Finite Group Databasebeta

← #317092

Subgroups — #317092

Order 512.

517 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 144 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#21733C211yesmaximal, characteristic
2#21715C211yesmaximal, characteristic
3#19118C212yesmaximal
4#10628C212yesmaximal
5#10017C212yesmaximal
6#9048C214yesmaximal
7#8780C211yesmaximal, characteristic
8#8659C212yesmaximal
9#3160C2211yescharacteristic
10#3104C2214yes—
11#2961C2212yes—
12C2×C8×Q8C2212yes—
13#2399C2211yescharacteristic
14#1995C2212yes—
15#1985C2212yes—
16#1946—21no—
17#1945—21no—
18#1940—21no—
19#1940—21no—
20#1848C2211yescharacteristic
21#1843C2211yescharacteristic
22#1773—21no—
23#1772—21no—
24C8×C4⋊C4—21no—
25C8×C4⋊C4—21no—
26#1726C2211yescharacteristic
27#1676—22no—
28#1674—22no—
29#1672—24no—
30#1564C2212yes—
31#1542C2214yes—
32#1487C2214yes—
33#1301C2214yes—
34#1281C2212yes—
35#1274C2212yes—
36#533C2312yes—
37C2×C4×Q8C2312yes—
38#500D412yes—
39#499—21no—
40#499—21no—
41#498D412yes—
42#484—24no—
43#477—24no—
44#462—24no—
45#451—24no—
46#446—44no—
47C8×Q8—22no—
48C8×Q8—22no—
49C8×Q8—22no—
50C8×Q8—22no—
51C4×Q16—44no—
52#431C2314yes—
53C2×#108—21no—
54C2×#108—21no—
55#427—21no—
56#427—21no—
57#424—22no—
58#423C2312yes—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63#405—22no—
64C2×#97D411yescharacteristic
65C2×#97D411yescharacteristic
66C2×#97C2311yescharacteristic
67C2×C4×C8—22no—
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#398—22no—
72#395—22no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75#376C2311yescharacteristic
76#339—22no—
77#339—22no—
78#336—22no—
79#336—22no—
80#335—22no—
81#334—22no—
82#333—24no—
83#332—24no—
84Q8⋊C8—44no—
85C22×Q8#4112yes—
86C2×Q16—44no—
87C2×M16—22no—
88C22×C8—22no—
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C4⋊Q8C2×D414yes—
94#125—22no—
95C22⋊Q8—24no—
96C4×Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—24no—
101#117—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4#4112yes—
106C2×C42—22no—
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2411yescharacteristic
110#108—42no—
111#108—42no—
112#107—22no—
113#107—22no—
114#107—44no—
115#106—22no—
116#106—22no—
117C4⋊C8—22no—
118C4⋊C8—24no—
119C4⋊C8—22no—
120C4⋊C8—24no—
121C4⋊C8—42no—
122C4⋊C8—44no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126#103—44no—
127#103—44no—
128C22⋊C8—24no—
129#97—22no—
130#97—22no—
131#97—41no—
132#97—42no—
133C4×C8—22no—
134C4×C8—24no—
135C4×C8—42no—
136C4×C8—42no—
137C4×C8—42no—
138C4×C8Q1614yescomplemented
139C4×C8C2×D411yescharacteristic
140C4×C8C2×D411yescharacteristic
141C4×C8#4111yescharacteristic
142C4×C8#4111yescharacteristic
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—22no—
148C2×Q8—22no—
149C2×Q8—24no—
150C2×Q8—22no—
151C2×Q8—44no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4C22×D411yescharacteristic
157C22×C4C2×#4112yes—
158Q16—88no—
159M16—42no—
160M16—42no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—44no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8#12111yescharacteristic
174C2×C8#12111yescharacteristic
175C2×C8#12111yescharacteristic
176C2×C8#12111yescharacteristic
177C2×C8C2×Q1612yes—
178C2×C8C2×Q1612yes—
179C2×C8#13712yes—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—44no—
189C22⋊C4—42no—
190C42—21no—
191C42—21no—
192C42—42no—
193C42—42no—
194C42—42no—
195C42C2×Q1612yes—
196C42#13712yes—
197C42C22×D411yescharacteristic
198C42C2×#4111yescharacteristic
199C42H(5,𝔽2)12yes—
200C23#52511yescharacteristic
201Q8—24no—
202Q8—24no—
203Q8—42no—
204Q8—42no—
205Q8—84no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4C2×#12111yescharacteristic
219C2×C4#53511yescharacteristic
220C2×C4#54612yes—
221C2×C4#55312yes—
222C2×C4C22×Q1611yescharacteristic
223C2×C4C2×#13711yescharacteristic
224C2×C4C2×#13711yescharacteristic
225C2×C4#57511yescharacteristic
226C8—22no—
227C8—21no—
228C8—21no—
229C8—41no—
230C8—84no—
231C8—82no—
232C8—82no—
233C8#46212yes—
234C8#46212yes—
235C22—22no—
236C22#253511yescharacteristic
237C22#310411yescharacteristic
238C22#311311yescharacteristic
239C4—22no—
240C4—22no—
241C4—42no—
242C4—42no—
243C4—44no—
244C4—82no—
245C4—84no—
246C4C2×#46211yescharacteristic
247C4#303711yescharacteristic
248C4#320911yescharacteristic
249C4#321611yescharacteristic
250C2—21no—
251C2—21no—
252C2#2816111yescharacteristic
253C2#2818012yes—
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2535
prime radicalC23#525
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
138. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#217332
#217152
#191182
#106282
#100172
#90482
#87802
#86592