Finite Group Databasebeta

← #143996

Subgroups — #143996

Order 512.

389 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 96 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
2C2×#1674C211yesmaximal, characteristic
3C2×#1670C211yesmaximal, characteristic
4#13263C211yesmaximal, characteristic
5C2×#1364C211yesmaximal, characteristic
6C2×#1362C211yesmaximal, characteristic
7#7154C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#494C2211yescharacteristic
10C2×#475—21no—
11C2×#462—21no—
12C2×C8×Q8C2211yescharacteristic
13#2241—22no—
14#2241—22no—
15C2×C4⋊C16C2211yescharacteristic
16C2×#358—22no—
17C2×C4×C16C2211yescharacteristic
18#1674—22no—
19#1674—22no—
20#1670—22no—
21#1670—22no—
22C2×#335C2211yescharacteristic
23C2×#335C2211yescharacteristic
24C2×#332—21no—
25C2×#332—21no—
26#1374—22no—
27#1364—22no—
28#1362—22no—
29C22×Q16—21no—
30C22×Q16—21no—
31C22×Q16—41no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4×Q8D411yescharacteristic
35C22×C16—21no—
36C22×C16D412yes—
37#500—21no—
38#500—21no—
39#500—21no—
40#500D412yes—
41#494—21no—
42#494—21no—
43#494—21no—
44#494D412yes—
45#475—24no—
46#475—42no—
47#462—24no—
48#462—42no—
49C8×Q8—21no—
50C8×Q8—21no—
51C8×Q8—21no—
52C8×Q8—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#106—41no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8D411yescharacteristic
60C2×#103—41no—
61C2×#103—41no—
62C2×C4×C8—21no—
63C2×C4×C8C2311yescharacteristic
64C4⋊C16—21no—
65C4⋊C16—21no—
66#358—28no—
67C4×C16—21no—
68C4×C16—21no—
69#335—21no—
70#335—21no—
71#335D412yes—
72#335D412yes—
73#332—24no—
74#332—24no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8D811yescharacteristic
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—44no—
81C2×Q16—41no—
82C2×Q16—41no—
83C2×Q16—42no—
84C2×Q16—42no—
85C2×Q16—82no—
86C2×Q16—82no—
87C2×Q16—82no—
88C22×C8—21no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8C2×D411yescharacteristic
92C22×C8#4111yescharacteristic
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4D811yescharacteristic
111C2×C42—21no—
112C2×C42C2×D411yescharacteristic
113C2×C16—22no—
114C2×C16—22no—
115C2×C16—22no—
116C2×C16—41no—
117C2×C16—42no—
118#107—22no—
119#107—22no—
120#107—41no—
121#107—41no—
122#106—44no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131#103—44no—
132#103—44no—
133C4×C8—41no—
134C4×C8—41no—
135C4×C8C2×D411yescharacteristic
136C4×C8C2×D411yescharacteristic
137C4×C8C2×D411yescharacteristic
138C4×C8C2×D411yescharacteristic
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—81no—
147C2×Q8—81no—
148C2×Q8—82no—
149C2×Q8—82no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8SD3212yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4#12111yescharacteristic
160C22×C4C2×D811yescharacteristic
161C22×C4#13511yescharacteristic
162Q16—44no—
163Q16—82no—
164Q16—82no—
165Q16—88no—
166Q16—81no—
167Q16—81no—
168C2×C8—22no—
169C2×C8—41no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—81no—
174C2×C8—81no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8#12011yescharacteristic
178C2×C8#12111yescharacteristic
179C2×C8#12111yescharacteristic
180C2×C8#12711yescharacteristic
181C2×C8#13512yes—
182C2×C8#13712yes—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C42—41no—
196C42—41no—
197C42#12011yescharacteristic
198C42#12011yescharacteristic
199C42AGL(1,ℤ/8)11yescharacteristic
200C42AGL(1,ℤ/8)11yescharacteristic
201C16—42no—
202C16—81no—
203C23#46611yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—41no—
207Q8—41no—
208Q8—82no—
209Q8—82no—
210Q8—82no—
211Q8—82no—
212Q8—161no—
213Q8—161no—
214Q8—162no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—162no—
224C2×C4—161no—
225C2×C4—161no—
226C2×C4#39211yescharacteristic
227C2×C4#44711yescharacteristic
228C2×C4#44711yescharacteristic
229C2×C4#44911yescharacteristic
230C2×C4#44911yescharacteristic
231C2×C4#46412yes—
232C2×C4#46911yescharacteristic
233C2×C4#47112yes—
234C2×C4C2×SD3211yescharacteristic
235C2×C4#50811yescharacteristic
236C2×C4#50911yescharacteristic
237C2×C4#51011yescharacteristic
238C8—21no—
239C8—21no—
240C8—82no—
241C8—82no—
242C8—81no—
243C8#45212yes—
244C8#49511yescharacteristic
245C8#49711yescharacteristic
246C22#168212yes—
247C22#199911yescharacteristic
248C22#221711yescharacteristic
249C22#222011yescharacteristic
250C22#222212yes—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—81no—
255C4—162no—
256C4—162no—
257C4#170212yes—
258C4#201511yescharacteristic
259C4#201911yescharacteristic
260C4#218511yescharacteristic
261C4#218611yescharacteristic
262C4#218711yescharacteristic
263C4#218911yescharacteristic
264C2#1026311yescharacteristic
265C2#1247812yes—
266C2#1248212yes—
267C2#1330511yescharacteristic
268C2#1331511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#466
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ 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×#22412
C2×#16742
C2×#16702
#132632
C2×#13642
C2×#13622
#71542