Finite Group Databasebeta

← #321593

Subgroups — #321593

Order 512.

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

normal subgroups only

This group has 714 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#28511C211yesmaximal, characteristic
4#21712C211yesmaximal, characteristic
5C2×#1704C214yesmaximal
6#10277C214yesmaximal
7#9692C211yesmaximal, characteristic
8#9691C211yesmaximal, characteristic
9C4×#334C211yesmaximal, characteristic
10C2×#487C2212yes—
11C2×#487C2212yes—
12C2×#477C2214yes—
13C2×#475C2214yes—
14C2×#446C2212yes—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2096—22no—
19#2095—22no—
20#1984C2212yes—
21#1956—22no—
22#1956—22no—
23#1956—21no—
24#1956—21no—
25#1939—22no—
26#1845—21no—
27#1844—21no—
28#1842C2211yescharacteristic
29#1770—22no—
30C4×C4⋊C8C2211yescharacteristic
31C4×C4⋊C8C2211yescharacteristic
32C42×C8C2211yescharacteristic
33#1704—216no—
34C2×#334C2212yes—
35C2×#333C2214yes—
36C2×#332C2214yes—
37#1277C2212yes—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×C4⋊Q8C2312yes—
43C2×C4×Q8C2312yes—
44#487—28no—
45#487—44no—
46#477—28no—
47#477—44no—
48#475—28no—
49#475—44no—
50#446—28no—
51C2×#107—24no—
52C2×#106—24no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#103—24no—
59C2×#103—24no—
60C2×#103—24no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#97—22no—
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8D412yes—
67C2×C4×C8D412yes—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391D412yes—
71#389—22no—
72#384—21no—
73#384—21no—
74#384—21no—
75#384—21no—
76#384—22no—
77#384—22no—
78#384D411yescharacteristic
79C4×C4⋊C4—22no—
80C43C2311yescharacteristic
81#336—24no—
82#336—21no—
83#336—21no—
84#334D418yes—
85#333—28no—
86#332—28no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8C2×D412yes—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8C2×D412yes—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—24no—
99C4⋊Q8—24no—
100C4⋊Q8—24no—
101C4⋊Q8—24no—
102C4⋊Q8—44no—
103C4⋊Q8—44no—
104C4⋊Q8—84no—
105C4×Q8—24no—
106C4×Q8—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C2×D412yes—
119C2×C42—22no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42#4112yes—
124C2×C42C2411yescharacteristic
125#107—48no—
126#106—48no—
127C4⋊C8—24no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—44no—
131C4⋊C8—44no—
132C4⋊C8C2×D414yes—
133C4⋊C8C2×D414yes—
134#103—48no—
135#103—48no—
136#103—48no—
137#103—44no—
138#103—44no—
139#97—44no—
140C4×C8—24no—
141C4×C8—24no—
142C4×C8—22no—
143C4×C8—42no—
144C4×C8—42no—
145C4×C8C2×D414yes—
146#95—24no—
147C2×Q8—24no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—24no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—44no—
154C2×Q8—44no—
155C2×Q8—44no—
156C2×Q8—84no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4#12112yes—
162C22×C4#12112yes—
163C22×C4C22×D411yescharacteristic
164C22×C4C22×D411yescharacteristic
165C22×C4C2×#4111yescharacteristic
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—24no—
170C2×C8—24no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—44no—
179C2×C8#12712yes—
180C2×C8#12712yes—
181C4⋊C4—24no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C4⋊C4—84no—
194C42—22no—
195C42—24no—
196C42—42no—
197C42—42no—
198C42#13514yes—
199C42#13714yes—
200C42#13712yes—
201C42#13712yes—
202C42C22×D412yes—
203C42H(5,𝔽2)12yes—
204C23C2×#12111yescharacteristic
205Q8—44no—
206Q8—44no—
207Q8—82no—
208Q8—82no—
209Q8—84no—
210Q8—164no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—84no—
223C2×C4#46414yes—
224C2×C4#46414yes—
225C2×C4#47014yes—
226C2×C4#47114yes—
227C2×C4C2×#12711yescharacteristic
228C2×C4#53511yescharacteristic
229C2×C4D4212yes—
230C2×C4#55012yes—
231C2×C4C2×#13512yes—
232C2×C4C2×#13712yes—
233C2×C4C2×#13711yescharacteristic
234C2×C4C2×#13711yescharacteristic
235C2×C4#57812yes—
236C8—22no—
237C8—22no—
238C8—42no—
239C8—44no—
240C8—44no—
241C8—82no—
242C22#254111yescharacteristic
243C22C2×#46412yes—
244C22C2×#47011yescharacteristic
245C22#305911yescharacteristic
246C22C2×#47112yes—
247C4—24no—
248C4—22no—
249C4—82no—
250C4—82no—
251C4—162no—
252C4—162no—
253C4#170418yescomplemented
254C4#314711yescharacteristic
255C4#315211yescharacteristic
256C4#321012yes—
257C4#327612yes—
258C4#327612yes—
259C2C2×#170412yes—
260C2#1921512yes—
261C2#2844411yescharacteristic
262C2#2848911yescharacteristic
263C2#2849411yescharacteristic
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
253. C4#1704semidirect

Maximal subgroups

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

subgroupindex
#332252
#285122
#285112
#217122
C2×#17042
#102772
#96922
#96912
C4×#3342