Finite Group Databasebeta

← #126347

Subgroups — #126347

Order 512.

309 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 66 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#31238C211yesmaximal, characteristic
2#13538C211yesmaximal, characteristic, complemented
3#13480C211yesmaximal, characteristic
4#10652C211yesmaximal, characteristic, complemented
5#7121C211yesmaximal, characteristic, complemented
6#7118C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8C2×#478—21no—
9#2454—21no—
10C2×#357—21no—
11#2067—21no—
12#2065—21no—
13#2059—21no—
14#1970C2211yescharacteristic
15#1948—21no—
16#1942C2211yescharacteristic
17#1939C2211yescharacteristic
18#1879—21no—
19#1845C2211yescharacteristic
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×D8D411yescharacteristic
28C2×#125—21no—
29C2×C4×D4—21no—
30C22×C16—21no—
31C22×C16—21no—
32#478—42no—
33#478—42no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×#107D411yescharacteristic
39C2×#106—21no—
40C2×#106—41no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×#102—21no—
45C2×#102—21no—
46C2×#102—21no—
47C2×C4×C8C2311yescharacteristic
48#399—41no—
49#396—41no—
50#390—21no—
51#390—21no—
52#389—21no—
53#389—21no—
54#385—41no—
55#384—21no—
56#380—41no—
57C4×C4⋊C4D411yescharacteristic
58#357—42no—
59#340—21no—
60#340—21no—
61#340—41no—
62#336—21no—
63C22×D4—21no—
64C23×C4—41no—
65C2×D8—21no—
66C2×D8—41no—
67C2×D8—41no—
68C2×D8SD1612yes—
69C22×C8—21no—
70C22×C8—41no—
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73#125—42no—
74#125—81no—
75C4×D4—42no—
76C4×D4—81no—
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⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4D811yescharacteristic
88C2×C4⋊C4D811yescharacteristic
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C42—21no—
95C2×C42C2×D411yescharacteristic
96C2×C16—41no—
97C2×C16—41no—
98C2×C16—41no—
99C2×C16—41no—
100C2×C16—41no—
101C2×C16—41no—
102#107—21no—
103#107—41no—
104#107—41no—
105#107—42no—
106#107—41no—
107#107—41no—
108#107SD1612yes—
109#106—42no—
110#106—82no—
111C4⋊C8—22no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116#102—42no—
117#102—41no—
118#102—41no—
119#102—81no—
120C4×C8—41no—
121#95—21no—
122#95—41no—
123#95—41no—
124C24—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—82no—
138C22×C4—81no—
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—42no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—81no—
154C2×C8—81no—
155C2×C8C2×SD1611yescharacteristic
156C2×C8C2×SD1611yescharacteristic
157C2×C8AGL(1,ℤ/8)11yescharacteristic
158C2×C8#13711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—161no—
175C4⋊C4—161no—
176C4⋊C4—161no—
177C22⋊C4—82no—
178C22⋊C4—81no—
179C22⋊C4—81no—
180C22⋊C4—161no—
181C22⋊C4—161no—
182C22⋊C4—161no—
183C42—21no—
184C42—21no—
185C42—41no—
186C42—41no—
187C16—81no—
188C16—81no—
189C23—42no—
190C23—81no—
191C23—81no—
192C23—81no—
193C23—81no—
194C23—81no—
195C23—81no—
196C23#44711yescharacteristic
197D4—42no—
198D4—42no—
199D4—81no—
200D4—81no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—164no—
222C2×C4—161no—
223C2×C4—161no—
224C2×C4—161no—
225C2×C4#44811yescharacteristic
226C2×C4#45011yescharacteristic
227C2×C4#45411yescharacteristic
228C2×C4#45511yescharacteristic
229C2×C4#45611yescharacteristic
230C2×C4#45811yescharacteristic
231C2×C4#50811yescharacteristic
232C2×C4#51011yescharacteristic
233C8—21no—
234C8—21no—
235C8—21no—
236C8—21no—
237C8—82no—
238C8—81no—
239C22—82no—
240C22—82no—
241C22—82no—
242C22—161no—
243C22—161no—
244C22—161no—
245C22—161no—
246C22#161011yescharacteristic
247C22#162711yescharacteristic
248C22#163911yescharacteristic
249C22#218911yescharacteristic
250C22#219011yescharacteristic
251C22#219211yescharacteristic
252C22#219411yescharacteristic
253C4—21no—
254C4—21no—
255C4—41no—
256C4—81no—
257C4—81no—
258C4—82no—
259C4—161no—
260C4—161no—
261C4—321no—
262C4#160811yescharacteristic
263C4#161611yescharacteristic
264C4#165111yescharacteristic
265C4#166411yescharacteristic
266C2—162no—
267C2#779111yescharacteristic
268C2#1229311yescharacteristic
269C2#1229411yescharacteristic
270C2#1235911yescharacteristic
271C2#1236011yescharacteristic
272C2#1237311yescharacteristic
273C2#1237811yescharacteristic
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. #13538C2semidirect
4. #10652C2semidirect
5. #7121C2semidirect
6. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#312382
#135382
#134802
#106522
#71212
#71182
#71122