Finite Group Databasebeta

← #141161

Subgroups — #141161

Order 512.

366 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 88 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#33215C211yesmaximal, characteristic
2#11724C212yesmaximal
3#11718C212yesmaximal
4#10803C211yesmaximal, characteristic
5#8602C211yesmaximal, characteristic
6C2×#500C411yescharacteristic, complemented
7C2×#405C2211yescharacteristic
8#2628C2212yes—
9#1993C412yescomplemented
10#1944—21no—
11#1942—21no—
12#1846C411yescharacteristic, complemented
13#1809—21no—
14#1808—21no—
15C2×#344—21no—
16#1742C2211yescharacteristic
17#1740C2211yescharacteristic
18#1729—21no—
19#1564—21no—
20#1458C2212yes—
21#1354—24no—
22#1352—24no—
23#1275—21no—
24C22×M16—21no—
25C22×M16D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8C2×C412yes—
28C2×#117—21no—
29C2×#117C2311yescharacteristic
30#500—41no—
31#500—41no—
32#431—22no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#106—21no—
37#405—41no—
38#405—41no—
39C2×#97C2×C411yescharacteristic
40C2×#97D411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#400—22no—
44#399—22no—
45#395—22no—
46#393—22no—
47#389—22no—
48#382C2×C412yes—
49#380—22no—
50#344—42no—
51#343—21no—
52#343D412yes—
53#342—21no—
54#342D412yes—
55#340—22no—
56#340—22no—
57#336—41no—
58#336—41no—
59#335—41no—
60#334—41no—
61#324—22no—
62#323—22no—
63C22×Q8—22no—
64C23×C4C2×D411yescharacteristic
65C2×M16—22no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16SD1612yes—
74C2×M16Q1612yes—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87#117—41no—
88#117—41no—
89#117—41no—
90#117—41no—
91#117—41no—
92#117—41no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4C2×D411yescharacteristic
99C2×C4⋊C4#4111yescharacteristic
100C2×C22⋊C4—22no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4C2×D411yescharacteristic
103C2×C22⋊C4#4111yescharacteristic
104C2×C42—41no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107#107—42no—
108#107—42no—
109#107—41no—
110#107—41no—
111#106—42no—
112C4⋊C8—42no—
113C22⋊C8—42no—
114#97—41no—
115#97—41no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—42no—
121#95—22no—
122#95—22no—
123#95—42no—
124#95—42no—
125C24#12111yescharacteristic
126C2×Q8—82no—
127C2×Q8—162no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C4×D411yescharacteristic
137C22×C4C4×D411yescharacteristic
138C22×C4#12111yescharacteristic
139C22×C4#12411yescharacteristic
140C22×C4#12711yescharacteristic
141C22×C4C2×SD1611yescharacteristic
142C22×C4C2×Q1611yescharacteristic
143C22×C4#13711yescharacteristic
144C22×C4#13711yescharacteristic
145M16—42no—
146M16—42no—
147M16—42no—
148M16—42no—
149M16—81no—
150M16—81no—
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—21no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—42no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8—82no—
168C2×C8#9911yescharacteristic
169C2×C8#9911yescharacteristic
170C2×C8#10312yes—
171C2×C8#10312yes—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—82no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C4⋊C4—162no—
182C22⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—162no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—81no—
192C42—81no—
193C23—21no—
194C23—21no—
195C23—41no—
196C23—41no—
197C23#39011yescharacteristic
198C23#46111yescharacteristic
199C23#46211yescharacteristic
200Q8—164no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4C2×#9911yescharacteristic
229C2×C4#41011yescharacteristic
230C2×C4C2×#10311yescharacteristic
231C2×C4#41911yescharacteristic
232C2×C4#42111yescharacteristic
233C2×C4#42111yescharacteristic
234C2×C4#46111yescharacteristic
235C2×C4#46211yescharacteristic
236C8—21no—
237C8—21no—
238C8—22no—
239C8—22no—
240C8—41no—
241C8—82no—
242C8—162no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—81no—
252C22#196311yescharacteristic
253C22#196411yescharacteristic
254C22#198111yescharacteristic
255C4—21no—
256C4—21no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—162no—
262C4—162no—
263C4—162no—
264C4#151111yescharacteristic
265C4#151511yescharacteristic
266C4#151811yescharacteristic
267C4#152011yescharacteristic
268C2—21no—
269C2—21no—
270C2—41no—
271C2—41no—
272C2#971411yescharacteristic
273C2#1323111yescharacteristic
274C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#500C4semidirect
9. #1993C4semidirect
12. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332152
#117242
#117182
#108032
#86022