Finite Group Databasebeta

← #125391

Subgroups — #125391

Order 512.

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

normal subgroups only

This group has 272 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#33275C211yesmaximal, characteristic
2#12596C211yesmaximal, characteristic, complemented
3#8137C211yesmaximal, characteristic, complemented
4#7127C211yesmaximal, characteristic, complemented
5#7126C211yesmaximal, characteristic, complemented
6#7110C211yesmaximal, characteristic
7#7110C211yesmaximal, characteristic
8#2644—21no—
9C2×#347—21no—
10#1972C411yescharacteristic
11#1972C2211yescharacteristic
12#1970C411yescharacteristic
13#1944C2211yescharacteristic
14#1943C411yescharacteristic
15#1941C411yescharacteristic
16#1936—21no—
17#1934—21no—
18#1934—21no—
19#1726C2211yescharacteristic
20#1450—21no—
21#1377C2211yescharacteristic
22#1377C2211yescharacteristic
23#1376—21no—
24#1275C2211yescharacteristic
25#1275C2211yescharacteristic
26#1274—21no—
27C22×D8C2×C411yescharacteristic
28C2×M32—21no—
29C2×M32—21no—
30#431—21no—
31#431—21no—
32C2×#107C2×C411yescharacteristic
33C2×#106C2×C411yescharacteristic
34C2×#106C2×C411yescharacteristic
35C2×#102—21no—
36C2×#102—21no—
37C2×#102—21no—
38C2×#97C2×C411yescharacteristic
39C2×#97D411yescharacteristic
40C2×#97C2311yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#397—41no—
45#397—41no—
46#397—41no—
47#396—41no—
48#390—21no—
49#390—21no—
50#390—21no—
51#389—21no—
52#389—21no—
53#389—21no—
54#376D411yescharacteristic
55#347—24no—
56#340—21no—
57#340—21no—
58#340—21no—
59#322—22no—
60C82—22no—
61C22×D4—21no—
62C2×D8—22no—
63C2×D8—41no—
64C2×D8—41no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42C2×D411yescharacteristic
84M32—22no—
85M32—22no—
86M32—41no—
87M32—41no—
88C2×C16—41no—
89C2×C16—41no—
90#107—41no—
91#106—22no—
92#106—41no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95#102—41no—
96#102—41no—
97#102—42no—
98#102—41no—
99#102—41no—
100C22⋊C8—41no—
101C22⋊C8—41no—
102#97—22no—
103#97—41no—
104#97—41no—
105C4×C8—21no—
106C4×C8—21no—
107C4×C8—22no—
108C4×C8—21no—
109C4×C8—21no—
110C4×C8—22no—
111C4×C8—22no—
112#95—21no—
113#95—21no—
114#95—41no—
115#95—41no—
116#95—41no—
117C24—41no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4AGL(1,ℤ/8)11yescharacteristic
131C22×C4AGL(1,ℤ/8)11yescharacteristic
132D8—82no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8#10411yescharacteristic
144C2×C8#10411yescharacteristic
145C2×C8#10412yes—
146C2×C8#13112yes—
147C2×C8#13511yescharacteristic
148C2×C8#13511yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—161no—
156C4⋊C4—161no—
157C4⋊C4—161no—
158C22⋊C4—161no—
159C22⋊C4—161no—
160C22⋊C4—161no—
161C22⋊C4—161no—
162C22⋊C4—161no—
163C22⋊C4—161no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C16—42no—
171C16—42no—
172C23—81no—
173C23—81no—
174C23—81no—
175C23—81no—
176C23—81no—
177C23—81no—
178C23—81no—
179C23#41811yescharacteristic
180D4—81no—
181D4—81no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4#40811yescharacteristic
204C2×C4#41011yescharacteristic
205C2×C4#41311yescharacteristic
206C2×C4#41611yescharacteristic
207C2×C4C2×#10411yescharacteristic
208C2×C4C2×#10411yescharacteristic
209C8—22no—
210C8—21no—
211C8—21no—
212C8—21no—
213C8—21no—
214C8—22no—
215C8—82no—
216C8—82no—
217C22—21no—
218C22—21no—
219C22—161no—
220C22—161no—
221C22—161no—
222C22—161no—
223C22—161no—
224C22—161no—
225C22—161no—
226C22#148411yescharacteristic
227C22#150511yescharacteristic
228C22#155311yescharacteristic
229C4—41no—
230C4—41no—
231C4—81no—
232C4—81no—
233C4—161no—
234C4—161no—
235C4—161no—
236C4#147311yescharacteristic
237C4#149411yescharacteristic
238C4#150011yescharacteristic
239C4#152411yescharacteristic
240C2—21no—
241C2—21no—
242C2—321no—
243C2#769211yescharacteristic
244C2#1176911yescharacteristic
245C2#1177011yescharacteristic
subgroupG/H
center, socleC22#1505
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12596C2semidirect
3. #8137C2semidirect
4. #7127C2semidirect
5. #7126C2semidirect

Maximal subgroups

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

subgroupindex
#332752
#125962
#81372
#71272
#71262
#71102