Finite Group Databasebeta

← #122838

Subgroups — #122838

Order 512.

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

normal subgroups only

This group has 297 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#28188C211yesmaximal, characteristic
2#11494C211yesmaximal, characteristic, complemented
3#8617C211yesmaximal, characteristic, complemented
4#7089C212yesmaximal, complemented
5#7088C212yesmaximal
6C2×C8×Q8C2211yescharacteristic
7C2×C4×D8C411yescharacteristic
8#2580—21no—
9#2163—21no—
10#2087C411yescharacteristic
11#2087C2211yescharacteristic
12#2054C411yescharacteristic
13#2054C2211yescharacteristic
14#1934—21no—
15#1934—21no—
16#1835C411yescharacteristic
17#1804—21no—
18#1801—21no—
19#1567—21no—
20#1322—21no—
21#1322C2212yes—
22#1292C2212yes—
23C22×D8C2×C411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×D4—21no—
26C8×Q8—41no—
27C8×Q8—41no—
28C4×D8—41no—
29C4×D8—41no—
30C2×#107C2×C411yescharacteristic
31#423—22no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8D411yescharacteristic
34C2×C4⋊C8C2311yescharacteristic
35C2×#102—21no—
36C2×#102—21no—
37C2×#102—21no—
38C2×#101—41no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#399—41no—
43#393—41no—
44#390—21no—
45#390—21no—
46#389—21no—
47#389—21no—
48#385—41no—
49#385—41no—
50#382—21no—
51C4⋊C16—41no—
52C22⋊C16—22no—
53#346—41no—
54#343—41no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340C2×C411yescharacteristic
59#340C2×C411yescharacteristic
60#335—41no—
61#335—41no—
62C22×Q8D811yescharacteristic
63C22×D4—21no—
64C23×C4—41no—
65C2×D8—41no—
66C2×D8—42no—
67C2×M16—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C4×Q8—41no—
73C4×Q8—41no—
74C4×D4—42no—
75C4×D4—42no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4D811yescharacteristic
82C2×C4⋊C4SD1611yescharacteristic
83C2×C4⋊C4SD1611yescharacteristic
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C42C22⋊C411yescharacteristic
90C2×C42C22⋊C411yescharacteristic
91C2×C42C2×D411yescharacteristic
92C2×C16—42no—
93#107—41no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97C4⋊C8—42no—
98C4⋊C8—42no—
99#102—42no—
100#102—42no—
101#102—41no—
102#102—41no—
103#101—82no—
104C4×C8—21no—
105C4×C8—21no—
106C4×C8—21no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—21no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114C24—41no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×D4—22no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—82no—
129C22×C4—161no—
130C22×C4—161no—
131C22×C4#10211yescharacteristic
132C22×C4#10211yescharacteristic
133C22×C4#10211yescharacteristic
134C22×C4#10211yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C2×D811yescharacteristic
137C22×C4C2×SD1611yescharacteristic
138D8—42no—
139D8—42no—
140M16—82no—
141M16—82no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—82no—
150C2×C8#9911yescharacteristic
151C2×C8#9911yescharacteristic
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—161no—
162C4⋊C4—161no—
163C22⋊C4—82no—
164C22⋊C4—82no—
165C22⋊C4—81no—
166C22⋊C4—81no—
167C22⋊C4—161no—
168C22⋊C4—161no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C42—21no—
174C42—21no—
175C16—82no—
176C23—42no—
177C23—81no—
178C23—81no—
179C23—81no—
180C23—81no—
181C23—81no—
182C23—81no—
183C23C2×#10211yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—81no—
189D4—81no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—164no—
216C2×C4C2×#9911yescharacteristic
217C2×C4#42111yescharacteristic
218C2×C4#50911yescharacteristic
219C2×C4#51011yescharacteristic
220C8—21no—
221C8—21no—
222C8—41no—
223C8—41no—
224C8—41no—
225C8—162no—
226C22—21no—
227C22—21no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22—161no—
232C22—161no—
233C22—161no—
234C22—161no—
235C22#151911yescharacteristic
236C22#213911yescharacteristic
237C22#214511yescharacteristic
238C4—21no—
239C4—21no—
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—41no—
246C4—161no—
247C4—161no—
248C4—161no—
249C4#212011yescharacteristic
250C4#213311yescharacteristic
251C2—21no—
252C2—21no—
253C2—162no—
254C2#1187011yescharacteristic
255C2#1187911yescharacteristic
256C2#1189011yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11494C2semidirect
3. #8617C2semidirect
4. #7089C2semidirect

Maximal subgroups

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

subgroupindex
#281882
#114942
#86172
#70892
#70882