Finite Group Databasebeta

← #145248

Subgroups — #145248

Order 512.

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

normal subgroups only

This group has 251 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#19194C211yesmaximal, characteristic
2#12028C211yesmaximal, characteristic, complemented
3#11973C211yesmaximal, characteristic, complemented
4#11970C211yesmaximal, characteristic, complemented
5#11632C211yesmaximal, characteristic
6#8630C211yesmaximal, characteristic
7#7290C211yesmaximal, characteristic, complemented
8#3246—21no—
9#3160C2211yescharacteristic
10#2931C2211yescharacteristic
11#2163C2211yescharacteristic
12#2162C2211yescharacteristic
13#1695—21no—
14#1693—21no—
15#1671—22no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18C2×#334C2211yescharacteristic
19#1550—21no—
20#1536—21no—
21#1368—22no—
22#1365—21no—
23#1364—21no—
24#1363—22no—
25D4⋊C16—22no—
26#1307—21no—
27C2×#131—21no—
28#533—21no—
29#532D411yescharacteristic
30#500D411yescharacteristic
31#500D411yescharacteristic
32#499—21no—
33#499—21no—
34#498—21no—
35#476—22no—
36#475—41no—
37#465—42no—
38#461—22no—
39#460—41no—
40#436—21no—
41#436—21no—
42C8×D4—21no—
43C8×D4—21no—
44#432—21no—
45#432—21no—
46#427—21no—
47#427—21no—
48#423D411yescharacteristic
49C2×C4⋊C8—21no—
50#408—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#334—21no—
61#334—21no—
62#334—21no—
63#334D412yes—
64#333—22no—
65#332—22no—
66#329—42no—
67C2×#41—21no—
68C2×SD16—42no—
69#131—82no—
70C2×M16—21no—
71C2×M16—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8#4111yescharacteristic
75C22×C8#4111yescharacteristic
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4⋊Q8SD1611yescharacteristic
79#127SD1611yescharacteristic
80#125—41no—
81#124—21no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84#121—21no—
85#121—21no—
86C4×Q8—41no—
87C4×Q8—41no—
88C4×D4—21no—
89C4×D4—21no—
90#117—41no—
91C2×C42C2×D411yescharacteristic
92C2×C16—22no—
93C2×C16—41no—
94#107—21no—
95#107—21no—
96#107—21no—
97#107—21no—
98#106—42no—
99#106—41no—
100#106—41no—
101C4⋊C8—21no—
102C4⋊C8—22no—
103C4⋊C8—21no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107#103—42no—
108#102—42no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112C22⋊C8—41no—
113#97—41no—
114#97—41no—
115C4×C8C2×D411yescharacteristic
116C4×C8C2×D411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8#4111yescharacteristic
119#41—81no—
120#41—81no—
121C2×Q8—21no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—41no—
128C22×C4—41no—
129C22×C4#12311yescharacteristic
130C22×C4#13511yescharacteristic
131C22×C4AGL(1,ℤ/8)11yescharacteristic
132SD16—162no—
133M16—82no—
134M16—81no—
135C2×C8—22no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—42no—
139C2×C8—82no—
140C2×C8—81no—
141C2×C8—81no—
142C2×C8—161no—
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
161C42C2×SD1611yescharacteristic
162C42AGL(1,ℤ/8)11yescharacteristic
163C16—42no—
164C16—81no—
165C23—41no—
166C23#48211yescharacteristic
167Q8—81no—
168Q8—81no—
169Q8—81no—
170Q8—161no—
171D4—42no—
172D4—42no—
173D4—81no—
174D4—82no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4#39411yescharacteristic
185C2×C4#44911yescharacteristic
186C2×C4#45011yescharacteristic
187C2×C4#46511yescharacteristic
188C2×C4#46911yescharacteristic
189C2×C4#48111yescharacteristic
190C8—21no—
191C8—21no—
192C8—21no—
193C8—82no—
194C8—81no—
195C8—161no—
196C8—161no—
197C8#47112yes—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22#203411yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—81no—
211C4—161no—
212C4—161no—
213C4#204611yescharacteristic
214C4#204911yescharacteristic
215C2—41no—
216C2—161no—
217C2#1044511yescharacteristic
218C2#1340011yescharacteristic
219C2#1340611yescharacteristic
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. #12028C2semidirect
3. #11973C2semidirect
4. #11970C2semidirect
7. #7290C2semidirect

Maximal subgroups

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

subgroupindex
#191942
#120282
#119732
#119702
#116322
#86302
#72902