Finite Group Databasebeta

← #318582

Subgroups — #318582

Order 512.

535 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 156 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#28297C212yesmaximal
2#19262C212yesmaximal
3#17977C211yesmaximal, characteristic
4C2×#1579C211yesmaximal, characteristic
5#10463C214yesmaximal
6#8826C211yesmaximal, characteristic
7#8683C212yesmaximal
8#8617C212yesmaximal
9#3170C2211yescharacteristic
10#3078C2214yes—
11#2960C2212yes—
12C2×C8×Q8C2212yes—
13C2×C4×Q16C2211yescharacteristic
14#2064—24no—
15#1979—22no—
16#1952—22no—
17#1848C2212yes—
18#1801C2214yes—
19#1684—24no—
20#1676—22no—
21#1670—22no—
22#1584—24no—
23#1579—24no—
24#1567C2211yescharacteristic
25#1567C2212yes—
26#1567C2212yes—
27C2×#334C2211yescharacteristic
28#1550C2214yes—
29#1493C2212yes—
30C2×Q8⋊C8C2212yes—
31C2×#322C2211yescharacteristic
32#1292C2214yes—
33C22×Q16D411yescharacteristic
34#533C2312yes—
35C2×C4×Q8C2312yes—
36#500D411yescharacteristic
37#494D411yescharacteristic
38#487D412yes—
39#479—44no—
40#477—24no—
41#475—24no—
42#467—22no—
43#446—22no—
44#446—22no—
45C8×Q8—22no—
46C8×Q8—22no—
47C8×Q8—22no—
48C8×Q8—22no—
49C8×Q8—22no—
50C8×Q8—22no—
51C4×Q16—22no—
52C4×Q16—22no—
53C4×Q16—22no—
54C2×#108—22no—
55C2×#108—22no—
56#428—24no—
57C2×#107D411yescharacteristic
58#424—22no—
59#423C2314yes—
60#423C2312yes—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63#419—24no—
64C2×#103D412yes—
65C2×#101—24no—
66#405—22no—
67C2×C4×C8—22no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#343—24no—
72#335—22no—
73#335—22no—
74#335—22no—
75#335—42no—
76#334—22no—
77#333—24no—
78#332—24no—
79Q8⋊C8—24no—
80Q8⋊C8—24no—
81#322D414yes—
82C22×Q8C2×D412yes—
83C2×Q16—22no—
84C2×Q16—22no—
85C2×Q16—22no—
86C2×Q16Q1614yescomplemented
87C2×M16—22no—
88C2×M16—24no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8C2×D411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22×C8#4112yes—
94C4⋊Q8C2×D412yes—
95C4⋊Q8C2×D412yes—
96#125—22no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×Q8—22no—
103C4×Q8—22no—
104C4×Q8—22no—
105#117—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4C2×D412yes—
108C2×C42—22no—
109C2×C42C2411yescharacteristic
110#108—44no—
111#108—82no—
112#107—22no—
113#107—22no—
114#107—44no—
115#106—44no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—24no—
120C4⋊C8—22no—
121C4⋊C8—24no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126#103—24no—
127#103—24no—
128#103—44no—
129#103—44no—
130#101—84no—
131#97—42no—
132C4×C8—21no—
133C4×C8—42no—
134C4×C8—42no—
135C4×C8—42no—
136C4×C8C2×D412yes—
137C4×C8C2×D412yes—
138C4×C8C2×D412yes—
139C4×C8C2×D411yescharacteristic
140C4×C8C2×D411yescharacteristic
141C4×C8#4112yes—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8C2×Q1614yes—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4C22×D411yescharacteristic
153C22×C4C2×#4111yescharacteristic
154C22×C42−1+411yescharacteristic
155Q16—28no—
156Q16—82no—
157M16—44no—
158M16—42no—
159M16—44no—
160M16—42no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—44no—
168C2×C8—81no—
169C2×C8—82no—
170C2×C8#12112yes—
171C2×C8#12112yes—
172C2×C8#12112yes—
173C2×C8#12112yes—
174C2×C8C2×Q1612yes—
175C2×C8#13512yes—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C22⋊C4—42no—
187C42—42no—
188C42—42no—
189C42—42no—
190C42C22×D411yescharacteristic
191C42C22×D411yescharacteristic
192C42C2×#4111yescharacteristic
193C42H(5,𝔽2)11yescharacteristic
194C23#54711yescharacteristic
195Q8—24no—
196Q8—24no—
197Q8—42no—
198Q8—42no—
199Q8—44no—
200Q8—82no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4C2×#12111yescharacteristic
211C2×C4C2×#12111yescharacteristic
212C2×C4#53711yescharacteristic
213C2×C4D4211yescharacteristic
214C2×C4#54711yescharacteristic
215C2×C4#55511yescharacteristic
216C2×C4C22×Q1611yescharacteristic
217C2×C4C2×#13511yescharacteristic
218C2×C4#57611yescharacteristic
219C2×C4#57811yescharacteristic
220C8—22no—
221C8—22no—
222C8—22no—
223C8—42no—
224C8—84no—
225C8—82no—
226C8—82no—
227C8#46714yes—
228C22—21no—
229C22—21no—
230C22#253711yescharacteristic
231C22#333811yescharacteristic
232C22#334211yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—44no—
237C4—82no—
238C4—82no—
239C4—82no—
240C4C2×#46711yescharacteristic
241C4#306111yescharacteristic
242C4#306912yes—
243C4D4×Q1611yescharacteristic
244C4#328711yescharacteristic
245C2—21no—
246C2—21no—
247C2#2822811yescharacteristic
248C2#2824411yescharacteristic
249C2#2825011yescharacteristic
subgroupG/H
center, socleC22#2537
commutator, Frattini, cosocleC2×C42C24
prime radicalC23#547
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
86. C2×Q16Q16semidirect

Maximal subgroups

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

subgroupindex
#282972
#192622
#179772
C2×#15792
#104632
#88262
#86832
#86172