Finite Group Databasebeta

← #139691

Subgroups — #139691

Order 512.

350 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 82 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#33497C211yesmaximal, characteristic, complemented
2#28188C211yesmaximal, characteristic
3#11663C212yesmaximal
4#8615C211yesmaximal, characteristic, complemented
5#7221C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×C4×D8C2211yescharacteristic
8#2580—21no—
9#2183—24no—
10#2163—21no—
11#2163C2212yes—
12#2162—21no—
13#2087C411yescharacteristic
14#2087C411yescharacteristic
15#2054C411yescharacteristic
16#2054C411yescharacteristic
17#1934—21no—
18#1934—21no—
19#1835C2211yescharacteristic
20#1804—21no—
21#1801—21no—
22#1566C2212yes—
23#1333—24no—
24#1292—21no—
25C22×D8C2×C411yescharacteristic
26C2×C4×Q8D411yescharacteristic
27C2×C4×D4—21no—
28C8×Q8—22no—
29C8×Q8—22no—
30C4×D8—22no—
31C4×D8—22no—
32C2×#107C2×C411yescharacteristic
33#423—22no—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8D411yescharacteristic
37C2×#102—21no—
38C2×#102—21no—
39C2×#102—21no—
40C2×#101—41no—
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44#399—41no—
45#393—41no—
46#390—21no—
47#390—21no—
48#389—21no—
49#389—21no—
50#385—41no—
51#385—41no—
52#382—21no—
53C4⋊C16—22no—
54C4⋊C16—22no—
55C4⋊C16—22no—
56C22⋊C16—22no—
57#346—22no—
58#346D414yes—
59C4×C16—22no—
60#343—41no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#334—22no—
67#334—22no—
68C22×Q8D811yescharacteristic
69C22×D4—21no—
70C23×C4—41no—
71C2×D8—22no—
72C2×D8—42no—
73C2×M16—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C4×Q8—22no—
79C4×Q8—22no—
80C4×D4—42no—
81C4×D4—42no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4D811yescharacteristic
88C2×C4⋊C4SD1611yescharacteristic
89C2×C4⋊C4SD1611yescharacteristic
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C16—24no—
99#107—22no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105#102—42no—
106#102—42no—
107#102—41no—
108#102—41no—
109#101—82no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8C2×D412yes—
115C4×C8#4112yes—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C24—41no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—41no—
124C2×D4—22no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—82no—
134C22×C4—161no—
135C22×C4—161no—
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143D8—24no—
144D8—42no—
145M16—82no—
146M16—82no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—82no—
154C2×C8C4×D412yes—
155C2×C8#12011yescharacteristic
156C2×C8#12311yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—161no—
167C4⋊C4—161no—
168C22⋊C4—82no—
169C22⋊C4—82no—
170C22⋊C4—82no—
171C22⋊C4—161no—
172C22⋊C4—161no—
173C42—21no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42#12112yes—
178C42#13512yes—
179C16—44no—
180C23—42no—
181C23—81no—
182C23—81no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23—81no—
187C23C2×#10211yescharacteristic
188Q8—44no—
189D4—42no—
190D4—42no—
191D4—81no—
192D4—81no—
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—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
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—164no—
217C2×C4#36012yes—
218C2×C4#38611yescharacteristic
219C2×C4#41611yescharacteristic
220C2×C4#46512yes—
221C2×C4#46612yes—
222C8—22no—
223C8—21no—
224C8—21no—
225C8—41no—
226C8—41no—
227C8—162no—
228C22—21no—
229C22—21no—
230C22—82no—
231C22—82no—
232C22—82no—
233C22—161no—
234C22—161no—
235C22—161no—
236C22—161no—
237C22#189311yescharacteristic
238C22C2×#36011yescharacteristic
239C22#214511yescharacteristic
240C4—22no—
241C4—22no—
242C4—21no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—161no—
248C4—161no—
249C4—161no—
250C4#189111yescharacteristic
251C4#191211yescharacteristic
252C2—21no—
253C2—21no—
254C2—162no—
255C2#948611yescharacteristic
256C2#1306511yescharacteristic
257C2#1307911yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33497C2semidirect
4. #8615C2semidirect
5. #7221C2semidirect

Maximal subgroups

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

subgroupindex
#334972
#281882
#116632
#86152
#72212