Finite Group Databasebeta

← #127339

Subgroups — #127339

Order 512.

637 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 180 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#21818C211yesmaximal, characteristic
2C2×#1801C212yesmaximal
3#7238C214yesmaximal, complemented
4C22×C4×Q8C2211yescharacteristic
5C2×#431C411yescharacteristic
6C2×#431C411yescharacteristic
7C2×#423—22no—
8C22×#101—22no—
9#1984—22no—
10#1984C412yes—
11#1984C2212yes—
12#1817—22no—
13#1801—22no—
14#1801—22no—
15#1801C414yes—
16#1801C2214yes—
17C2×#343C414yes—
18C23×Q8D411yescharacteristic
19C22×M16—24no—
20C2×C4×Q8—42no—
21C2×C4×Q8—42no—
22C2×C4×Q8C2×C412yes—
23C2×C4×Q8C2×C411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×Q8D412yes—
26C2×C4×Q8Q811yescharacteristic
27C2×C4×Q8C2311yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4Q811yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D412yes—
32#431—42no—
33#431—42no—
34#423—24no—
35#423—24no—
36#423—42no—
37#423—42no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—42no—
40C2×C4⋊C8—42no—
41C2×C4⋊C8C2×C412yes—
42C2×#101—24no—
43C2×#101—24no—
44C2×#101—44no—
45C2×#101—44no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C4×C8—21no—
49C2×C4×C8C2×C412yes—
50#343—24no—
51#343C2×C418yes—
52#336—24no—
53#336—24no—
54C22×Q8—42no—
55C22×Q8—42no—
56C22×Q8C22⋊C412yes—
57C22×Q8C4⋊C411yescharacteristic
58C22×Q8D812yes—
59C22×Q8SD1614yes—
60C22×Q8Q1612yes—
61C22×Q8C2×D411yescharacteristic
62C23×C4C4211yescharacteristic
63C23×C4C22⋊C412yes—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C4⋊C411yescharacteristic
66C23×C4C4⋊C412yes—
67C2×M16—28no—
68C2×M16—44no—
69C2×M16—48no—
70C2×M16—48no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—44no—
74C4×Q8—42no—
75C4×Q8—42no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×Q8—42no—
79C4×Q8—42no—
80C4×Q8—42no—
81C4×Q8—42no—
82C4×Q8—41no—
83C4×Q8—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4C4212yes—
88C2×C4⋊C4C22⋊C412yes—
89C2×C4⋊C4C4⋊C412yes—
90C2×C4⋊C4C4⋊C412yes—
91C2×C4⋊C4C4⋊C411yescharacteristic
92C2×C4⋊C4C22×C412yes—
93C2×C4⋊C4C2×Q811yescharacteristic
94C2×C42—22no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42C4211yescharacteristic
101C2×C42C22⋊C412yes—
102C2×C42C22⋊C412yes—
103C2×C42C22⋊C411yescharacteristic
104C2×C42C4⋊C412yes—
105C2×C42C4⋊C411yescharacteristic
106C2×C42C22×C411yescharacteristic
107C2×C42C2×D412yes—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—44no—
111C4⋊C8—44no—
112C4⋊C8—44no—
113C4⋊C8—44no—
114#101—48no—
115#101—44no—
116#101—44no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C4×C8—42no—
120C4×C8—42no—
121C24#9511yescharacteristic
122C2×Q8—42no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—42no—
126C2×Q8—41no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—41no—
132C2×Q8#10214yes—
133C2×Q8#10314yes—
134C2×Q8#10612yes—
135C2×Q8#10712yes—
136C2×Q8C2×D811yescharacteristic
137C2×Q8C2×SD1612yes—
138C2×Q8C2×Q1611yescharacteristic
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—82no—
149C22×C4—81no—
150C22×C4#9511yescharacteristic
151C22×C4#9512yes—
152C22×C4#9512yes—
153C22×C4#9511yescharacteristic
154C22×C4#9511yescharacteristic
155C22×C4#9912yes—
156C22×C4#9912yes—
157C22×C4#10214yes—
158C22×C4#10314yes—
159C22×C4#10612yes—
160C22×C4#10712yes—
161C22×C4C2×C4211yescharacteristic
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4C2×C22⋊C412yes—
164C22×C4C2×C4⋊C411yescharacteristic
165C22×C4C2×C4⋊C412yes—
166M16—48no—
167M16—48no—
168M16—88no—
169C2×C8—48no—
170C2×C8—44no—
171C2×C8—44no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—88no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—22no—
193C42—22no—
194C42—22no—
195C42—22no—
196C23—41no—
197C23—41no—
198C23H(3,ℤ/4)12yes—
199C23#34012yes—
200C23#34212yes—
201C23C2×#9511yescharacteristic
202Q8—44no—
203Q8—44no—
204Q8—42no—
205Q8—42no—
206Q8—41no—
207Q8—42no—
208Q8—41no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4#34014yes—
239C2×C4#34012yes—
240C2×C4#34212yes—
241C2×C4C2×#9911yescharacteristic
242C2×C4C2×#9911yescharacteristic
243C2×C4C2×#10212yes—
244C2×C4C2×#10312yes—
245C2×C4C2×#10611yescharacteristic
246C2×C4C2×#10711yescharacteristic
247C8—82no—
248C8—88no—
249C8—82no—
250C22—21no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—42no—
255C22—42no—
256C22#128914yes—
257C22C2×H(3,ℤ/4)11yescharacteristic
258C22C2×#34011yescharacteristic
259C22C2×#34211yescharacteristic
260C4—21no—
261C4—22no—
262C4—22no—
263C4—24no—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—42no—
270C4—41no—
271C4—41no—
272C4#213312yes—
273C2—24no—
274C2—21no—
275C2—21no—
276C2C2×#128911yescharacteristic
277C2#1257911yescharacteristic
278C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7238C2semidirect

Maximal subgroups

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

subgroupindex
#218182
C2×#18012
#72382