Finite Group Databasebeta

← #145296

Subgroups — #145296

Order 512.

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

normal subgroups only

This group has 263 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#19264C211yesmaximal, characteristic
2#12021C211yesmaximal, characteristic, complemented
3#11985C211yesmaximal, characteristic
4#11675C211yesmaximal, characteristic
5#11607C211yesmaximal, characteristic, complemented
6#8623C211yesmaximal, characteristic, complemented
7#7246C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#3112—21no—
10C2×C8×Q8C2211yescharacteristic
11#2150C2211yescharacteristic
12C4×M32C2211yescharacteristic
13#1678—22no—
14#1676—21no—
15#1668—21no—
16#1568C2211yescharacteristic
17#1568C2211yescharacteristic
18#1568C2211yescharacteristic
19#1551—21no—
20#1467—21no—
21#1368—22no—
22#1364—21no—
23#1361—21no—
24Q8⋊C16—21no—
25Q8⋊C16—21no—
26#1331—21no—
27#1331—21no—
28#1290—21no—
29#532—21no—
30C2×C4×Q8D411yescharacteristic
31C2×M32—21no—
32C2×M32—21no—
33#500D411yescharacteristic
34#493D411yescharacteristic
35#486—21no—
36#480—41no—
37#477—22no—
38#466—21no—
39#466—21no—
40#460—22no—
41#448—41no—
42C8×Q8—21no—
43C8×Q8—21no—
44C8×Q8—21no—
45C8×Q8—21no—
46#440—42no—
47C4×D8—21no—
48C4×D8—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8D411yescharacteristic
54C2×C4×C8—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#333—22no—
68#331—22no—
69#324—42no—
70C2×#41—41no—
71C22×Q8SD1611yescharacteristic
72C2×SD16—42no—
73C2×D8—21no—
74C2×D8—21no—
75C22×C8—21no—
76C22×C8#4111yescharacteristic
77C22×C8#4111yescharacteristic
78C4⋊Q8—21no—
79#127—21no—
80#124—41no—
81#121—41no—
82#121—41no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×D4—41no—
88C4×D4—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4SD1611yescharacteristic
91C2×C42—21no—
92C2×C42C2×D411yescharacteristic
93M32—42no—
94M32—81no—
95C2×C16—21no—
96C2×C16—21no—
97C2×C16—41no—
98#107—21no—
99#107—21no—
100#107—42no—
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—
118C4×C8—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—42no—
128C2×Q8—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—81no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—81no—
136C22×C4#12311yescharacteristic
137C22×C4C2×SD1611yescharacteristic
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139SD16—84no—
140D8—41no—
141D8—41no—
142D8—81no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8#12111yescharacteristic
152C2×C8#12111yescharacteristic
153C2×C8#12311yescharacteristic
154C2×C8#12411yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C22⋊C4—81no—
162C22⋊C4—81no—
163C42—41no—
164C42—41no—
165C42#12011yescharacteristic
166C42#12111yescharacteristic
167C42#13511yescharacteristic
168C42AGL(1,ℤ/8)11yescharacteristic
169C16—22no—
170C16—41no—
171C16—81no—
172C23—81no—
173C23#48111yescharacteristic
174Q8—22no—
175Q8—22no—
176Q8—41no—
177Q8—41no—
178Q8—161no—
179D4—82no—
180D4—81no—
181D4—81no—
182D4—161no—
183D4—162no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—162no—
195C2×C4#39411yescharacteristic
196C2×C4#44911yescharacteristic
197C2×C4#45011yescharacteristic
198C2×C4#46511yescharacteristic
199C2×C4#46911yescharacteristic
200C2×C4#48211yescharacteristic
201C8—21no—
202C8—21no—
203C8—21no—
204C8—82no—
205C8—81no—
206C8—161no—
207C8—161no—
208C8#47111yescharacteristic
209C8#47311yescharacteristic
210C22—21no—
211C22—21no—
212C22—21no—
213C22—161no—
214C22—161no—
215C22—161no—
216C22#203411yescharacteristic
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—42no—
223C4—81no—
224C4—161no—
225C4#205011yescharacteristic
226C4#205211yescharacteristic
227C2—41no—
228C2—321no—
229C2#1045911yescharacteristic
230C2#1339511yescharacteristic
231C2#1340411yescharacteristic
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 ◁ C2×C8×Q8 ◁ 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. #11607C2semidirect
6. #8623C2semidirect
7. #7246C2semidirect

Maximal subgroups

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

subgroupindex
#192642
#120212
#119852
#116752
#116072
#86232
#72462