Finite Group Databasebeta

← #321594

Subgroups — #321594

Order 512.

756 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 198 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#33225C211yesmaximal, characteristic
2#28512C211yesmaximal, characteristic
3#28440C211yesmaximal, characteristic
4#21712C211yesmaximal, characteristic
5C2×#1700C214yesmaximal
6#10272C214yesmaximal
7#9694C211yesmaximal, characteristic
8#9692C211yesmaximal, characteristic
9C4×#335C211yesmaximal, characteristic
10C2×#494—21no—
11C2×#487C2212yes—
12C2×#487C2212yes—
13C2×#475C2214yes—
14C2×#462C2214yes—
15C2×#446C2212yes—
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2096—22no—
20#2016—22no—
21#1984C2212yes—
22#1956—22no—
23#1956—22no—
24#1956—21no—
25#1956—21no—
26#1939—22no—
27#1844—21no—
28#1842C2211yescharacteristic
29#1770—22no—
30C4×C4⋊C8C2211yescharacteristic
31C4×C4⋊C8C2211yescharacteristic
32C42×C8C2211yescharacteristic
33#1700—216no—
34C2×#335C2212yes—
35C2×#332C2214yes—
36C2×#332C2214yes—
37#1277C2212yes—
38C22×Q16—24no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8C2312yes—
44C2×C4⋊Q8C2312yes—
45C2×C4×Q8C2312yes—
46#494—44no—
47#487—28no—
48#487—44no—
49#475—28no—
50#475—44no—
51#462—28no—
52#462—44no—
53#446—28no—
54C2×#106—24no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—24no—
65C2×#97—22no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8D412yes—
69C2×C4×C8D412yes—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391D412yes—
73#389—22no—
74#384—21no—
75#384—21no—
76#384—22no—
77#384—22no—
78#384—22no—
79#384D411yescharacteristic
80C4×C4⋊C4—22no—
81C43C2311yescharacteristic
82#336—24no—
83#336—21no—
84#336—21no—
85#335D418yes—
86#332—28no—
87#332—28no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8C2×D412yes—
91C2×Q16—48no—
92C2×Q16—48no—
93C2×Q16—84no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8C2×D412yes—
98C22×C8C2×D411yescharacteristic
99C22×C8C2×D411yescharacteristic
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—24no—
103C4⋊Q8—24no—
104C4⋊Q8—24no—
105C4⋊Q8—24no—
106C4⋊Q8—44no—
107C4⋊Q8—42no—
108C4⋊Q8—44no—
109C4⋊Q8—42no—
110C4⋊Q8—82no—
111C4×Q8—24no—
112C4×Q8—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4C2×D412yes—
125C2×C42—22no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42#4112yes—
130C2×C42C2411yescharacteristic
131#106—48no—
132C4⋊C8—24no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—44no—
136C4⋊C8—44no—
137C4⋊C8C2×D414yes—
138C4⋊C8C2×D414yes—
139#103—48no—
140#103—48no—
141#103—48no—
142#103—44no—
143#103—44no—
144#97—44no—
145C4×C8—24no—
146C4×C8—24no—
147C4×C8—22no—
148C4×C8—42no—
149C4×C8—42no—
150C4×C8C2×D414yes—
151#95—24no—
152C2×Q8—24no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—24no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—42no—
161C2×Q8—44no—
162C2×Q8—42no—
163C2×Q8—44no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4#12112yes—
169C22×C4#12112yes—
170C22×C4C22×D411yescharacteristic
171C22×C4C22×D411yescharacteristic
172C22×C4C2×#4111yescharacteristic
173Q16—816no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—24no—
178C2×C8—24no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8#12712yes—
188C2×C8#12712yes—
189C4⋊C4—24no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—84no—
199C4⋊C4—84no—
200C4⋊C4—84no—
201C4⋊C4—84no—
202C42—22no—
203C42—24no—
204C42—42no—
205C42—42no—
206C42#13514yes—
207C42AGL(1,ℤ/8)12yes—
208C42#13714yes—
209C42#13712yes—
210C42C22×D412yes—
211C42H(5,𝔽2)12yes—
212C23C2×#12111yescharacteristic
213Q8—44no—
214Q8—44no—
215Q8—82no—
216Q8—82no—
217Q8—84no—
218Q8—84no—
219Q8—84no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—84no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4#46414yes—
234C2×C4#46414yes—
235C2×C4#46814yes—
236C2×C4#47314yes—
237C2×C4C2×#12711yescharacteristic
238C2×C4#53511yescharacteristic
239C2×C4D4212yes—
240C2×C4#55012yes—
241C2×C4C2×#13512yes—
242C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
243C2×C4C2×#13712yes—
244C2×C4C2×#13711yescharacteristic
245C2×C4#57712yes—
246C8—22no—
247C8—22no—
248C8—42no—
249C8—44no—
250C8—44no—
251C8—82no—
252C22#254111yescharacteristic
253C22C2×#46412yes—
254C22C2×#46811yescharacteristic
255C22#305711yescharacteristic
256C22C2×#47312yes—
257C4—24no—
258C4—22no—
259C4—82no—
260C4—82no—
261C4—162no—
262C4—162no—
263C4#170018yescomplemented
264C4#314711yescharacteristic
265C4#315211yescharacteristic
266C4#321012yes—
267C4#327212yes—
268C4#327812yes—
269C2C2×#170012yes—
270C2#1921312yes—
271C2#2844411yescharacteristic
272C2#2849011yescharacteristic
273C2#2849811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
263. C4#1700semidirect

Maximal subgroups

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

subgroupindex
#332252
#285122
#284402
#217122
C2×#17002
#102722
#96942
#96922
C4×#3352