Finite Group Databasebeta

← #134641

Subgroups — #134641

Order 512.

374 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 92 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#21694C211yesmaximal, characteristic
2#12884C211yesmaximal, characteristic, complemented
3#11926C212yesmaximal
4#8780C211yesmaximal, characteristic, complemented
5#7278C212yesmaximal, complemented
6C2×#495C2211yescharacteristic
7C2×#442C411yescharacteristic
8C2×#441C411yescharacteristic
9C2×C4×SD16—21no—
10#2326—21no—
11#2150C2212yes—
12#1972C411yescharacteristic
13#1971C411yescharacteristic
14#1968—21no—
15#1959—21no—
16#1844C2211yescharacteristic
17#1726C2211yescharacteristic
18#1564C2212yes—
19#1376—21no—
20#1376—21no—
21#1360—24no—
22#1281—21no—
23#1274—21no—
24C22×Q16C2×C411yescharacteristic
25C22×SD16—21no—
26C22×D8C2×C411yescharacteristic
27C2×#124—21no—
28C2×C4×Q8—21no—
29C2×C4×D4—21no—
30C2×M32—22no—
31#495—21no—
32#495—21no—
33#495D412yes—
34#495Q812yes—
35#442—42no—
36#441—42no—
37C4×SD16—44no—
38#431—22no—
39C2×#106—21no—
40C2×#106C2×C411yescharacteristic
41C2×#106C2×C411yescharacteristic
42C2×#103—21no—
43C2×#103—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×#97C2×C411yescharacteristic
47C2×#97C2×C411yescharacteristic
48C2×#97D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8C2311yescharacteristic
52#391—21no—
53#390—21no—
54#388—42no—
55#385—42no—
56#384—21no—
57#376D411yescharacteristic
58C4⋊C16—22no—
59C4⋊C16—22no—
60#335—22no—
61#334—22no—
62C22×Q8—21no—
63C22×D4—21no—
64C23×C4—41no—
65C2×Q16—21no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16C4⋊C412yes—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—42no—
72C2×SD16—42no—
73C2×D8—21no—
74C2×D8—41no—
75C2×D8—41no—
76C2×D8C4⋊C412yes—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#124—22no—
82#124—22no—
83#124—42no—
84C4×Q8—42no—
85C4×Q8—42no—
86C4×D4—42no—
87C4×D4—42no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—42no—
95C2×C42—41no—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22⋊C411yescharacteristic
98C2×C42C2×D411yescharacteristic
99M32—42no—
100M32—42no—
101C2×C16—42no—
102#106—42no—
103#106—41no—
104#106—41no—
105C4⋊C8—42no—
106#103—42no—
107#103—42no—
108#102—42no—
109#102—42no—
110C22⋊C8—42no—
111#97—22no—
112#97—22no—
113#97—41no—
114C4×C8—22no—
115C4×C8—22no—
116C4×C8—42no—
117C4×C8C2×D411yescharacteristic
118C4×C8C2×Q811yescharacteristic
119C4×C8#4111yescharacteristic
120C4×C8#4111yescharacteristic
121#95—22no—
122#95—42no—
123C24—41no—
124C2×Q8—22no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×D4—22no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—82no—
142C22×C4—162no—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4AGL(1,ℤ/8)12yes—
145Q16—42no—
146Q16—42no—
147SD16—44no—
148SD16—44no—
149D8—42no—
150D8—42no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—82no—
160C2×C8C2×C4⋊C411yescharacteristic
161C2×C8#11711yescharacteristic
162C2×C8#12111yescharacteristic
163C2×C8C22⋊Q811yescharacteristic
164C2×C8#13512yes—
165C2×C8AGL(1,ℤ/8)12yes—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—82no—
171C4⋊C4—162no—
172C22⋊C4—44no—
173C22⋊C4—44no—
174C22⋊C4—82no—
175C22⋊C4—162no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—82no—
180C42#10012yes—
181C42#12111yescharacteristic
182C42C22⋊Q811yescharacteristic
183C42#13512yes—
184C16—82no—
185C23—42no—
186C23—82no—
187C23—82no—
188C23—81no—
189C23—81no—
190C23#41811yescharacteristic
191Q8—42no—
192Q8—42no—
193Q8—81no—
194Q8—81no—
195D4—42no—
196D4—42no—
197D4—81no—
198D4—81no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—164no—
212C2×C4#38011yescharacteristic
213C2×C4C2×#10011yescharacteristic
214C2×C4#41311yescharacteristic
215C2×C4#41611yescharacteristic
216C2×C4#41611yescharacteristic
217C2×C4#41811yescharacteristic
218C2×C4#46312yes—
219C2×C4#47812yes—
220C8—21no—
221C8—21no—
222C8—41no—
223C8—41no—
224C8—162no—
225C8#46312yes—
226C8#47812yes—
227C22—22no—
228C22—84no—
229C22—82no—
230C22—161no—
231C22—161no—
232C22—162no—
233C22#155311yescharacteristic
234C22#155511yescharacteristic
235C22#180411yescharacteristic
236C4—22no—
237C4—22no—
238C4—42no—
239C4—81no—
240C4—82no—
241C4—161no—
242C4—161no—
243C4#154311yescharacteristic
244C4#155311yescharacteristic
245C4#180011yescharacteristic
246C4#180411yescharacteristic
247C2—21no—
248C2—21no—
249C2—162no—
250C2#859211yescharacteristic
251C2#1282812yes—
subgroupG/H
center, socleC22#1804
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12884C2semidirect
4. #8780C2semidirect
5. #7278C2semidirect

Maximal subgroups

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

subgroupindex
#216942
#128842
#119262
#87802
#72782