Finite Group Databasebeta

← #145247

Subgroups — #145247

Order 512.

264 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C2×#1695C211yesmaximal, characteristic
2#12028C211yesmaximal, characteristic, complemented
3#11974C211yesmaximal, characteristic, complemented
4#11974C211yesmaximal, characteristic, complemented
5#11636C211yesmaximal, characteristic
6#8624C211yesmaximal, characteristic
7#7294C211yesmaximal, characteristic, complemented
8C2×#500C2211yescharacteristic
9C2×#476—21no—
10C2×#436C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#1695—22no—
14#1695—22no—
15#1568C2211yescharacteristic
16#1568C2211yescharacteristic
17C2×#334C2211yescharacteristic
18C2×#333—21no—
19#1466—21no—
20#1368—22no—
21#1364—22no—
22#1364—22no—
23#1330—22no—
24#1290—21no—
25C22×M16—21no—
26C2×C4⋊Q8—21no—
27C2×C4×D4D411yescharacteristic
28#500—21no—
29#500—21no—
30#500—21no—
31#500D412yes—
32#476—24no—
33#476—42no—
34#436—21no—
35#436—21no—
36#436—21no—
37#436—21no—
38#432—21no—
39#432—21no—
40C2×#107—21no—
41C2×#107—21no—
42C2×#106—41no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#102—41no—
46C2×C22⋊C8—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#334—21no—
57#334—21no—
58#334—21no—
59#334D412yes—
60#333—24no—
61#323—42no—
62C22×Q8—41no—
63C22×D4SD1611yescharacteristic
64C23×C4—21no—
65C2×M16—41no—
66C2×M16—41no—
67C2×M16—82no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8#4111yescharacteristic
71C22×C8#4111yescharacteristic
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75C4⋊Q8—41no—
76C4×D4—21no—
77C4×D4—21no—
78C4×D4—21no—
79C4×D4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4SD1611yescharacteristic
83C2×C22⋊C4—21no—
84C2×C42C2×D411yescharacteristic
85C2×C16—22no—
86C2×C16—41no—
87#107—22no—
88#107—22no—
89#107—41no—
90#107—41no—
91#106—44no—
92C4⋊C8—21no—
93C4⋊C8—22no—
94C4⋊C8—21no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—41no—
98#102—44no—
99C22⋊C8—41no—
100C22⋊C8—41no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103#97—41no—
104#97—41no—
105C4×C8C2×D411yescharacteristic
106C4×C8C2×D411yescharacteristic
107C4×C8C2×D411yescharacteristic
108C4×C8#4111yescharacteristic
109C24—21no—
110C2×Q8—42no—
111C2×Q8—81no—
112C2×Q8—162no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—42no—
116C22×C4—21no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—81no—
120C22×C4—81no—
121C22×C4—81no—
122C22×C4#12311yescharacteristic
123C22×C4C2×SD1611yescharacteristic
124C22×C4AGL(1,ℤ/8)11yescharacteristic
125M16—82no—
126M16—82no—
127C2×C8—22no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—42no—
131C2×C8—82no—
132C2×C8—81no—
133C2×C8—81no—
134C2×C8—81no—
135C2×C8#12111yescharacteristic
136C2×C8#12111yescharacteristic
137C2×C8#12311yescharacteristic
138C2×C8#12411yescharacteristic
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—81no—
144C4⋊C4—81no—
145C22⋊C4—41no—
146C22⋊C4—41no—
147C42#12011yescharacteristic
148C42#12111yescharacteristic
149C42#13511yescharacteristic
150C42AGL(1,ℤ/8)11yescharacteristic
151C16—42no—
152C16—81no—
153C23—41no—
154C23—42no—
155C23—81no—
156C23—81no—
157C23#48111yescharacteristic
158Q8—162no—
159Q8—162no—
160D4—42no—
161D4—81no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—82no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—82no—
172C2×C4—162no—
173C2×C4#39411yescharacteristic
174C2×C4#44911yescharacteristic
175C2×C4#45011yescharacteristic
176C2×C4#46511yescharacteristic
177C2×C4#46911yescharacteristic
178C2×C4#48211yescharacteristic
179C8—21no—
180C8—21no—
181C8—21no—
182C8—82no—
183C8—81no—
184C8—161no—
185C8—161no—
186C8#47112yes—
187C22—21no—
188C22—21no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—162no—
193C22#203411yescharacteristic
194C22#223912yes—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—81no—
201C4—162no—
202C4#204611yescharacteristic
203C4#204911yescharacteristic
204C2—22no—
205C2—161no—
206C2#1044511yescharacteristic
207C2#1339911yescharacteristic
208C2#1340411yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12028C2semidirect
3. #11974C2semidirect
4. #11974C2semidirect
7. #7294C2semidirect

Maximal subgroups

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

subgroupindex
C2×#16952
#120282
#119742
#116362
#86242
#72942