Finite Group Databasebeta

← #141525

Subgroups — #141525

Order 512.

296 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2241C211yesmaximal, characteristic
2#12022C211yesmaximal, characteristic
3#11972C211yesmaximal, characteristic
4#11962C211yesmaximal, characteristic
5C16×M16C211yesmaximal, characteristic
6#9801C211yesmaximal, characteristic
7#9798C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#494C2211yescharacteristic
10#2241—21no—
11#2241—21no—
12#2241—22no—
13#2162C2211yescharacteristic
14C2×#358C414yes—
15C2×C4×C16C2211yescharacteristic
16#1997—21no—
17#1997—21no—
18#1567C2211yescharacteristic
19#1567C2211yescharacteristic
20#1549—21no—
21#1548—21no—
22C8×M16C2211yescharacteristic
23#1371—22no—
24#1365—22no—
25#1362—22no—
26#1312—21no—
27C8×C16—21no—
28C22×Q16—21no—
29C22×Q16C2×C412yes—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C22×C16C2×C412yes—
33#500—21no—
34#500—21no—
35#500—21no—
36#500—21no—
37#494—21no—
38#494—21no—
39#494—21no—
40#494—21no—
41#431D411yescharacteristic
42C2×#107—21no—
43C2×#107C2×C412yes—
44#423—21no—
45#423—21no—
46C4×M16D411yescharacteristic
47C2×C4×C8C2311yescharacteristic
48C4⋊C16D412yes—
49#358—24no—
50#358—24no—
51C22⋊C16—21no—
52C4×C16—21no—
53C4×C16D411yescharacteristic
54C4×C16D411yescharacteristic
55C4×C16D412yes—
56#343—42no—
57#343—42no—
58#335—21no—
59#335—21no—
60#335—21no—
61#335—21no—
62#333—42no—
63#332—42no—
64#322—21no—
65C82—21no—
66C22×Q8—22no—
67C22×Q8—41no—
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—44no—
71C2×Q16—41no—
72C2×Q16—41no—
73C2×Q16—82no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16D812yes—
77C22×C8C22×C411yescharacteristic
78C22×C8#4111yescharacteristic
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C42C2×D411yescharacteristic
92C2×C16—22no—
93C2×C16—42no—
94C2×C16C22⋊C412yes—
95C2×C16C22⋊C412yes—
96#107—22no—
97#107—22no—
98#107—41no—
99#107—41no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8D812yes—
105C22⋊C8—21no—
106#97—21no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8C2×D411yescharacteristic
110C4×C8C2×D411yescharacteristic
111C4×C8C2×D411yescharacteristic
112C4×C8#4111yescharacteristic
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—82no—
116C2×Q8—82no—
117C2×Q8—81no—
118C2×Q8—81no—
119C2×Q8—162no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4C4×D411yescharacteristic
123C22×C4C2×D811yescharacteristic
124C22×C4#13511yescharacteristic
125Q16—44no—
126Q16—82no—
127Q16—82no—
128Q16—82no—
129M16—22no—
130M16—82no—
131M16—82no—
132M16Q3214yes—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—81no—
138C2×C8—81no—
139C2×C8C2×C22⋊C411yescharacteristic
140C2×C8C4×D411yescharacteristic
141C2×C8#12111yescharacteristic
142C2×C8#12411yescharacteristic
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—81no—
146C4⋊C4—81no—
147C4⋊C4—81no—
148C4⋊C4—81no—
149C4⋊C4—81no—
150C4⋊C4—81no—
151C42#12111yescharacteristic
152C42#12711yescharacteristic
153C42C2×D811yescharacteristic
154C42#13511yescharacteristic
155C16—22no—
156C16—22no—
157C16—42no—
158C23C4×D811yescharacteristic
159Q8—82no—
160Q8—82no—
161Q8—162no—
162Q8—162no—
163Q8—162no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—162no—
170C2×C4—162no—
171C2×C4#39011yescharacteristic
172C2×C4C4×D811yescharacteristic
173C2×C4#46611yescharacteristic
174C2×C4#46611yescharacteristic
175C2×C4#49311yescharacteristic
176C2×C4#49511yescharacteristic
177C2×C4C2×Q3212yes—
178C2×C4#50812yes—
179C8—24no—
180C8—21no—
181C8—21no—
182C8—42no—
183C8—161no—
184C8—161no—
185C8#41312yes—
186C8#47112yes—
187C22—21no—
188C22—21no—
189C22#197011yescharacteristic
190C22C4×Q3212yes—
191C4—21no—
192C4—21no—
193C4—162no—
194C4—162no—
195C4#196711yescharacteristic
196C4#197711yescharacteristic
197C4#221612yes—
198C4#222312yes—
199C4#224711yescharacteristic
200C4#224911yescharacteristic
201C2—22no—
202C2#979611yescharacteristic
203C2#1323611yescharacteristic
204C2#1324611yescharacteristic
subgroupG/H
center, socleC22#1970
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23C4×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ 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×#22412
#120222
#119722
#119622
C16×M162
#98012
#97982