Finite Group Databasebeta

← #135293

Subgroups — #135293

Order 512.

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

normal subgroups only

This group has 570 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
1#20660C211yesmaximal, characteristic
2#12681C211yesmaximal, characteristic
3#10717C211yesmaximal, characteristic
4#7254C214yesmaximal, complemented
5C23×M16C2211yescharacteristic
6C2×#427C411yescharacteristic, complemented
7C2×#427C411yescharacteristic, complemented
8#2280—21no—
9#2280—21no—
10C2×#349—21no—
11C2×#349C2212yes—
12#2112—21no—
13#1809—21no—
14#1809—21no—
15#1809C412yescomplemented
16#1809C2212yes—
17#1735—21no—
18C2×H(3,ℤ/4)—21no—
19C2×H(3,ℤ/4)C2212yes—
20C24×C4D411yescharacteristic
21C22×M16—24no—
22C22×M16—42no—
23C22×M16—42no—
24C22×M16C2×C411yescharacteristic
25C22×M16D411yescharacteristic
26C22×M16D411yescharacteristic
27C22×M16C2311yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8D411yescharacteristic
30C2×#117—22no—
31C2×#117—21no—
32C2×#117—21no—
33C2×M32—22no—
34#427—42no—
35#427—42no—
36C2×#107—21no—
37C2×#107C2×C412yes—
38C2×#106—21no—
39C2×#106C2×C412yes—
40#381—41no—
41#381—41no—
42#380—22no—
43#380—22no—
44#380—42no—
45#379—41no—
46#379—41no—
47C4×C22⋊C4—24no—
48C4×C22⋊C4—41no—
49C4×C22⋊C4—41no—
50#349—24no—
51#349—42no—
52C22⋊C16—42no—
53#340—22no—
54#340—22no—
55H(3,ℤ/4)—24no—
56H(3,ℤ/4)—42no—
57C25D811yescharacteristic
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4SD1611yescharacteristic
68C23×C4SD1611yescharacteristic
69C23×C4C2×D411yescharacteristic
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—416no—
82C2×M16#4114yes—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22×C411yescharacteristic
92C22×C8C2×D411yescharacteristic
93#117—42no—
94#117—42no—
95#117—42no—
96#117—42no—
97#117—42no—
98#117—42no—
99#117—42no—
100#117—42no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—42no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C42—24no—
111C2×C42—41no—
112C2×C42—41no—
113M32—42no—
114M32—42no—
115C2×C16—42no—
116#107—42no—
117#107—42no—
118#106—42no—
119#106—42no—
120#95—44no—
121#95—42no—
122#95—42no—
123C24—22no—
124C24—22no—
125C24—41no—
126C24—82no—
127C24#10211yescharacteristic
128C24#10211yescharacteristic
129C24C2×D811yescharacteristic
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4#9912yes—
161C22×C4#10211yescharacteristic
162C22×C4#10211yescharacteristic
163C22×C4#10412yes—
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4#12312yes—
166C22×C4C2×SD1611yescharacteristic
167C22×C4#13712yes—
168M16—44no—
169M16—44no—
170M16—44no—
171M16—44no—
172C2×C8—28no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—44no—
183C2×C8#9912yes—
184C2×C8#11712yes—
185C2×C8#12312yes—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—84no—
195C22⋊C4—84no—
196C22⋊C4—88no—
197C22⋊C4—82no—
198C22⋊C4—82no—
199C42—44no—
200C42—44no—
201C42—82no—
202C42—82no—
203C16—82no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—42no—
212C23—44no—
213C23—42no—
214C23—42no—
215C23—44no—
216C23—82no—
217C23—84no—
218C23—84no—
219C23#32712yes—
220C23C2×#10211yescharacteristic
221C23#48012yes—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—84no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—88no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4#32712yes—
253C2×C4#38111yescharacteristic
254C2×C4C2×#9911yescharacteristic
255C2×C4C2×#9911yescharacteristic
256C2×C4#41911yescharacteristic
257C2×C4C2×#10411yescharacteristic
258C2×C4#42111yescharacteristic
259C2×C4#48312yes—
260C8—22no—
261C8—22no—
262C8—22no—
263C8—41no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—41no—
275C22—42no—
276C22—81no—
277C22—81no—
278C22—88no—
279C22C2×#32711yescharacteristic
280C22#151911yescharacteristic
281C22#182311yescharacteristic
282C4—22no—
283C4—22no—
284C4—41no—
285C4—162no—
286C4—164no—
287C4—162no—
288C4#150811yescharacteristic
289C4#151111yescharacteristic
290C4#181911yescharacteristic
291C4#182611yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—42no—
296C2—81no—
297C2#871211yescharacteristic
298C2#1168512yes—
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7254C2semidirect
6. C2×#427C4semidirect
7. C2×#427C4semidirect
15. #1809C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#206602
#126812
#107172
#72542