Finite Group Databasebeta

← #145298

Subgroups — #145298

Order 512.

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

normal subgroups only

This group has 255 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#19341C211yesmaximal, characteristic
2#12021C211yesmaximal, characteristic, complemented
3#11982C211yesmaximal, characteristic
4#11675C211yesmaximal, characteristic
5#11606C211yesmaximal, characteristic, complemented
6#8633C211yesmaximal, characteristic, complemented
7#7242C211yesmaximal, characteristic, complemented
8#3216—21no—
9#3169C2211yescharacteristic
10#2960C2211yescharacteristic
11C2×C4⋊C16C2211yescharacteristic
12C4×M32C2211yescharacteristic
13#1702—22no—
14#1674—21no—
15#1668—21no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1568C2211yescharacteristic
19#1557—21no—
20#1537—21no—
21#1368—22no—
22#1365—21no—
23#1361—21no—
24Q8⋊C16—21no—
25Q8⋊C16—21no—
26#1331—21no—
27#1331—21no—
28#1307—21no—
29#533D411yescharacteristic
30#532—21no—
31C2×M32—21no—
32C22×C16—21no—
33#500D411yescharacteristic
34#497—41no—
35#493D411yescharacteristic
36#489—41no—
37#486—21no—
38#466—21no—
39#466—21no—
40#462—22no—
41#460—22no—
42#452—42no—
43#446—21no—
44#446—21no—
45C8×Q8—21no—
46C8×Q8—21no—
47C4×D8—21no—
48C4×D8—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#424—21no—
53#423D411yescharacteristic
54#405—21no—
55C2×C4×C8C2311yescharacteristic
56C4⋊C16—21no—
57C4⋊C16—21no—
58#346—21no—
59C4×C16D411yescharacteristic
60C4×C16D411yescharacteristic
61#335—21no—
62#335—21no—
63#335—21no—
64#334—21no—
65#334—21no—
66#334D412yes—
67#332—22no—
68#331—22no—
69#330—42no—
70C2×#41—41no—
71C2×Q16—42no—
72C2×SD16—42no—
73C2×D8—21no—
74C2×D8—21no—
75C2×M16—21no—
76C22×C8#4111yescharacteristic
77C22×C8#4111yescharacteristic
78C4⋊Q8—21no—
79C4⋊Q8SD1611yescharacteristic
80C4⋊Q8SD1611yescharacteristic
81#127—21no—
82#125—21no—
83#124—41no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86#121—41no—
87#121—41no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×D4—41no—
91C4×D4—41no—
92#117—21no—
93C2×C42C2×D411yescharacteristic
94M32—42no—
95C2×C16—21no—
96C2×C16—21no—
97C2×C16—41no—
98C2×C16—81no—
99#107—21no—
100#107—21no—
101C4⋊C8—21no—
102C4⋊C8—22no—
103C4⋊C8—21no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108#103—42no—
109#102—42no—
110#102—41no—
111#102—41no—
112C22⋊C8—41no—
113C22⋊C8—41no—
114C22⋊C8—41no—
115#97—41no—
116#97—41no—
117#97—41no—
118#97—41no—
119C4×C8—41no—
120C4×C8C2×D411yescharacteristic
121C4×C8C2×D411yescharacteristic
122C4×C8C2×D411yescharacteristic
123C4×C8#4111yescharacteristic
124#41—162no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—81no—
132C22×C4—81no—
133C22×C4#12311yescharacteristic
134C22×C4#13511yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136Q16—162no—
137SD16—162no—
138D8—41no—
139D8—41no—
140D8—81no—
141M16—81no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8#12111yescharacteristic
150C2×C8#12111yescharacteristic
151C2×C8#12311yescharacteristic
152C2×C8#12411yescharacteristic
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C22⋊C4—41no—
162C22⋊C4—81no—
163C22⋊C4—81no—
164C42—41no—
165C42#12011yescharacteristic
166C42#12111yescharacteristic
167C42C2×SD1611yescharacteristic
168C42AGL(1,ℤ/8)11yescharacteristic
169C16—22no—
170C16—41no—
171C16—81no—
172C23—81no—
173C23#48211yescharacteristic
174Q8—41no—
175Q8—41no—
176Q8—42no—
177Q8—82no—
178D4—82no—
179D4—81no—
180D4—81no—
181D4—161no—
182D4—162no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—162no—
191C2×C4#39411yescharacteristic
192C2×C4#44911yescharacteristic
193C2×C4#45011yescharacteristic
194C2×C4#46511yescharacteristic
195C2×C4#46911yescharacteristic
196C2×C4#48111yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—82no—
201C8—81no—
202C8—161no—
203C8—161no—
204C8#47111yescharacteristic
205C8#47311yescharacteristic
206C22—21no—
207C22—21no—
208C22—21no—
209C22—161no—
210C22—161no—
211C22—161no—
212C22#203411yescharacteristic
213C4—21no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—81no—
219C4—81no—
220C4—161no—
221C4#205011yescharacteristic
222C4#205211yescharacteristic
223C2—41no—
224C2—321no—
225C2#1045911yescharacteristic
226C2#1339611yescharacteristic
227C2#1340611yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
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. #12021C2semidirect
5. #11606C2semidirect
6. #8633C2semidirect
7. #7242C2semidirect

Maximal subgroups

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

subgroupindex
#193412
#120212
#119822
#116752
#116062
#86332
#72422