Finite Group Databasebeta

← #149647

Subgroups — #149647

Order 512.

448 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 82 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×#1857C211yesmaximal, characteristic
2C2×#1319C211yesmaximal, characteristic
3#7595C211yesmaximal, characteristic
4#7376C214yesmaximal, complemented
5C2×#519C411yescharacteristic
6C2×#428—21no—
7C22×#101C411yescharacteristic
8C2×#391C2211yescharacteristic
9#1857—21no—
10#1857—21no—
11#1857C412yes—
12#1857C2212yes—
13#1817—21no—
14C2×#339—21no—
15C2×#324—21no—
16#1319—21no—
17#1319C2212yes—
18#1272—21no—
19#1272C2212yes—
20C23×Q8C2×C411yescharacteristic
21C22×M16—21no—
22#519—21no—
23#519—42no—
24#519—41no—
25#519C2×C412yes—
26C2×C4×Q8—41no—
27C2×C4×Q8—41no—
28C2×#117—21no—
29C2×#117—41no—
30C22×C4⋊C4D411yescharacteristic
31C22×C42D411yescharacteristic
32#428—22no—
33#428—22no—
34#428—42no—
35C2×#107—41no—
36C2×#106—41no—
37C2×#101—21no—
38C2×#101—22no—
39C2×#101C814yescomplemented
40C2×#101C2×C412yes—
41C2×C22⋊C8—41no—
42C2×C22⋊C8—41no—
43#391—41no—
44#391C2×C411yescharacteristic
45#391D411yescharacteristic
46#391D411yescharacteristic
47#391C2311yescharacteristic
48C2×#95—41no—
49#339—22no—
50#339—22no—
51#336—22no—
52#336—41no—
53#324—22no—
54#324—41no—
55C22×Q8—21no—
56C22×Q8—22no—
57C22×Q8—41no—
58C22×Q8—41no—
59C22×Q8—42no—
60C22×Q8C22⋊C411yescharacteristic
61C22×Q8C2×C812yes—
62C22×Q8M1612yes—
63C22×Q8C22×C411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C2×M16—24no—
69C2×M16—22no—
70C2×M16—42no—
71C2×M16—41no—
72C22×C8—41no—
73C22×C8—42no—
74C22×C8—81no—
75C4×Q8—42no—
76C4×Q8—42no—
77C4×Q8—42no—
78C4×Q8—42no—
79C4×Q8—44no—
80C4×Q8—44no—
81#117—22no—
82#117—22no—
83#117—41no—
84#117—42no—
85#117—42no—
86#117—41no—
87#117—42no—
88#117—42no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4C22⋊C411yescharacteristic
97C2×C4⋊C4C2×D411yescharacteristic
98C2×C22⋊C4—41no—
99C2×C42—21no—
100C2×C42—41no—
101C2×C42—42no—
102C2×C42—42no—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C2×D411yescharacteristic
105#107—42no—
106#107—42no—
107#106—42no—
108#106—42no—
109#101—24no—
110#101—22no—
111#101C2×C814yescomplemented
112C22⋊C8—42no—
113C22⋊C8—81no—
114C22⋊C8—82no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119C24#9911yescharacteristic
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—42no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—42no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—41no—
136C2×Q8C22⋊C812yes—
137C2×Q8C22×C811yescharacteristic
138C2×Q8C2×M1611yescharacteristic
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—81no—
160C22×C4—82no—
161C22×C4C22⋊C812yes—
162C22×C4#9911yescharacteristic
163C22×C4#10112yes—
164C22×C4C2×C22⋊C411yescharacteristic
165M16—24no—
166M16—44no—
167M16—42no—
168C2×C8—42no—
169C2×C8—44no—
170C2×C8—82no—
171C2×C8—82no—
172C2×C8—82no—
173C2×C8—82no—
174C2×C8—162no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—81no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C42—41no—
189C42—41no—
190C42—42no—
191C42—42no—
192C42—44no—
193C42—44no—
194C42—81no—
195C23—41no—
196C23—41no—
197C23#32412yes—
198C23AΣL(1,𝔽16)11yescharacteristic
199C23AΣL(1,𝔽16)11yescharacteristic
200C23#35212yes—
201C23C2×#9911yescharacteristic
202Q8—42no—
203Q8—41no—
204Q8—41no—
205Q8—44no—
206Q8—44no—
207Q8—42no—
208Q8—42no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—84no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—161no—
245C2×C4#32412yes—
246C2×C4C2×C22⋊C811yescharacteristic
247C2×C4C2×#10111yescharacteristic
248C8—44no—
249C8—162no—
250C8—162no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22#132612yes—
260C22#132612yes—
261C22C2×#32411yescharacteristic
262C22C2×AΣL(1,𝔽16)11yescharacteristic
263C22C2×#35211yescharacteristic
264C4—21no—
265C4—22no—
266C4—21no—
267C4—42no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—81no—
272C4—81no—
273C4—82no—
274C4—81no—
275C4—82no—
276C2—21no—
277C2—22no—
278C2—22no—
279C2—21no—
280C2C2×#132611yescharacteristic
281C2#1356911yescharacteristic
282C2#1357311yescharacteristic
subgroupG/H
center, socleC22C2×AΣL(1,𝔽16)
commutatorC2×Q8C22×C8
Frattini, cosocle#391C23
prime radicalC24#99
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7376C2semidirect
39. C2×#101C8semidirect
111. #101C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#18572
C2×#13192
#75952
#73762