Finite Group Databasebeta

← #145241

Subgroups — #145241

Order 512.

265 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 64 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#19192C211yesmaximal, characteristic
2#12036C211yesmaximal, characteristic, complemented
3#11972C211yesmaximal, characteristic, complemented
4#11971C211yesmaximal, characteristic, complemented
5#11630C211yesmaximal, characteristic
6#8638C211yesmaximal, characteristic
7#7291C211yesmaximal, characteristic, complemented
8#3206—21no—
9C2×#500C2211yescharacteristic
10#2931C2211yescharacteristic
11#2162C2211yescharacteristic
12#2162C2211yescharacteristic
13#1693—22no—
14#1673—21no—
15#1673—21no—
16#1567C2211yescharacteristic
17#1567C2211yescharacteristic
18#1564C2211yescharacteristic
19#1560—21no—
20#1549—21no—
21#1372—21no—
22#1371—21no—
23#1365—22no—
24#1364—22no—
25D4⋊C16—22no—
26#1309—21no—
27C2×#131—21no—
28#532D411yescharacteristic
29C2×C4⋊Q8—21no—
30#500—21no—
31#500—21no—
32#500—21no—
33#500D412yes—
34#492—41no—
35#487—41no—
36#461—22no—
37#461—22no—
38#448—42no—
39#436—21no—
40#436—21no—
41C8×D4—21no—
42C8×D4—21no—
43#431—21no—
44C2×#107—21no—
45C2×#107—21no—
46#423—21no—
47#423—21no—
48#423D411yescharacteristic
49#408—21no—
50C2×C4×C8C2311yescharacteristic
51C4⋊C16—21no—
52C4⋊C16D412yes—
53C22⋊C16—21no—
54C22⋊C16—21no—
55C4×C16—21no—
56C4×C16—21no—
57#335—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#335D411yescharacteristic
62#335D411yescharacteristic
63#334—21no—
64#333—41no—
65#332—22no—
66#332—22no—
67#331—41no—
68C2×#41—21no—
69C22×Q8—41no—
70C2×SD16—42no—
71C2×SD16—42no—
72#131—82no—
73C2×M16—21no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C22×C8—41no—
78C22×C8#4111yescharacteristic
79C22×C8#4111yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8D811yescharacteristic
85#127D811yescharacteristic
86#124—21no—
87#121—21no—
88#121—21no—
89C4×D4—21no—
90C4×D4—21no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C42C2×D411yescharacteristic
94C2×C16—22no—
95C2×C16—41no—
96#107—22no—
97#107—22no—
98#107—41no—
99#107—41no—
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8—21no—
103C4⋊C8—21no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108#103—42no—
109#103—42no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112C22⋊C8—41no—
113C4×C8—41no—
114C4×C8C2×D411yescharacteristic
115C4×C8C2×D411yescharacteristic
116C4×C8C2×D411yescharacteristic
117C4×C8#4111yescharacteristic
118#41—81no—
119#41—81no—
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8—81no—
123C2×Q8—161no—
124C2×Q8—161no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C22×C4—41no—
130C22×C4—81no—
131C22×C4#12311yescharacteristic
132C22×C4#13511yescharacteristic
133C22×C4AGL(1,ℤ/8)11yescharacteristic
134SD16—162no—
135SD16—162no—
136M16—82no—
137M16—81no—
138M16—161no—
139M16—161no—
140C2×C8—22no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—82no—
146C2×C8—81no—
147C2×C8—81no—
148C2×C8#12111yescharacteristic
149C2×C8#12111yescharacteristic
150C2×C8#12311yescharacteristic
151C2×C8#12411yescharacteristic
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C42#12011yescharacteristic
161C42#12111yescharacteristic
162C42C2×D811yescharacteristic
163C42#13711yescharacteristic
164C16—42no—
165C16—81no—
166C23—41no—
167C23#48211yescharacteristic
168Q8—81no—
169Q8—82no—
170Q8—82no—
171Q8—162no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—41no—
176D4—81no—
177D4—82no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—82no—
188C2×C4—162no—
189C2×C4#39411yescharacteristic
190C2×C4#44711yescharacteristic
191C2×C4#45211yescharacteristic
192C2×C4#46611yescharacteristic
193C2×C4#46811yescharacteristic
194C2×C4#48011yescharacteristic
195C8—21no—
196C8—21no—
197C8—21no—
198C8—81no—
199C8—81no—
200C8—81no—
201C8—161no—
202C8—161no—
203C8#47112yes—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22#203311yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—81no—
215C4—162no—
216C4#204611yescharacteristic
217C4#204911yescharacteristic
218C4#219211yescharacteristic
219C4#219411yescharacteristic
220C4#222211yescharacteristic
221C4#222311yescharacteristic
222C2—41no—
223C2—161no—
224C2#1044411yescharacteristic
225C2#1338711yescharacteristic
226C2#1339411yescharacteristic
subgroupG/H
center, socleC22#2033
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. #12036C2semidirect
3. #11972C2semidirect
4. #11971C2semidirect
7. #7291C2semidirect

Maximal subgroups

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

subgroupindex
#191922
#120362
#119722
#119712
#116302
#86382
#72912