Finite Group Databasebeta

← #347824

Subgroups — #347824

Order 512.

1056 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 182 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×#2975C211yesmaximal, characteristic
2#32953C212yesmaximal
3#29808C211yesmaximal, characteristic, complemented
4C2×#1689C212yesmaximal, complemented
5#9227C214yesmaximal, complemented
6#9132C212yesmaximal, complemented
7#8656C212yesmaximal, complemented
8#8598C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#494C2211yescharacteristic, complemented
11C2×#474C2214yes—
12#2975—28no—
13C2×#436—22no—
14C2×#436C2212yes—
15C2×#436C2212yes—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#407—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#1689—28no—
30C2×#334C2212yescomplemented
31C2×#332C2214yescomplemented
32#1296C2214yescomplemented
33#1277C2212yescomplemented
34C23×D4D411yescharacteristic
35C24×C4—21no—
36C22×Q16—22no—
37C22×M16—22no—
38C22×M16—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—
53#436—216no—
54#436—24no—
55#436—24no—
56#436—24no—
57#436D418yes—
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—
67#407—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#334—24no—
86#332—28no—
87C25—21no—
88C22×Q8—22no—
89C22×D4—24no—
90C22×D4—21no—
91C22×D4—24no—
92C22×D4—42no—
93C22×D4C2×D412yes—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—84no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4#4111yescharacteristic
103C2×Q16—44no—
104C2×Q16—84no—
105C2×M16—28no—
106C2×M16—28no—
107C2×M16—84no—
108C2×M16—84no—
109C22×C8—22no—
110C22×C8—24no—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—24no—
114C4⋊Q8—24no—
115C22⋊Q8—84no—
116C4×D4—24no—
117C4×D4—28no—
118C4×D4—24no—
119C4×D4—24no—
120C4×D4—24no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4C2×D414yes—
124C4×D4#4114yes—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—42no—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42C2411yescharacteristic
144#107—48no—
145C4⋊C8—24no—
146C4⋊C8—22no—
147C4⋊C8—24no—
148C4⋊C8—24no—
149C4⋊C8C2×D414yes—
150#103—84no—
151#102—48no—
152#102—48no—
153C22⋊C8—24no—
154C22⋊C8—28no—
155C22⋊C8—28no—
156C22⋊C8—24no—
157C22⋊C8—44no—
158C4×C8—24no—
159C4×C8—22no—
160C4×C8C2×D414yes—
161#95—42no—
162#95—42no—
163C24—24no—
164C24—22no—
165C24—42no—
166C24—42no—
167C24—82no—
168C24—82no—
169C24#12111yescharacteristic
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×D4—24no—
175C2×D4—22no—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—416no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4#12114yes—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—28no—
187C22×C4—21no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—84no—
199C22×C4—88no—
200C22×C4—88no—
201C22×C4—82no—
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#13714yes—
206C22×C4C22×D411yescharacteristic
207C22×C4C22×D411yescharacteristic
208C22×C4C2×#4111yescharacteristic
209Q16—88no—
210M16—88no—
211M16—88no—
212C2×C8—24no—
213C2×C8—22no—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8—44no—
217C2×C8—44no—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8#12014yes—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—82no—
228C4⋊C4—84no—
229C4⋊C4#12114yes—
230C22⋊C4—28no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—84no—
234C42—21no—
235C42—22no—
236C42—21no—
237C42—22no—
238C42C22×D412yes—
239C42C2×#4112yes—
240C23—21no—
241C23—22no—
242C23—21no—
243C23—22no—
244C23—24no—
245C23—48no—
246C23—44no—
247C23—42no—
248C23—42no—
249C23—44no—
250C23—88no—
251C23—84no—
252C23—84no—
253C23—88no—
254C23#47012yes—
255C23C2×#12111yescharacteristic
256Q8—84no—
257Q8—84no—
258D4—28no—
259D4—48no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—28no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—48no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—88no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4#47012yes—
286C2×C4C2×#12011yescharacteristic
287C2×C4C2×#12112yes—
288C2×C4#52512yes—
289C2×C4#52511yescharacteristic
290C2×C4C2×#13712yes—
291C2×C4#57512yes—
292C2×C4#57614yes—
293C8—24no—
294C8—42no—
295C8—44no—
296C8—84no—
297C22—22no—
298C22—24no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—816no—
306C22—84no—
307C22—84no—
308C22#284711yescharacteristic
309C22#304112yes—
310C22C2×#47011yescharacteristic
311C22#305411yescharacteristic
312C22#306912yes—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—44no—
317C4—41no—
318C4—162no—
319C4—162no—
320C4#300812yes—
321C4#305614yes—
322C4#312212yes—
323C2—42no—
324C2—84no—
325C2#1942312yes—
326C2#1942412yes—
327C2#3291111yescharacteristic
328C2#3293711yescharacteristic
329C2#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×#1689C2semidirect
5. #9227C2semidirect
6. #9132C2semidirect
7. #8656C2semidirect
8. #8598C2semidirect
10. C2×#494C22semidirect
22. #1956C22semidirect
25. #1834C22semidirect
30. C2×#334C22semidirect
31. C2×#332C22semidirect
32. #1296C22semidirect
33. #1277C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29752
#329532
#298082
C2×#16892
#92272
#91322
#86562
#85982