Finite Group Databasebeta

← #449435

Subgroups — #449435

Order 512.

704 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 182 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#23560C211yesmaximal, characteristic
2#20760C211yesmaximal, characteristic
3#11393C212yesmaximal
4#11281C212yesmaximal
5#11262C212yesmaximal
6#11236C212yesmaximal
7#10900C212yesmaximal
8#10893C211yesmaximal, characteristic
9#10882C212yesmaximal
10#3458C2211yescharacteristic
11#2911C2212yes—
12C2×#424C2212yes—
13#2873C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2285C2211yescharacteristic
16#2285C2211yescharacteristic
17#2061—22no—
18#2060—22no—
19#1835—22no—
20#1834—22no—
21C2×#344C2212yes—
22#1742C2211yescharacteristic
23#1740C2212yes—
24#1740C2211yescharacteristic
25C2×#341C2212yes—
26#1735C2212yes—
27#1735C2212yes—
28#1735C2212yes—
29#1733C2212yes—
30#1733C2212yes—
31C2×#339C2212yes—
32C2×#339C2212yes—
33C2×H(3,ℤ/4)C2212yes—
34#1729C2212yes—
35C22×M16C2312yes—
36C22×M16C2312yes—
37C23×C8C2312yes—
38#544—22no—
39#544—24no—
40#543—41no—
41#533—42no—
42#533—41no—
43C2×#117C2312yes—
44C2×#117C2312yes—
45C2×#117C2311yescharacteristic
46C2×#117C2311yescharacteristic
47C2×#117C2312yes—
48C22×C42C2311yescharacteristic
49#428—24no—
50#428—24no—
51#427—42no—
52#424—24no—
53#424—24no—
54C2×C4⋊C8—42no—
55#405—42no—
56C4×M16—41no—
57C2×C4×C8—24no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#384—21no—
61#384D412yes—
62#382—21no—
63#382D412yes—
64#380—41no—
65#380—41no—
66C4×C4⋊C4—41no—
67C4×C4⋊C4—41no—
68#344—24no—
69#343—22no—
70#342—41no—
71#342—42no—
72#341—24no—
73#340—24no—
74#340—24no—
75#340—24no—
76#339—22no—
77#339—22no—
78#339—42no—
79#339—42no—
80#339Q818yescomplemented
81H(3,ℤ/4)—24no—
82#336—42no—
83C23×C4—21no—
84C23×C4C2411yescharacteristic
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16C2×Q814yes—
96C2×M16#4114yes—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8#4114yes—
106C4⋊Q8—44no—
107C4⋊Q8—42no—
108#125—44no—
109#125—42no—
110C22⋊Q8—44no—
111C22⋊Q8—48no—
112C4×Q8—44no—
113#117—22no—
114#117—22no—
115#117—22no—
116#117—22no—
117#117—22no—
118#117—22no—
119#117—22no—
120#117—22no—
121#117—22no—
122#117—22no—
123#117—22no—
124#117—22no—
125#117—22no—
126#117—22no—
127#117—22no—
128#117C2×Q814yes—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4#4114yes—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C42—22no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—22no—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×Q814yes—
151#107—44no—
152#106—44no—
153C4⋊C8—48no—
154#97—42no—
155C4×C8—24no—
156C4×C8—24no—
157C4×C8—42no—
158#95—42no—
159#95—42no—
160C24H(5,𝔽2)11yescharacteristic
161C2×Q8—44no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4C22×D411yescharacteristic
172C22×C4C22×Q812yes—
173C22×C4C2×#4112yes—
174C22×C4C2×#4112yes—
175C22×C4C2×#4112yes—
176C22×C4H(5,𝔽2)11yescharacteristic
177C22×C4H(5,𝔽2)12yes—
178C22×C42−1+412yes—
179M16—24no—
180M16—24no—
181M16—24no—
182M16—24no—
183M16—42no—
184M16—42no—
185M16—42no—
186M16—42no—
187C2×C8—24no—
188C2×C8—24no—
189C2×C8—24no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—84no—
211C4⋊C4—84no—
212C22⋊C4—24no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C42—21no—
221C42—21no—
222C42—24no—
223C42—24no—
224C42—42no—
225C42—42no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—42no—
230C42—41no—
231C42—42no—
232C42—42no—
233C42C4⋊Q812yes—
234C42C4⋊Q812yes—
235C23—42no—
236C23#53411yescharacteristic
237C23#53812yes—
238C23#55412yes—
239C23#55511yescharacteristic
240C23#56311yescharacteristic
241Q8—164no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—48no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—84no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4—82no—
257C2×C4#52511yescharacteristic
258C2×C4#52912yes—
259C2×C4C2×C4⋊Q811yescharacteristic
260C2×C4#53312yes—
261C2×C4#54212yes—
262C2×C4#54212yes—
263C2×C4#54412yes—
264C2×C4#54612yes—
265C2×C4#54812yes—
266C2×C4#55112yes—
267C2×C4#55111yescharacteristic
268C2×C4#55212yes—
269C2×C4#55412yes—
270C2×C4#55412yes—
271C2×C4#55712yes—
272C2×C4H(3,𝔽4)11yescharacteristic
273C8—44no—
274C8—44no—
275C8—44no—
276C22—22no—
277C22—24no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22#257412yes—
282C22#258311yescharacteristic
283C22#268411yescharacteristic
284C22#273111yescharacteristic
285C22#277412yes—
286C4—24no—
287C4—42no—
288C4—84no—
289C4—82no—
290C4—82no—
291C4—84no—
292C4—162no—
293C4#247712yes—
294C4#252012yes—
295C4#255012yes—
296C4#276312yes—
297C2—21no—
298C2—22no—
299C2—21no—
300C2—22no—
301C2#1594411yescharacteristic
302C2#3128112yes—
subgroupG/H
centerC2×C4H(3,𝔽4)
commutator, Frattini, cosocleC23×C4C24
socleC22#2684
prime radicalC24H(5,𝔽2)
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
80. #339Q8semidirect

Maximal subgroups

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

subgroupindex
#235602
#207602
#113932
#112812
#112622
#112362
#109002
#108932
#108822