Finite Group Databasebeta

← #126299

Subgroups — #126299

Order 512.

294 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 64 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#32652C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic, complemented
3#13480C211yesmaximal, characteristic
4#10617C211yesmaximal, characteristic, complemented
5#7128C211yesmaximal, characteristic, complemented
6#7121C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8#2580—21no—
9C2×#357—21no—
10#2092—21no—
11#2085—21no—
12#2065—21no—
13#1984C2211yescharacteristic
14#1970C2211yescharacteristic
15#1942C2211yescharacteristic
16#1879—21no—
17#1862—21no—
18#1845C2211yescharacteristic
19#1803—21no—
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×D8D411yescharacteristic
28C2×C4×Q8D411yescharacteristic
29C2×C4×D4—21no—
30C22×C16—21no—
31C22×C16—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#106—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#103—41no—
41C2×#102—21no—
42C2×#102—21no—
43C2×#102—21no—
44C2×C4×C8C2311yescharacteristic
45#399—41no—
46#393—41no—
47#390—21no—
48#390—21no—
49#389—21no—
50#389—21no—
51#385—41no—
52#385—41no—
53#384—21no—
54#382—21no—
55#357—42no—
56#340—21no—
57#340—21no—
58#340—41no—
59#336—21no—
60#336—21no—
61C22×Q8D811yescharacteristic
62C22×D4—21no—
63C23×C4—41no—
64C2×D8—21no—
65C2×D8—41no—
66C2×D8—41no—
67C2×D8SD1612yes—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C4×Q8—41no—
73C4×Q8—41no—
74C4×D4—81no—
75C4×D4—81no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4D811yescharacteristic
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C42—21no—
92C2×C42C2×D411yescharacteristic
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96C2×C16—41no—
97C2×C16—41no—
98C2×C16—41no—
99#107—21no—
100#107—21no—
101#107—41no—
102#107—41no—
103#107—42no—
104#107—41no—
105#107—41no—
106#106—81no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#103—82no—
111#102—41no—
112#102—41no—
113#102—81no—
114#102—81no—
115C4×C8—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C24—41no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×D4—22no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—82no—
137C22×C4—81no—
138C22×C4—161no—
139C22×C4—161no—
140C22×C4#12011yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4AGL(1,ℤ/8)11yescharacteristic
143D8—42no—
144D8—81no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—81no—
153C2×C8—81no—
154C2×C8—81no—
155C2×C8C2×SD1611yescharacteristic
156C2×C8#13511yescharacteristic
157C2×C8#13711yescharacteristic
158C2×C8#13711yescharacteristic
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—161no—
172C4⋊C4—161no—
173C4⋊C4—161no—
174C22⋊C4—82no—
175C22⋊C4—82no—
176C22⋊C4—81no—
177C22⋊C4—81no—
178C22⋊C4—161no—
179C22⋊C4—161no—
180C42—21no—
181C42—21no—
182C42—41no—
183C42—41no—
184C16—81no—
185C16—81no—
186C23—42no—
187C23—81no—
188C23—81no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23—81no—
193C23#44711yescharacteristic
194Q8—42no—
195Q8—81no—
196D4—42no—
197D4—42no—
198D4—81no—
199D4—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—164no—
221C2×C4—161no—
222C2×C4—161no—
223C2×C4—161no—
224C2×C4#44811yescharacteristic
225C2×C4#45211yescharacteristic
226C2×C4#45411yescharacteristic
227C2×C4#45511yescharacteristic
228C2×C4#45511yescharacteristic
229C2×C4#45811yescharacteristic
230C2×C4#50811yescharacteristic
231C2×C4#50911yescharacteristic
232C8—21no—
233C8—21no—
234C8—21no—
235C8—21no—
236C8—81no—
237C8—161no—
238C22—82no—
239C22—82no—
240C22—82no—
241C22—161no—
242C22—161no—
243C22—161no—
244C22—161no—
245C22#160711yescharacteristic
246C22#162711yescharacteristic
247C22#164111yescharacteristic
248C22#218611yescharacteristic
249C22#219111yescharacteristic
250C22#219211yescharacteristic
251C22#219411yescharacteristic
252C4—21no—
253C4—21no—
254C4—41no—
255C4—81no—
256C4—81no—
257C4—161no—
258C4—161no—
259C4—161no—
260C4—321no—
261C4#160811yescharacteristic
262C4#161511yescharacteristic
263C4#164311yescharacteristic
264C4#166211yescharacteristic
265C2—162no—
266C2#778711yescharacteristic
267C2#1228011yescharacteristic
268C2#1228411yescharacteristic
269C2#1235511yescharacteristic
270C2#1236211yescharacteristic
271C2#1238111yescharacteristic
272C2#1238811yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8D4
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13537C2semidirect
4. #10617C2semidirect
5. #7128C2semidirect
6. #7121C2semidirect

Maximal subgroups

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

subgroupindex
#326522
#135372
#134802
#106172
#71282
#71212
#71122