Finite Group Databasebeta

← #150831

Subgroups — #150831

Order 512.

379 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 64 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#28449C211yesmaximal, characteristic
2#12798C211yesmaximal, characteristic
3#11476C211yesmaximal, characteristic
4#7403C214yesmaximal
5C2×#497—21no—
6C2×#462—21no—
7C2×#461—21no—
8#2868C411yescharacteristic, complemented
9C4×#124C2211yescharacteristic
10C2×#354—21no—
11#2016C412yescomplemented
12#2013C2212yes—
13#1960—21no—
14#1799—21no—
15#1767C411yescharacteristic, complemented
16#1461—21no—
17#1393C2212yes—
18#1324—21no—
19#1318C2212yes—
20C22×Q16—22no—
21C22×SD16—22no—
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2×C412yes—
25C2×#124D411yescharacteristic
26C2×C4×Q8D411yescharacteristic
27C2×C4×D4D411yescharacteristic
28#497—42no—
29#497—42no—
30#462—42no—
31#462—42no—
32#461—42no—
33#461—42no—
34#431—41no—
35C2×C4⋊C8—21no—
36C2×#103—22no—
37C2×#103—22no—
38C2×#99—22no—
39C2×#97—21no—
40#391C2×C411yescharacteristic
41#390C2311yescharacteristic
42#388—41no—
43#384—41no—
44#384—41no—
45#382D411yescharacteristic
46#380—41no—
47C4×C22⋊C4—41no—
48C43C2×C411yescharacteristic
49#354—42no—
50#342—41no—
51#336C2×C412yes—
52#324—22no—
53#324—41no—
54C22×Q8—22no—
55C22×Q8C22⋊C411yescharacteristic
56C22×D4C2×D411yescharacteristic
57C23×C4—21no—
58C2×Q16—44no—
59C2×Q16—44no—
60C2×Q16—82no—
61C2×SD16—44no—
62C2×SD16—42no—
63C2×SD16—42no—
64C2×SD16—82no—
65C22×C8—22no—
66C4⋊Q8—42no—
67C4⋊Q8—41no—
68C4⋊Q8—41no—
69C4⋊Q8—41no—
70C4⋊Q8—41no—
71C4⋊Q8—42no—
72C4⋊Q8—82no—
73#124—41no—
74#124D812yes—
75#124SD1612yes—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×D4—41no—
79C4×D4—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4C22⋊C411yescharacteristic
87C2×C4⋊C4C2×D411yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—42no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22×C411yescharacteristic
98C4⋊C8—22no—
99C4⋊C8—22no—
100C4⋊C8—81no—
101#103—44no—
102#103—44no—
103#99—42no—
104#99—42no—
105C22⋊C8—42no—
106#97—42no—
107C4×C8—81no—
108#95—41no—
109#95—41no—
110#95—81no—
111C24—21no—
112C2×Q8—21no—
113C2×Q8—41no—
114C2×Q8—41no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8#10212yes—
122C2×D4—21no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4C2×D811yescharacteristic
126C2×D4C2×SD1611yescharacteristic
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—42no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4#9911yescharacteristic
138C22×C4#9911yescharacteristic
139C22×C4C2×C22⋊C411yescharacteristic
140Q16—88no—
141SD16—84no—
142SD16—84no—
143C2×C8—24no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—82no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—81no—
150C4⋊C4—81no—
151C4⋊C4—81no—
152C4⋊C4—84no—
153C4⋊C4—84no—
154C4⋊C4—81no—
155C4⋊C4—81no—
156C4⋊C4—162no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—81no—
160C22⋊C4—82no—
161C22⋊C4—82no—
162C22⋊C4—82no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—41no—
167C42—42no—
168C42—42no—
169C42—41no—
170C42—41no—
171C42#10212yes—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23—81no—
177C23—81no—
178C23C2×#9911yescharacteristic
179Q8—42no—
180Q8—81no—
181Q8—82no—
182Q8—82no—
183Q8—82no—
184Q8—82no—
185D4—41no—
186D4—41no—
187D4—81no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—82no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—161no—
211C2×C4—162no—
212C2×C4—162no—
213C2×C4—162no—
214C2×C4#35311yescharacteristic
215C2×C4#35311yescharacteristic
216C2×C4C2×#10211yescharacteristic
217C2×C4#42111yescharacteristic
218C8—44no—
219C8—162no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22—161no—
229C22#151911yescharacteristic
230C22#212211yescharacteristic
231C22C2×#35311yescharacteristic
232C4—21no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—42no—
237C4—41no—
238C4—42no—
239C4—81no—
240C4—81no—
241C4—162no—
242C4—322no—
243C4#140712yes—
244C2—21no—
245C2—21no—
246C2—81no—
247C2—81no—
248C2#1187711yescharacteristic
249C2C2×#140711yescharacteristic
250C2#1358611yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #2868C4semidirect
11. #2016C4semidirect
15. #1767C4semidirect

Maximal subgroups

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

subgroupindex
#284492
#127982
#114762
#74032