Finite Group Databasebeta

← #145222

Subgroups — #145222

Order 512.

279 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 72 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×#1670C211yesmaximal, characteristic
2#12032C211yesmaximal, characteristic
3#11962C211yesmaximal, characteristic
4#11962C211yesmaximal, characteristic
5#11637C211yesmaximal, characteristic
6#8617C211yesmaximal, characteristic
7#7288C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9C2×#475—21no—
10C2×C8×Q8C2211yescharacteristic
11#2162C2211yescharacteristic
12#2162C2211yescharacteristic
13#1801—21no—
14#1670—22no—
15#1670—22no—
16#1567C2211yescharacteristic
17#1567C2211yescharacteristic
18C2×#335C2211yescharacteristic
19C2×#332—21no—
20#1371—22no—
21#1362—22no—
22#1362—22no—
23Q8⋊C16—22no—
24#1292—21no—
25C22×Q16—21no—
26C22×Q16—21no—
27C2×C4⋊Q8—21no—
28C2×C4×Q8D411yescharacteristic
29#494—21no—
30#494—21no—
31#494—21no—
32#494D412yes—
33#475—24no—
34#475—42no—
35C8×Q8—21no—
36C8×Q8—21no—
37C8×Q8—21no—
38C8×Q8—21no—
39C2×#106—41no—
40#423—21no—
41#423—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#103—41no—
46C2×#101—41no—
47C2×C4×C8—21no—
48C2×C4×C8C2311yescharacteristic
49C4⋊C16—21no—
50C4⋊C16D412yes—
51C22⋊C16—21no—
52C22⋊C16—21no—
53C4×C16—21no—
54C4×C16—21no—
55#343—41no—
56#335—21no—
57#335—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#335D412yes—
62#332—24no—
63C22×Q8—41no—
64C22×Q8D811yescharacteristic
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—82no—
70C2×Q16—82no—
71C2×M16—41no—
72C2×M16—41no—
73C22×C8—21no—
74C22×C8—41no—
75C22×C8#4111yescharacteristic
76C22×C8#4111yescharacteristic
77C4⋊Q8—22no—
78C4⋊Q8—41no—
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4×Q8—21no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4D811yescharacteristic
89C2×C42—21no—
90C2×C42C2×D411yescharacteristic
91C2×C16—22no—
92C2×C16—41no—
93#106—44no—
94C4⋊C8—21no—
95C4⋊C8—22no—
96C4⋊C8—21no—
97C4⋊C8—41no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104#103—44no—
105#101—82no—
106C4×C8—41no—
107C4×C8—41no—
108C4×C8C2×D411yescharacteristic
109C4×C8C2×D411yescharacteristic
110C4×C8C2×D411yescharacteristic
111C4×C8#4111yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8—82no—
119C2×Q8—82no—
120C2×Q8—81no—
121C2×Q8Q3212yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—81no—
125C22×C4#12311yescharacteristic
126C22×C4C2×D811yescharacteristic
127C22×C4#13711yescharacteristic
128Q16—82no—
129Q16—82no—
130Q16—82no—
131Q16—82no—
132M16—81no—
133M16—81no—
134M16—81no—
135M16—81no—
136C2×C8—22no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—42no—
140C2×C8—81no—
141C2×C8—81no—
142C2×C8—81no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8#12111yescharacteristic
146C2×C8#12111yescharacteristic
147C2×C8#12311yescharacteristic
148C2×C8#12411yescharacteristic
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C42—41no—
158C42—41no—
159C42#12011yescharacteristic
160C42#12111yescharacteristic
161C42#13511yescharacteristic
162C42AGL(1,ℤ/8)11yescharacteristic
163C16—42no—
164C16—81no—
165C23#48011yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—82no—
171Q8—82no—
172Q8—162no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—81no—
181C2×C4—82no—
182C2×C4—162no—
183C2×C4#39411yescharacteristic
184C2×C4#44711yescharacteristic
185C2×C4#45211yescharacteristic
186C2×C4#46611yescharacteristic
187C2×C4#46811yescharacteristic
188C2×C4#48211yescharacteristic
189C2×C4C2×Q3211yescharacteristic
190C2×C4#50811yescharacteristic
191C2×C4#50911yescharacteristic
192C2×C4#51011yescharacteristic
193C8—21no—
194C8—21no—
195C8—21no—
196C8—81no—
197C8—82no—
198C8—161no—
199C8—161no—
200C8#47112yes—
201C22—21no—
202C22—21no—
203C22#203311yescharacteristic
204C22#223611yescharacteristic
205C22#223711yescharacteristic
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—81no—
211C4—162no—
212C4#204611yescharacteristic
213C4#204811yescharacteristic
214C4#218611yescharacteristic
215C4#218811yescharacteristic
216C4#221611yescharacteristic
217C4#222011yescharacteristic
218C2—21no—
219C2—21no—
220C2#1044211yescharacteristic
221C2#1338311yescharacteristic
222C2#1339011yescharacteristic
subgroupG/H
center, socleC22#2033
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#480
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ 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×#16702
#120322
#119622
#116372
#86172
#72882