Finite Group Databasebeta

← #145226

Subgroups — #145226

Order 512.

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

normal subgroups only

This group has 290 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#19203C211yesmaximal, characteristic
2#12036C211yesmaximal, characteristic, complemented
3#11961C211yesmaximal, characteristic
4#11961C211yesmaximal, characteristic
5#11638C211yesmaximal, characteristic, complemented
6#8639C211yesmaximal, characteristic, complemented
7#7291C211yesmaximal, characteristic, complemented
8#3208—21no—
9C2×#493C2211yescharacteristic
10#2960C2211yescharacteristic
11#2162C2211yescharacteristic
12#2162C2211yescharacteristic
13#1688—22no—
14#1668—21no—
15#1668—21no—
16#1567C2211yescharacteristic
17#1567C2211yescharacteristic
18#1564C2211yescharacteristic
19#1561—21no—
20#1545—21no—
21#1372—21no—
22#1371—21no—
23#1361—22no—
24#1361—22no—
25Q8⋊C16—22no—
26#1309—21no—
27C22×D8—21no—
28C22×D8—21no—
29#533D411yescharacteristic
30C2×#127—21no—
31#493—21no—
32#493—21no—
33#493—21no—
34#493D412yes—
35#492—41no—
36#486—41no—
37#460—22no—
38#460—22no—
39#450—42no—
40#446—21no—
41#446—21no—
42C8×Q8—21no—
43C8×Q8—21no—
44#431—21no—
45#424—21no—
46#423—21no—
47#423—21no—
48#423D411yescharacteristic
49#405—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#332—41no—
65#332—41no—
66#331—22no—
67#331—22no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4—41no—
71C2×SD16—42no—
72C2×SD16—42no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—41no—
76C2×D8—41no—
77C2×D8—82no—
78C2×D8—82no—
79C2×M16—21no—
80C2×M16—41no—
81C2×M16—41no—
82C22×C8#4111yescharacteristic
83C22×C8#4111yescharacteristic
84C4⋊Q8D811yescharacteristic
85C4⋊Q8D811yescharacteristic
86#127—22no—
87#127—41no—
88#127—41no—
89#127—41no—
90#125—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95#117—21no—
96C2×C42C2×D411yescharacteristic
97C2×C16—22no—
98C2×C16—41no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8—21no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109#102—42no—
110#102—42no—
111C22⋊C8—41no—
112#97—41no—
113C4×C8—41no—
114C4×C8—41no—
115C4×C8C2×D411yescharacteristic
116C4×C8C2×D411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8#4111yescharacteristic
119C24—81no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—81no—
128C2×D4—81no—
129C2×D4—81no—
130C2×D4—161no—
131C2×D4—161no—
132C2×D4—161no—
133C2×D4—161no—
134C22×C4#12311yescharacteristic
135C22×C4#13511yescharacteristic
136C22×C4AGL(1,ℤ/8)11yescharacteristic
137SD16—162no—
138SD16—162no—
139D8—82no—
140D8—82no—
141D8—82no—
142D8—82no—
143M16—81no—
144M16—161no—
145M16—161no—
146C2×C8—22no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8#12111yescharacteristic
154C2×C8#12111yescharacteristic
155C2×C8#12311yescharacteristic
156C2×C8#12411yescharacteristic
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C22⋊C4—41no—
164C42—41no—
165C42#12011yescharacteristic
166C42#12111yescharacteristic
167C42C2×D811yescharacteristic
168C42#13711yescharacteristic
169C16—42no—
170C16—81no—
171C23—82no—
172C23—162no—
173C23—162no—
174C23—162no—
175C23#48211yescharacteristic
176Q8—41no—
177Q8—41no—
178Q8—41no—
179Q8—41no—
180D4—82no—
181D4—82no—
182D4—82no—
183D4—82no—
184D4—162no—
185D4—162no—
186D4—162no—
187D4—161no—
188D4—161no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4#39411yescharacteristic
198C2×C4#44711yescharacteristic
199C2×C4#45211yescharacteristic
200C2×C4#46611yescharacteristic
201C2×C4#46811yescharacteristic
202C2×C4#48011yescharacteristic
203C8—21no—
204C8—21no—
205C8—21no—
206C8—81no—
207C8—81no—
208C8—81no—
209C8—161no—
210C8—161no—
211C8#47112yes—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—162no—
216C22—162no—
217C22—162no—
218C22—324no—
219C22#203311yescharacteristic
220C4—21no—
221C4—21no—
222C4—21no—
223C4—81no—
224C4—81no—
225C4#204611yescharacteristic
226C4#204811yescharacteristic
227C4#219211yescharacteristic
228C4#219411yescharacteristic
229C4#222111yescharacteristic
230C4#222311yescharacteristic
231C2—41no—
232C2—322no—
233C2#1044211yescharacteristic
234C2#1338711yescharacteristic
235C2#1339311yescharacteristic
subgroupG/H
center, socleC22#2033
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC2×#493C22
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. #12036C2semidirect
5. #11638C2semidirect
6. #8639C2semidirect
7. #7291C2semidirect

Maximal subgroups

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

subgroupindex
#192032
#120362
#119612
#116382
#86392
#72912