Finite Group Databasebeta

← #153067

Subgroups — #153067

Order 512.

360 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 66 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#29145C211yesmaximal, characteristic
2#12797C211yesmaximal, characteristic
3C4×#353C211yesmaximal, characteristic
4#7504C214yesmaximal
5#2862C411yescharacteristic, complemented
6C4×#127C2211yescharacteristic
7C2×#353C2212yes—
8C2×#353C2212yes—
9#2023—21no—
10#2023—21no—
11#1961C2212yes—
12#1836C412yescomplemented
13#1804—21no—
14#1804—21no—
15#1799—21no—
16#1767C411yescharacteristic, complemented
17C4×#99—21no—
18#1393—21no—
19#1393—21no—
20C2×C4⋊Q8—21no—
21C2×#127C2311yescharacteristic
22C2×C4×D4—41no—
23C2×C4×D4D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25C2×C4⋊C8—21no—
26C2×#102—22no—
27C2×#102—22no—
28C2×#99—22no—
29C2×#99—22no—
30C2×#97C2×C412yes—
31#391—41no—
32#391—41no—
33#390—41no—
34#390D411yescharacteristic
35#390D411yescharacteristic
36#384C2×C411yescharacteristic
37#382—21no—
38#382C2×C412yes—
39#380—41no—
40C4×C22⋊C4—41no—
41C43C2×C411yescharacteristic
42#353—22no—
43#353—24no—
44#353Q814yescomplemented
45#342—41no—
46#342—41no—
47#340—22no—
48#340—22no—
49#336—21no—
50C22×Q8—41no—
51C22×D4—41no—
52C22×D4C2×D411yescharacteristic
53C22×D4C2×D411yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C22×C8—22no—
57C4⋊Q8—82no—
58#127—41no—
59#127C2×Q812yes—
60#127#4112yes—
61C4×D4—41no—
62C4×D4—41no—
63C4×D4—41no—
64C4×D4—41no—
65C4×D4—84no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—81no—
70C2×C4⋊C4C22⋊C411yescharacteristic
71C2×C4⋊C4C22⋊C411yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—42no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—81no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22×C411yescharacteristic
84C4⋊C8—41no—
85C4⋊C8—41no—
86#102—82no—
87#102—82no—
88#99—24no—
89#99—42no—
90#99—42no—
91#99—42no—
92#97—42no—
93#95—42no—
94#95—41no—
95#95—41no—
96#95—81no—
97#95—81no—
98C24—21no—
99C24—21no—
100C2×Q8—82no—
101C2×Q8—82no—
102C2×Q8—82no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—41no—
106C2×D4—41no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—82no—
112C2×D4—82no—
113C2×D4C22⋊Q812yes—
114C2×D4#12411yescharacteristic
115C2×D4C4⋊Q811yescharacteristic
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—42no—
125C22×C4—82no—
126C22×C4—82no—
127C22×C4—82no—
128C22×C4—82no—
129C22×C4#9911yescharacteristic
130C22×C4#9911yescharacteristic
131C22×C4C2×C22⋊C411yescharacteristic
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—42no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—82no—
140C4⋊C4—82no—
141C4⋊C4—162no—
142C22⋊C4—44no—
143C22⋊C4—81no—
144C22⋊C4—81no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C22⋊C4—82no—
148C22⋊C4—82no—
149C22⋊C4—82no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—41no—
154C42—42no—
155C42—81no—
156C42—162no—
157C42C4×Q812yes—
158C23—22no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—81no—
166C23—81no—
167C23—81no—
168C23—81no—
169C23C2×#9911yescharacteristic
170Q8—164no—
171D4—42no—
172D4—41no—
173D4—41no—
174D4—81no—
175D4—81no—
176D4—84no—
177D4—84no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—84no—
198C2×C4—82no—
199C2×C4—161no—
200C2×C4—161no—
201C2×C4—162no—
202C2×C4—162no—
203C2×C4—162no—
204C2×C4—162no—
205C2×C4—162no—
206C2×C4#35411yescharacteristic
207C2×C4#35411yescharacteristic
208C2×C4#39111yescharacteristic
209C2×C4#41011yescharacteristic
210C8—82no—
211C22—21no—
212C22—21no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22—82no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22—161no—
224C22—161no—
225C22#199611yescharacteristic
226C22C2×#35411yescharacteristic
227C22#212811yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—42no—
234C4—41no—
235C4—81no—
236C4—81no—
237C4—162no—
238C4—164no—
239C4—322no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—81no—
244C2—81no—
245C2#1329911yescharacteristic
246C2#1359411yescharacteristic
247C2#1359511yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, prime radical, cosocleC2×#127C23
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G
LECSG ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2862C4semidirect
12. #1836C4semidirect
16. #1767C4semidirect
44. #353Q8semidirect

Maximal subgroups

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

subgroupindex
#291452
#127972
C4×#3532
#75042