Finite Group Databasebeta

← #145243

Subgroups — #145243

Order 512.

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

normal subgroups only

This group has 266 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#19105C211yesmaximal, characteristic
2#12031C211yesmaximal, characteristic, complemented
3#11973C211yesmaximal, characteristic, complemented
4#11970C211yesmaximal, characteristic, complemented
5#11631C211yesmaximal, characteristic
6#8620C211yesmaximal, characteristic
7#7289C211yesmaximal, characteristic, complemented
8#3160C2211yescharacteristic
9#3101—21no—
10C2×C8×D4C2211yescharacteristic
11#2163C2211yescharacteristic
12#2162C2211yescharacteristic
13#1675—21no—
14#1673—21no—
15#1671—22no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1564C2211yescharacteristic
19#1542—21no—
20#1466—21no—
21#1368—22no—
22#1365—21no—
23#1364—21no—
24#1363—22no—
25D4⋊C16—22no—
26#1290—21no—
27C23×C8—21no—
28#533—21no—
29C2×C4×D4D411yescharacteristic
30#500D411yescharacteristic
31#500D411yescharacteristic
32#499—21no—
33#499—21no—
34#498—21no—
35#483—41no—
36#476—22no—
37#461—22no—
38#450—41no—
39C4×SD16—42no—
40C8×D4—21no—
41C8×D4—21no—
42C8×D4—21no—
43C8×D4—21no—
44#432—21no—
45#432—21no—
46#431—21no—
47#427—21no—
48#427—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C22⋊C8—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16D412yes—
54C22⋊C16—21no—
55C22⋊C16—21no—
56#346—21no—
57C4×C16—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#334—21no—
62#334—21no—
63#334D412yes—
64#333—22no—
65#332—22no—
66#323—42no—
67C22×D4SD1611yescharacteristic
68C23×C4—21no—
69C2×SD16—42no—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—82no—
74C22×C8#4111yescharacteristic
75C22×C8#4111yescharacteristic
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78#125—41no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81C4×Q8—41no—
82C4×Q8—41no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4—21no—
86C4×D4—21no—
87#117—41no—
88C2×C4⋊C4SD1611yescharacteristic
89C2×C22⋊C4—21no—
90C2×C42C2×D411yescharacteristic
91C2×C16—22no—
92C2×C16—41no—
93#107—21no—
94#107—21no—
95#107—21no—
96#107—21no—
97#106—42no—
98#106—41no—
99#106—41no—
100C4⋊C8—21no—
101C4⋊C8—22no—
102C4⋊C8—21no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105#103—42no—
106#102—42no—
107C22⋊C8—41no—
108C22⋊C8—41no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112#97—41no—
113#97—41no—
114C4×C8—41no—
115C4×C8C2×D411yescharacteristic
116C4×C8C2×D411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8#4111yescharacteristic
119C24—21no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—42no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4#12311yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133SD16—84no—
134C2×C8—22no—
135C2×C8—41no—
136C2×C8—41no—
137C2×C8—42no—
138C2×C8—82no—
139C2×C8—82no—
140C2×C8—81no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8#12111yescharacteristic
144C2×C8#12111yescharacteristic
145C2×C8#12311yescharacteristic
146C2×C8#12411yescharacteristic
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C22⋊C4—41no—
156C22⋊C4—41no—
157C22⋊C4—81no—
158C42—81no—
159C42#12011yescharacteristic
160C42#12111yescharacteristic
161C42#13511yescharacteristic
162C42AGL(1,ℤ/8)11yescharacteristic
163C16—42no—
164C16—81no—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—81no—
169C23—81no—
170C23#48111yescharacteristic
171Q8—82no—
172Q8—161no—
173D4—22no—
174D4—22no—
175D4—41no—
176D4—41no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—82no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4#39411yescharacteristic
189C2×C4#44911yescharacteristic
190C2×C4#45011yescharacteristic
191C2×C4#46511yescharacteristic
192C2×C4#46911yescharacteristic
193C2×C4#48211yescharacteristic
194C8—21no—
195C8—21no—
196C8—21no—
197C8—82no—
198C8—81no—
199C8—161no—
200C8—161no—
201C8#47112yes—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—42no—
206C22—82no—
207C22—82no—
208C22—81no—
209C22—81no—
210C22#203411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—81no—
217C4—161no—
218C4—161no—
219C4#204611yescharacteristic
220C4#204911yescharacteristic
221C2—41no—
222C2—82no—
223C2#1044511yescharacteristic
224C2#1339511yescharacteristic
225C2#1340111yescharacteristic
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×C8×D4 ◁ 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. #11973C2semidirect
4. #11970C2semidirect
7. #7289C2semidirect

Maximal subgroups

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

subgroupindex
#191052
#120312
#119732
#119702
#116312
#86202
#72892