Finite Group Databasebeta

← #145245

Subgroups — #145245

Order 512.

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

normal subgroups only

This group has 252 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#19193C211yesmaximal, characteristic
2#12031C211yesmaximal, characteristic, complemented
3#11975C211yesmaximal, characteristic, complemented
4#11974C211yesmaximal, characteristic, complemented
5#11635C211yesmaximal, characteristic
6#8634C211yesmaximal, characteristic
7#7293C211yesmaximal, characteristic, complemented
8#3227—21no—
9C2×#500C2211yescharacteristic
10#2931C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#1695—22no—
14#1675—22no—
15#1568C2211yescharacteristic
16#1568C2211yescharacteristic
17#1564C2211yescharacteristic
18#1548—21no—
19#1536—21no—
20#1368—22no—
21#1365—22no—
22#1364—22no—
23#1330—22no—
24#1307—21no—
25C2×#131—21no—
26#532D411yescharacteristic
27C2×C4⋊Q8—21no—
28#500—21no—
29#500—21no—
30#500—21no—
31#500D412yes—
32#498—41no—
33#486—41no—
34#481—42no—
35#476—24no—
36#436—21no—
37#436—21no—
38C8×D4—21no—
39C8×D4—21no—
40#432—21no—
41#432—21no—
42#431—21no—
43C2×#107—21no—
44C2×#107—21no—
45#423D411yescharacteristic
46#408—21no—
47C2×C4×C8C2311yescharacteristic
48C4⋊C16—21no—
49C4⋊C16D412yes—
50C22⋊C16—21no—
51C22⋊C16—21no—
52#346—21no—
53#346—21no—
54#335—21no—
55#335—21no—
56#335—21no—
57#334—21no—
58#334—21no—
59#334D412yes—
60#333—24no—
61#329—42no—
62C2×#41—21no—
63C22×Q8—41no—
64#131—82no—
65C2×M16—21no—
66C2×M16—41no—
67C22×C8—41no—
68C22×C8#4111yescharacteristic
69C22×C8#4111yescharacteristic
70C4⋊Q8—22no—
71C4⋊Q8—41no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74C4⋊Q8SD1611yescharacteristic
75#127SD1611yescharacteristic
76#124—21no—
77#121—21no—
78#121—21no—
79C4×D4—21no—
80C4×D4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C42C2×D411yescharacteristic
84C2×C16—22no—
85C2×C16—41no—
86#107—22no—
87#107—22no—
88#107—41no—
89#107—41no—
90#106—44no—
91C4⋊C8—21no—
92C4⋊C8—22no—
93C4⋊C8—21no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96#102—44no—
97C22⋊C8—41no—
98C22⋊C8—41no—
99C22⋊C8—41no—
100C22⋊C8—41no—
101C22⋊C8—41no—
102#97—41no—
103#97—41no—
104C4×C8—41no—
105C4×C8C2×D411yescharacteristic
106C4×C8C2×D411yescharacteristic
107C4×C8C2×D411yescharacteristic
108C4×C8#4111yescharacteristic
109#41—81no—
110#41—81no—
111C2×Q8—21no—
112C2×Q8—42no—
113C2×Q8—81no—
114C2×Q8—162no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—41no—
119C22×C4—41no—
120C22×C4—81no—
121C22×C4#12311yescharacteristic
122C22×C4#13511yescharacteristic
123C22×C4AGL(1,ℤ/8)11yescharacteristic
124M16—82no—
125M16—81no—
126C2×C8—22no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—42no—
130C2×C8—82no—
131C2×C8—81no—
132C2×C8—81no—
133C2×C8#12111yescharacteristic
134C2×C8#12111yescharacteristic
135C2×C8#12311yescharacteristic
136C2×C8#12411yescharacteristic
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—81no—
142C4⋊C4—81no—
143C22⋊C4—41no—
144C22⋊C4—41no—
145C42#12011yescharacteristic
146C42#12111yescharacteristic
147C42C2×SD1611yescharacteristic
148C42AGL(1,ℤ/8)11yescharacteristic
149C16—42no—
150C16—81no—
151C23—41no—
152C23#48211yescharacteristic
153Q8—81no—
154Q8—162no—
155Q8—162no—
156D4—42no—
157D4—81no—
158D4—81no—
159D4—82no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—82no—
168C2×C4—162no—
169C2×C4#39411yescharacteristic
170C2×C4#44911yescharacteristic
171C2×C4#45011yescharacteristic
172C2×C4#46511yescharacteristic
173C2×C4#46911yescharacteristic
174C2×C4#48111yescharacteristic
175C8—21no—
176C8—21no—
177C8—21no—
178C8—82no—
179C8—81no—
180C8—161no—
181C8—161no—
182C8#47112yes—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22#203411yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—81no—
196C4—162no—
197C4#204611yescharacteristic
198C4#204911yescharacteristic
199C2—41no—
200C2—161no—
201C2#1044511yescharacteristic
202C2#1339711yescharacteristic
203C2#1340211yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#532D4
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12031C2semidirect
3. #11975C2semidirect
4. #11974C2semidirect
7. #7293C2semidirect

Maximal subgroups

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

subgroupindex
#191932
#120312
#119752
#119742
#116352
#86342
#72932