Finite Group Databasebeta

← #303013

Subgroups — #303013

Order 512.

1079 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1079 conjugacy classes of subgroups — too many to draw. Only its 180 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#29319C212yesmaximal, complemented
3#28514C211yesmaximal, characteristic
4#10574C212yesmaximal, complemented
5#10573C212yesmaximal, complemented
6#9868C212yesmaximal, complemented
7#8924C212yesmaximal
8#8920C212yesmaximal
9#8231C211yesmaximal, characteristic
10C2×D4×Q8—22no—
11C2×#478—22no—
12#2859C2211yescharacteristic
13#2683—22no—
14#2551C2212yes—
15#2538—22no—
16#2452—24no—
17#2409—22no—
18#2366C2212yes—
19C4×C4⋊Q8C2212yes—
20C4×#127C2211yescharacteristic
21#2092—22no—
22#2084—22no—
23#2057—22no—
24#1957C2211yescharacteristic
25#1957C2211yescharacteristic
26#1948C2214yes—
27#1947C2214yes—
28#1943—24no—
29#1845C2211yescharacteristic
30#1844C2211yescharacteristic
31#1767C2212yes—
32C4×C4⋊C8C2211yescharacteristic
33C2×#331C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C23×Q8—22no—
38D4×Q8—416no—
39C2×C4⋊Q8—24no—
40C2×C4⋊Q8D412yes—
41C2×#127C2311yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—28no—
45C2×C4×Q8—22no—
46C2×C4×D4—24no—
47C2×C4×D4—22no—
48C2×C4×D4D412yes—
49#478—48no—
50C2×#107—22no—
51C2×#107—24no—
52C2×#106—22no—
53C2×#106—24no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2314yes—
56C2×#102—22no—
57C2×#102—22no—
58C2×#97—22no—
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2311yescharacteristic
61#396—28no—
62#395—24no—
63#393—24no—
64#391—22no—
65#390—24no—
66#390—22no—
67#390D412yes—
68#389—24no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389D414yes—
73#386—24no—
74#385—24no—
75#384D412yes—
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79#382—21no—
80#380—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2314yes—
83C43C2311yescharacteristic
84#340—24no—
85#340—24no—
86#336—22no—
87#336—21no—
88#331Q818yescomplemented
89C22×Q8—24no—
90C22×Q8—22no—
91C22×Q8—416no—
92C22×Q8—84no—
93C22×D4—22no—
94C22×D4C2×D412yes—
95C23×C4—24no—
96C23×C4—22no—
97C22×C8—24no—
98C22×C8—21no—
99C22×C8—21no—
100C4⋊Q8—28no—
101C4⋊Q8—48no—
102C4⋊Q8—48no—
103#127—22no—
104#127C2×Q814yes—
105#125—44no—
106C22⋊Q8—432no—
107C22⋊Q8—416no—
108C4×Q8—48no—
109C4×D4—48no—
110C4×D4—44no—
111C4×D4—48no—
112C4×D4—48no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4C2×D414yes—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4C2×D412yes—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—22no—
132C2×C42—24no—
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2411yescharacteristic
138#107—44no—
139#107—48no—
140#106—44no—
141#106—84no—
142C4⋊C8—24no—
143C4⋊C8—24no—
144C4⋊C8C2×Q818yescomplemented
145#102—44no—
146#102—44no—
147#97—82no—
148C4×C8—22no—
149C4×C8—22no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—42no—
154C24—22no—
155C2×Q8—28no—
156C2×Q8—44no—
157C2×Q8—416no—
158C2×Q8—44no—
159C2×Q8—84no—
160C2×Q8—832no—
161C2×D4—22no—
162C2×D4—28no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4C4⋊Q814yes—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—48no—
173C22×C4—416no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—84no—
178C22×C4#12011yescharacteristic
179C22×C4#12012yes—
180C22×C4#12011yescharacteristic
181C22×C4C22×D411yescharacteristic
182C22×C4C22×D412yes—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—28no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—44no—
191C2×C8—44no—
192C4⋊C4—28no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—48no—
199C4⋊C4—416no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4AGL(1,ℤ/8)18yes—
210C22⋊C4—48no—
211C22⋊C4—416no—
212C22⋊C4—82no—
213C22⋊C4—84no—
214C42—24no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—41no—
219C42—44no—
220C42—44no—
221C42C22×Q812yes—
222C422−1+412yes—
223C23—24no—
224C23—44no—
225C23—44no—
226C23—42no—
227C23—42no—
228C23C2×#12011yescharacteristic
229Q8—416no—
230Q8—816no—
231D4—44no—
232D4—48no—
233D4—48no—
234D4—82no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—28no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—88no—
252C2×C4—816no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4#45312yes—
258C2×C4C2×C4⋊Q811yescharacteristic
259C2×C4#53611yescharacteristic
260C2×C4D4×Q814yes—
261C2×C4C2×AGL(1,ℤ/8)14yes—
262C2×C4#57712yes—
263C2×C4#57812yes—
264C8—42no—
265C8—42no—
266C8—48no—
267C22—44no—
268C22—48no—
269C22—82no—
270C22—82no—
271C22—84no—
272C22#243411yescharacteristic
273C22#300312yes—
274C22#301312yes—
275C22C2×#45311yescharacteristic
276C22#302211yescharacteristic
277C4—22no—
278C4—21no—
279C4—21no—
280C4—41no—
281C4—48no—
282C4—42no—
283C4—82no—
284C4—84no—
285C4—161no—
286C4—161no—
287C4#338414yes—
288C4#338614yes—
289C2—84no—
290C2#1852411yescharacteristic
291C2#1868811yescharacteristic
292C2#1886212yes—
293C2#2582411yescharacteristic
294C2#2584412yes—
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. #29319C2semidirect
4. #10574C2semidirect
5. #10573C2semidirect
6. #9868C2semidirect
88. #331Q8semidirect
144. C4⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#293192
#285142
#105742
#105732
#98682
#89242
#89202
#82312