Finite Group Databasebeta

← #126320

Subgroups — #126320

Order 512.

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

normal subgroups only

This group has 302 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#13527C211yesmaximal, characteristic
3#13505C211yesmaximal, characteristic, complemented
4#10618C211yesmaximal, characteristic, complemented
5#7128C211yesmaximal, characteristic, complemented
6#7121C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8#2580—21no—
9C2×#367—21no—
10C2×#366—21no—
11C2×#357—21no—
12#2092—21no—
13#2085—21no—
14#2065—21no—
15#1984C2211yescharacteristic
16#1970C2211yescharacteristic
17#1942C2211yescharacteristic
18#1879—21no—
19#1862—21no—
20#1845C2211yescharacteristic
21#1803—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#367—42no—
56#366—42no—
57#357—42no—
58#340—21no—
59#340—21no—
60#340—41no—
61#336—21no—
62#336—21no—
63C22×Q8D811yescharacteristic
64C22×D4—21no—
65C23×C4—41no—
66C2×D8—21no—
67C2×D8—41no—
68C2×D8—41no—
69C2×D8Q1612yes—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4×Q8—41no—
75C4×Q8—41no—
76C4×D4—81no—
77C4×D4—81no—
78C2×C4⋊C4—21no—
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⋊C4D811yescharacteristic
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C42—21no—
94C2×C42C2×D411yescharacteristic
95C2×C16—22no—
96C2×C16—22no—
97C2×C16—41no—
98C2×C16—41no—
99C2×C16—41no—
100C2×C16—41no—
101#107—21no—
102#107—21no—
103#107—41no—
104#107—41no—
105#107—41no—
106#107—41no—
107#107—41no—
108#107—41no—
109#106—81no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113#103—82no—
114#102—41no—
115#102—41no—
116#102—81no—
117#102—81no—
118C4×C8—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123C24—41no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—41no—
127C2×Q8—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—81no—
141C22×C4—161no—
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—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—81no—
156C2×C8—81no—
157C2×C8—81no—
158C2×C8C2×Q1611yescharacteristic
159C2×C8#13511yescharacteristic
160C2×C8AGL(1,ℤ/8)11yescharacteristic
161C2×C8#13711yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—161no—
175C4⋊C4—161no—
176C4⋊C4—161no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C22⋊C4—81no—
180C22⋊C4—81no—
181C22⋊C4—161no—
182C22⋊C4—161no—
183C42—21no—
184C42—21no—
185C42—41no—
186C42—41no—
187C16—42no—
188C16—42no—
189C23—42no—
190C23—81no—
191C23—81no—
192C23—81no—
193C23—81no—
194C23—81no—
195C23—81no—
196C23#44711yescharacteristic
197Q8—42no—
198Q8—81no—
199D4—42no—
200D4—42no—
201D4—81no—
202D4—81no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—164no—
224C2×C4—161no—
225C2×C4—161no—
226C2×C4—161no—
227C2×C4#44911yescharacteristic
228C2×C4#45111yescharacteristic
229C2×C4#45311yescharacteristic
230C2×C4#45511yescharacteristic
231C2×C4#45611yescharacteristic
232C2×C4#45811yescharacteristic
233C2×C4#50811yescharacteristic
234C2×C4#51011yescharacteristic
235C8—21no—
236C8—21no—
237C8—21no—
238C8—21no—
239C8—81no—
240C8—161no—
241C22—82no—
242C22—82no—
243C22—82no—
244C22—161no—
245C22—161no—
246C22—161no—
247C22—161no—
248C22#160711yescharacteristic
249C22#162111yescharacteristic
250C22#164911yescharacteristic
251C22#218911yescharacteristic
252C22#219011yescharacteristic
253C22#219211yescharacteristic
254C22#219411yescharacteristic
255C4—21no—
256C4—21no—
257C4—41no—
258C4—81no—
259C4—81no—
260C4—161no—
261C4—161no—
262C4—161no—
263C4—321no—
264C4#160911yescharacteristic
265C4#161411yescharacteristic
266C4#164411yescharacteristic
267C4#166111yescharacteristic
268C2—162no—
269C2#778911yescharacteristic
270C2#1228311yescharacteristic
271C2#1228411yescharacteristic
272C2#1233311yescharacteristic
273C2#1233611yescharacteristic
274C2#1241111yescharacteristic
275C2#1241411yescharacteristic
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
3. #13505C2semidirect
4. #10618C2semidirect
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
#135272
#135052
#106182
#71282
#71212
#71122