Finite Group Databasebeta

← #138492

Subgroups — #138492

Order 512.

329 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 80 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#29879C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic
3#11969C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic
5#7221C212yesmaximal, complemented
6C2×#499C2211yescharacteristic
7#2920—21no—
8C2×#431C2211yescharacteristic
9#2851—21no—
10#2163C2212yes—
11#2088—21no—
12#2087C411yescharacteristic
13#2086C411yescharacteristic
14#2077—21no—
15#2076C411yescharacteristic
16#2075C411yescharacteristic
17#1991—21no—
18#1835C2211yescharacteristic
19#1572—24no—
20#1567—21no—
21#1566C2212yes—
22#1363—24no—
23#1322—21no—
24#1314—21no—
25C2×#125—21no—
26C22×C42D411yescharacteristic
27#499—22no—
28#499—22no—
29#432—22no—
30#432—41no—
31#431—22no—
32#431—22no—
33#431—22no—
34#431—41no—
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×#106—21no—
38#423—22no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41#407—41no—
42C2×C22⋊C8C2×C411yescharacteristic
43C2×C22⋊C8D411yescharacteristic
44C4×M16—41no—
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#389—21no—
48#389—21no—
49#387—41no—
50#387—41no—
51#385—41no—
52#385—41no—
53#382—21no—
54#380—21no—
55#380—21no—
56C4⋊C16—22no—
57C22⋊C16—22no—
58#346—22no—
59#340—21no—
60#340—21no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#335—24no—
67#334—22no—
68#334D414yes—
69C23×C4C22⋊C411yescharacteristic
70C23×C4D811yescharacteristic
71C23×C4SD1611yescharacteristic
72C2×M16—42no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22×C411yescharacteristic
76C22×C8C2×D411yescharacteristic
77#125—41no—
78#125—41no—
79#125—41no—
80#125—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42D811yescharacteristic
97C2×C42SD1611yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C16—42no—
100#107—22no—
101#107—22no—
102#106—42no—
103C4⋊C8—22no—
104C4⋊C8—22no—
105C4⋊C8—24no—
106C4⋊C8—42no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111#97—42no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—42no—
115C4×C8C2×D412yes—
116C4×C8#4112yes—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123C24#10211yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—81no—
139C22×C4#10211yescharacteristic
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4C2×D811yescharacteristic
144C22×C4C2×SD1611yescharacteristic
145M16—82no—
146M16—82no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—44no—
154C2×C8C4×D412yes—
155C2×C8#12112yes—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C22⋊C4—82no—
171C22⋊C4—82no—
172C42—21no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42#12011yescharacteristic
179C42#12311yescharacteristic
180C42AGL(1,ℤ/8)11yescharacteristic
181C42#13711yescharacteristic
182C16—82no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—41no—
188C23C2×#10211yescharacteristic
189C2×C4—21no—
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—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
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—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4#38611yescharacteristic
221C2×C4#41711yescharacteristic
222C2×C4#44711yescharacteristic
223C2×C4#44811yescharacteristic
224C2×C4#48011yescharacteristic
225C2×C4#48111yescharacteristic
226C8—22no—
227C8—22no—
228C8—41no—
229C8—41no—
230C8—84no—
231C22—21no—
232C22—21no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22#187411yescharacteristic
239C22#214511yescharacteristic
240C22#214511yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—161no—
250C4—161no—
251C4—161no—
252C4—161no—
253C4#190911yescharacteristic
254C4#191511yescharacteristic
255C2—21no—
256C2—21no—
257C2—81no—
258C2#919511yescharacteristic
259C2#1299811yescharacteristic
260C2#1300211yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11969C2semidirect
5. #7221C2semidirect

Maximal subgroups

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

subgroupindex
#298792
#178152
#119692
#114602
#72212