Finite Group Databasebeta

← #142152

Subgroups — #142152

Order 512.

532 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 112 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
2C4×#343C211yesmaximal, characteristic
3#8558C211yesmaximal, characteristic
4#7251C214yesmaximal
5C2×C4×M16—21no—
6C2×#400—21no—
7C2×#384C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1797—21no—
10C2×#343—21no—
11C2×#343C2212yes—
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×C4⋊C8—41no—
31C2×C22⋊C8—22no—
32C4×M16—24no—
33C4×M16—42no—
34C2×#97—41no—
35C2×C4×C8—41no—
36#400—44no—
37#384—41no—
38#384—41no—
39C4×C4⋊C4—41no—
40C4×C4⋊C4—41no—
41C2×#95—22no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95C2×C412yes—
47C2×#95C2×C411yescharacteristic
48C2×#95D411yescharacteristic
49#343—24no—
50#343—22no—
51#343Q814yes—
52#336—41no—
53#324—24no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22⋊C411yescharacteristic
60C23×C4C22×C411yescharacteristic
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C2×M16—24no—
64C2×M16—24no—
65C2×M16—24no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C22×C8—42no—
70C22×C8—42no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
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—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—82no—
88C2×C4⋊C4C2×Q812yes—
89C2×C4⋊C4#4112yes—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—42no—
98C2×C42—42no—
99C2×C42SD1612yes—
100C2×C42Q1612yes—
101C4⋊C8—82no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104#97—44no—
105C4×C8—44no—
106#95—41no—
107#95—41no—
108#95—42no—
109#95—42no—
110#95—44no—
111#95—44no—
112#95—82no—
113#95—82no—
114C24C2×C22⋊C411yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
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—21no—
131C22×C4—21no—
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—41no—
141C22×C4—41no—
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—42no—
155C22×C4—42no—
156C22×C4#9911yescharacteristic
157C22×C4#9911yescharacteristic
158C22×C4#10312yes—
159C22×C4#10312yes—
160C22×C4#10412yes—
161C22×C4#10412yes—
162C22×C4C4×Q812yes—
163C22×C4C22⋊Q812yes—
164C22×C4#12411yescharacteristic
165C22×C4C4⋊Q811yescharacteristic
166C22×C4C2×SD1611yescharacteristic
167C22×C4C2×Q1611yescharacteristic
168C22×C4AGL(1,ℤ/8)12yes—
169M16—28no—
170M16—44no—
171M16—44no—
172M16—44no—
173C2×C8—44no—
174C2×C8—44no—
175C2×C8—44no—
176C2×C8—82no—
177C2×C8—82no—
178C2×C8—82no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—82no—
183C4⋊C4—84no—
184C4⋊C4—84no—
185C4⋊C4—84no—
186C4⋊C4—82no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—42no—
192C42—42no—
193C42—42no—
194C42—81no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23#39111yescharacteristic
200C23C2×#9911yescharacteristic
201C23#41011yescharacteristic
202C23C2×#10311yescharacteristic
203C23#41811yescharacteristic
204C23C2×#10411yescharacteristic
205C23C2×#10411yescharacteristic
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4#32812yes—
260C2×C4#32812yes—
261C2×C4#35212yes—
262C2×C4#35512yes—
263C2×C4#47612yes—
264C2×C4#47712yes—
265C8—48no—
266C8—162no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22C2×#32811yescharacteristic
280C22#150311yescharacteristic
281C22#150511yescharacteristic
282C22#150711yescharacteristic
283C22#198711yescharacteristic
284C22#199411yescharacteristic
285C22#199611yescharacteristic
286C22C2×#35211yescharacteristic
287C22#212211yescharacteristic
288C22C2×#35511yescharacteristic
289C22#213511yescharacteristic
290C4—22no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—41no—
296C4—42no—
297C4—41no—
298C4—81no—
299C4—162no—
300C4—162no—
301C4#135314yes—
302C2—21no—
303C2—21no—
304C2—21no—
305C2—21no—
306C2#990811yescharacteristic
307C2C2×#135311yescharacteristic
308C2#1172411yescharacteristic
309C2#1178011yescharacteristic
310C2#1178111yescharacteristic
311C2#1329211yescharacteristic
312C2#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
C4×#3432
#85582
#72512