Finite Group Databasebeta

← #143291

Subgroups — #143291

Order 512.

390 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2245C211yesmaximal, characteristic
2#28680C211yesmaximal, characteristic
3C2×#1690C211yesmaximal, characteristic
4#13253C211yesmaximal, characteristic
5C2×#1363C211yesmaximal, characteristic
6C2×#1338C211yesmaximal, characteristic
7#7178C211yesmaximal, characteristic
8C2×#499C2211yescharacteristic
9C2×#494C2211yescharacteristic
10C2×#475—21no—
11C2×#446C2211yescharacteristic
12#2584—21no—
13#2245—22no—
14#2245—22no—
15C2×C4⋊C16C2211yescharacteristic
16C2×#358—21no—
17C2×#358—21no—
18C2×#358—22no—
19C2×#346C2211yescharacteristic
20#2087—21no—
21#2086—21no—
22#1989—21no—
23#1943—22no—
24C4×#107C2211yescharacteristic
25#1690—22no—
26#1690—22no—
27C2×#334C2211yescharacteristic
28C2×#332—21no—
29#1363—22no—
30#1338—22no—
31C22×Q16—21no—
32C22×Q16—21no—
33C2×C4⋊Q8—21no—
34C2×#125—21no—
35C2×C4×Q8D411yescharacteristic
36C22×C16—21no—
37C22×C16—21no—
38#499—21no—
39#499—21no—
40#499—21no—
41#499—21no—
42#494—21no—
43#494—21no—
44#494—21no—
45#494D412yes—
46#475—24no—
47#475—42no—
48#446—21no—
49#446—21no—
50#446—21no—
51#446—21no—
52C2×#107D411yescharacteristic
53C2×#107D411yescharacteristic
54C2×#107D412yes—
55C2×#106—21no—
56C2×#106—41no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8D411yescharacteristic
60C2×#103—41no—
61C2×#97—21no—
62C2×C4×C8C2311yescharacteristic
63#398—41no—
64#395—41no—
65#389—22no—
66#389—22no—
67#389—42no—
68C4×C4⋊C4—21no—
69C4⋊C16—21no—
70C4⋊C16—21no—
71#358—24no—
72#358—24no—
73#358—44no—
74#346—21no—
75#346D412yes—
76#340—22no—
77#340—22no—
78#340—21no—
79#334—21no—
80#334D412yes—
81#332—24no—
82C22×Q8—41no—
83C22×Q8D811yescharacteristic
84C2×Q16—22no—
85C2×Q16—22no—
86C2×Q16—44no—
87C2×Q16—41no—
88C2×Q16—41no—
89C2×Q16—42no—
90C2×Q16—42no—
91C22×C8—21no—
92C22×C8—41no—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C4⋊Q8—22no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99#125—41no—
100#125—41no—
101#125—41no—
102#125—41no—
103C4×Q8—21no—
104C4×Q8—21no—
105C4×Q8—21no—
106C4×Q8—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4D811yescharacteristic
118C2×C42—21no—
119C2×C42—41no—
120C2×C42C2×D411yescharacteristic
121C2×C16—22no—
122C2×C16—42no—
123C2×C16—41no—
124C2×C16—42no—
125#107—22no—
126#107—21no—
127#107—21no—
128#107—21no—
129#107—21no—
130#107SD1614yes—
131#106—44no—
132#106—41no—
133#106—41no—
134C4⋊C8—21no—
135C4⋊C8—22no—
136C4⋊C8—21no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140#103—44no—
141#97—41no—
142#97—41no—
143C4×C8C2×D411yescharacteristic
144C4×C8C2×D411yescharacteristic
145C4×C8C2×D411yescharacteristic
146C4×C8#4111yescharacteristic
147#95—42no—
148#95—42no—
149#95—41no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—81no—
155C2×Q8—82no—
156C2×Q8—82no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—81no—
163C22×C4#12011yescharacteristic
164C22×C4C2×D811yescharacteristic
165C22×C4AGL(1,ℤ/8)11yescharacteristic
166Q16—44no—
167Q16—44no—
168Q16—82no—
169Q16—81no—
170Q16—81no—
171C2×C8—22no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—82no—
176C2×C8#12011yescharacteristic
177C2×C8#12111yescharacteristic
178C2×C8#12111yescharacteristic
179C2×C8#12711yescharacteristic
180C2×C8C2×SD1612yes—
181C2×C8#13712yes—
182C4⋊C4—21no—
183C4⋊C4—24no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—42no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—84no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—162no—
202C42—41no—
203C42—41no—
204C42—161no—
205C42#12011yescharacteristic
206C42#12111yescharacteristic
207C42#13511yescharacteristic
208C42AGL(1,ℤ/8)11yescharacteristic
209C16—42no—
210C16—81no—
211C23#44711yescharacteristic
212Q8—42no—
213Q8—81no—
214Q8—82no—
215Q8—82no—
216Q8—162no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—162no—
235C2×C4#39211yescharacteristic
236C2×C4#44711yescharacteristic
237C2×C4#44812yes—
238C2×C4#44911yescharacteristic
239C2×C4#44911yescharacteristic
240C2×C4#45412yes—
241C2×C4#46611yescharacteristic
242C2×C4#46911yescharacteristic
243C8—21no—
244C8—21no—
245C8—82no—
246C8—81no—
247C8#46112yes—
248C8#49211yescharacteristic
249C8#49711yescharacteristic
250C22#162712yes—
251C22#199911yescharacteristic
252C22#219112yes—
253C22#219411yescharacteristic
254C22#219511yescharacteristic
255C4—21no—
256C4—21no—
257C4—21no—
258C4—21no—
259C4—81no—
260C4—84no—
261C4—81no—
262C4—161no—
263C4—162no—
264C4—161no—
265C4#162612yes—
266C4#200311yescharacteristic
267C4#201311yescharacteristic
268C2#1005111yescharacteristic
269C2#1236212yes—
270C2#1236312yes—
271C2#1330811yescharacteristic
272C2#1331711yescharacteristic
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×#446 ◁ 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×#22452
#286802
C2×#16902
#132532
C2×#13632
C2×#13382
#71782