Finite Group Databasebeta

← #133969

Subgroups — #133969

Order 512.

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

normal subgroups only

This group has 737 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#35644C211yesmaximal, characteristic
2#12862C214yesmaximal, complemented
3#8550C212yesmaximal
4C2×#544C411yescharacteristic
5C22×C22⋊Q8C2211yescharacteristic
6#2561—22no—
7#2503—22no—
8#2492—24no—
9#2489—24no—
10#2310C2212yes—
11#2280C412yes—
12#2274C411yescharacteristic
13#1784C2214yes—
14#1463—22no—
15#1462—22no—
16#1394—24no—
17#1394—24no—
18C23×Q8C2311yescharacteristic
19C24×C4C2×C411yescharacteristic
20#544—44no—
21C2×C4⋊Q8—22no—
22C2×#125—22no—
23C2×C22⋊Q8—24no—
24C2×C22⋊Q8—22no—
25C2×C22⋊Q8—44no—
26C2×#117C2×C412yes—
27C2×#117C2×C412yes—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D412yes—
30C22×C22⋊C4D412yes—
31C2×#101—28no—
32#407—24no—
33#400—28no—
34#397—24no—
35#393—24no—
36#391—24no—
37#391—24no—
38#391D414yes—
39#391D414yes—
40#388—28no—
41#382—24no—
42#381—22no—
43#381—24no—
44#380—22no—
45#380—22no—
46#379—22no—
47C4×C4⋊C4—24no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4—24no—
50#324—44no—
51#323—44no—
52C25C22⋊C411yescharacteristic
53C22×Q8—22no—
54C22×Q8—42no—
55C22×Q8—44no—
56C22×Q8C2×D414yes—
57C23×C4—22no—
58C23×C4—42no—
59C23×C4—44no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C412yes—
62C23×C4C22×C411yescharacteristic
63C23×C4C2×D412yes—
64C2×M16—44no—
65C4⋊Q8—48no—
66#125—48no—
67C22⋊Q8—416no—
68C22⋊Q8—44no—
69C22⋊Q8—44no—
70#117—44no—
71#117—44no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4C22⋊C412yes—
81C2×C4⋊C4C22⋊C412yes—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—42no—
89C2×C42—22no—
90C2×C42—24no—
91C2×C42C22⋊C414yes—
92#101—416no—
93C22⋊C8—44no—
94C22⋊C8—44no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—28no—
99#95—28no—
100C24—21no—
101C24—22no—
102C24—42no—
103C24—42no—
104C24—81no—
105C24C2×C22⋊C411yescharacteristic
106C2×Q8—28no—
107C2×Q8—44no—
108C2×Q8—44no—
109C2×Q8—44no—
110C2×Q8—88no—
111C2×Q8—84no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—44no—
127C22×C4—42no—
128C22×C4—88no—
129C22×C4—82no—
130C22×C4#9914yes—
131C22×C4C2×C22⋊C412yes—
132C22×C4#12014yes—
133M16—88no—
134C2×C8—84no—
135C4⋊C4—28no—
136C4⋊C4—48no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—44no—
140C4⋊C4—48no—
141C4⋊C4—44no—
142C22⋊C4—48no—
143C22⋊C4—48no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—84no—
147C42—44no—
148C42—44no—
149C42—84no—
150C23—22no—
151C23—22no—
152C23—24no—
153C23—41no—
154C23—41no—
155C23—42no—
156C23—41no—
157C23—42no—
158C23—41no—
159C23—42no—
160C23—82no—
161C23—82no—
162C23—88no—
163C23#37911yescharacteristic
164C23C2×#9912yes—
165Q8—88no—
166Q8—164no—
167C2×C4—28no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—28no—
173C2×C4—44no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—44no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—44no—
183C2×C4—42no—
184C2×C4—88no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4#41014yes—
190C8—164no—
191C22—21no—
192C22—21no—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—44no—
197C22—44no—
198C22—42no—
199C22—82no—
200C22—84no—
201C22—84no—
202C22—81no—
203C22—81no—
204C22#178111yescharacteristic
205C22#178212yes—
206C4—24no—
207C4—44no—
208C4—44no—
209C4—84no—
210C4—81no—
211C4—82no—
212C4—82no—
213C2—22no—
214C2—42no—
215C2—82no—
216C2#854411yescharacteristic
217C2#1270911yescharacteristic
218C2#1271011yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12862C2semidirect

Maximal subgroups

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

subgroupindex
#356442
#128622
#85502