Finite Group Databasebeta

← #190063

Subgroups — #190063

Order 512.

1034 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 230 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#34155C211yesmaximal, characteristic
2#21800C211yesmaximal, characteristic
3#21794C211yesmaximal, characteristic
4#8739C214yesmaximal, complemented
5#8738C214yesmaximal, complemented
6#7941C212yesmaximal, complemented
7#7940C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2372—22no—
12#2369—21no—
13#2366—21no—
14#2363—24no—
15#2297C2214yes—
16#2293C414yescomplemented
17#2283C2212yes—
18C42×D4C414yescomplemented
19C4×#117C2212yes—
20#1967—22no—
21#1966—21no—
22#1965—21no—
23#1964—24no—
24#1785C2212yes—
25#1785C2212yes—
26#1782C2214yes—
27#1284C2214yes—
28#1284C2214yes—
29#1284C2218yes—
30C22×#41C2312yes—
31#532—22no—
32#532—22no—
33#532—22no—
34#532—22no—
35C2×C4⋊Q8D411yescharacteristic
36C2×#127D411yescharacteristic
37C2×#124D412yes—
38C2×#121D414yes—
39C2×C4×D4—24no—
40C2×C4×D4C2×C414yes—
41C2×#117—22no—
42C2×#117C2×C414yes—
43C22×C42C2×C414yes—
44C22×C42C2311yescharacteristic
45#432—22no—
46#432—22no—
47#431—22no—
48#431—22no—
49#423—21no—
50#423—21no—
51C2×#101—22no—
52C2×#101—22no—
53C2×#100—22no—
54C2×#100—22no—
55C2×#99—24no—
56C2×#99—24no—
57#407C2311yescharacteristic
58#407C2311yescharacteristic
59#407C2312yes—
60C4×M16—21no—
61C4×M16—21no—
62#391—24no—
63#390—24no—
64#390—24no—
65#389—24no—
66#388—28no—
67#386—24no—
68#386—24no—
69#383—22no—
70#382—22no—
71#381C2314yes—
72#380—22no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2×C412yes—
77C4×C4⋊C4C2×C412yes—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—28no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4C2×C418yes—
82C4×C22⋊C4C2314yes—
83#376—22no—
84#376—22no—
85C43C2×C414yes—
86C2×#41—24no—
87C2×#41—24no—
88C2×#41—82no—
89C2×#41—82no—
90C22×Q8C2×D412yes—
91C22×D4C22⋊C414yes—
92C22×D4C2×D414yes—
93C22×D4C2×D412yes—
94C23×C4—24no—
95C23×C4C22×C414yes—
96C23×C4C2×D412yes—
97C23×C4C2411yescharacteristic
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—22no—
101C4⋊Q8—21no—
102C4⋊Q8—21no—
103C4⋊Q8—22no—
104#127—21no—
105#127—21no—
106#127—22no—
107#124—22no—
108#124—22no—
109#124—24no—
110#121—24no—
111#121—24no—
112#121—24no—
113#121—24no—
114C4×D4—24no—
115C4×D4—24no—
116C4×D4—24no—
117C4×D4—24no—
118C4×D4—416no—
119#117—48no—
120#117—84no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4C22⋊C414yes—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4C22⋊C418yes—
131C2×C42—24no—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42—42no—
135C2×C42—84no—
136C2×C42C22×C414yes—
137C2×C42C22×C414yes—
138C2×C42C22×C412yes—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4112yes—
142C2×C42#4112yes—
143C2×C42#4112yes—
144C2×C42#4112yes—
145C4⋊C8—42no—
146C4⋊C8—42no—
147#101—82no—
148#101—82no—
149#100—82no—
150#100—82no—
151#99—84no—
152#99—84no—
153C22⋊C8—24no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156#97—41no—
157#97—41no—
158C4×C8—41no—
159C4×C8—41no—
160#95—22no—
161#95—22no—
162#95—28no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—42no—
167#95—42no—
168C24C2×C22⋊C414yes—
169C24C22×D411yescharacteristic
170#41—88no—
171#41—88no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—48no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—42no—
185C2×D4—42no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—24no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—416no—
195C22×C4—44no—
196C22×C4—84no—
197C22×C4C2×C22⋊C414yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4#12312yes—
201C22×C4#12312yes—
202C22×C4#12312yes—
203C22×C4#12312yes—
204C22×C4C23×C411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C2×#4112yes—
207C22×C4C2×#4112yes—
208M16—41no—
209M16—41no—
210M16—42no—
211M16—42no—
212M16—41no—
213M16—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—42no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—44no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C22⋊C4—28no—
224C22⋊C4—28no—
225C22⋊C4—48no—
226C22⋊C4—416no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—84no—
230C42—44no—
231C42—44no—
232C42—48no—
233C42—48no—
234C42—84no—
235C42—84no—
236C42#11712yes—
237C42#11712yes—
238C42#11712yes—
239C42#11712yes—
240C42#12311yescharacteristic
241C42#12311yescharacteristic
242C42#12311yescharacteristic
243C42#12311yescharacteristic
244C42#12312yes—
245C42#12312yes—
246C23—21no—
247C23—21no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—42no—
252C23—48no—
253C23C22×C22⋊C411yescharacteristic
254C23C2×#12312yes—
255Q8—82no—
256Q8—82no—
257D4—48no—
258D4—48no—
259D4—82no—
260D4—82no—
261D4—88no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—28no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—416no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—816no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4#38112yes—
289C2×C4#38112yes—
290C2×C4#38112yes—
291C2×C4#38112yes—
292C2×C4C2×#11712yes—
293C2×C4C2×#12311yescharacteristic
294C2×C4C2×#12311yescharacteristic
295C8—81no—
296C8—81no—
297C8—82no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—44no—
303C22—48no—
304C22—81no—
305C22—81no—
306C22—84no—
307C22—84no—
308C22C2×#38111yescharacteristic
309C22#288311yescharacteristic
310C22#288311yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—48no—
318C4—44no—
319C4—84no—
320C4—164no—
321C4#182714yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—84no—
326C2#2069011yescharacteristic
327C2C2×#182712yes—
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8739C2semidirect
5. #8738C2semidirect
6. #7941C2semidirect
7. #7940C2semidirect
16. #2293C4semidirect
18. C42×D4C4semidirect

Maximal subgroups

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

subgroupindex
#341552
#218002
#217942
#87392
#87382
#79412
#79402