Finite Group Databasebeta

← #149688

Subgroups — #149688

Order 512.

363 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 70 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#21702C211yesmaximal, characteristic
2#11477C211yesmaximal, characteristic
3#11476C211yesmaximal, characteristic
4#7372C212yesmaximal, complemented
5#7353C212yesmaximal, complemented
6C2×#497C411yescharacteristic
7C2×#441—21no—
8C2×#440—21no—
9C4×C4⋊Q8C411yescharacteristic
10C4×#124C2211yescharacteristic
11#1969—21no—
12#1968—21no—
13#1960C2211yescharacteristic
14#1959C411yescharacteristic
15#1847C2211yescharacteristic
16C4×#97C411yescharacteristic
17#1461—21no—
18#1459—21no—
19#1324C2212yes—
20#1324C2212yes—
21#1319—21no—
22#1318—21no—
23C22×Q16—21no—
24C22×SD16—21no—
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×#124C2×C411yescharacteristic
27C2×C4×Q8—21no—
28C2×C4×Q8D411yescharacteristic
29C2×C4×D4D411yescharacteristic
30#497—22no—
31#497C814yescomplemented
32#441—42no—
33#441—42no—
34#440—42no—
35#440—42no—
36#431—41no—
37C2×#107—21no—
38C2×#106—21no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#103—21no—
42C2×#102—21no—
43C2×C22⋊C8—41no—
44C2×#97—21no—
45C2×#97C2×C411yescharacteristic
46C2×#97C2×C411yescharacteristic
47C2×C4×C8—21no—
48#391—21no—
49#391D411yescharacteristic
50#390D411yescharacteristic
51#389—41no—
52#388—41no—
53#384C2×C411yescharacteristic
54#382C2311yescharacteristic
55C4×C4⋊C4—41no—
56C4×C22⋊C4—41no—
57C43C2×C411yescharacteristic
58#324—22no—
59#324—22no—
60#324—41no—
61#323—41no—
62C22×Q8—21no—
63C22×Q8C22⋊C411yescharacteristic
64C22×D4C22⋊C411yescharacteristic
65C23×C4—21no—
66C2×Q16—24no—
67C2×Q16—42no—
68C2×Q16—42no—
69C2×SD16—24no—
70C2×SD16—42no—
71C2×SD16—42no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—81no—
75C4⋊Q8—21no—
76C4⋊Q8—41no—
77C4⋊Q8C2×C812yescomplemented
78#124—41no—
79#124C2×C812yes—
80#124M1612yes—
81C4×Q8—41no—
82C4×Q8—41no—
83C4×Q8—41no—
84C4×Q8—41no—
85C4×Q8—41no—
86C4×Q8—41no—
87C4×D4—41no—
88C4×D4—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4C2×D411yescharacteristic
95C2×C4⋊C4C2×D411yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22×C411yescharacteristic
106#107—42no—
107#106—42no—
108C4⋊C8—81no—
109#103—42no—
110#103—42no—
111#103—42no—
112#102—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—81no—
116#97—22no—
117#97—22no—
118#97—21no—
119#97—41no—
120#97C2×C812yescomplemented
121C4×C8—42no—
122C4×C8—81no—
123#95—41no—
124#95—41no—
125#95—42no—
126C24—21no—
127C2×Q8—21no—
128C2×Q8—22no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8C22⋊C812yes—
137C2×D4—21no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4C22⋊C812yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—82no—
151C22×C4#9911yescharacteristic
152C22×C4#9911yescharacteristic
153C22×C4C2×C22⋊C411yescharacteristic
154Q16—48no—
155SD16—48no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—82no—
163C2×C8—82no—
164C2×C8—162no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—81no—
172C4⋊C4—82no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—81no—
178C22⋊C4—82no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—41no—
183C42—42no—
184C42—41no—
185C42—41no—
186C42—81no—
187C42—81no—
188C42—81no—
189C42C22×C811yescharacteristic
190C42C2×M1611yescharacteristic
191C23—22no—
192C23—41no—
193C23—41no—
194C23—81no—
195C23—81no—
196C23C2×#9911yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—81no—
201Q8—81no—
202Q8—81no—
203D4—42no—
204D4—81no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—161no—
231C2×C4#35311yescharacteristic
232C2×C4#35311yescharacteristic
233C2×C4C2×C22⋊C811yescharacteristic
234C2×C4#41311yescharacteristic
235C8—24no—
236C8—42no—
237C8—162no—
238C8—162no—
239C22—21no—
240C22—21no—
241C22—42no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22—161no—
246C22#145911yescharacteristic
247C22C2×#35311yescharacteristic
248C22#213211yescharacteristic
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—41no—
254C4—41no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4—81no—
259C4—81no—
260C4—82no—
261C4—161no—
262C2—21no—
263C2—21no—
264C2—82no—
265C2#1147511yescharacteristic
266C2#1357811yescharacteristic
267C2#1358011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC42C22×C8
Frattini, cosocle#382C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7372C2semidirect
5. #7353C2semidirect
31. #497C8semidirect
77. C4⋊Q8C2×C8semidirect
120. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#217022
#114772
#114762
#73722
#73532