Finite Group Databasebeta

← #139586

Subgroups — #139586

Order 512.

384 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 90 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#33487C211yesmaximal, characteristic
2#27662C211yesmaximal, characteristic
3#12867C212yesmaximal, complemented
4#8609C211yesmaximal, characteristic
5#7230C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×#107C2211yescharacteristic
8#2487—21no—
9#2168—24no—
10#2163—21no—
11#2112—21no—
12C2×C22⋊C16C2212yes—
13#2087C412yescomplemented
14#2076C412yescomplemented
15#1942—21no—
16#1824—21no—
17#1808—21no—
18#1808C2212yes—
19#1808C2211yescharacteristic
20#1799—21no—
21#1380—24no—
22#1289—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4×D4D411yescharacteristic
26C22×C4⋊C4—21no—
27C2×M32—22no—
28C22×C16D414yes—
29C8×D4—22no—
30C8×D4—22no—
31C2×#107—22no—
32C2×#107—22no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—22no—
36C2×#107C2×C412yes—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×#99—41no—
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C22⋊C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#400—41no—
43#399—41no—
44#389—22no—
45#386—41no—
46#385—41no—
47#380—22no—
48#380—21no—
49#380—22no—
50C4⋊C16—22no—
51C22⋊C16—22no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#346—22no—
55#342—41no—
56#340—22no—
57#340—22no—
58#340—22no—
59#340C2×C412yes—
60C22×D4D811yescharacteristic
61C23×C4—41no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—42no—
67C22×C8—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D412yes—
72C22×C8C2×D411yescharacteristic
73C22×C8#4112yes—
74C4×D4—22no—
75C4×D4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4SD1611yescharacteristic
84C2×C22⋊C4—41no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4D811yescharacteristic
88C2×C22⋊C4SD1611yescharacteristic
89C2×C42C22⋊C411yescharacteristic
90M32—82no—
91C2×C16—24no—
92C2×C16—44no—
93#107—24no—
94#107—42no—
95#107—41no—
96#107—41no—
97#107—44no—
98C4⋊C8—22no—
99#99—82no—
100C22⋊C8—22no—
101C22⋊C8—22no—
102C4×C8—21no—
103C4×C8—21no—
104#95—42no—
105#95—42no—
106#95—41no—
107#95—42no—
108C24#10211yescharacteristic
109C24C2×D811yescharacteristic
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—82no—
123C22×C4—82no—
124C22×C4—162no—
125C22×C4#10211yescharacteristic
126C22×C4#10211yescharacteristic
127C22×C4#10211yescharacteristic
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4#12112yes—
130C22×C4C2×SD1611yescharacteristic
131C22×C4#13512yes—
132C2×C8—24no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—41no—
137C2×C8—42no—
138C2×C8—42no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—81no—
142C2×C8C4×D412yes—
143C2×C8#12011yescharacteristic
144C2×C8#12311yescharacteristic
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C4⋊C4—162no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—81no—
157C22⋊C4—81no—
158C22⋊C4—82no—
159C22⋊C4—82no—
160C22⋊C4—162no—
161C42—21no—
162C42—21no—
163C16—44no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C2×#10211yescharacteristic
172C23#46612yes—
173C23#50911yescharacteristic
174C23#51011yescharacteristic
175C23#51012yes—
176D4—42no—
177D4—42no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—164no—
201C2×C4—162no—
202C2×C4—162no—
203C2×C4—162no—
204C2×C4#38611yescharacteristic
205C2×C4#41611yescharacteristic
206C2×C4#46512yes—
207C8—21no—
208C8—22no—
209C8—41no—
210C8—41no—
211C8#45812yes—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—22no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22—42no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22#189311yescharacteristic
225C22#213911yescharacteristic
226C22#214111yescharacteristic
227C22#221912yes—
228C22#222012yes—
229C4—21no—
230C4—22no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—162no—
235C4—162no—
236C4—322no—
237C4#189011yescharacteristic
238C4#190911yescharacteristic
239C2—21no—
240C2—22no—
241C2—21no—
242C2—22no—
243C2—81no—
244C2#947011yescharacteristic
245C2#1306211yescharacteristic
246C2#1307411yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12867C2semidirect
5. #7230C2semidirect
13. #2087C4semidirect
14. #2076C4semidirect

Maximal subgroups

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

subgroupindex
#334872
#276622
#128672
#86092
#72302