Finite Group Databasebeta

← #137910

Subgroups — #137910

Order 512.

333 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 90 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#33497C211yesmaximal, characteristic, complemented
2#22211C211yesmaximal, characteristic
3#11634C212yesmaximal
4#11493C211yesmaximal, characteristic, complemented
5#7144C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×#436C2211yescharacteristic
8#2398C411yescharacteristic
9#2183—24no—
10#2163—21no—
11#2163C2212yes—
12#2162—21no—
13#2162C2212yes—
14#1985C411yescharacteristic
15#1955C411yescharacteristic
16#1923—21no—
17#1923—21no—
18#1851—21no—
19#1851—21no—
20#1843C411yescharacteristic
21#1746C2211yescharacteristic
22#1330—24no—
23#1322—21no—
24#1322—21no—
25C22×M16—21no—
26C2×C4×Q8C2×C411yescharacteristic
27C2×C4×D4C2×C411yescharacteristic
28C8×Q8—22no—
29C8×Q8—22no—
30#436—22no—
31#436—22no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8D411yescharacteristic
37C2×C4⋊C8D411yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×#97C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#400C811yescharacteristic
46#400C811yescharacteristic
47#397—21no—
48#394C811yescharacteristic
49#394C811yescharacteristic
50#393—21no—
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53#376C2×C411yescharacteristic
54C4⋊C16—22no—
55C4⋊C16—22no—
56C4⋊C16—22no—
57C4⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16—22no—
60#346—22no—
61#346D414yes—
62C4×C16—22no—
63C4×C16—22no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×Q8M1611yescharacteristic
69C22×D4C2×C811yescharacteristic
70C23×C4—21no—
71C2×M16—42no—
72C2×M16—81no—
73C2×M16—81no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C4×Q8—22no—
81C4×Q8—22no—
82C4×D4—22no—
83C4×D4—22no—
84C2×C4⋊C4C2×C811yescharacteristic
85C2×C4⋊C4C2×C811yescharacteristic
86C2×C4⋊C4C2×C811yescharacteristic
87C2×C4⋊C4M1611yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22×C411yescharacteristic
97C2×C16—24no—
98C2×C16—42no—
99C4⋊C8—22no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—81no—
104C4⋊C8—81no—
105C22⋊C8—41no—
106C22⋊C8—41no—
107C22⋊C8—81no—
108C22⋊C8—81no—
109#97—22no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8C2×D412yes—
115C4×C8#4112yes—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95C2×C811yescharacteristic
122#95C2×C811yescharacteristic
123C24—21no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—41no—
127C2×D4—22no—
128C2×D4—81no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4C22⋊C811yescharacteristic
139C22×C4C22⋊C811yescharacteristic
140C22×C4C22⋊C811yescharacteristic
141C22×C4C22⋊C811yescharacteristic
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4C22×C811yescharacteristic
144C22×C4C2×M1611yescharacteristic
145M16—84no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8#12011yescharacteristic
159C2×C8#12112yes—
160C2×C8#12311yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—81no—
169C22⋊C4—81no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—41no—
175C42—41no—
176C42C4×D412yes—
177C42#13112yes—
178C16—44no—
179C16—82no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23C2×C22⋊C811yescharacteristic
188Q8—44no—
189D4—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4#35012yes—
218C2×C4#38611yescharacteristic
219C2×C4#40811yescharacteristic
220C2×C4C8×D412yes—
221C2×C4#43512yes—
222C8—22no—
223C8—21no—
224C8—21no—
225C8—41no—
226C8—41no—
227C8—81no—
228C8—81no—
229C22—21no—
230C22—21no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22#186911yescharacteristic
239C22C2×#35011yescharacteristic
240C22#211711yescharacteristic
241C4—21no—
242C4—22no—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—81no—
249C4—81no—
250C4—81no—
251C4#191011yescharacteristic
252C4#191211yescharacteristic
253C2—21no—
254C2—21no—
255C2—161no—
256C2#910211yescharacteristic
257C2#1297811yescharacteristic
258C2#1298911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33497C2semidirect
4. #11493C2semidirect
5. #7144C2semidirect

Maximal subgroups

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

subgroupindex
#334972
#222112
#116342
#114932
#71442