Finite Group Databasebeta

← #136102

Subgroups — #136102

Order 512.

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

normal subgroups only

This group has 374 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#23357C211yesmaximal, characteristic
2#11475C211yesmaximal, characteristic
3#8963C211yesmaximal, characteristic
4#8860C212yesmaximal, complemented
5#7285C212yesmaximal, complemented
6#2975C2211yescharacteristic
7C4×#127C2211yescharacteristic
8#1870—22no—
9#1870—22no—
10#1870—21no—
11#1870C2212yes—
12#1855—22no—
13#1839—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—42no—
28#431—21no—
29#431—21no—
30C2×#108—42no—
31#407—22no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8D412yes—
37#390—22no—
38#390C2×C411yescharacteristic
39#390Q811yescharacteristic
40#384C2×C411yescharacteristic
41C43D411yescharacteristic
42#341—42no—
43#336—21no—
44#336—22no—
45#336—21no—
46#336—22no—
47D4⋊C8—42no—
48#324C2×C414yes—
49#323—22no—
50#323—22no—
51C22×D4—22no—
52C22×D4C4⋊C411yescharacteristic
53C22×D4C2×Q811yescharacteristic
54C23×C4—21no—
55C23×C4C2×D412yes—
56C2×M16—22no—
57C2×M16—22no—
58C2×M16—42no—
59C2×M16—82no—
60C2×M16—82no—
61C2×M16—81no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—42no—
65#127—22no—
66#127—41no—
67C4×D4—41no—
68C4×D4—41no—
69C4×D4—41no—
70C4×D4—41no—
71C4×D4—82no—
72C4×D4—82no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4C4⋊C411yescharacteristic
75C2×C4⋊C4C22×C411yescharacteristic
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4#4112yes—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42C4⋊C411yescharacteristic
83C2×C42C4⋊C411yescharacteristic
84C2×C42C2×D411yescharacteristic
85#108—84no—
86C4⋊C8—22no—
87C4⋊C8—41no—
88C22⋊C8—24no—
89C22⋊C8—42no—
90C22⋊C8—42no—
91C22⋊C8—41no—
92C22⋊C8—42no—
93C22⋊C8—41no—
94C22⋊C8—82no—
95C22⋊C8—82no—
96C22⋊C8—81no—
97C4×C8—41no—
98C4×C8—41no—
99C24—21no—
100C24C22⋊Q812yes—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4—44no—
110C2×D4—82no—
111C2×D4—82no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4C2×C4⋊C411yescharacteristic
124C22×C4C4×D412yes—
125C22×C4#12011yescharacteristic
126C22×C4#12311yescharacteristic
127M16—44no—
128M16—84no—
129M16—82no—
130M16—82no—
131M16—82no—
132C2×C8—44no—
133C2×C8—42no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—82no—
141C2×C8—82no—
142C2×C8—81no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—82no—
146C22⋊C4—42no—
147C22⋊C4—81no—
148C22⋊C4—82no—
149C22⋊C4—82no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—41no—
154C42—42no—
155C42—81no—
156C42#10812yescomplemented
157C23—22no—
158C23—22no—
159C23—22no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23#38511yescharacteristic
168D4—42no—
169D4—81no—
170D4—81no—
171D4—81no—
172D4—88no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—82no—
194C2×C4C2×#10811yescharacteristic
195C2×C4#43011yescharacteristic
196C2×C4B(4,𝔽2)11yescharacteristic
197C2×C4#45811yescharacteristic
198C8—82no—
199C8—84no—
200C8—82no—
201C8—82no—
202C8—161no—
203C22—21no—
204C22—21no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22#185511yescharacteristic
216C22#185811yescharacteristic
217C22#186711yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—41no—
224C4—81no—
225C4—81no—
226C4#220012yes—
227C2—21no—
228C2—21no—
229C2—82no—
230C2—161no—
231C2#886011yescharacteristic
232C2#1296611yescharacteristic
233C2#1297411yescharacteristic
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
156. C42#108semidirect

Maximal subgroups

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

subgroupindex
#233572
#114752
#89632
#88602
#72852