Finite Group Databasebeta

← #101781

Subgroups — #101781

Order 512.

398 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 84 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#28720C211yesmaximal, characteristic
2#23272C211yesmaximal, characteristic
3#8583C211yesmaximal, characteristic
4#7680C212yesmaximal, complemented
5#7027C212yesmaximal, complemented
6C2×#488—21no—
7#2859C2211yescharacteristic
8C4×#124C2211yescharacteristic
9#2072C412yes—
10#2071C412yes—
11#1991—21no—
12#1985—21no—
13#1923—22no—
14#1870—21no—
15#1805—21no—
16#1804—21no—
17#1767C2212yes—
18C4×C4⋊C8C2211yescharacteristic
19#1753—22no—
20#1744C2212yes—
21C2×#329—21no—
22#1295—21no—
23#1271—22no—
24#1271—22no—
25C22×M16—21no—
26C2×#125—21no—
27C2×#124D411yescharacteristic
28C2×C4×Q8D411yescharacteristic
29C2×C4×D4D411yescharacteristic
30#488—44no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×#103—22no—
35C2×#102—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—21no—
38C2×#97—22no—
39C2×#97—21no—
40C2×C4×C8—21no—
41C2×C4×C8—22no—
42#391C2×C411yescharacteristic
43#390C2×C411yescharacteristic
44#389—22no—
45#389—22no—
46#388—22no—
47#384C2×C412yes—
48#382—21no—
49#382D411yescharacteristic
50C4×C22⋊C4—22no—
51C43C2311yescharacteristic
52#340—22no—
53#340—22no—
54#336—22no—
55#336—22no—
56#336—22no—
57#336C2×C412yes—
58#336D412yes—
59#336D412yes—
60#329—42no—
61C22×Q8C22⋊C411yescharacteristic
62C22×D4C22⋊C411yescharacteristic
63C23×C4—21no—
64C2×M16—44no—
65C2×M16—81no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—42no—
70#125—41no—
71#125—42no—
72#125—41no—
73#124—21no—
74#124—22no—
75#124—21no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×D4—41no—
79C4×D4—41no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4C22⋊C411yescharacteristic
86C2×C4⋊C4C22⋊C411yescharacteristic
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C42—21no—
91C2×C42C22×C411yescharacteristic
92C2×C42C2×D411yescharacteristic
93C2×C42C2×D412yes—
94C2×C42C2×D411yescharacteristic
95C2×C42#4112yes—
96C4⋊C8—24no—
97C4⋊C8—42no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—81no—
101#103—44no—
102#102—44no—
103C22⋊C8—44no—
104C22⋊C8—81no—
105#97—42no—
106#97—42no—
107#97—81no—
108C4×C8—44no—
109C4×C8—42no—
110#95—22no—
111#95—22no—
112#95—41no—
113#95—41no—
114C24—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—41no—
122C2×D4—41no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C4×D412yes—
133C22×C4#12011yescharacteristic
134C22×C4#12112yes—
135C22×C4#12311yescharacteristic
136M16—84no—
137C2×C8—24no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—44no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—82no—
148C2×C8—82no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—81no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—81no—
164C42—22no—
165C42—22no—
166C42—21no—
167C42—21no—
168C42—42no—
169C42—41no—
170C42—41no—
171C42—41no—
172C42AGL(1,ℤ/8)11yescharacteristic
173C42AGL(1,ℤ/8)12yes—
174C42#13711yescharacteristic
175C23—42no—
176C23—41no—
177C23—41no—
178C23—42no—
179C23—41no—
180C23#38611yescharacteristic
181Q8—42no—
182Q8—81no—
183D4—42no—
184D4—81no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4#41711yescharacteristic
208C2×C4#41811yescharacteristic
209C2×C4#44312yes—
210C2×C4#44412yes—
211C2×C4#45511yescharacteristic
212C2×C4#45511yescharacteristic
213C2×C4#46812yes—
214C2×C4#46912yes—
215C8—44no—
216C8—82no—
217C8—82no—
218C8—84no—
219C22—82no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22—82no—
224C22#187612yes—
225C22#188511yescharacteristic
226C22#189511yescharacteristic
227C22#190612yes—
228C22#191511yescharacteristic
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—22no—
236C4—22no—
237C4—41no—
238C4—81no—
239C4—81no—
240C4—81no—
241C4—162no—
242C2—161no—
243C2#906711yescharacteristic
244C2#921012yes—
245C2#926612yes—
246C2#940111yescharacteristic
247C2#943911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7680C2semidirect
5. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#287202
#232722
#85832
#76802
#70272