Finite Group Databasebeta

← #143363

Subgroups — #143363

Order 512.

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

normal subgroups only

This group has 415 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
2#28737C211yesmaximal, characteristic
3C2×#1674C211yesmaximal, characteristic
4#13275C211yesmaximal, characteristic
5C2×#1362C211yesmaximal, characteristic
6C2×Q8⋊C16C211yesmaximal, characteristic
7#7171C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#494C2211yescharacteristic
10C2×#462—21no—
11C2×C8×Q8C2211yescharacteristic
12#2568—21no—
13#2241—22no—
14#2241—22no—
15C2×C4⋊C16C2211yescharacteristic
16C2×#358—22no—
17C2×C4×C16C2211yescharacteristic
18#2056—22no—
19#1981—21no—
20#1969—22no—
21C4×#107C2211yescharacteristic
22#1674—22no—
23#1674—22no—
24C2×#335C2211yescharacteristic
25C2×#332—21no—
26#1380—22no—
27#1362—22no—
28Q8⋊C16—22no—
29C22×Q16—21no—
30C22×Q16—41no—
31C22×Q16D412yes—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4×Q8D411yescharacteristic
35C22×C16—21no—
36C22×C16—21no—
37#500—21no—
38#500—21no—
39#500—21no—
40#500D412yes—
41#494—21no—
42#494—21no—
43#494—21no—
44#494—21no—
45#462—24no—
46#462—42no—
47C8×Q8—21no—
48C8×Q8—21no—
49C8×Q8—21no—
50C8×Q8—21no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107D412yes—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8D411yescharacteristic
57C2×#103—24no—
58C2×#103—41no—
59C2×C4×C8—21no—
60C2×C4×C8C2311yescharacteristic
61#400—42no—
62#391—24no—
63#391—41no—
64#382—41no—
65C4×C4⋊C4—21no—
66C4⋊C16—21no—
67C4⋊C16—21no—
68#358—28no—
69C4×C16—21no—
70C4×C16D412yes—
71#340—21no—
72#335—21no—
73#335D412yes—
74#332—24no—
75C22×Q8—22no—
76C22×Q8—41no—
77C22×Q8D811yescharacteristic
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—44no—
81C2×Q16—44no—
82C2×Q16—41no—
83C2×Q16—41no—
84C2×Q16—42no—
85C2×Q16—42no—
86C2×Q16—82no—
87C2×Q16—82no—
88C22×C8—21no—
89C22×C8—41no—
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92C4⋊Q8—22no—
93C4⋊Q8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4D811yescharacteristic
111C2×C42—21no—
112C2×C42—41no—
113C2×C42C2×D411yescharacteristic
114C2×C16—22no—
115C2×C16—42no—
116C2×C16—41no—
117C2×C16—42no—
118#107—22no—
119#107—22no—
120#107—22no—
121#107—22no—
122#107—41no—
123#107—41no—
124C4⋊C8—21no—
125C4⋊C8—22no—
126C4⋊C8—21no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130#103—44no—
131#103—84no—
132C4×C8—41no—
133C4×C8—41no—
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D411yescharacteristic
136C4×C8C2×D411yescharacteristic
137C4×C8#4111yescharacteristic
138#95—44no—
139#95—44no—
140#95—41no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×Q8SD3212yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—81no—
162C22×C4#12011yescharacteristic
163C22×C4C2×D811yescharacteristic
164C22×C4AGL(1,ℤ/8)11yescharacteristic
165Q16—44no—
166Q16—82no—
167Q16—88no—
168Q16—82no—
169Q16—81no—
170Q16—81no—
171C2×C8—22no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—42no—
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—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C42—41no—
198C42—41no—
199C42—81no—
200C42—81no—
201C42#12011yescharacteristic
202C42#12111yescharacteristic
203C42#13511yescharacteristic
204C42AGL(1,ℤ/8)11yescharacteristic
205C16—42no—
206C16—81no—
207C23#44711yescharacteristic
208Q8—22no—
209Q8—22no—
210Q8—41no—
211Q8—41no—
212Q8—84no—
213Q8—82no—
214Q8—82no—
215Q8—82no—
216Q8—82no—
217Q8—161no—
218Q8—161no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—82no—
226C2×C4—84no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—161no—
232C2×C4—161no—
233C2×C4#39211yescharacteristic
234C2×C4#44711yescharacteristic
235C2×C4#44911yescharacteristic
236C2×C4#44911yescharacteristic
237C2×C4#45212yes—
238C2×C4#45512yes—
239C2×C4#46611yescharacteristic
240C2×C4#46911yescharacteristic
241C2×C4C2×SD3211yescharacteristic
242C2×C4#50811yescharacteristic
243C2×C4#50911yescharacteristic
244C2×C4#51011yescharacteristic
245C8—21no—
246C8—21no—
247C8—82no—
248C8—81no—
249C8#46412yes—
250C8#49511yescharacteristic
251C8#49711yescharacteristic
252C22#164112yes—
253C22#199911yescharacteristic
254C22#218511yescharacteristic
255C22#218611yescharacteristic
256C22#219512yes—
257C4—21no—
258C4—21no—
259C4—42no—
260C4—81no—
261C4—162no—
262C4—162no—
263C4—162no—
264C4#164012yes—
265C4#200411yescharacteristic
266C4#201511yescharacteristic
267C4#218711yescharacteristic
268C4#218911yescharacteristic
269C4#221711yescharacteristic
270C4#222011yescharacteristic
271C2#1006311yescharacteristic
272C2#1238212yes—
273C2#1238312yes—
274C2#1330511yescharacteristic
275C2#1332211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
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
#287372
C2×#16742
#132752
C2×#13622
C2×Q8⋊C162
#71712