Finite Group Databasebeta

← #136100

Subgroups — #136100

Order 512.

427 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 68 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#23176C211yesmaximal, characteristic
2#11475C211yesmaximal, characteristic
3#8963C211yesmaximal, characteristic
4#8860C212yesmaximal, complemented
5#7285C212yesmaximal, complemented
6#2975C2211yescharacteristic
7C2×#436—22no—
8C4×#127C2211yescharacteristic
9#1954—22no—
10#1870—21no—
11#1870C2212yes—
12#1855—22no—
13C4×C4⋊C8—21no—
14#1478—21no—
15#1460C2211yescharacteristic
16#1459C2212yes—
17#1325C412yes—
18#1318C412yes—
19#1298—22no—
20#1276—21no—
21C22×M16—21no—
22C22×M16D412yes—
23C2×#127Q811yescharacteristic
24C2×C4×D4—22no—
25C2×C4×D4D411yescharacteristic
26C2×C4×D4C2311yescharacteristic
27#436—41no—
28#436—41no—
29#436—48no—
30#431—21no—
31#431—21no—
32C2×#108—42no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35#407—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8D412yes—
41C2×C4×C8—22no—
42#390—22no—
43#390C2×C411yescharacteristic
44#390Q811yescharacteristic
45#384C2×C411yescharacteristic
46C43D411yescharacteristic
47#341—42no—
48#336—22no—
49D4⋊C8—41no—
50D4⋊C8—41no—
51#324C2×C414yes—
52#323—22no—
53#323—22no—
54C22×D4—22no—
55C22×D4C4⋊C411yescharacteristic
56C22×D4C2×Q811yescharacteristic
57C23×C4—21no—
58C23×C4C2×D412yes—
59C2×M16—22no—
60C2×M16—22no—
61C2×M16—44no—
62C2×M16—44no—
63C2×M16—42no—
64C2×M16—81no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—42no—
68#127—22no—
69#127—41no—
70C4×D4—41no—
71C4×D4—41no—
72C4×D4—44no—
73C4×D4—44no—
74C4×D4—41no—
75C4×D4—41no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4C4⋊C411yescharacteristic
78C2×C4⋊C4C22×C411yescharacteristic
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4#4112yes—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C4⋊C411yescharacteristic
86C2×C42C4⋊C411yescharacteristic
87C2×C42C2×D411yescharacteristic
88#108—84no—
89C4⋊C8—24no—
90C4⋊C8—21no—
91C4⋊C8—21no—
92C4⋊C8—42no—
93C4⋊C8—41no—
94C22⋊C8—24no—
95C22⋊C8—44no—
96C22⋊C8—44no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—41no—
100C22⋊C8—42no—
101C22⋊C8—41no—
102C22⋊C8—81no—
103C4×C8—44no—
104C4×C8—41no—
105C4×C8—41no—
106C24—21no—
107C24C22⋊Q812yes—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—41no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—44no—
117C2×D4—44no—
118C2×D4—44no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4C2×C4⋊C411yescharacteristic
131C22×C4C4×D412yes—
132C22×C4#12011yescharacteristic
133C22×C4#12311yescharacteristic
134M16—44no—
135M16—82no—
136M16—84no—
137M16—82no—
138M16—82no—
139C2×C8—24no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—44no—
144C2×C8—42no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—42no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8—81no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—44no—
154C22⋊C4—42no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C22⋊C4—81no—
158C42—22no—
159C42—22no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—41no—
164C42—42no—
165C42#10812yescomplemented
166C23—22no—
167C23—22no—
168C23—22no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23#38511yescharacteristic
177D4—42no—
178D4—81no—
179D4—81no—
180D4—81no—
181D4—88no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4C2×#10811yescharacteristic
205C2×C4#43011yescharacteristic
206C2×C4B(4,𝔽2)11yescharacteristic
207C2×C4#45811yescharacteristic
208C8—44no—
209C8—82no—
210C8—84no—
211C8—81no—
212C8—81no—
213C8—161no—
214C22—21no—
215C22—21no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—82no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22#185511yescharacteristic
227C22#185811yescharacteristic
228C22#186711yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—42no—
234C4—42no—
235C4—41no—
236C4—81no—
237C4#219611yescharacteristic
238C4#219811yescharacteristic
239C2—21no—
240C2—21no—
241C2—82no—
242C2—161no—
243C2#886011yescharacteristic
244C2#1296411yescharacteristic
245C2#1297511yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC2×C4×D4C23
socleC22#1867
prime radicalC2×#127Q8
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8860C2semidirect
5. #7285C2semidirect
165. C42#108semidirect

Maximal subgroups

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

subgroupindex
#231762
#114752
#89632
#88602
#72852