋊C411yescharacteristic
71C2×C4⋊C4C2×D411yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—42no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82C4⋊C8—41no—
83C4⋊C8—41no—
84#102—82no—
85#102—82no—
86#99—42no—
87#99—42no—
88C22⋊C8—42no—
89C22⋊C8—81no—
90#97—42no—
91#95—42no—
92#95—81no—
93C24—21no—
94C24—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—82no—
104C2×D4—82no—
105C2×D4#10312yes—
106C2×D4C2×SD1611yescharacteristic
107C2×D4C2×Q1611yescharacteristic
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—42no—
117C22×C4—82no—
118C22×C4—82no—
119C22×C4—82no—
120C22×C4#9911yescharacteristic
121C22×C4#9911yescharacteristic
122C22×C4C2×C22⋊C411yescharacteristic
123C2×C8—42no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8—82no—
127C2×C8—162no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—82no—
133C4⋊C4—84no—
134C4⋊C4—82no—
135C4⋊C4—82no—
136C4⋊C4—162no—
137C22⋊C4—81no—
138C22⋊C4—81no—
139C22⋊C4—82no—
140C22⋊C4—82no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—41no—
147C42—42no—
148C42—81no—
149C42#10312yes—
150C23—21no—
151C23—22no—
152C23—21no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—81no—
158C23—81no—
159C23—81no—
160C23—81no—
161C23C2×#9911yescharacteristic
162D4—42no—
163D4—41no—
164D4—41no—
165D4—81no—
166D4—81no—
167D4—84no—
168D4—84no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
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—82no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—161no—
190C2×C4—161no—
191C2×C4—162no—
192C2×C4—162no—
193C2×C4—162no—
194C2×C4#35411yescharacteristic
195C2×C4#35411yescharacteristic
196C2×C4C2×#10311yescharacteristic
197C2×C4#42111yescharacteristic
198C8—82no—
199C8—162no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—42no—
204C22—41no—
205C22—81no—
206C22—82no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—161no—
213C22—161no—
214C22#152011yescharacteristic
215C22#212211yescharacteristic
216C22C2×#35411yescharacteristic
217C4—21no—
218C4—21no—
219C4—21no—
220C4—42no—
221C4—41no—
222C4—81no—
223C4—81no—
224C4—81no—
225C4—162no—
226C4—322no—
227C4#141212yes—
228C2—21no—
229C2—21no—
230C2—81no—
231C2—81no—
232C2—82no—
233C2#1190011yescharacteristic
234C2C2×#141211yescharacteristic
235C2#1360411yescharacteristic
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. #7428C2semidirect
5. #2859C4semidirect
8. #2025C4semidirect
15. #1767C4semidirect

Maximal subgroups

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

subgroupindex
#291492
#127972
#114752
#74282