Finite Group Databasebeta

← #145341

Subgroups — #145341

Order 512.

242 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 60 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#19287C211yesmaximal, characteristic
2#12029C211yesmaximal, characteristic
3#11950C211yesmaximal, characteristic
4#11865C211yesmaximal, characteristic
5#11495C211yesmaximal, characteristic
6#8665C211yesmaximal, characteristic
7#7219C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3078—21no—
10C2×#446C2211yescharacteristic
11#2163C2211yescharacteristic
12#2162C2211yescharacteristic
13#1837C2211yescharacteristic
14#1704—22no—
15#1696—21no—
16#1690—21no—
17#1566C2211yescharacteristic
18#1564C2211yescharacteristic
19#1550—21no—
20#1487—21no—
21#1368—22no—
22#1364—21no—
23#1362—21no—
24#1350—22no—
25#1322—22no—
26#1301—21no—
27#533—21no—
28C2×C4×Q8D411yescharacteristic
29#500D411yescharacteristic
30#494D411yescharacteristic
31#487—21no—
32#479—42no—
33#477—22no—
34#475—22no—
35#467—21no—
36#467—21no—
37#446—21no—
38#446—21no—
39#446—21no—
40#446—21no—
41C4×Q16—21no—
42C4×Q16—21no—
43#431—21no—
44#428—41no—
45#423—21no—
46#423—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49#419—41no—
50C2×#97—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16—21no—
54C22⋊C16—21no—
55C22⋊C16D412yes—
56#346—21no—
57C4×C16—21no—
58#341—21no—
59#341—21no—
60#335—21no—
61#334—21no—
62#334—21no—
63#334D412yes—
64#333—22no—
65#332—22no—
66Q8⋊C8—42no—
67C22×Q8SD1611yescharacteristic
68C2×Q16—21no—
69C2×Q16—21no—
70C2×M16—41no—
71C2×M16—41no—
72C22×C8—21no—
73C22×C8C2×D411yescharacteristic
74C22×C8#4111yescharacteristic
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77#125—41no—
78C22⋊Q8—41no—
79C22⋊Q8—41no—
80C4×Q8—21no—
81C4×Q8—21no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—41no—
85C4×Q8—41no—
86#117—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4SD1611yescharacteristic
89C2×C42—21no—
90C2×C42C2×D411yescharacteristic
91C2×C16—22no—
92C2×C16—41no—
93#107—21no—
94#107—21no—
95#107—42no—
96#106—42no—
97C4⋊C8—21no—
98C4⋊C8—22no—
99C4⋊C8—21no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106#103—42no—
107#103—42no—
108#103—41no—
109#103—41no—
110C22⋊C8—41no—
111#97—41no—
112#97—41no—
113C4×C8—41no—
114C4×C8C2×D411yescharacteristic
115C4×C8C2×D411yescharacteristic
116C4×C8#4111yescharacteristic
117C4×C8#4111yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—42no—
121C2×Q8—41no—
122C2×Q8—41no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4#12111yescharacteristic
126C22×C4C2×SD1611yescharacteristic
127C22×C4#13511yescharacteristic
128Q16—81no—
129Q16—81no—
130M16—82no—
131M16—81no—
132M16—81no—
133C2×C8—41no—
134C2×C8—41no—
135C2×C8—41no—
136C2×C8—42no—
137C2×C8—81no—
138C2×C8—81no—
139C2×C8—81no—
140C2×C8#12111yescharacteristic
141C2×C8#12111yescharacteristic
142C2×C8#12311yescharacteristic
143C2×C8#12411yescharacteristic
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—81no—
153C4⋊C4—81no—
154C22⋊C4—81no—
155C42—41no—
156C42—41no—
157C42—81no—
158C42#12011yescharacteristic
159C42#12311yescharacteristic
160C42AGL(1,ℤ/8)11yescharacteristic
161C42AGL(1,ℤ/8)11yescharacteristic
162C16—42no—
163C16—81no—
164C23#46511yescharacteristic
165Q8—42no—
166Q8—42no—
167Q8—81no—
168Q8—81no—
169Q8—161no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4#39411yescharacteristic
181C2×C4#44911yescharacteristic
182C2×C4#45011yescharacteristic
183C2×C4#46911yescharacteristic
184C2×C4#48111yescharacteristic
185C2×C4#48211yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8—21no—
190C8—82no—
191C8—81no—
192C8—161no—
193C8—161no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22#203411yescharacteristic
198C4—21no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—81no—
203C4—81no—
204C4—161no—
205C4—161no—
206C4#204911yescharacteristic
207C4#205311yescharacteristic
208C4#223912yes—
209C2—41no—
210C2#1046711yescharacteristic
211C2#1339911yescharacteristic
212C2#1340611yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#465
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ 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
#192872
#120292
#119502
#118652
#114952
#86652
#72192