Finite Group Databasebeta

← #347764

Subgroups — #347764

Order 512.

1132 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1132 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
1C2×#2972C211yesmaximal, characteristic
2#32953C212yesmaximal
3#29808C211yesmaximal, characteristic, complemented
4C2×#1669C212yesmaximal, complemented
5#9219C214yesmaximal, complemented
6#9132C212yesmaximal, complemented
7#8648C212yesmaximal, complemented
8#8599C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#494C2211yescharacteristic, complemented
11C2×#474C2214yes—
12#2972—28no—
13C2×C8×D4—22no—
14C2×C8×D4C2212yes—
15C2×C8×D4C2212yes—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C22×C22⋊C8—22no—
19#2857C2212yes—
20#2056—22no—
21#2001—22no—
22#1956C2212yescomplemented
23#1862—24no—
24#1852—22no—
25#1834C2211yescharacteristic, complemented
26#1802C2214yes—
27#1797C2212yes—
28#1786—22no—
29#1669—28no—
30C2×#335C2212yescomplemented
31C2×#332C2214yescomplemented
32#1296C2214yescomplemented
33#1277C2212yescomplemented
34C23×D4D411yescharacteristic
35C24×C4—21no—
36C22×Q16—22no—
37C23×C8—22no—
38C23×C8—22no—
39C2×C4⋊Q8C2312yes—
40C2×C22⋊Q8—22no—
41C2×C4×D4—24no—
42C2×C4×D4—24no—
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C22×C4⋊C4D411yescharacteristic
48C22×C22⋊C4—21no—
49C22×C42C2311yescharacteristic
50#494—24no—
51#474—28no—
52#474—28no—
53C8×D4—216no—
54C8×D4—24no—
55C8×D4—24no—
56C8×D4—24no—
57C8×D4D418yes—
58#431—24no—
59#431—24no—
60C2×#107—24no—
61C2×C4⋊C8D412yes—
62C2×C4⋊C8C2314yes—
63C2×C4⋊C8C2312yes—
64C2×#103—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×C22⋊C8—216no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8D412yes—
73C2×C4×C8—21no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#389—22no—
78#389—22no—
79#384C2312yes—
80#380—22no—
81#380—22no—
82#340—22no—
83#340—24no—
84#336—22no—
85#335—24no—
86#332—28no—
87C25—21no—
88C22×Q8—22no—
89C22×D4—24no—
90C22×D4—24no—
91C22×D4—21no—
92C22×D4—22no—
93C22×D4Q1614yescomplemented
94C22×D4C2×D412yes—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—84no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×Q16—44no—
105C2×Q16—84no—
106C22×C8—22no—
107C22×C8—28no—
108C22×C8—28no—
109C22×C8—24no—
110C22×C8—84no—
111C22×C8—84no—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C4⋊Q8—24no—
115C4⋊Q8—24no—
116C22⋊Q8—84no—
117C4×D4—24no—
118C4×D4—28no—
119C4×D4—24no—
120C4×D4—24no—
121C4×D4—24no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4C2×D414yes—
125C4×D4#4114yes—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D412yes—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—42no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42C2411yescharacteristic
145#107—48no—
146C4⋊C8—24no—
147C4⋊C8—22no—
148C4⋊C8—24no—
149C4⋊C8—24no—
150C4⋊C8C2×D414yes—
151#103—84no—
152#102—48no—
153#102—48no—
154C22⋊C8—24no—
155C22⋊C8—28no—
156C22⋊C8—24no—
157C22⋊C8—28no—
158C22⋊C8—44no—
159C4×C8—24no—
160C4×C8—22no—
161C4×C8C2×D414yes—
162#95—42no—
163#95—42no—
164C24—24no—
165C24—24no—
166C24—22no—
167C24—24no—
168C24—82no—
169C24—82no—
170C24#12111yescharacteristic
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—216no—
178C2×D4—216no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—24no—
183C2×D4#12114yes—
184C2×D4C2×Q1614yes—
185C2×D4#13514yes—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—28no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—84no—
203C22×C4—88no—
204C22×C4—88no—
205C22×C4—82no—
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4C2×Q1612yes—
210C22×C4#13512yes—
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214Q16—88no—
215C2×C8—24no—
216C2×C8—22no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—44no—
223C2×C8—88no—
224C2×C8—88no—
225C2×C8#12014yes—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C4⋊C4#12114yes—
235C22⋊C4—28no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—84no—
239C42—21no—
240C42—22no—
241C42—21no—
242C42—22no—
243C42C22×D412yes—
244C42C2×#4112yes—
245C23—216no—
246C23—24no—
247C23—21no—
248C23—22no—
249C23—24no—
250C23—21no—
251C23—24no—
252C23—22no—
253C23—44no—
254C23—44no—
255C23—88no—
256C23—84no—
257C23—84no—
258C23—88no—
259C23#46712yes—
260C23C2×#12111yescharacteristic
261Q8—84no—
262Q8—84no—
263D4—216no—
264D4—24no—
265D4—24no—
266D4—24no—
267D4#46718yes—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—88no—
286C2×C4—84no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4#46712yes—
294C2×C4C2×#12011yescharacteristic
295C2×C4C2×#12112yes—
296C2×C4#52512yes—
297C2×C4#52511yescharacteristic
298C2×C4C22×Q1611yescharacteristic
299C2×C4C2×#13512yes—
300C2×C4C2×#13511yescharacteristic
301C2×C4#57614yes—
302C8—24no—
303C8—42no—
304C8—44no—
305C8—84no—
306C22—22no—
307C22—28no—
308C22—24no—
309C22—41no—
310C22—44no—
311C22—41no—
312C22—44no—
313C22—48no—
314C22—42no—
315C22—48no—
316C22—816no—
317C22#284711yescharacteristic
318C22#304112yes—
319C22C2×#46711yescharacteristic
320C22#305011yescharacteristic
321C22#306912yes—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—44no—
326C4—41no—
327C4—162no—
328C4—162no—
329C4#300812yes—
330C4C2×#46712yes—
331C4#305012yes—
332C4#312212yes—
333C2—42no—
334C2—48no—
335C2#1938712yes—
336C2#1938812yes—
337C2#3291111yescharacteristic
338C2#3293111yescharacteristic
339C2#3294611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
3. #29808C2semidirect
4. C2×#1669C2semidirect
5. #9219C2semidirect
6. #9132C2semidirect
7. #8648C2semidirect
8. #8599C2semidirect
10. C2×#494C22semidirect
22. #1956C22semidirect
25. #1834C22semidirect
30. C2×#335C22semidirect
31. C2×#332C22semidirect
32. #1296C22semidirect
33. #1277C22semidirect
93. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#329532
#298082
C2×#16692
#92192
#91322
#86482
#85992