Finite Group Databasebeta

← #101779

Subgroups — #101779

Order 512.

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

normal subgroups only

This group has 392 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#29142C211yesmaximal, characteristic
2#23272C211yesmaximal, characteristic
3#8583C211yesmaximal, characteristic
4#7682C212yesmaximal, complemented
5#7027C212yesmaximal, complemented
6#2862C2211yescharacteristic
7C4×#124C2211yescharacteristic
8#2024C412yes—
9#2023C412yes—
10#1985—21no—
11#1958—21no—
12#1923—22no—
13#1870—21no—
14#1835—21no—
15#1805—21no—
16#1804—21no—
17#1767C2212yes—
18C4×C4⋊C8C2211yescharacteristic
19#1753—22no—
20#1744C2212yes—
21C2×#329—21no—
22#1295—21no—
23#1272—22no—
24#1272—22no—
25C22×M16—21no—
26C2×C4⋊Q8C2×C412yes—
27C2×#124D411yescharacteristic
28C2×C4×Q8D411yescharacteristic
29C2×C4×D4D411yescharacteristic
30C2×C4⋊C8—22no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×#103—22no—
34C2×#102—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—21no—
37C2×#97—22no—
38C2×#97—21no—
39C2×C4×C8—21no—
40C2×C4×C8—22no—
41#391—22no—
42#391—22no—
43#391C2×C411yescharacteristic
44#390C2×C411yescharacteristic
45#388—22no—
46#382—21no—
47#382—21no—
48#382D411yescharacteristic
49C4×C22⋊C4—22no—
50C43C2311yescharacteristic
51#340—22no—
52#340—22no—
53#336—22no—
54#336—22no—
55#336—22no—
56#336C2×C412yes—
57#336D412yes—
58#336D412yes—
59#329—42no—
60C22×Q8—22no—
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—
70C4⋊Q8—82no—
71#124—21no—
72#124—21no—
73#124—22no—
74C4×Q8—41no—
75C4×Q8—41no—
76C4×D4—41no—
77C4×D4—41no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4C22⋊C411yescharacteristic
82C2×C4⋊C4C22⋊C411yescharacteristic
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C42—21no—
87C2×C42C22×C411yescharacteristic
88C2×C42C2×D411yescharacteristic
89C2×C42C2×D412yes—
90C2×C42C2×D411yescharacteristic
91C2×C42#4112yes—
92C4⋊C8—24no—
93C4⋊C8—42no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—81no—
97#103—82no—
98#102—82no—
99C22⋊C8—44no—
100C22⋊C8—81no—
101#97—42no—
102#97—42no—
103#97—81no—
104C4×C8—44no—
105C4×C8—41no—
106C4×C8—41no—
107#95—22no—
108#95—22no—
109#95—41no—
110#95—42no—
111#95—41no—
112C24—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—82no—
120C2×Q8—82no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—41no—
124C2×D4—41no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—42no—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C4×D412yes—
135C22×C4#12011yescharacteristic
136C22×C4#12112yes—
137C22×C4#12311yescharacteristic
138M16—84no—
139C2×C8—24no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—44no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—82no—
150C2×C8—82no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—81no—
156C4⋊C4—82no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—81no—
162C42—22no—
163C42—22no—
164C42—21no—
165C42—21no—
166C42—42no—
167C42—41no—
168C42—41no—
169C42—41no—
170C42AGL(1,ℤ/8)11yescharacteristic
171C42#13711yescharacteristic
172C42#13712yes—
173C23—42no—
174C23—41no—
175C23—41no—
176C23—42no—
177C23—41no—
178C23#38611yescharacteristic
179Q8—42no—
180Q8—81no—
181Q8—162no—
182Q8—162no—
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#41911yescharacteristic
209C2×C4#44312yes—
210C2×C4#44412yes—
211C2×C4#45511yescharacteristic
212C2×C4#45511yescharacteristic
213C2×C4#46812yes—
214C2×C4#47012yes—
215C8—44no—
216C8—82no—
217C8—82no—
218C8—84no—
219C22—82no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22—82no—
224C22#187612yes—
225C22#188511yescharacteristic
226C22#189411yescharacteristic
227C22#190612yes—
228C22#191411yescharacteristic
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#920912yes—
245C2#926512yes—
246C2#939911yescharacteristic
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. #7682C2semidirect
5. #7027C2semidirect

Maximal subgroups

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

subgroupindex
#291422
#232722
#85832
#76822
#70272