Finite Group Databasebeta

← #238151

Subgroups — #238151

Order 512.

970 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 202 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#37643C211yesmaximal, characteristic, complemented
2#21893C212yesmaximal
3#21195C211yesmaximal, characteristic
4#10355C214yesmaximal, complemented
5#10354C214yesmaximal, complemented
6#8041C211yesmaximal, characteristic, complemented
7#7897C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#423C2212yes—
10C2×#416—22no—
11C22×C4×C8C2211yescharacteristic
12#2845—22no—
13#2629—24no—
14#2629—22no—
15#2552—22no—
16#2552—22no—
17#2552C2214yes—
18#2285C2212yes—
19#2285C2211yescharacteristic
20#1987C2214yes—
21#1982C2214yes—
22#1958—22no—
23#1957C2212yes—
24#1956C2212yes—
25#1895—24no—
26#1802—22no—
27#1802—22no—
28#1296C2214yes—
29#1279C2218yes—
30C22×#41—21no—
31C22×M16—22no—
32C23×C8—21no—
33#532—24no—
34#532—22no—
35#532Q814yes—
36C2×C4⋊Q8C2311yescharacteristic
37C2×#127C2311yescharacteristic
38C2×#125—22no—
39C2×#125—24no—
40C2×#124—21no—
41C2×#121—22no—
42C2×C4×D4—22no—
43C2×#117—22no—
44C2×#117D412yes—
45C22×C42C2311yescharacteristic
46#423—24no—
47#423—24no—
48C2×C4⋊C8—22no—
49C2×C4⋊C8C2314yes—
50#416—48no—
51C2×#103—24no—
52C2×#102—24no—
53C2×#102—24no—
54C2×#102—24no—
55C2×C4×C8—24no—
56C2×C4×C8—21no—
57C2×C4×C8C2312yes—
58#398—28no—
59#398—28no—
60#398—28no—
61#396—28no—
62#389—24no—
63#389—24no—
64#389—24no—
65#384—22no—
66#384—24no—
67#384D412yes—
68#384D412yes—
69#384C2312yes—
70#382—22no—
71#382D412yes—
72#380—22no—
73#380—22no—
74#380D414yes—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4C2314yes—
78#340—24no—
79#340—24no—
80C2×#41—24no—
81C2×#41—44no—
82C2×#41—44no—
83C2×#41—42no—
84C22×Q8—21no—
85C22×D4—22no—
86C22×D4—21no—
87C22×D4—41no—
88C23×C4—41no—
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C2×M16—44no—
92C2×M16—82no—
93C2×M16—82no—
94C22×C8—24no—
95C22×C8—22no—
96C22×C8—24no—
97C22×C8—42no—
98C22×C8—42no—
99C4⋊Q8—22no—
100C4⋊Q8—21no—
101C4⋊Q8C2×Q812yes—
102#127—22no—
103#127—21no—
104#127C2×Q812yes—
105#125—84no—
106#125—84no—
107#125—84no—
108#124—24no—
109#124—42no—
110#121—28no—
111#121—44no—
112C4×D4—28no—
113C4×D4—44no—
114#117—44no—
115#117—82no—
116#117—82no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4C2×D414yes—
128C2×C4⋊C4C2×D414yes—
129C2×C4⋊C4C2×D414yes—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—42no—
133C2×C42—22no—
134C2×C42—24no—
135C2×C42C2×D412yes—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×Q812yes—
138C2×C42#4114yes—
139C2×C42#4112yes—
140C2×C42C2411yescharacteristic
141C4⋊C8—24no—
142C4⋊C8—24no—
143C4⋊C8—44no—
144#103—48no—
145#102—48no—
146#102—84no—
147#102—84no—
148C4×C8—24no—
149C4×C8—42no—
150#95—28no—
151#95—24no—
152#95—24no—
153#95—24no—
154C24—41no—
155C24#12011yescharacteristic
156#41—416no—
157#41—88no—
158C2×Q8—22no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—41no—
162C2×D4—24no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—44no—
167C2×D4—42no—
168C2×D4—41no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—82no—
174C2×D4—82no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—41no—
185C22×C4—44no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4#12012yes—
189C22×C4#12011yescharacteristic
190C22×C4C22⋊Q814yes—
191C22×C4C22⋊Q812yes—
192C22×C4#12414yes—
193C22×C4#12412yes—
194C22×C4C22×D411yescharacteristic
195C22×C4C22×D412yes—
196M16—88no—
197C2×C8—24no—
198C2×C8—42no—
199C2×C8—48no—
200C2×C8—42no—
201C2×C8—44no—
202C2×C8—44no—
203C2×C8—44no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—88no—
220C4⋊C4—88no—
221C4⋊C4—88no—
222C22⋊C4—48no—
223C22⋊C4—82no—
224C22⋊C4—84no—
225C42—44no—
226C42—44no—
227C42—44no—
228C42C22⋊Q814yes—
229C42C22⋊Q812yes—
230C42#12412yes—
231C42#12414yes—
232C42C22×Q811yescharacteristic
233C42C2×#4112yes—
234C42C2×#4111yescharacteristic
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—82no—
241C23—82no—
242C23—81no—
243C23—81no—
244C23#39312yes—
245C23C2×#12011yescharacteristic
246Q8—44no—
247Q8—82no—
248D4—44no—
249D4—44no—
250D4—44no—
251D4—82no—
252D4—88no—
253D4—82no—
254D4—82no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—88no—
278C2×C4#39314yes—
279C2×C4#39312yes—
280C2×C4#45312yes—
281C2×C4#45412yes—
282C2×C4C2×C22⋊Q811yescharacteristic
283C2×C4C2×C22⋊Q812yes—
284C2×C4C2×#12411yescharacteristic
285C2×C4C2×#12412yes—
286C2×C4#57612yes—
287C2×C4#57712yes—
288C2×C4#57712yes—
289C2×C4#57812yes—
290C8—44no—
291C8—84no—
292C22—24no—
293C22—22no—
294C22—41no—
295C22—41no—
296C22—42no—
297C22—84no—
298C22—82no—
299C22—161no—
300C22—161no—
301C22—162no—
302C22C2×#39311yescharacteristic
303C22#301212yes—
304C22#301312yes—
305C22C2×#45311yescharacteristic
306C22C2×#45411yescharacteristic
307C4—24no—
308C4—21no—
309C4—21no—
310C4—84no—
311C4—82no—
312C4—84no—
313C4—84no—
314C4#202714yes—
315C4#308312yes—
316C4#308412yes—
317C4#313612yes—
318C4#313712yes—
319C2—42no—
320C2—162no—
321C2#1868712yes—
322C2#1869611yescharacteristic
323C2#1869811yescharacteristic
324C2#2345812yes—
325C2C2×#202711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37643C2semidirect
4. #10355C2semidirect
5. #10354C2semidirect
6. #8041C2semidirect
7. #7897C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#218932
#211952
#103552
#103542
#80412
#78972