Finite Group Databasebeta

← #101883

Subgroups — #101883

Order 512.

430 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 78 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#28718C211yesmaximal, characteristic
2#23356C211yesmaximal, characteristic
3#7702C212yesmaximal, complemented
4#7637C211yesmaximal, characteristic
5#7023C212yesmaximal, complemented
6C2×#486C412yes—
7#2862C2211yescharacteristic
8C4×#127C2211yescharacteristic
9#2073—21no—
10#2073—21no—
11#1990C412yes—
12#1870—22no—
13#1870—21no—
14#1869—22no—
15#1869—21no—
16#1839C2212yes—
17#1804—21no—
18#1804—21no—
19C4×C4⋊C8C2211yescharacteristic
20#1745C2212yes—
21#1272—22no—
22#1272—22no—
23#1270—21no—
24#1270—21no—
25C22×M16—21no—
26C23×C8—21no—
27C2×C4⋊Q8C2×C412yes—
28C2×#127C2×C411yescharacteristic
29C2×C4×D4—22no—
30C2×C4×D4D411yescharacteristic
31C2×C4×D4D411yescharacteristic
32#486—44no—
33C2×C4⋊C8—21no—
34C2×#102—22no—
35C2×#102—22no—
36C2×C22⋊C8—21no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—21no—
40C2×#97—22no—
41C2×C4×C8C2×C412yes—
42#391—22no—
43#391—22no—
44#390—22no—
45#390D411yescharacteristic
46#390D411yescharacteristic
47#384C2×C411yescharacteristic
48#382—21no—
49#382—21no—
50C43C2311yescharacteristic
51#340—22no—
52#340—22no—
53#336—21no—
54#336—22no—
55#336—22no—
56#336—22no—
57#336—22no—
58#336D412yes—
59#336D412yes—
60C22×Q8—22no—
61C22×D4—22no—
62C22×D4C22⋊C411yescharacteristic
63C22×D4C22⋊C411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C2×M16—82no—
67C2×M16—81no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—42no—
72C22×C8—82no—
73C22×C8—81no—
74C4⋊Q8—42no—
75C4⋊Q8—42no—
76#127—22no—
77#127—41no—
78C4×D4—41no—
79C4×D4—41no—
80C4×D4—41no—
81C4×D4—41no—
82C4×D4—82no—
83C4×D4—82no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4C22⋊C411yescharacteristic
88C2×C4⋊C4C22⋊C411yescharacteristic
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42C22×C411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D412yes—
97C2×C42#4112yes—
98C4⋊C8—41no—
99C4⋊C8—41no—
100#102—44no—
101#102—44no—
102C22⋊C8—82no—
103C22⋊C8—82no—
104C22⋊C8—81no—
105C22⋊C8—81no—
106#97—24no—
107#97—42no—
108C4×C8—42no—
109#95—41no—
110#95—42no—
111#95—41no—
112C24—21no—
113C24—21no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—82no—
117C2×Q8—82no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—44no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—82no—
129C2×D4#10012yes—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4C2×C22⋊C411yescharacteristic
142C22×C4C4×D412yes—
143C22×C4#12011yescharacteristic
144C22×C4#12112yes—
145C22×C4#12311yescharacteristic
146M16—84no—
147C2×C8—44no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8—84no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C22⋊C4—81no—
167C22⋊C4—81no—
168C22⋊C4—82no—
169C22⋊C4—82no—
170C42—21no—
171C42—21no—
172C42—22no—
173C42—22no—
174C42—21no—
175C42—21no—
176C42—41no—
177C42—42no—
178C23—22no—
179C23—42no—
180C23—41no—
181C23—42no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—41no—
188C23#38611yescharacteristic
189Q8—162no—
190Q8—162no—
191D4—42no—
192D4—42no—
193D4—81no—
194D4—84no—
195D4—84no—
196D4—81no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4C2×#10011yescharacteristic
221C2×C4#41311yescharacteristic
222C2×C4#44312yes—
223C2×C4#44412yes—
224C2×C4#45511yescharacteristic
225C2×C4#45511yescharacteristic
226C8—82no—
227C8—82no—
228C8—82no—
229C8—84no—
230C22—44no—
231C22—42no—
232C22—82no—
233C22—81no—
234C22—81no—
235C22—82no—
236C22—82no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#187612yes—
241C22#188511yescharacteristic
242C22#190011yescharacteristic
243C22#190412yes—
244C22#190911yescharacteristic
245C4—22no—
246C4—22no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—41no—
253C4—81no—
254C4—81no—
255C4—162no—
256C2—82no—
257C2—161no—
258C2#906711yescharacteristic
259C2#924312yes—
260C2#927912yes—
261C2#942511yescharacteristic
262C2#945111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7702C2semidirect
5. #7023C2semidirect

Maximal subgroups

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

subgroupindex
#287182
#233562
#77022
#76372
#70232