Finite Group Databasebeta

← #302853

Subgroups — #302853

Order 512.

1129 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1129 conjugacy classes of subgroups — too many to draw. Only its 208 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#59817C211yesmaximal, characteristic
2#28689C212yesmaximal, complemented
3#28508C211yesmaximal, characteristic
4#10573C214yesmaximal, complemented
5#9865C212yesmaximal, complemented
6#8920C214yesmaximal
7C4×#331C211yesmaximal, characteristic
8C2×D4×Q8—22no—
9C2×#500—22no—
10C2×#474—22no—
11#2868C2211yescharacteristic
12#2683—22no—
13#2551C2212yes—
14#2538—22no—
15#2452—24no—
16#2409—22no—
17#2366C2212yes—
18C4×C4⋊Q8C2212yes—
19C4×#127C2211yescharacteristic
20#2093—22no—
21#2083—24no—
22#1990—22no—
23#1957C2212yes—
24#1947C2218yes—
25#1845C2212yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#331C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C23×Q8—22no—
32D4×Q8—48no—
33D4×Q8—48no—
34C2×C4⋊Q8—24no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8D412yes—
38C2×#127C2311yescharacteristic
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—28no—
41C2×C4×Q8—22no—
42C2×C4×D4—24no—
43C2×C4×D4—22no—
44C2×C4×D4D412yes—
45#500—48no—
46#474—48no—
47C2×#107—28no—
48C2×#107—24no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8C2314yes—
51C2×#102—24no—
52C2×C4×C8—22no—
53C2×C4×C8C2312yes—
54#396—28no—
55#395—24no—
56#393—24no—
57#391—22no—
58#390—24no—
59#390—22no—
60#390D412yes—
61#389—24no—
62#389—22no—
63#389D414yes—
64#386—24no—
65#385—24no—
66#384—22no—
67#384—21no—
68#384D412yes—
69#384C2312yes—
70#384C2311yescharacteristic
71#380—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4C2314yes—
74C43C2311yescharacteristic
75#340—28no—
76#336—22no—
77#336—21no—
78#331Q818yescomplemented
79C22×Q8—24no—
80C22×Q8—22no—
81C22×Q8—416no—
82C22×Q8—41no—
83C22×Q8—84no—
84C22×D4—22no—
85C22×D4C2×D412yes—
86C23×C4—24no—
87C23×C4—22no—
88C22×C8—24no—
89C22×C8—22no—
90C4⋊Q8—24no—
91C4⋊Q8—44no—
92C4⋊Q8—48no—
93C4⋊Q8—48no—
94C4⋊Q8—44no—
95C4⋊Q8—44no—
96C4⋊Q8D818yes—
97#127—22no—
98#127C2×Q814yes—
99C22⋊Q8—432no—
100C22⋊Q8—416no—
101C4×Q8—44no—
102C4×Q8—44no—
103C4×D4—44no—
104C4×D4—48no—
105C4×D4—48no—
106C4×D4—44no—
107C4×D4—44no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D412yes—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—22no—
125C2×C42—24no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2411yescharacteristic
131#107—48no—
132#107—416no—
133C4⋊C8—24no—
134C4⋊C8—24no—
135C4⋊C8C2×Q818yescomplemented
136#102—48no—
137C4×C8—24no—
138C4×C8—44no—
139#95—24no—
140#95—24no—
141#95—24no—
142C24—22no—
143C2×Q8—28no—
144C2×Q8—416no—
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—84no—
148C2×Q8—832no—
149C2×Q8—84no—
150C2×Q8—82no—
151C2×D4—22no—
152C2×D4—28no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—44no—
158C2×D4C4⋊Q814yes—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—416no—
163C22×C4—48no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—84no—
167C22×C4#12011yescharacteristic
168C22×C4#12012yes—
169C22×C4#12011yescharacteristic
170C22×C4C22×D411yescharacteristic
171C22×C4C22×D412yes—
172C2×C8—24no—
173C2×C8—28no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—44no—
177C2×C8—44no—
178C4⋊C4—24no—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—42no—
182C4⋊C4—416no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C4⋊C4—84no—
193C4⋊C4C2×D818yes—
194C22⋊C4—416no—
195C22⋊C4—48no—
196C22⋊C4—82no—
197C22⋊C4—84no—
198C42—24no—
199C42—22no—
200C42—41no—
201C42—44no—
202C42—44no—
203C42C2×D814yes—
204C42AGL(1,ℤ/8)14yes—
205C42C22×Q812yes—
206C422−1+412yes—
207C23—24no—
208C23—44no—
209C23—44no—
210C23—42no—
211C23—42no—
212C23C2×#12011yescharacteristic
213Q8—416no—
214Q8—816no—
215Q8—164no—
216D4—44no—
217D4—48no—
218D4—48no—
219D4—82no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—816no—
235C2×C4—88no—
236C2×C4—84no—
237C2×C4—82no—
238C2×C4#44714yes—
239C2×C4#44714yes—
240C2×C4#45312yes—
241C2×C4C2×C4⋊Q811yescharacteristic
242C2×C4#53611yescharacteristic
243C2×C4D4×Q814yes—
244C2×C4C22×D812yes—
245C2×C4C2×AGL(1,ℤ/8)12yes—
246C2×C4#57814yes—
247C8—44no—
248C8—48no—
249C22—44no—
250C22—48no—
251C22—82no—
252C22—82no—
253C22—84no—
254C22#243411yescharacteristic
255C22C2×#44712yes—
256C22#301112yes—
257C22C2×#45311yescharacteristic
258C22#302211yescharacteristic
259C4—21no—
260C4—21no—
261C4—24no—
262C4—41no—
263C4—48no—
264C4—82no—
265C4—84no—
266C4—162no—
267C4#161914yes—
268C4Q8×D814yes—
269C4#338614yes—
270C2—84no—
271C2C2×#161911yescharacteristic
272C2#1871911yescharacteristic
273C2#1892212yes—
274C2#2582411yescharacteristic
275C2#2583812yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
2. #28689C2semidirect
4. #10573C2semidirect
5. #9865C2semidirect
78. #331Q8semidirect
135. C4⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#286892
#285082
#105732
#98652
#89202
C4×#3312