Finite Group Databasebeta

← #321436

Subgroups — #321436

Order 512.

816 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 226 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#33310C211yesmaximal, characteristic
2#28748C212yesmaximal
3#28687C212yesmaximal
4C4×#494C211yesmaximal, characteristic
5C2×#1716C212yesmaximal
6#10317C212yesmaximal
7#10316C212yesmaximal
8#9841C212yesmaximal
9#8260C211yesmaximal, characteristic
10C2×#498C2212yes—
11C2×#498C2214yes—
12C2×#498C2212yes—
13C2×#494C2211yescharacteristic
14C2×#487C2214yes—
15C2×C4×Q16—22no—
16#2868C2212yes—
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2084—24no—
20#2043—24no—
21#1990C2214yes—
22#1981—24no—
23#1969—22no—
24#1956C2211yescharacteristic
25#1956C2214yes—
26#1956C2211yescharacteristic
27#1948—24no—
28#1845C2211yescharacteristic
29#1844—22no—
30#1844—22no—
31C4×C4⋊C8C2212yes—
32C42×C8C2211yescharacteristic
33#1716—28no—
34C2×#322C2212yes—
35#1273C2212yes—
36C22×Q16—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8C2314yes—
40C2×C4⋊Q8C2311yescharacteristic
41C2×C4⋊Q8C2311yescharacteristic
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44#498—28no—
45#498—24no—
46#498—24no—
47#498—44no—
48#498—44no—
49#494—22no—
50#494D414yes—
51#487—28no—
52#487—44no—
53C4×Q16—48no—
54C2×#107—22no—
55C2×#106—28no—
56C2×#106—24no—
57C2×#106—24no—
58C2×C4⋊C8D414yes—
59C2×#103—28no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—28no—
63C2×C4×C8D412yes—
64C2×C4×C8D412yes—
65C2×C4×C8C2314yes—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#391—21no—
70#389—22no—
71#389—22no—
72#384—22no—
73#384—22no—
74#384—22no—
75#384—22no—
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C43C2311yescharacteristic
81#340—28no—
82#336D414yes—
83#322—24no—
84C22×Q8—21no—
85C22×Q8—21no—
86C22×Q8—24no—
87C2×Q16—28no—
88C2×Q16—44no—
89C2×Q16—44no—
90C22×C8C2×D418yes—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93C4⋊Q8—24no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—24no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—24no—
100C4⋊Q8—24no—
101C4⋊Q8—48no—
102C4⋊Q8—84no—
103C4⋊Q8C2×D412yes—
104C4⋊Q8C2×D412yes—
105C4×Q8—44no—
106C4×Q8—44no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D412yes—
127C2×C42C2411yescharacteristic
128#107—44no—
129#106—416no—
130#106—48no—
131#106—48no—
132C4⋊C8—28no—
133#103—416no—
134#103—44no—
135#103—44no—
136#103—88no—
137C4×C8—24no—
138C4×C8—24no—
139C4×C8—24no—
140C4×C8—22no—
141C4×C8SD1618yes—
142C4×C8C2×D412yes—
143C4×C8#4112yes—
144#95—42no—
145#95—42no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—48no—
157C2×Q8—48no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—44no—
163C22×C4#12711yescharacteristic
164C22×C4#12712yes—
165C22×C4#12711yescharacteristic
166C22×C4C22×D411yescharacteristic
167C22×C4C22×D412yes—
168Q16—416no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—28no—
172C2×C8—28no—
173C2×C8—28no—
174C2×C8#12114yes—
175C2×C8C2×SD1614yes—
176C2×C8C2×SD1614yes—
177C2×C8#13714yes—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—84no—
195C4⋊C4—84no—
196C42—22no—
197C42—24no—
198C42—82no—
199C42—82no—
200C42C2×SD1614yes—
201C42#13718yes—
202C42C22×D411yescharacteristic
203C42C2×#4111yescharacteristic
204C42H(5,𝔽2)12yes—
205C23C2×#12711yescharacteristic
206Q8—44no—
207Q8—44no—
208Q8—82no—
209Q8—82no—
210Q8—88no—
211Q8—88no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—84no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—84no—
227C2×C4—84no—
228C2×C4#49212yes—
229C2×C4#49212yes—
230C2×C4#49714yes—
231C2×C4#49714yes—
232C2×C4C2×#12111yescharacteristic
233C2×C4#53511yescharacteristic
234C2×C4D4212yes—
235C2×C4#55012yes—
236C2×C4C22×SD1612yes—
237C2×C4C2×#13714yes—
238C2×C4C2×#13712yes—
239C2×C4#57714yes—
240C8—24no—
241C8—48no—
242C8#46118yes—
243C22#254111yescharacteristic
244C22C2×#49211yescharacteristic
245C22C2×#49712yes—
246C22#315211yescharacteristic
247C22#315412yes—
248C4—22no—
249C4—24no—
250C4—82no—
251C4—82no—
252C4—164no—
253C4—161no—
254C4—161no—
255C4#171614yescomplemented
256C4C2×#46112yes—
257C4#320612yes—
258C4#327814yes—
259C4#340014yes—
260C2C2×#171611yescharacteristic
261C2#1965411yescharacteristic
262C2#1968012yes—
263C2#2843811yescharacteristic
264C2#2847312yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
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
255. C4#1716semidirect

Maximal subgroups

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

subgroupindex
#333102
#287482
#286872
C4×#4942
C2×#17162
#103172
#103162
#98412
#82602