Finite Group Databasebeta

← #122840

Subgroups — #122840

Order 512.

307 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 68 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#27662C211yesmaximal, characteristic
2#11466C211yesmaximal, characteristic
3#8614C211yesmaximal, characteristic
4#7089C214yesmaximal, complemented
5C2×C8×D4C2211yescharacteristic
6C22×#107C411yescharacteristic
7#2487—21no—
8#2112—21no—
9#2087C2212yes—
10#2076C412yes—
11#1942—21no—
12#1824—21no—
13#1808—21no—
14#1808C411yescharacteristic
15#1800—21no—
16#1566—21no—
17#1322C2212yes—
18#1314—21no—
19#1292C2212yes—
20C23×C8C2×C411yescharacteristic
21C23×C8D411yescharacteristic
22C2×C4×D4D411yescharacteristic
23C22×C4⋊C4—21no—
24C2×M32—41no—
25C8×D4—41no—
26C8×D4—41no—
27C2×#107—21no—
28C2×#107—21no—
29C2×#107—41no—
30C2×#107—41no—
31C2×#107C2×C412yes—
32#423—22no—
33C2×C4⋊C8C2311yescharacteristic
34C2×#100—41no—
35C2×C22⋊C8C2×C411yescharacteristic
36C2×C22⋊C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38#400—41no—
39#399—41no—
40#389—22no—
41#386—41no—
42#385—41no—
43#380—22no—
44#380—21no—
45C22⋊C16—22no—
46C22⋊C16—41no—
47#343—41no—
48#340—22no—
49#340—22no—
50#340C2×C412yes—
51#334—41no—
52#334—41no—
53C22×D4D811yescharacteristic
54C23×C4—41no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C411yescharacteristic
57C2×M16—42no—
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—42no—
63C22×C8—42no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4D811yescharacteristic
77C2×C22⋊C4—41no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4SD1611yescharacteristic
80C2×C22⋊C4SD1611yescharacteristic
81C2×C42C2×D411yescharacteristic
82M32—82no—
83C2×C16—42no—
84#107—42no—
85#107—41no—
86#107—41no—
87#107—42no—
88C4⋊C8—41no—
89C4⋊C8—42no—
90C4⋊C8—42no—
91#100—82no—
92C22⋊C8—41no—
93C22⋊C8—41no—
94C4×C8—21no—
95C4×C8—21no—
96#95—42no—
97#95—41no—
98#95—42no—
99C24#10211yescharacteristic
100C24#10211yescharacteristic
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—41no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—82no—
115C22×C4—162no—
116C22×C4#10211yescharacteristic
117C22×C4#10211yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C2×D811yescharacteristic
120C22×C4C2×SD1611yescharacteristic
121M16—82no—
122M16—82no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—81no—
134C2×C8—82no—
135C2×C8#9911yescharacteristic
136C2×C8#9911yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—82no—
143C4⋊C4—162no—
144C22⋊C4—41no—
145C22⋊C4—41no—
146C22⋊C4—82no—
147C22⋊C4—162no—
148C42—21no—
149C42—21no—
150C16—82no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23#36012yes—
161C23C2×#10211yescharacteristic
162D4—44no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—81no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—164no—
186C2×C4C2×#9911yescharacteristic
187C2×C4#42111yescharacteristic
188C2×C4#51012yes—
189C8—21no—
190C8—41no—
191C8—41no—
192C8—41no—
193C8—162no—
194C8#35212yes—
195C22—21no—
196C22—21no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—81no—
207C22#151911yescharacteristic
208C22#214111yescharacteristic
209C22C2×#36011yescharacteristic
210C4—21no—
211C4—22no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—162no—
216C4—162no—
217C4C2×#35211yescharacteristic
218C4#213511yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2—81no—
223C2#1187011yescharacteristic
224C2#1188111yescharacteristic
225C2#1188911yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7089C2semidirect

Maximal subgroups

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

subgroupindex
#276622
#114662
#86142
#70892