Finite Group Databasebeta

← #163238

Subgroups — #163238

Order 512.

984 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 210 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
1C2×#2861C211yesmaximal, characteristic
2#20778C212yesmaximal, complemented
3#7716C218yesmaximal, complemented
4#7629C212yesmaximal, complemented
5#7626C212yesmaximal
6C22×#125C411yescharacteristic
7C22×#124C411yescharacteristic
8#2861—22no—
9#2861—22no—
10#2861C414yes—
11#2861C2214yes—
12C2×#391C411yescharacteristic
13C2×#391C2211yescharacteristic
14C2×#390C411yescharacteristic
15C2×#390C2211yescharacteristic
16C2×#388—22no—
17C2×#388—22no—
18C2×C43C2211yescharacteristic
19#1839C2214yescomplemented
20#1831C2214yescomplemented
21#1817—22no—
22#1797—22no—
23#1767—22no—
24#1749C2214yes—
25#1745—22no—
26#1272C2218yescomplemented
27#1270C2218yes—
28C23×Q8C2×C411yescharacteristic
29C23×D4C2×C411yescharacteristic
30C2×#125—22no—
31C2×#125—44no—
32C2×#125C2×C414yes—
33C2×#124—22no—
34C2×#124—44no—
35C2×#124C2×C414yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D412yes—
47C22×C42D412yes—
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×#97—24no—
52#391—21no—
53#391—21no—
54#391—22no—
55#391C2×C412yes—
56#391C2×C414yes—
57#391C2312yes—
58#390—21no—
59#390—21no—
60#390—22no—
61#390C2×C412yes—
62#390C2×C414yes—
63#390C2312yes—
64#388—24no—
65#388—24no—
66#388—24no—
67#388—24no—
68#388—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72C2×#95—22no—
73C2×#95—22no—
74C43C2×C412yes—
75C43D412yes—
76C43D412yes—
77C43C2312yes—
78#336—24no—
79#336—24no—
80#336—24no—
81#336C2318yes—
82C25—21no—
83C22×Q8—21no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8C22×C412yes—
88C22×D4—21no—
89C22×D4—42no—
90C22×D4—42no—
91C22×D4—42no—
92C22×D4C22×C412yes—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D412yes—
101C22×C8—28no—
102#125—48no—
103#125—48no—
104#124—48no—
105#124—48no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4C22×C412yes—
115C2×C4⋊C4C22×C412yes—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—48no—
123C2×C22⋊C4—48no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—44no—
126C2×C42—24no—
127C2×C42—42no—
128C2×C42—44no—
129C2×C42—42no—
130C2×C42—44no—
131C2×C42C22⋊C412yes—
132C2×C42C22⋊C412yes—
133C2×C42C22⋊C412yes—
134C2×C42C22⋊C412yes—
135C2×C42C22⋊C412yes—
136C2×C42C22⋊C411yescharacteristic
137C2×C42C22⋊C411yescharacteristic
138C2×C42C22×C412yes—
139C2×C42C22×C412yes—
140C2×C42C22×C411yescharacteristic
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2411yescharacteristic
147C4⋊C8—48no—
148C22⋊C8—48no—
149#97—48no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—24no—
154C24—22no—
155C24—44no—
156C24—42no—
157C24—42no—
158C24—42no—
159C24—44no—
160C24C2×C22⋊C411yescharacteristic
161C2×Q8—24no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—88no—
168C2×D4—24no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—88no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4#10014yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C22⋊C412yes—
204C22×C4C23×C411yescharacteristic
205C22×C4C22×D412yes—
206C2×C8—48no—
207C2×C8—48no—
208C2×C8—48no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—48no—
214C4⋊C4—48no—
215C22⋊C4—48no—
216C22⋊C4—48no—
217C22⋊C4—48no—
218C22⋊C4—48no—
219C22⋊C4—48no—
220C22⋊C4—48no—
221C42—24no—
222C42—24no—
223C42—24no—
224C42—24no—
225C42—24no—
226C42—24no—
227C42—41no—
228C42—41no—
229C42—42no—
230C42—42no—
231C42—42no—
232C42—42no—
233C42—41no—
234C42—42no—
235C42—42no—
236C42—41no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—44no—
244C23—44no—
245C23—84no—
246C23—88no—
247C23—84no—
248C23—88no—
249C23—84no—
250C23C2×#10012yes—
251C23#42114yes—
252C23C22×C22⋊C411yescharacteristic
253Q8—84no—
254Q8—84no—
255D4—84no—
256D4—84no—
257C2×C4—28no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4#41012yes—
293C2×C4C2×#10014yes—
294C2×C4#56812yes—
295C2×C4#57514yes—
296C8—88no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—22no—
301C22—84no—
302C22—82no—
303C22—84no—
304C22—82no—
305C22—82no—
306C22—168no—
307C22#153614yes—
308C22#287911yescharacteristic
309C22C2×#41011yescharacteristic
310C22C22×#10011yescharacteristic
311C22#289612yes—
312C22C2×#42112yes—
313C4—24no—
314C4—24no—
315C4—22no—
316C4—22no—
317C4—42no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—82no—
322C4—82no—
323C4—82no—
324C2—24no—
325C2—162no—
326C2#1739311yescharacteristic
327C2#1749712yes—
328C2#1757912yes—
329C2C2×#153611yescharacteristic
330C2#1775411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #7716C2semidirect
4. #7629C2semidirect
19. #1839C22semidirect
20. #1831C22semidirect
26. #1272C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#207782
#77162
#76292
#76262