Finite Group Databasebeta

← #126092

Subgroups — #126092

Order 512.

302 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 62 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#32529C211yesmaximal, characteristic
2#13452C211yesmaximal, characteristic, complemented
3#13363C211yesmaximal, characteristic
4#10545C211yesmaximal, characteristic, complemented
5#7121C211yesmaximal, characteristic, complemented
6#7118C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9#2513—21no—
10C2×#367—21no—
11C2×#357—21no—
12#2092—21no—
13#2065—21no—
14#2054—21no—
15#1981—21no—
16#1970C2211yescharacteristic
17#1949—21no—
18#1942C2211yescharacteristic
19#1939C2211yescharacteristic
20#1925—21no—
21C2×#332—21no—
22#1380—21no—
23#1380—21no—
24#1374C2211yescharacteristic
25#1374C2211yescharacteristic
26#1277C2211yescharacteristic
27C22×D8D411yescharacteristic
28C2×C4⋊Q8—21no—
29C22×C16—21no—
30C22×C16—21no—
31#500—42no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#106—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×#103—41no—
42C2×#103—41no—
43C2×#102—21no—
44C2×#102—21no—
45C2×#102—21no—
46C2×C4×C8C2311yescharacteristic
47#397—41no—
48#397—41no—
49#390—21no—
50#390—21no—
51#390—21no—
52#389—21no—
53#388—41no—
54#388—41no—
55#382—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4D411yescharacteristic
58#367—42no—
59#357—42no—
60#340—21no—
61#340—21no—
62#336—21no—
63#332—42no—
64C22×Q8—41no—
65C22×D4—21no—
66C2×D8—21no—
67C2×D8—21no—
68C2×D8—41no—
69C2×D8—41no—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4D811yescharacteristic
85C2×C4⋊C4D811yescharacteristic
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C42—21no—
93C2×C42—41no—
94C2×C42C2×D411yescharacteristic
95C2×C16—22no—
96C2×C16—41no—
97C2×C16—41no—
98C2×C16—41no—
99C2×C16—41no—
100C2×C16—41no—
101#107—22no—
102#107—22no—
103#107—21no—
104#107—21no—
105#107—41no—
106#107—41no—
107#107—41no—
108#107—41no—
109#106—81no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115#103—82no—
116#103—82no—
117#102—41no—
118#102—41no—
119#102—81no—
120#102—81no—
121C4×C8—41no—
122#95—21no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127C24—41no—
128C2×Q8—42no—
129C2×Q8—81no—
130C2×Q8—81no—
131C2×Q8—81no—
132C2×Q8—81no—
133C2×Q8—81no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—81no—
146C22×C4#12011yescharacteristic
147C22×C4C2×D811yescharacteristic
148C22×C4AGL(1,ℤ/8)11yescharacteristic
149D8—42no—
150D8—81no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—81no—
159C2×C8—81no—
160C2×C8—81no—
161C2×C8#13511yescharacteristic
162C2×C8#13511yescharacteristic
163C2×C8#13711yescharacteristic
164C2×C8#13711yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—161no—
179C22⋊C4—82no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C22⋊C4—161no—
183C22⋊C4—161no—
184C22⋊C4—161no—
185C22⋊C4—161no—
186C42—21no—
187C42—21no—
188C42—41no—
189C42—41no—
190C42—161no—
191C16—42no—
192C16—81no—
193C23—81no—
194C23—81no—
195C23—81no—
196C23—81no—
197C23—81no—
198C23—81no—
199C23—81no—
200C23#44711yescharacteristic
201Q8—82no—
202Q8—82no—
203Q8—161no—
204Q8—161no—
205D4—81no—
206D4—81no—
207D4—81no—
208D4—81no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—161no—
231C2×C4—161no—
232C2×C4#45211yescharacteristic
233C2×C4#45211yescharacteristic
234C2×C4#45411yescharacteristic
235C2×C4#45511yescharacteristic
236C2×C4#45611yescharacteristic
237C2×C4B(4,𝔽2)11yescharacteristic
238C2×C4#50811yescharacteristic
239C2×C4#50911yescharacteristic
240C8—21no—
241C8—21no—
242C8—21no—
243C8—21no—
244C8—81no—
245C8—161no—
246C22—161no—
247C22—161no—
248C22—161no—
249C22—161no—
250C22—161no—
251C22—161no—
252C22—161no—
253C22#159811yescharacteristic
254C22#164111yescharacteristic
255C22#164711yescharacteristic
256C22#218611yescharacteristic
257C22#219011yescharacteristic
258C22#219411yescharacteristic
259C22#219411yescharacteristic
260C4—21no—
261C4—21no—
262C4—41no—
263C4—81no—
264C4—81no—
265C4—161no—
266C4—161no—
267C4—162no—
268C4—161no—
269C4#160311yescharacteristic
270C4#160611yescharacteristic
271C4#163811yescharacteristic
272C4#166411yescharacteristic
273C2—321no—
274C2#776511yescharacteristic
275C2#1216011yescharacteristic
276C2#1216711yescharacteristic
277C2#1238111yescharacteristic
278C2#1239011yescharacteristic
279C2#1239511yescharacteristic
280C2#1240211yescharacteristic
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 ◁ #1939 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13452C2semidirect
4. #10545C2semidirect
5. #7121C2semidirect
6. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#325292
#134522
#133632
#105452
#71212
#71182
#71122