Finite Group Databasebeta

← #233157

Subgroups — #233157

Order 512.

1342 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 274 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×#3209C212yesmaximal
2C22×#500C211yesmaximal, characteristic
3#21300C211yesmaximal, characteristic
4C2×#1549C212yesmaximal
5#10265C218yesmaximal, complemented
6#7872C211yesmaximal, characteristic
7C2×#533C2211yescharacteristic
8C22×C4⋊Q8C2212yes—
9#3209—28no—
10#3209—28no—
11C2×#500—22no—
12C2×#500—28no—
13C2×#500C2214yes—
14C2×#494—22no—
15C2×#487—22no—
16C2×#462C2218yes—
17C2×#451—24no—
18C2×#431C2212yes—
19C22×#107—22no—
20C2×#423C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#1984—22no—
23#1984C2214yes—
24#1729—22no—
25#1549—28no—
26C2×#332C2218yes—
27#1277C2214yes—
28C23×Q8—22no—
29C22×Q16—28no—
30C22×M16—21no—
31C23×C8D412yes—
32#533—24no—
33#533—24no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8D411yescharacteristic
41C2×C4⋊Q8D418yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8C2314yes—
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8D412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8C2312yes—
49C2×#117—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C42C2311yescharacteristic
53#500—216no—
54#500—216no—
55#494—216no—
56#487—216no—
57#462—216no—
58#462—216no—
59#451—232no—
60#431—24no—
61#431D418yes—
62C2×#107—216no—
63C2×#107—28no—
64#423—24no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2314yes—
68C2×#103—28no—
69C2×C22⋊C8—22no—
70C2×C4×C8—24no—
71C2×C4×C8—22no—
72C2×C4×C8D412yes—
73C2×C4×C8C2312yes—
74#336—28no—
75#332—216no—
76C22×Q8—21no—
77C22×Q8—28no—
78C22×Q8—24no—
79C22×Q8—44no—
80C22×Q8—42no—
81C22×Q8—88no—
82C22×Q8C2×D412yes—
83C23×C4—42no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4#4111yescharacteristic
87C2×Q16—232no—
88C2×Q16—816no—
89C2×M16—42no—
90C2×M16—44no—
91C22×C8—24no—
92C22×C8—24no—
93C22×C8—24no—
94C22×C8—22no—
95C22×C8Q1618yescomplemented
96C22×C8C2×D414yes—
97C4⋊Q8—28no—
98C4⋊Q8—24no—
99C4⋊Q8—28no—
100C4⋊Q8—24no—
101C4⋊Q8—28no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—44no—
106C4⋊Q8—44no—
107C4⋊Q8—44no—
108C4⋊Q8C2×D418yes—
109#125—44no—
110C22⋊Q8—28no—
111C22⋊Q8—44no—
112C4×Q8—24no—
113C4×Q8—24no—
114C4×Q8—42no—
115C4×Q8—42no—
116#117—42no—
117#117—42no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4C2×D412yes—
132C2×C22⋊C4—22no—
133C2×C42—22no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D414yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D414yes—
138C2×C42#4111yescharacteristic
139C2×C42C2411yescharacteristic
140#107—232no—
141C4⋊C8—24no—
142C4⋊C8—24no—
143C4⋊C8—42no—
144C4⋊C8C2×D418yes—
145#103—232no—
146C22⋊C8—28no—
147C4×C8—24no—
148C4×C8—28no—
149C4×C8—24no—
150C24#12111yescharacteristic
151C2×Q8—28no—
152C2×Q8—24no—
153C2×Q8—28no—
154C2×Q8—44no—
155C2×Q8—42no—
156C2×Q8—48no—
157C2×Q8—44no—
158C2×Q8—416no—
159C2×Q8—816no—
160C2×Q8—88no—
161C2×Q8—816no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—44no—
167C22×C4—48no—
168C22×C4—88no—
169C22×C4#12014yes—
170C22×C4#12012yes—
171C22×C4#12114yes—
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12712yes—
176C22×C4C2×Q1614yes—
177C22×C4#13714yes—
178C22×C4C22×D411yescharacteristic
179C22×C4C22×D411yescharacteristic
180C22×C4C2×#4111yescharacteristic
181Q16—832no—
182M16—84no—
183C2×C8—24no—
184C2×C8—24no—
185C2×C8—216no—
186C2×C8—28no—
187C2×C8—48no—
188C2×C8—42no—
189C2×C8—44no—
190C2×C8C2×Q1618yes—
191C4⋊C4—24no—
192C4⋊C4—28no—
193C4⋊C4—28no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—48no—
198C4⋊C4—48no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C22⋊C4—44no—
204C42—42no—
205C42—42no—
206C42#12012yes—
207C42#12014yes—
208C42#12114yes—
209C42#12712yes—
210C42C22×D412yes—
211C42C22×D412yes—
212C23—22no—
213C23—22no—
214C23#39212yes—
215C23#46214yes—
216C23C2×#12111yescharacteristic
217Q8—48no—
218Q8—416no—
219Q8—816no—
220Q8—88no—
221Q8—84no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—28no—
225C2×C4—28no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—48no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—816no—
232C2×C4—88no—
233C2×C4#39212yes—
234C2×C4#39212yes—
235C2×C4#39212yes—
236C2×C4#45118yes—
237C2×C4#46214yes—
238C2×C4#47314yes—
239C2×C4#49414yes—
240C2×C4#49714yes—
241C2×C4C2×#12011yescharacteristic
242C2×C4C2×#12012yes—
243C2×C4C2×#12112yes—
244C2×C4C2×#12711yescharacteristic
245C2×C4C22×Q1612yes—
246C2×C4C2×#13712yes—
247C2×C4#57614yes—
248C8—28no—
249C8—48no—
250C8—82no—
251C22—28no—
252C22—22no—
253C22—22no—
254C22#201614yes—
255C22C2×#39211yescharacteristic
256C22C2×#46212yes—
257C22#305812yes—
258C22C2×#47312yes—
259C4—22no—
260C4—22no—
261C4—82no—
262C4—88no—
263C4—82no—
264C4#201614yes—
265C4#202014yes—
266C4C2×#45112yes—
267C4#300814yes—
268C4C2×#49411yescharacteristic
269C4C2×#49711yescharacteristic
270C2—24no—
271C2#1920814yes—
272C2#2338711yescharacteristic
273C2C2×#201611yescharacteristic
274C2C2×#202011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10265C2semidirect
95. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C22×#5002
#213002
C2×#15492
#102652
#78722