Finite Group Databasebeta

← #318598

Subgroups — #318598

Order 512.

519 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 148 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#19287C212yesmaximal
3#17977C211yesmaximal, characteristic
4C2×#1593C211yesmaximal, characteristic
5#10462C214yesmaximal
6#8825C211yesmaximal, characteristic
7#8683C212yesmaximal
8#8619C212yesmaximal
9#3170C2211yescharacteristic
10#3078C2214yes—
11#2960C2212yes—
12C2×#446C2212yes—
13C2×C4×Q16C2211yescharacteristic
14#2064—24no—
15#1979—22no—
16#1952—22no—
17#1848C2212yes—
18#1801C2214yes—
19#1704—24no—
20#1696—22no—
21#1690—22no—
22#1593—24no—
23#1584—24no—
24#1567C2211yescharacteristic
25#1566C2212yes—
26#1566C2212yes—
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—
45#446—22no—
46#446—22no—
47#446—22no—
48#446—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×#97—22no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#343—24no—
72#335—22no—
73#334—22no—
74#334—22no—
75#334—22no—
76#334—42no—
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—24no—
86C2×M16—22no—
87C2×M16—24no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92C22×C8#4112yes—
93C4⋊Q8C2×D412yes—
94C4⋊Q8C2×D412yes—
95#125—22no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×Q8—22no—
103C4×Q8—22no—
104#117—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4C2×D412yes—
107C2×C42—22no—
108C2×C42C2411yescharacteristic
109#108—44no—
110#108—82no—
111#107—22no—
112#107—22no—
113#107—44no—
114#106—44no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—24no—
119C4⋊C8—22no—
120C4⋊C8—24no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125#103—24no—
126#103—24no—
127#103—44no—
128#103—44no—
129#101—84no—
130#97—42no—
131#97—42no—
132#97—42no—
133C4×C8—21no—
134C4×C8—42no—
135C4×C8C2×D412yes—
136C4×C8C2×D412yes—
137C4×C8C2×D412yes—
138C4×C8C2×D411yescharacteristic
139C4×C8C2×D411yescharacteristic
140C4×C8#4112yes—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—24no—
146C2×Q8—24no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4C22×D411yescharacteristic
150C22×C4C2×#4111yescharacteristic
151C22×C42−1+411yescharacteristic
152Q16—44no—
153Q16—82no—
154M16—44no—
155M16—42no—
156M16—44no—
157M16—42no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—81no—
165C2×C8—82no—
166C2×C8#12112yes—
167C2×C8#12112yes—
168C2×C8#12112yes—
169C2×C8#12112yes—
170C2×C8#13714yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C22⋊C4—42no—
182C42—42no—
183C42—42no—
184C42—42no—
185C42C22×D411yescharacteristic
186C42C22×D411yescharacteristic
187C42C2×#4111yescharacteristic
188C42H(5,𝔽2)11yescharacteristic
189C23#54711yescharacteristic
190Q8—44no—
191Q8—42no—
192Q8—42no—
193Q8—44no—
194Q8—82no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—82no—
203C2×C4C2×#12111yescharacteristic
204C2×C4C2×#12111yescharacteristic
205C2×C4#53711yescharacteristic
206C2×C4D4211yescharacteristic
207C2×C4#54711yescharacteristic
208C2×C4#55511yescharacteristic
209C2×C4C2×#13712yes—
210C2×C4#57712yes—
211C8—22no—
212C8—22no—
213C8—22no—
214C8—42no—
215C8—84no—
216C8—82no—
217C8—82no—
218C8#47014yes—
219C22—21no—
220C22—21no—
221C22#253711yescharacteristic
222C22#335612yes—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—82no—
227C4—82no—
228C4—82no—
229C4—82no—
230C4C2×#47011yescharacteristic
231C4#306011yescharacteristic
232C4#306912yes—
233C4#327812yes—
234C2—22no—
235C2#2823211yescharacteristic
236C2#2824411yescharacteristic
237C2#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

Maximal subgroups

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

subgroupindex
#282972
#192872
#179772
C2×#15932
#104622
#88252
#86832
#86192