Finite Group Databasebeta

← #149903

Subgroups — #149903

Order 512.

350 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 56 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#27918C211yesmaximal, characteristic
2#11511C211yesmaximal, characteristic, complemented
3C2×#1324C211yesmaximal, characteristic, complemented
4#7372C212yesmaximal, complemented
5#7352C212yesmaximal, complemented
6C2×#481—21no—
7C2×#480—21no—
8#2856C411yescharacteristic
9C2×#382C2211yescharacteristic
10#2092—21no—
11#2092—21no—
12#1961C411yescharacteristic
13#1961C2211yescharacteristic
14#1847C411yescharacteristic
15#1847C2211yescharacteristic
16#1749C411yescharacteristic
17#1463—21no—
18C2×#324—21no—
19#1324—21no—
20#1324—21no—
21#1324C2212yes—
22#1324C2212yes—
23C2×#127C2×C411yescharacteristic
24C2×#121—21no—
25C2×#121—41no—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C42C2×C411yescharacteristic
29#481—42no—
30#481—42no—
31#480—42no—
32#480—42no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#106—21no—
36C2×#106—21no—
37C2×#102—21no—
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41#407—41no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—41no—
44C2×#97C2×C411yescharacteristic
45C2×#97C2×C411yescharacteristic
46#390—21no—
47#390—41no—
48#384C2×C411yescharacteristic
49#382—41no—
50#382C2×C411yescharacteristic
51#382D411yescharacteristic
52#382D411yescharacteristic
53#382C2311yescharacteristic
54C2×#95—41no—
55#336—21no—
56#324—22no—
57#324—22no—
58#324—41no—
59#324—41no—
60C22×D4—21no—
61C22×D4—41no—
62C22×D4—41no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C2×M16—81no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—81no—
71#127—22no—
72#127—41no—
73#121—41no—
74#121—41no—
75#121—41no—
76#121—41no—
77#121—84no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4C22⋊C411yescharacteristic
87C2×C4⋊C4C22⋊C411yescharacteristic
88C2×C4⋊C4C2×D411yescharacteristic
89C2×C4⋊C4C2×D411yescharacteristic
90C2×C22⋊C4—42no—
91C2×C22⋊C4—41no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—41no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22×C411yescharacteristic
97#107—42no—
98#107—42no—
99#106—42no—
100#106—42no—
101#102—42no—
102#102—42no—
103#102—42no—
104#102—42no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—81no—
110C22⋊C8—81no—
111#97—22no—
112#97—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117C24—41no—
118C24#9911yescharacteristic
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—82no—
128C2×D4—84no—
129C2×D4—81no—
130C2×D4—81no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—81no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4#9911yescharacteristic
153C22×C4C2×C22⋊C411yescharacteristic
154M16—162no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—82no—
162C2×C8—82no—
163C2×C8—162no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—82no—
173C22⋊C4—81no—
174C22⋊C4—81no—
175C22⋊C4—84no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—42no—
180C42#13112yes—
181C23—21no—
182C23—22no—
183C23—21no—
184C23—41no—
185C23—82no—
186C23—82no—
187C23—81no—
188C23—81no—
189C23—81no—
190C23#35311yescharacteristic
191C23#35311yescharacteristic
192C23C2×#9911yescharacteristic
193D4—81no—
194D4—81no—
195D4—81no—
196D4—81no—
197D4—164no—
198D4—162no—
199D4—162no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—161no—
229C2×C4—161no—
230C2×C4#40811yescharacteristic
231C2×C4#41311yescharacteristic
232C8—42no—
233C8—42no—
234C8—162no—
235C8—162no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—42no—
244C22—42no—
245C22—162no—
246C22—162no—
247C22—161no—
248C22—161no—
249C22—161no—
250C22#140612yes—
251C22#146811yescharacteristic
252C22C2×#35311yescharacteristic
253C22#213311yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4—81no—
261C4—81no—
262C4—81no—
263C4—161no—
264C2—21no—
265C2—21no—
266C2—22no—
267C2—41no—
268C2—321no—
269C2#1154911yescharacteristic
270C2C2×#140611yescharacteristic
271C2#1357811yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radical#2856C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11511C2semidirect
3. C2×#1324C2semidirect
4. #7372C2semidirect
5. #7352C2semidirect

Maximal subgroups

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

subgroupindex
#279182
#115112
C2×#13242
#73722
#73522