Finite Group Databasebeta

← #145233

Subgroups — #145233

Order 512.

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

normal subgroups only

This group has 261 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#19205C211yesmaximal, characteristic
2#12028C211yesmaximal, characteristic, complemented
3#11963C211yesmaximal, characteristic
4#11960C211yesmaximal, characteristic
5#11640C211yesmaximal, characteristic, complemented
6#8631C211yesmaximal, characteristic, complemented
7#7290C211yesmaximal, characteristic, complemented
8#3246—21no—
9#3147C2211yescharacteristic
10#2960C2211yescharacteristic
11#2163C2211yescharacteristic
12#2162C2211yescharacteristic
13#1690—21no—
14#1688—21no—
15#1666—22no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18C2×#334C2211yescharacteristic
19#1547—21no—
20#1537—21no—
21#1368—22no—
22#1362—21no—
23#1361—21no—
24#1360—22no—
25Q8⋊C16—22no—
26#1307—21no—
27C2×#135—21no—
28C2×#135—21no—
29#533D411yescharacteristic
30#532—21no—
31#495—21no—
32#495—21no—
33#494D411yescharacteristic
34#493D411yescharacteristic
35#492—21no—
36#476—41no—
37#475—22no—
38#465—42no—
39#461—41no—
40#460—22no—
41#446—21no—
42#446—21no—
43C8×Q8—21no—
44C8×Q8—21no—
45#432—21no—
46#432—21no—
47#424—21no—
48#423D411yescharacteristic
49C2×C4⋊C8—21no—
50#405—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#332—22no—
65#331—22no—
66#330—42no—
67C2×#41—41no—
68#135—82no—
69#135—82no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×SD16—42no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×D8—21no—
76C2×D8—21no—
77C2×M16—21no—
78C22×C8—41no—
79C22×C8#4111yescharacteristic
80C22×C8#4111yescharacteristic
81C4⋊Q8—21no—
82C4⋊Q8SD1611yescharacteristic
83C4⋊Q8SD1611yescharacteristic
84#127—21no—
85#125—21no—
86#124—41no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89#121—41no—
90#121—41no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×D4—41no—
94C4×D4—41no—
95#117—21no—
96C2×C42C2×D411yescharacteristic
97C2×C16—22no—
98C2×C16—41no—
99#106—42no—
100C4⋊C8—21no—
101C4⋊C8—22no—
102C4⋊C8—21no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108#103—42no—
109#102—42no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112#97—41no—
113#97—41no—
114#97—41no—
115C4×C8—41no—
116C4×C8C2×D411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120#41—82no—
121#41—82no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—81no—
129C22×C4—81no—
130C22×C4#12311yescharacteristic
131C22×C4#13511yescharacteristic
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133Q16—81no—
134Q16—81no—
135SD16—82no—
136SD16—82no—
137SD16—162no—
138D8—81no—
139D8—81no—
140M16—81no—
141C2×C8—22no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—42no—
145C2×C8—81no—
146C2×C8—81no—
147C2×C8—161no—
148C2×C8#12111yescharacteristic
149C2×C8#12111yescharacteristic
150C2×C8#12311yescharacteristic
151C2×C8#12411yescharacteristic
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C22⋊C4—41no—
161C22⋊C4—81no—
162C22⋊C4—81no—
163C42—41no—
164C42#12011yescharacteristic
165C42#12111yescharacteristic
166C42C2×SD1611yescharacteristic
167C42AGL(1,ℤ/8)11yescharacteristic
168C16—42no—
169C16—81no—
170C23—81no—
171C23#48211yescharacteristic
172Q8—42no—
173Q8—42no—
174Q8—81no—
175Q8—81no—
176D4—81no—
177D4—81no—
178D4—81no—
179D4—81no—
180D4—161no—
181D4—162no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—81no—
189C2×C4—162no—
190C2×C4#39411yescharacteristic
191C2×C4#44911yescharacteristic
192C2×C4#45011yescharacteristic
193C2×C4#46511yescharacteristic
194C2×C4#46911yescharacteristic
195C2×C4#48111yescharacteristic
196C8—21no—
197C8—21no—
198C8—21no—
199C8—82no—
200C8—81no—
201C8—161no—
202C8—161no—
203C8#47112yes—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—161no—
208C22—161no—
209C22—161no—
210C22#203411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—81no—
217C4—81no—
218C4—161no—
219C4#204611yescharacteristic
220C4#204811yescharacteristic
221C2—41no—
222C2—321no—
223C2#1044311yescharacteristic
224C2#1340011yescharacteristic
225C2#1340611yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3147C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ 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
5. #11640C2semidirect
6. #8631C2semidirect
7. #7290C2semidirect

Maximal subgroups

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

subgroupindex
#192052
#120282
#119632
#119602
#116402
#86312
#72902