Finite Group Databasebeta

← #142057

Subgroups — #142057

Order 512.

488 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 94 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#15152C211yesmaximal, characteristic
2#10802C211yesmaximal, characteristic
3#8153C211yesmaximal, characteristic
4#7251C214yesmaximal
5C2×#400—21no—
6C2×#384C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8#1749—21no—
9C2×#343—21no—
10C2×#343C2212yes—
11#1728—21no—
12C2×#324C2212yes—
13#1445—21no—
14#1442C412yes—
15#1441C2212yes—
16C4⋊#95—21no—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic
19#1280—21no—
20C22×M16—22no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2×C412yes—
25C22×C4⋊C4D411yescharacteristic
26C22×C4⋊C4C2311yescharacteristic
27C22×C42C2×C411yescharacteristic
28C22×C42D411yescharacteristic
29C22×C42D411yescharacteristic
30C2×C22⋊C8—22no—
31C2×#97—41no—
32#400—44no—
33#384—41no—
34#384—41no—
35C4×C4⋊C4—41no—
36C4×C4⋊C4—41no—
37C2×#95—22no—
38C2×#95—21no—
39C2×#95—22no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95C2×C412yes—
43C2×#95C2×C411yescharacteristic
44C2×#95D411yescharacteristic
45#343—24no—
46#343—24no—
47#336—41no—
48#336—41no—
49#336—41no—
50#324—22no—
51#324Q814yes—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D411yescharacteristic
60C23×C4C2×D411yescharacteristic
61C2×M16—24no—
62C2×M16—42no—
63C2×M16—42no—
64C2×M16—42no—
65C2×M16—42no—
66C2×M16—42no—
67C22×C8—42no—
68C22×C8—42no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—82no—
86C2×C4⋊C4C2×Q812yes—
87C2×C4⋊C4#4112yes—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42—41no—
96C2×C42—42no—
97C2×C42—42no—
98C22⋊C8—24no—
99C22⋊C8—42no—
100#97—82no—
101#95—41no—
102#95—41no—
103#95—42no—
104#95—42no—
105#95—44no—
106#95—44no—
107#95—82no—
108#95—82no—
109C24C2×C22⋊C411yescharacteristic
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4#9911yescharacteristic
155C22×C4#9911yescharacteristic
156C22×C4#10412yes—
157C22×C4C4×Q812yes—
158C22×C4C22⋊Q812yes—
159C22×C4#12411yescharacteristic
160C22×C4C4⋊Q811yescharacteristic
161C22×C4AGL(1,ℤ/8)12yes—
162C22×C4#13712yes—
163M16—44no—
164M16—44no—
165M16—82no—
166M16—82no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—82no—
170C2×C8—82no—
171C2×C8—82no—
172C2×C8—82no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—84no—
177C4⋊C4—82no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—82no—
181C42—42no—
182C42—42no—
183C42—42no—
184C42—42no—
185C42—42no—
186C42—81no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23#39111yescharacteristic
192C23C2×#9911yescharacteristic
193C23#41011yescharacteristic
194C23#41811yescharacteristic
195C23#41911yescharacteristic
196C23C2×#10411yescharacteristic
197C23#42111yescharacteristic
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4#35212yes—
252C2×C4#35512yes—
253C2×C4#48912yes—
254C2×C4#50112yes—
255C8—84no—
256C8—84no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22#150611yescharacteristic
270C22#150711yescharacteristic
271C22#151111yescharacteristic
272C22#151411yescharacteristic
273C22#198611yescharacteristic
274C22#199311yescharacteristic
275C22#199611yescharacteristic
276C22C2×#35211yescharacteristic
277C22#212211yescharacteristic
278C22C2×#35511yescharacteristic
279C22#213511yescharacteristic
280C4—22no—
281C4—22no—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—81no—
288C4—81no—
289C4—162no—
290C4—162no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—21no—
295C2#988811yescharacteristic
296C2#1178011yescharacteristic
297C2#1178111yescharacteristic
298C2#1181411yescharacteristic
299C2#1181911yescharacteristic
300C2#1329511yescharacteristic
301C2#1330111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#151522
#108022
#81532
#72512