Finite Group Databasebeta

← #145438

Subgroups — #145438

Order 512.

267 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#1696C211yesmaximal, characteristic
2#13290C211yesmaximal, characteristic
3#13268C211yesmaximal, characteristic
4#12000C211yesmaximal, characteristic
5#11940C211yesmaximal, characteristic
6#8663C211yesmaximal, characteristic
7#7280C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#477—21no—
10C2×#446C2211yescharacteristic
11#2150C2211yescharacteristic
12C4×M32C2211yescharacteristic
13#1848C2211yescharacteristic
14#1696—22no—
15#1696—22no—
16#1564C2211yescharacteristic
17C2×#334C2211yescharacteristic
18C2×#333—21no—
19#1487—21no—
20#1387—21no—
21#1387—21no—
22#1377—21no—
23#1377—21no—
24#1368—22no—
25#1364—22no—
26#1301—21no—
27C2×C4⋊Q8—21no—
28C2×C4×Q8D411yescharacteristic
29C2×M32—21no—
30C2×M32D411yescharacteristic
31C2×M32D411yescharacteristic
32#500—21no—
33#500—21no—
34#500—21no—
35#500D412yes—
36#477—24no—
37#477—42no—
38#446—21no—
39#446—21no—
40#446—21no—
41#446—21no—
42#431—21no—
43C2×#108—21no—
44C2×#108—21no—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107—41no—
48#423—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D411yescharacteristic
52C2×#103—41no—
53C2×#97—21no—
54C2×C4×C8C2311yescharacteristic
55C4⋊C16—21no—
56C4⋊C16—21no—
57#346—21no—
58C4×C16—21no—
59#335—21no—
60#334—21no—
61#334—21no—
62#334D412yes—
63#333—24no—
64Q8⋊C8—42no—
65C22×Q8—41no—
66C22×Q8SD1611yescharacteristic
67C2×M16—41no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8C2×D411yescharacteristic
71C22×C8#4111yescharacteristic
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75C4⋊Q8—41no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×Q8—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4SD1611yescharacteristic
84C2×C42—21no—
85C2×C42C2×D411yescharacteristic
86M32—21no—
87M32—21no—
88M32—21no—
89M32—41no—
90M32—41no—
91M32Q1612yes—
92C2×C16—21no—
93C2×C16—21no—
94C2×C16—41no—
95#108—41no—
96#108—41no—
97#108—81no—
98#107—22no—
99#107—22no—
100#107—44no—
101#107—41no—
102#107—41no—
103C4⋊C8—21no—
104C4⋊C8—22no—
105C4⋊C8—21no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112#103—44no—
113C22⋊C8—41no—
114#97—41no—
115#97—41no—
116C4×C8—41no—
117C4×C8C2×D411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120C4×C8#4111yescharacteristic
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—81no—
126C2×Q8—162no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—81no—
130C22×C4#12111yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132C22×C4#13511yescharacteristic
133M16—81no—
134M16—81no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—41no—
138C2×C8—42no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—82no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8#12111yescharacteristic
145C2×C8#12111yescharacteristic
146C2×C8#12311yescharacteristic
147C2×C8#12411yescharacteristic
148C2×C8C2×Q1611yescharacteristic
149C2×C8#13511yescharacteristic
150C2×C8#13711yescharacteristic
151C2×C8#13711yescharacteristic
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C42—41no—
161C42—41no—
162C42#12011yescharacteristic
163C42#12311yescharacteristic
164C42AGL(1,ℤ/8)11yescharacteristic
165C42AGL(1,ℤ/8)11yescharacteristic
166C16—22no—
167C16—41no—
168C16—81no—
169C23#46511yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—162no—
173Q8—162no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—82no—
181C2×C4—162no—
182C2×C4#39411yescharacteristic
183C2×C4#44911yescharacteristic
184C2×C4#45011yescharacteristic
185C2×C4#46211yescharacteristic
186C2×C4#46411yescharacteristic
187C2×C4#46911yescharacteristic
188C2×C4#47111yescharacteristic
189C2×C4#47311yescharacteristic
190C2×C4#48111yescharacteristic
191C2×C4#48211yescharacteristic
192C8—21no—
193C8—21no—
194C8—82no—
195C8—81no—
196C8—161no—
197C8—161no—
198C8#46211yescharacteristic
199C8#46411yescharacteristic
200C8#46711yescharacteristic
201C8#47011yescharacteristic
202C22—21no—
203C22—21no—
204C22#167211yescharacteristic
205C22#168011yescharacteristic
206C22#203411yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—81no—
211C4—81no—
212C4—162no—
213C4#169411yescharacteristic
214C4#170211yescharacteristic
215C4#205211yescharacteristic
216C4#205611yescharacteristic
217C4#223912yes—
218C2—21no—
219C2—21no—
220C2#1047911yescharacteristic
221C2#1339911yescharacteristic
222C2#1340211yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#465
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#16962
#132902
#132682
#120002
#119402
#86632
#72802