Finite Group Databasebeta

← #139019

Subgroups — #139019

Order 512.

281 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 76 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#33802C211yesmaximal, characteristic
2#18072C211yesmaximal, characteristic
3#11984C211yesmaximal, characteristic
4C2×#1363C211yesmaximal, characteristic
5#9792C211yesmaximal, characteristic
6#7294C211yesmaximal, characteristic
7#7210C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9C2×#499C2211yescharacteristic
10#2962—21no—
11#2255—21no—
12#2245—21no—
13#2236C412yes—
14#2235C412yes—
15#2229—22no—
16#2163C2211yescharacteristic
17C2×C4⋊C16C2211yescharacteristic
18#1593—21no—
19#1587—22no—
20#1579—21no—
21#1568C2211yescharacteristic
22C2×#334C2211yescharacteristic
23#1528—21no—
24#1481—21no—
25#1470—21no—
26#1451C2211yescharacteristic
27#1364—21no—
28#1363—21no—
29#1363—21no—
30#1362—21no—
31#533—21no—
32C2×#125—21no—
33C22×C16—21no—
34#500D411yescharacteristic
35#499—21no—
36#499—21no—
37#499—21no—
38#499—21no—
39#494D411yescharacteristic
40#487—21no—
41#467—21no—
42#467C2×C412yes—
43#446—22no—
44#446—42no—
45C8×Q8—22no—
46C4×Q16—21no—
47C4×Q16D412yes—
48#432—21no—
49#431D411yescharacteristic
50C2×#107C2×C411yescharacteristic
51C2×#107C2×C411yescharacteristic
52C2×#106—21no—
53#424—41no—
54C2×C4⋊C8—21no—
55#405—41no—
56C4×M16D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#367—22no—
59#366—22no—
60C4⋊C16—21no—
61C4⋊C16—21no—
62C4⋊C16—21no—
63#358—22no—
64#358—22no—
65C22⋊C16C2×C412yes—
66#346—21no—
67#335—21no—
68#334—21no—
69#334—21no—
70#334D411yescharacteristic
71#334D411yescharacteristic
72#330—42no—
73Q8⋊C8—22no—
74Q8⋊C8—22no—
75#324—42no—
76#322—21no—
77#322—21no—
78C2×Q16—21no—
79C2×Q16C22⋊C412yes—
80C2×M16—21no—
81C22×C8—41no—
82C22×C8C22×C411yescharacteristic
83C22×C8#4111yescharacteristic
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86#125—41no—
87#125—41no—
88#125—41no—
89#125—41no—
90#125—41no—
91C22⋊Q8—22no—
92C22⋊Q8—41no—
93C4×Q8—22no—
94C4×Q8—41no—
95#117—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C42C2×D411yescharacteristic
101C2×C16—22no—
102C2×C16—41no—
103C2×C16—81no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107C22⋊C412yes—
110#106—41no—
111#106—41no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#103—22no—
120#103—22no—
121C22⋊C8—21no—
122C22⋊C8—41no—
123#97—42no—
124#97—41no—
125#97D812yes—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—42no—
129C4×C8C2×D411yescharacteristic
130C4×C8C2×D411yescharacteristic
131C4×C8C2×D411yescharacteristic
132C4×C8#4111yescharacteristic
133C2×Q8—22no—
134C2×Q8—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4C4×D411yescharacteristic
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140Q16—42no—
141Q16—41no—
142Q16—41no—
143M16—42no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—82no—
152C2×C8—81no—
153C2×C8C2×C22⋊C411yescharacteristic
154C2×C8C4×D411yescharacteristic
155C2×C8#12111yescharacteristic
156C2×C8#12311yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C22⋊C4—42no—
172C42—42no—
173C42#12011yescharacteristic
174C42#12111yescharacteristic
175C42C2×D811yescharacteristic
176C42#13511yescharacteristic
177C16—42no—
178C16—81no—
179C23#44211yescharacteristic
180Q8—42no—
181Q8—42no—
182Q8—161no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4—161no—
193C2×C4#38611yescharacteristic
194C2×C4C4×D811yescharacteristic
195C2×C4#44711yescharacteristic
196C2×C4#44911yescharacteristic
197C2×C4#45911yescharacteristic
198C2×C4#46311yescharacteristic
199C8—21no—
200C8—21no—
201C8—21no—
202C8—42no—
203C8—81no—
204C8—81no—
205C8—81no—
206C8—161no—
207C8#47111yescharacteristic
208C8#47311yescharacteristic
209C22—21no—
210C22—21no—
211C22#187911yescharacteristic
212C22#218012yes—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—82no—
219C4—82no—
220C4—161no—
221C4—161no—
222C4#190811yescharacteristic
223C4#191511yescharacteristic
224C2—22no—
225C2#927111yescharacteristic
226C2#1302111yescharacteristic
227C2#1303811yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#442
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#338022
#180722
#119842
C2×#13632
#97922
#72942
#72102