Finite Group Databasebeta

← #173940

Subgroups — #173940

Order 512.

1035 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1035 conjugacy classes of subgroups — too many to draw. Only its 176 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#59822C211yesmaximal, characteristic
2#29139C212yesmaximal, complemented
3#28440C211yesmaximal, characteristic, complemented
4#7758C218yesmaximal, complemented
5#7571C211yesmaximal, characteristic, complemented
6#7569C212yesmaximal, complemented
7C2×#555—21no—
8C2×#494—21no—
9C2×#462—22no—
10#2868C2211yescharacteristic
11#2733C2214yes—
12#2548—22no—
13#2541—22no—
14#2537—21no—
15#2414—24no—
16#2354—22no—
17C4×C4⋊Q8C2211yescharacteristic
18C4×#127C2212yes—
19#2023C2218yes—
20#2016C2214yes—
21#1957—22no—
22#1956—21no—
23#1834—22no—
24#1802—24no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2212yes—
27#1272C2214yes—
28#1270C2218yes—
29C22×Q16—24no—
30#555—44no—
31#555—44no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8C2312yes—
36C2×#127—22no—
37C2×#127D412yes—
38C2×#124—24no—
39C2×#121—24no—
40C2×#121—24no—
41C2×C4×Q8D412yes—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4D414yes—
45C2×#117—24no—
46#494—44no—
47#462—48no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—22no—
50C2×#103—24no—
51C2×#102—28no—
52C2×C4×C8—22no—
53C2×C4×C8—21no—
54#394—24no—
55#394—24no—
56#392—28no—
57#391C2312yes—
58#390—24no—
59#390—24no—
60#390—22no—
61#390—22no—
62#390C2314yes—
63#389—22no—
64#388—28no—
65#384—22no—
66#384—22no—
67#384D412yes—
68#384D411yescharacteristic
69#380—24no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C43C2311yescharacteristic
73#340—28no—
74#336C2312yes—
75#336C2314yes—
76C22×Q8—22no—
77C22×Q8C2×D412yes—
78C22×D4—24no—
79C22×D4—24no—
80C22×D4—22no—
81C22×D4—22no—
82C22×D4C2×D414yes—
83C23×C4—24no—
84C2×Q16—48no—
85C2×Q16—48no—
86C2×Q16—84no—
87C22×C8—22no—
88C22×C8—24no—
89C4⋊Q8—22no—
90C4⋊Q8—44no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—82no—
94C4⋊Q8D814yes—
95#127—24no—
96#127—24no—
97#127—84no—
98#124—48no—
99#124—48no—
100#121—416no—
101#121—88no—
102C4×Q8—44no—
103C4×D4—48no—
104C4×D4—44no—
105C4×D4—44no—
106C4×D4—44no—
107C4×D4—44no—
108#117—416no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D412yes—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—24no—
125C2×C42—24no—
126C2×C42—22no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42C2411yescharacteristic
132C4⋊C8—24no—
133C4⋊C8—44no—
134#103—48no—
135#102—88no—
136C4×C8—42no—
137C4×C8—44no—
138#95—24no—
139#95—24no—
140C24—24no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—44no—
145C2×Q8—44no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8C2×D814yes—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—44no—
152C2×D4—44no—
153C2×D4—48no—
154C2×D4—44no—
155C2×D4—44no—
156C2×D4—44no—
157C2×D4—44no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—88no—
161C2×D4—84no—
162C2×D4—84no—
163C2×D4—84no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—48no—
168C22×C4—48no—
169C22×C4—48no—
170C22×C4—42no—
171C22×C4#12011yescharacteristic
172C22×C4#12012yes—
173C22×C4#12011yescharacteristic
174C22×C4C22×D411yescharacteristic
175C22×C4C22×D412yes—
176Q16—816no—
177C2×C8—24no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—44no—
183C4⋊C4—28no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C4⋊C4—84no—
193C4⋊C4—82no—
194C22⋊C4—416no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—84no—
200C22⋊C4—88no—
201C42—22no—
202C42—24no—
203C42—41no—
204C42—48no—
205C42—44no—
206C42C2×D812yes—
207C42#13514yes—
208C42AGL(1,ℤ/8)12yes—
209C42#13714yes—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23C2×#12011yescharacteristic
218Q8—44no—
219Q8—82no—
220Q8—84no—
221Q8—84no—
222D4—48no—
223D4—84no—
224D4—88no—
225D4—88no—
226D4—168no—
227D4—168no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—816no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4#44714yes—
247C2×C4#45214yes—
248C2×C4#45214yes—
249C2×C4#45312yes—
250C2×C4#45514yes—
251C2×C4C22×D811yescharacteristic
252C2×C4C2×#13512yes—
253C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
254C2×C4C2×#13712yes—
255C8—44no—
256C8—84no—
257C22—84no—
258C22—84no—
259C22—84no—
260C22—84no—
261C22—84no—
262C22—84no—
263C22—84no—
264C22C2×#44711yescharacteristic
265C22C2×#45212yes—
266C22C2×#45311yescharacteristic
267C22C2×#45512yes—
268C22#302711yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—44no—
275C4—84no—
276C4—82no—
277C4—162no—
278C4#163414yes—
279C4#164118yes—
280C2—164no—
281C2#1838711yescharacteristic
282C2#1840612yes—
283C2C2×#163411yescharacteristic
284C2C2×#164112yes—
285C2#1907411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#59822C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29139C2semidirect
3. #28440C2semidirect
4. #7758C2semidirect
5. #7571C2semidirect
6. #7569C2semidirect

Maximal subgroups

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

subgroupindex
#598222
#291392
#284402
#77582
#75712
#75692