Finite Group Databasebeta

← #122679

Subgroups — #122679

Order 512.

379 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C2×#2087C211yesmaximal, characteristic
2#11440C211yesmaximal, characteristic
3#8002C211yesmaximal, characteristic
4#7089C214yesmaximal, complemented
5#2920—21no—
6C22×#107C411yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8C2×#389—21no—
9#2112—21no—
10#2087—21no—
11#2087—21no—
12#2087C412yes—
13#2087C2212yes—
14C2×#340—21no—
15C2×#340C411yescharacteristic
16#1322—21no—
17#1322C2212yes—
18#1292—21no—
19#1292C2212yes—
20C23×C8C2×C411yescharacteristic
21C23×C8D411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—41no—
25C22×C42D411yescharacteristic
26C2×M32—41no—
27#432—22no—
28#432—41no—
29#431—22no—
30#431—41no—
31C2×#107—21no—
32C2×#107—41no—
33C2×#107—41no—
34C2×#107C2×C412yes—
35#423—22no—
36C2×C4⋊C8—41no—
37C2×C4⋊C8—41no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8D411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×C4⋊C8C2311yescharacteristic
42#407—41no—
43C4×M16—41no—
44#389—22no—
45#389—22no—
46#389—41no—
47#389—41no—
48C2×#95—41no—
49C22⋊C16—22no—
50C22⋊C16—41no—
51#340—21no—
52#340—22no—
53#340—22no—
54#340C2×C412yes—
55C23×C4—41no—
56C23×C4—41no—
57C23×C4C22⋊C411yescharacteristic
58C23×C4D811yescharacteristic
59C23×C4SD1611yescharacteristic
60C2×M16—42no—
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—42no—
66C22×C8—42no—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22×C411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C2×C4⋊C4—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—84no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42C22⋊C411yescharacteristic
89C2×C42D811yescharacteristic
90C2×C42SD1611yescharacteristic
91C2×C42C2×D411yescharacteristic
92M32—82no—
93C2×C16—42no—
94#107—42no—
95#107—42no—
96C4⋊C8—24no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104#97—42no—
105C4×C8—42no—
106#95—42no—
107#95—42no—
108C24#10211yescharacteristic
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—82no—
122C22×C4—82no—
123C22×C4—82no—
124C22×C4—82no—
125C22×C4—81no—
126C22×C4—81no—
127C22×C4#9911yescharacteristic
128C22×C4#9911yescharacteristic
129C22×C4#10211yescharacteristic
130C22×C4#10211yescharacteristic
131C22×C4#10211yescharacteristic
132C22×C4#10411yescharacteristic
133C22×C4#10411yescharacteristic
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4C2×D811yescharacteristic
136C22×C4C2×SD1611yescharacteristic
137M16—82no—
138M16—82no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—44no—
151C2×C8—81no—
152C4⋊C4—24no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—84no—
158C4⋊C4—84no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C16—82no—
170C23—22no—
171C23—22no—
172C23—21no—
173C23—21no—
174C23#32711yescharacteristic
175C23#32711yescharacteristic
176C23C2×#10211yescharacteristic
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—44no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—88no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—162no—
205C2×C4—162no—
206C2×C4#32711yescharacteristic
207C2×C4#32711yescharacteristic
208C2×C4#35212yes—
209C2×C4#35612yes—
210C2×C4C2×#9911yescharacteristic
211C2×C4C2×#10411yescharacteristic
212C2×C4#51012yes—
213C8—41no—
214C8—41no—
215C8—41no—
216C8—41no—
217C8—84no—
218C22—24no—
219C22—24no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22#134112yes—
227C22#134612yes—
228C22C2×#32711yescharacteristic
229C22#214111yescharacteristic
230C22#214511yescharacteristic
231C4—21no—
232C4—21no—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—84no—
238C4—162no—
239C4#134612yes—
240C4C2×#35211yescharacteristic
241C4C2×#35611yescharacteristic
242C2—21no—
243C2—22no—
244C2—22no—
245C2—21no—
246C2C2×#134111yescharacteristic
247C2C2×#134611yescharacteristic
248C2#1170611yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7089C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
#114402
#80022
#70892