Finite Group Databasebeta

← #166121

Subgroups — #166121

Order 512.

795 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 795 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2809C211yesmaximal, characteristic
2#27614C212yesmaximal
3#23732C211yesmaximal, characteristic
4#7831C214yesmaximal, complemented
5#7802C214yesmaximal, complemented
6#7607C212yesmaximal
7#7595C211yesmaximal, characteristic
8C2×#428—21no—
9C2×#419—21no—
10#2809—22no—
11#2809—24no—
12#2809C2214yes—
13C2×#400—21no—
14C2×#398—22no—
15C2×#391—21no—
16C2×#391C2211yescharacteristic
17C2×#389C2212yes—
18C2×#387—22no—
19#2285C2211yescharacteristic
20#2100C2214yes—
21#2097—22no—
22#2096C2212yes—
23#2095C2212yes—
24#2083C2214yes—
25#1921—24no—
26#1852—22no—
27#1824—22no—
28#1817C2212yes—
29#1728C2211yescharacteristic
30#1272C2214yes—
31#1271C2218yes—
32C23×Q8D411yescharacteristic
33C22×M16—21no—
34C2×#117—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D412yes—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42C2311yescharacteristic
43#428—44no—
44C2×#107—24no—
45C2×#107—22no—
46C2×#106—22no—
47#419—44no—
48C2×#103—22no—
49C2×#103—22no—
50C2×C22⋊C8—22no—
51#400—24no—
52#400—24no—
53#398—28no—
54#398—28no—
55#391—24no—
56#391—24no—
57#391—22no—
58#391D412yes—
59#391C2312yes—
60#389—22no—
61#389—22no—
62#389D414yes—
63#389C2314yes—
64#387—28no—
65#387—28no—
66#384C2311yescharacteristic
67#384C2311yescharacteristic
68#382—21no—
69#380—22no—
70C4×C4⋊C4—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76#340—24no—
77#340—24no—
78#340—24no—
79#336—22no—
80#336—21no—
81#336C2314yes—
82#336C2312yes—
83C22×Q8—21no—
84C22×Q8—22no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8Q1614yes—
88C22×Q8C2×D412yes—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C2×M16—24no—
96C2×M16—42no—
97C2×M16—42no—
98C22×C8—22no—
99C22×C8—24no—
100#117—42no—
101#117—42no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—48no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D412yes—
122C2×C22⋊C4—41no—
123C2×C42—24no—
124C2×C42—42no—
125C2×C42—41no—
126C2×C42—82no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2411yescharacteristic
130#107—48no—
131#107—44no—
132#106—44no—
133#103—44no—
134#103—44no—
135C22⋊C8—44no—
136#95—24no—
137#95—28no—
138#95—28no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—42no—
144C24#12011yescharacteristic
145C2×Q8—22no—
146C2×Q8—24no—
147C2×Q8—42no—
148C2×Q8—48no—
149C2×Q8—42no—
150C2×Q8—84no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8C2×Q1614yes—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4#12012yes—
177C22×C4#12011yescharacteristic
178C22×C4C2×Q1612yes—
179C22×C4#13712yes—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D412yes—
182M16—48no—
183C2×C8—24no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—44no—
187C2×C8—44no—
188C2×C8—44no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—48no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—44no—
200C4⋊C4—48no—
201C4⋊C4—44no—
202C4⋊C4—81no—
203C4⋊C4—82no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C22⋊C4—82no—
207C42—42no—
208C42—41no—
209C42—41no—
210C42—82no—
211C42—84no—
212C23—22no—
213C23—22no—
214C23#45112yes—
215C23#45312yes—
216C23#45812yes—
217C23C2×#12011yescharacteristic
218Q8—44no—
219Q8—82no—
220Q8—84no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—48no—
234C2×C4—48no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—84no—
246C2×C4—88no—
247C2×C4—88no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4#45112yes—
253C2×C4C22×Q1611yescharacteristic
254C2×C4C2×#13711yescharacteristic
255C2×C4#57514yes—
256C8—44no—
257C8—84no—
258C22—22no—
259C22—24no—
260C22—22no—
261C22—24no—
262C22#161414yes—
263C22C2×#45111yescharacteristic
264C22#300512yes—
265C22C2×#45311yescharacteristic
266C22#302012yes—
267C22C2×#45811yescharacteristic
268C4—21no—
269C4—21no—
270C4—22no—
271C4—24no—
272C4—44no—
273C4—42no—
274C4—82no—
275C4—84no—
276C4—84no—
277C4—161no—
278C4—161no—
279C2—24no—
280C2C2×#161411yescharacteristic
281C2#1813912yes—
282C2#1854711yescharacteristic
283C2#1877912yes—
284C2#1899411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7831C2semidirect
5. #7802C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
#276142
#237322
#78312
#78022
#76072
#75952