Finite Group Databasebeta

← #144006

Subgroups — #144006

Order 512.

350 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 82 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#19209C211yesmaximal, characteristic
3#19204C211yesmaximal, characteristic
4#13266C211yesmaximal, characteristic
5#11940C211yesmaximal, characteristic
6#11932C211yesmaximal, characteristic
7#7155C211yesmaximal, characteristic
8#3230—21no—
9#3229—21no—
10C2×#500C2211yescharacteristic
11C2×#494C2211yescharacteristic
12#2960C2211yescharacteristic
13#2241—22no—
14#2241—22no—
15#2150C2211yescharacteristic
16C2×#358—22no—
17C2×C4×C16C2211yescharacteristic
18#1696—22no—
19#1690—22no—
20#1676—22no—
21#1670—22no—
22#1564C2211yescharacteristic
23#1564C2211yescharacteristic
24#1548—21no—
25#1546—21no—
26#1379—22no—
27#1364—22no—
28#1362—22no—
29C22×Q16—21no—
30C22×Q16—21no—
31#533D411yescharacteristic
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×M32—21no—
35C22×C16D412yes—
36#500—21no—
37#500—21no—
38#500—21no—
39#500—41no—
40#500D412yes—
41#498—41no—
42#494—21no—
43#494—21no—
44#494—21no—
45#494D412yes—
46#487—41no—
47#487—41no—
48#484—42no—
49#483—42no—
50#477—24no—
51#475—24no—
52#446—21no—
53#446—21no—
54C8×Q8—21no—
55C8×Q8—21no—
56#431—21no—
57#431—21no—
58C2×#107—21no—
59C2×#107—21no—
60#424—21no—
61#423D411yescharacteristic
62#405—21no—
63C2×C4×C8C2311yescharacteristic
64C4⋊C16—21no—
65C4⋊C16—21no—
66#358—28no—
67C4×C16—21no—
68C4×C16—21no—
69#335D412yes—
70#335D412yes—
71#334—21no—
72#334—21no—
73#333—24no—
74#332—24no—
75C22×Q8—41no—
76C22×Q8—41no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×Q16—42no—
82C2×Q16—42no—
83C2×Q16—82no—
84C2×M16—21no—
85C22×C8C2×D411yescharacteristic
86C22×C8#4111yescharacteristic
87C4⋊Q8—22no—
88C4⋊Q8—22no—
89C4⋊Q8—41no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8D811yescharacteristic
96C4⋊Q8D811yescharacteristic
97#125—21no—
98C22⋊Q8—21no—
99C22⋊Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102#117—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C42C2×D411yescharacteristic
108M32—42no—
109C2×C16—22no—
110C2×C16—22no—
111C2×C16—22no—
112C2×C16—41no—
113#107—22no—
114#107—22no—
115#107—44no—
116#107—41no—
117#107—41no—
118#106—44no—
119C4⋊C8—21no—
120C4⋊C8—22no—
121C4⋊C8—21no—
122C4⋊C8—22no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125#103—44no—
126#103—44no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129#97—41no—
130C4×C8—41no—
131C4×C8—41no—
132C4×C8—41no—
133C4×C8C2×D411yescharacteristic
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D411yescharacteristic
136C4×C8C2×D411yescharacteristic
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8—82no—
144C2×Q8—82no—
145C2×Q8—162no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4#12111yescharacteristic
149C22×C4AGL(1,ℤ/8)11yescharacteristic
150C22×C4AGL(1,ℤ/8)11yescharacteristic
151Q16—44no—
152Q16—82no—
153Q16—82no—
154Q16—81no—
155Q16—81no—
156M16—42no—
157C2×C8—22no—
158C2×C8—41no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8#12011yescharacteristic
162C2×C8#12111yescharacteristic
163C2×C8#12111yescharacteristic
164C2×C8#12711yescharacteristic
165C2×C8#13512yes—
166C2×C8#13712yes—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C22⋊C4—41no—
182C42—41no—
183C42#12011yescharacteristic
184C42#12011yescharacteristic
185C42C2×D811yescharacteristic
186C42#13511yescharacteristic
187C16—42no—
188C16—81no—
189C23#46911yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—82no—
193Q8—82no—
194Q8—162no—
195Q8—162no—
196Q8—162no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—162no—
204C2×C4—162no—
205C2×C4#39211yescharacteristic
206C2×C4#44711yescharacteristic
207C2×C4#44711yescharacteristic
208C2×C4#44911yescharacteristic
209C2×C4#44911yescharacteristic
210C2×C4#46412yes—
211C2×C4#46611yescharacteristic
212C2×C4#47112yes—
213C8—21no—
214C8—21no—
215C8—82no—
216C8—82no—
217C8—81no—
218C8#45212yes—
219C8#49511yescharacteristic
220C8#49711yescharacteristic
221C22—21no—
222C22—21no—
223C22#170212yes—
224C22#199911yescharacteristic
225C4—21no—
226C4—21no—
227C4—81no—
228C4—81no—
229C4—162no—
230C4—162no—
231C4#168212yes—
232C4#201511yescharacteristic
233C4#201911yescharacteristic
234C4#219312yes—
235C4#219512yes—
236C2—22no—
237C2#1026311yescharacteristic
238C2#1332011yescharacteristic
239C2#1332211yescharacteristic
subgroupG/H
center, socleC22#1999
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#469
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ 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
#192092
#192042
#132662
#119402
#119322
#71552