Finite Group Databasebeta

← #125682

Subgroups — #125682

Order 512.

350 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 64 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#32460C211yesmaximal, characteristic
2#13409C212yesmaximal, complemented
3#10274C211yesmaximal, characteristic
4#7121C212yesmaximal, complemented
5#7102C211yesmaximal, characteristic
6C2×#494C2211yescharacteristic
7C2×#474—21no—
8#2487—21no—
9C2×#358—22no—
10C2×#358—22no—
11#2087—21no—
12#2076—22no—
13#1954C2211yescharacteristic
14#1942C2212yes—
15#1852—22no—
16C2×#332—21no—
17#1374C2212yes—
18#1277C2211yescharacteristic
19C22×Q16—22no—
20C2×C4⋊Q8—21no—
21C2×C4×D4D411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C16—22no—
24#494—42no—
25#474—42no—
26#474—42no—
27C2×#107—21no—
28C2×#107—41no—
29C2×#107D412yes—
30C2×#107D412yes—
31C2×C4⋊C8—21no—
32C2×C4⋊C8D411yescharacteristic
33C2×#102—41no—
34C2×C22⋊C8—22no—
35C2×C4×C8C2311yescharacteristic
36#400—41no—
37#399—41no—
38#389—22no—
39#386—41no—
40#385—41no—
41#380—21no—
42#380—22no—
43#358—44no—
44#358—44no—
45#340—21no—
46#340—22no—
47#340—22no—
48#332—42no—
49C22×Q8—41no—
50C22×D4D811yescharacteristic
51C23×C4—21no—
52C23×C4—41no—
53C2×Q16—24no—
54C2×Q16—42no—
55C2×Q16—42no—
56C2×Q16—42no—
57C2×Q16—42no—
58C2×Q16—42no—
59C22×C8—21no—
60C22×C8—41no—
61C22×C8C2×D412yes—
62C4⋊Q8—22no—
63C4⋊Q8—41no—
64C4⋊Q8—41no—
65C4⋊Q8—41no—
66C4×D4—41no—
67C4×D4—41no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—81no—
75C2×C4⋊C4—81no—
76C2×C4⋊C4D811yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—42no—
80C2×C42C2×D411yescharacteristic
81C2×C16—42no—
82C2×C16—42no—
83C2×C16—42no—
84#107—22no—
85#107—22no—
86#107—44no—
87#107—42no—
88#107—81no—
89C4⋊C8—22no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92#102—44no—
93C22⋊C8—42no—
94C22⋊C8—42no—
95C4×C8—41no—
96#95—42no—
97#95—41no—
98#95—42no—
99C24—21no—
100C2×Q8—42no—
101C2×Q8—81no—
102C2×Q8—82no—
103C2×Q8—82no—
104C2×D4—21no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4Q3212yes—
108C22×C4—21no—
109C22×C4—42no—
110C22×C4—41no—
111C22×C4—42no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—82no—
116C22×C4—81no—
117C22×C4—162no—
118C22×C4#12011yescharacteristic
119C22×C4C2×D811yescharacteristic
120C22×C4AGL(1,ℤ/8)11yescharacteristic
121Q16—44no—
122Q16—44no—
123Q16—82no—
124Q16—82no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—41no—
130C2×C8—81no—
131C2×C8—81no—
132C2×C8#13512yes—
133C2×C8AGL(1,ℤ/8)12yes—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—81no—
142C4⋊C4—81no—
143C4⋊C4—82no—
144C4⋊C4—162no—
145C22⋊C4—81no—
146C22⋊C4—82no—
147C22⋊C4—162no—
148C42—21no—
149C42—21no—
150C16—82no—
151C23—22no—
152C23—41no—
153C23—42no—
154C23—41no—
155C23—42no—
156C23#44711yescharacteristic
157Q8—84no—
158Q8—161no—
159Q8—161no—
160D4—42no—
161D4—81no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—82no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—164no—
179C2×C4—162no—
180C2×C4#44912yes—
181C2×C4#45311yescharacteristic
182C2×C4#45412yes—
183C2×C4B(4,𝔽2)11yescharacteristic
184C2×C4C2×Q3211yescharacteristic
185C2×C4#51011yescharacteristic
186C8—22no—
187C8—22no—
188C8—82no—
189C8—81no—
190C22—44no—
191C22—42no—
192C22—81no—
193C22—82no—
194C22—81no—
195C22#159511yescharacteristic
196C22#162512yes—
197C22#218811yescharacteristic
198C22#219111yescharacteristic
199C22#219412yes—
200C4—21no—
201C4—21no—
202C4—41no—
203C4—81no—
204C4—162no—
205C4—162no—
206C4—162no—
207C4#159712yes—
208C4#162011yescharacteristic
209C4#165711yescharacteristic
210C2—82no—
211C2#773611yescharacteristic
212C2#1210611yescharacteristic
213C2#1211111yescharacteristic
214C2#1234612yes—
215C2#1235012yes—
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #13409C2semidirect
4. #7121C2semidirect

Maximal subgroups

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

subgroupindex
#324602
#134092
#102742
#71212
#71022