Finite Group Databasebeta

← #126408

Subgroups — #126408

Order 512.

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

normal subgroups only

This group has 317 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#13526C211yesmaximal, characteristic
3#13506C211yesmaximal, characteristic, complemented
4#10649C211yesmaximal, characteristic, complemented
5#7121C211yesmaximal, characteristic, complemented
6#7118C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8C2×#478—21no—
9#2454—21no—
10C2×#367—21no—
11C2×#366—21no—
12C2×#357—21no—
13#2067—21no—
14#2065—21no—
15#2059—21no—
16#1970C2211yescharacteristic
17#1947—21no—
18#1942C2211yescharacteristic
19#1939C2211yescharacteristic
20#1879—21no—
21#1845C2211yescharacteristic
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×#107—41no—
37C2×#107D411yescharacteristic
38C2×#107D411yescharacteristic
39C2×#107D411yescharacteristic
40C2×#106—21no—
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#367—42no—
59#366—42no—
60#357—42no—
61#340—21no—
62#340—21no—
63#340—41no—
64#336—21no—
65C22×D4—21no—
66C23×C4—41no—
67C2×D8—21no—
68C2×D8—41no—
69C2×D8—41no—
70C2×D8Q1612yes—
71C22×C8—21no—
72C22×C8—41no—
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75#125—42no—
76#125—81no—
77C4×D4—42no—
78C4×D4—81no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4D811yescharacteristic
90C2×C4⋊C4D811yescharacteristic
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C42—21no—
97C2×C42C2×D411yescharacteristic
98C2×C16—22no—
99C2×C16—22no—
100C2×C16—41no—
101C2×C16—41no—
102C2×C16—41no—
103C2×C16—41no—
104#107—21no—
105#107—41no—
106#107—41no—
107#107—41no—
108#107—41no—
109#107—41no—
110#107—41no—
111#107—82no—
112#107D812yes—
113#106—42no—
114C4⋊C8—22no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#102—42no—
120#102—41no—
121#102—41no—
122#102—81no—
123C4×C8—41no—
124#95—21no—
125#95—41no—
126#95—41no—
127C24—41no—
128C2×D4—22no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—81no—
142C22×C4—161no—
143C22×C4#12011yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4AGL(1,ℤ/8)11yescharacteristic
146D8—42no—
147D8—81no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—81no—
157C2×C8—81no—
158C2×C8C2×D811yescharacteristic
159C2×C8C2×Q1611yescharacteristic
160C2×C8AGL(1,ℤ/8)11yescharacteristic
161C2×C8#13711yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—161no—
178C4⋊C4—161no—
179C4⋊C4—161no—
180C22⋊C4—82no—
181C22⋊C4—81no—
182C22⋊C4—81no—
183C22⋊C4—161no—
184C22⋊C4—161no—
185C22⋊C4—161no—
186C42—21no—
187C42—21no—
188C42—41no—
189C42—41no—
190C16—42no—
191C16—42no—
192C23—42no—
193C23—81no—
194C23—81no—
195C23—81no—
196C23—81no—
197C23—81no—
198C23—81no—
199C23#44711yescharacteristic
200D4—42no—
201D4—42no—
202D4—81no—
203D4—81no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—164no—
225C2×C4—161no—
226C2×C4—161no—
227C2×C4—161no—
228C2×C4#44711yescharacteristic
229C2×C4#45111yescharacteristic
230C2×C4#45311yescharacteristic
231C2×C4#45611yescharacteristic
232C2×C4#45611yescharacteristic
233C2×C4#45811yescharacteristic
234C2×C4#50811yescharacteristic
235C2×C4#50911yescharacteristic
236C8—21no—
237C8—21no—
238C8—21no—
239C8—21no—
240C8—82no—
241C8—81no—
242C22—82no—
243C22—82no—
244C22—82no—
245C22—161no—
246C22—161no—
247C22—161no—
248C22—161no—
249C22#161011yescharacteristic
250C22#161911yescharacteristic
251C22#164911yescharacteristic
252C22#218611yescharacteristic
253C22#219111yescharacteristic
254C22#219211yescharacteristic
255C22#219411yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—81no—
260C4—81no—
261C4—82no—
262C4—161no—
263C4—161no—
264C4—321no—
265C4#161011yescharacteristic
266C4#161411yescharacteristic
267C4#164911yescharacteristic
268C4#166011yescharacteristic
269C2—162no—
270C2#779711yescharacteristic
271C2#1229011yescharacteristic
272C2#1229411yescharacteristic
273C2#1231711yescharacteristic
274C2#1232611yescharacteristic
275C2#1240711yescharacteristic
276C2#1241211yescharacteristic
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
3. #13506C2semidirect
4. #10649C2semidirect
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
#135262
#135062
#106492
#71212
#71182
#71122