Finite Group Databasebeta

← #121966

Subgroups — #121966

Order 512.

638 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 138 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#13856C211yesmaximal, characteristic
2C2×#1285C212yesmaximal
3C2×#1285C212yesmaximal
4#7910C211yesmaximal, characteristic
5#7910C211yesmaximal, characteristic
6C2×#432C2212yes—
7C2×#432C2212yes—
8C2×C4×C4⋊C4C412yes—
9C2×C4×C4⋊C4C412yes—
10C2×C4×C4⋊C4C2211yescharacteristic
11C2×C4×C4⋊C4C2211yescharacteristic
12C2×#376C2211yescharacteristic
13#1444—21no—
14#1444—22no—
15#1444—21no—
16#1285—24no—
17#1285—24no—
18#1280—21no—
19#1280—22no—
20#1280—21no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C42C2×C412yes—
26C22×C42C2×C411yescharacteristic
27C22×C42C2×C412yes—
28C22×C42C2×C411yescharacteristic
29C22×C42C2311yescharacteristic
30#432—22no—
31#432—22no—
32#432—22no—
33#432—22no—
34#432—22no—
35#432—22no—
36#432D414yes—
37#432Q814yes—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×#97—22no—
43C2×#97—22no—
44C4×C4⋊C4—22no—
45C4×C4⋊C4—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—44no—
49C4×C4⋊C4—44no—
50#376—22no—
51#376—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95D411yescharacteristic
57C2×#95D412yes—
58C2×#95D411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4C22⋊C412yes—
63C23×C4C22⋊C412yes—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C22×C8—42no—
68C22×C8—42no—
69C2×C4⋊C4—44no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—44no—
73C2×C4⋊C4—44no—
74C2×C4⋊C4—44no—
75C2×C4⋊C4—44no—
76C2×C4⋊C4—44no—
77C2×C4⋊C4—44no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C42—24no—
82C2×C42—24no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—24no—
88C2×C42—42no—
89C2×C42—42no—
90C2×C42—42no—
91C2×C42—42no—
92C2×C42C2×D412yes—
93C2×C42C2×Q812yes—
94C2×C42#4112yes—
95C2×C42#4112yes—
96C2×C42#4112yes—
97C2×C42#4112yes—
98C4⋊C8—24no—
99C4⋊C8—24no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C22⋊C8—24no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106#97—24no—
107#97—24no—
108#97—42no—
109#97—42no—
110#95—22no—
111#95—22no—
112#95—24no—
113#95—44no—
114#95—44no—
115#95—42no—
116#95—41no—
117#95—41no—
118C24C2×C22⋊C411yescharacteristic
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4#10014yes—
150C22×C4#10114yes—
151C22×C4#11711yescharacteristic
152C22×C4#11711yescharacteristic
153C22×C4C4×D412yes—
154C22×C4C4×Q812yes—
155C22×C4#12112yes—
156C22×C4C22⋊Q812yes—
157C22×C4#12312yes—
158C22×C4#12312yes—
159C22×C4#12411yescharacteristic
160C22×C4#12411yescharacteristic
161C22×C4#12711yescharacteristic
162C22×C4C4⋊Q811yescharacteristic
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—84no—
166C2×C8—82no—
167C2×C8—82no—
168C4⋊C4—48no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—416no—
172C4⋊C4—416no—
173C42—28no—
174C42—28no—
175C42—82no—
176C42—84no—
177C42—82no—
178C42C22⋊Q812yes—
179C42C22⋊Q812yes—
180C42#12312yes—
181C42#12312yes—
182C42#12512yes—
183C42#12512yes—
184C42#12612yes—
185C42#12612yes—
186C23—21no—
187C23—22no—
188C23—21no—
189C23#38111yescharacteristic
190C23#39011yescharacteristic
191C23#39111yescharacteristic
192C23C2×#10012yes—
193C23C2×#10112yes—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—82no—
235C2×C4#38712yes—
236C2×C4#38712yes—
237C2×C4#39712yes—
238C2×C4#39812yes—
239C2×C4#39912yes—
240C2×C4#40012yes—
241C8—84no—
242C8—84no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#144511yescharacteristic
252C22#182112yes—
253C22#182711yescharacteristic
254C22#182712yes—
255C22#182711yescharacteristic
256C22#196511yescharacteristic
257C22#196611yescharacteristic
258C22#199712yes—
259C4—22no—
260C4—22no—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—22no—
265C4—44no—
266C4—44no—
267C4—82no—
268C4—84no—
269C4—82no—
270C2—21no—
271C2—21no—
272C2—22no—
273C2#874212yes—
274C2#874312yes—
275C2#1134711yescharacteristic
276C2#1134811yescharacteristic
277C2#1135411yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#138562
C2×#12852
#79102