Finite Group Databasebeta

← #142922

Subgroups — #142922

Order 512.

487 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 92 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×#2236C212yesmaximal
2#30242C211yesmaximal, characteristic
3C2×#1348C212yesmaximal
4#11377C211yesmaximal, characteristic
5#7183C211yesmaximal, characteristic
6C2×#467—21no—
7C2×#467C2212yes—
8C22×#107C2211yescharacteristic
9#2857—21no—
10#2236—24no—
11#2236—24no—
12C2×#366—22no—
13C2×#358—22no—
14C2×C22⋊C16C2212yes—
15#2095—22no—
16#2016—22no—
17C2×#340—21no—
18C2×#340C2211yescharacteristic
19C2×#336C2211yescharacteristic
20C4⋊#95—21no—
21#1348—24no—
22C22×Q16—21no—
23C22×Q16D412yes—
24C23×C8D411yescharacteristic
25C23×C8C2311yescharacteristic
26C2×C4⋊Q8—21no—
27C2×C22⋊Q8—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C42D411yescharacteristic
31C22×C16—22no—
32#467—22no—
33#467—22no—
34#467—22no—
35#467—22no—
36#467—42no—
37#467—42no—
38C2×#107—22no—
39C2×#107—22no—
40C2×#107—21no—
41C2×#107—21no—
42C2×#107D412yes—
43C2×#107D412yes—
44C2×#103—22no—
45C2×#103—22no—
46#391—22no—
47#389—42no—
48#384—21no—
49#380—42no—
50C2×#95—42no—
51#366—28no—
52#358—28no—
53C22⋊C16—22no—
54C22⋊C16D414yes—
55#340—22no—
56#340—42no—
57#336—22no—
58C22×Q8—22no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4D812yes—
62C23×C4C2×D411yescharacteristic
63C2×Q16—22no—
64C2×Q16—22no—
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—42no—
68C2×Q16—81no—
69C2×Q16—81no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8D812yes—
77C22×C8C2×D412yes—
78C22×C8C2×D411yescharacteristic
79C22×C8C2×D412yes—
80C22×C8#4111yescharacteristic
81C4⋊Q8—81no—
82C4⋊Q8—81no—
83C22⋊Q8—42no—
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86C22⋊Q8—42no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C16—24no—
109C2×C16—42no—
110C2×C16—42no—
111#107—24no—
112#107—22no—
113#107—22no—
114#107—42no—
115#107—42no—
116#103—44no—
117#103—42no—
118#103—42no—
119#95—44no—
120#95—84no—
121C24C2×D811yescharacteristic
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4—81no—
143C22×C4—82no—
144C22×C4—81no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4—82no—
148C22×C4Q3214yes—
149C22×C4#12011yescharacteristic
150C22×C4#12111yescharacteristic
151C22×C4#12711yescharacteristic
152C22×C4C2×D811yescharacteristic
153C22×C4AGL(1,ℤ/8)11yescharacteristic
154C22×C4AGL(1,ℤ/8)11yescharacteristic
155Q16—44no—
156Q16—82no—
157Q16—84no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—41no—
162C2×C8—42no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—81no—
168C2×C8#12012yes—
169C2×C8#12112yes—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—81no—
181C4⋊C4—82no—
182C4⋊C4—81no—
183C4⋊C4—82no—
184C22⋊C4—82no—
185C22⋊C4—82no—
186C42—41no—
187C42—41no—
188C42—81no—
189C16—44no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23#44711yescharacteristic
194C23#45911yescharacteristic
195C23#49311yescharacteristic
196C23C2×Q3212yes—
197C23#51012yes—
198Q8—82no—
199Q8—82no—
200Q8—82no—
201Q8—82no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—84no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—161no—
227C2×C4—162no—
228C2×C4—161no—
229C2×C4—161no—
230C2×C4—162no—
231C2×C4—161no—
232C2×C4#39211yescharacteristic
233C2×C4#44711yescharacteristic
234C2×C4#45511yescharacteristic
235C2×C4#45611yescharacteristic
236C2×C4#45811yescharacteristic
237C2×C4#45811yescharacteristic
238C2×C4#45911yescharacteristic
239C2×C4#49611yescharacteristic
240C8—22no—
241C8—22no—
242C8—42no—
243C22—22no—
244C22—22no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—22no—
251C22#160711yescharacteristic
252C22#161011yescharacteristic
253C22#201111yescharacteristic
254C22#218812yes—
255C22#219012yes—
256C22#220912yes—
257C22#224711yescharacteristic
258C22#225111yescharacteristic
259C4—21no—
260C4—21no—
261C4—44no—
262C4—162no—
263C4—162no—
264C4—162no—
265C4—162no—
266C4#160711yescharacteristic
267C4#161011yescharacteristic
268C4#200711yescharacteristic
269C4#202211yescharacteristic
270C2—21no—
271C2—22no—
272C2—21no—
273C2#997011yescharacteristic
274C2#1228212yes—
275C2#1229212yes—
276C2#1333011yescharacteristic
277C2#1334111yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×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×#22362
#302422
C2×#13482
#113772
#71832