Finite Group Databasebeta

← #347822

Subgroups — #347822

Order 512.

1026 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1026 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×#2983C211yesmaximal, characteristic, complemented
2#32903C212yesmaximal, complemented
3#29806C211yesmaximal, characteristic
4C2×#1688C212yesmaximal, complemented
5#9230C214yesmaximal, complemented
6#9134C212yesmaximal, complemented
7#8657C212yesmaximal, complemented
8#8598C211yesmaximal, characteristic, complemented
9C22×C4×Q8C2211yescharacteristic
10C2×#493C2211yescharacteristic
11C2×#460C2214yes—
12C2×#448—22no—
13#2983—28no—
14C2×#446—22no—
15C2×#446C2212yes—
16C2×#446C2212yes—
17C2×#432—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2856C2212yes—
21#2054—22no—
22#1999—22no—
23#1957C2212yes—
24#1880—24no—
25#1874—22no—
26#1834C2211yescharacteristic
27#1803C2214yes—
28#1797C2212yes—
29#1688—28no—
30C2×#334C2212yes—
31C2×#331C2214yes—
32#1296C2214yes—
33#1277C2212yes—
34C23×Q8D411yescharacteristic
35C22×SD16—24no—
36C22×D8—22no—
37C2×#127C2312yes—
38C2×#121—22no—
39C2×#121—22no—
40C2×#121—22no—
41C2×C4×Q8—24no—
42C2×C4×Q8—24no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8D412yes—
46C2×C4×Q8C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42C2311yescharacteristic
51#493—24no—
52#460—28no—
53#460—28no—
54#448—48no—
55#446—216no—
56#446—24no—
57#446—24no—
58#446—24no—
59#446D418yescomplemented
60#432—216no—
61#431—24no—
62#431—24no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8D412yes—
66C2×C4⋊C8C2314yes—
67C2×C4⋊C8C2312yes—
68C2×#103—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8D412yes—
74C2×#97—22no—
75C2×#97—22no—
76C2×C4×C8—21no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#390—22no—
80#390—22no—
81#390—22no—
82#384C2312yes—
83#340—22no—
84#340—24no—
85#336—22no—
86#334—24no—
87#331—28no—
88C22×Q8—24no—
89C22×Q8—21no—
90C22×Q8—24no—
91C22×Q8—42no—
92C22×Q8C2×D412yes—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—42no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C2×SD16—48no—
103C2×SD16—48no—
104C2×SD16—48no—
105C2×D8—44no—
106C2×D8—84no—
107C22×C8—22no—
108C22×C8—24no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111#127—24no—
112#127—24no—
113#121—44no—
114#121—44no—
115#121—84no—
116#121—84no—
117C4×Q8—28no—
118C4×Q8—24no—
119C4×Q8—24no—
120C4×Q8—24no—
121C4×Q8—24no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8C2×D414yes—
125C4×Q8#4114yes—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C2×D412yes—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—22no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42C2411yescharacteristic
148C4⋊C8—28no—
149C4⋊C8—28no—
150C4⋊C8—24no—
151C4⋊C8—22no—
152C4⋊C8—24no—
153C4⋊C8—24no—
154C4⋊C8C2×D414yes—
155#103—48no—
156#102—48no—
157#102—84no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8—44no—
161#97—28no—
162#97—28no—
163C4×C8—24no—
164C4×C8—22no—
165C4×C8C2×D414yes—
166C24—42no—
167C24#12111yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C2×Q8—24no—
171C2×Q8—24no—
172C2×Q8—416no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8#12114yes—
176C2×D4—42no—
177C2×D4—44no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—82no—
186C2×D4—82no—
187C2×D4—84no—
188C2×D4—84no—
189C2×D4—84no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4AGL(1,ℤ/8)14yes—
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C2×#4111yescharacteristic
212SD16—816no—
213D8—88no—
214C2×C8—24no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8#12014yes—
223C4⋊C4—24no—
224C4⋊C4—28no—
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—82no—
232C4⋊C4#12114yes—
233C22⋊C4—84no—
234C22⋊C4—84no—
235C22⋊C4—84no—
236C42—28no—
237C42—21no—
238C42—24no—
239C42—22no—
240C42—24no—
241C42—21no—
242C42—22no—
243C42C22×D412yes—
244C42C2×#4112yes—
245C23—21no—
246C23—22no—
247C23—21no—
248C23—22no—
249C23—82no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23#46912yes—
257C23C2×#12111yescharacteristic
258Q8—28no—
259Q8—48no—
260Q8—44no—
261D4—84no—
262D4—84no—
263D4—84no—
264D4—84no—
265D4—164no—
266D4—168no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—48no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—84no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#46912yes—
289C2×C4C2×#12011yescharacteristic
290C2×C4C2×#12112yes—
291C2×C4#52512yes—
292C2×C4#52511yescharacteristic
293C2×C4C2×AGL(1,ℤ/8)12yes—
294C2×C4#57512yes—
295C2×C4#57714yes—
296C8—24no—
297C8—42no—
298C8—44no—
299C8—84no—
300C22—22no—
301C22—24no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—162no—
306C22—162no—
307C22—162no—
308C22—162no—
309C22—162no—
310C22—162no—
311C22—162no—
312C22#284711yescharacteristic
313C22#304312yes—
314C22C2×#46911yescharacteristic
315C22#305411yescharacteristic
316C22#306812yes—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—44no—
321C4—41no—
322C4—44no—
323C4—162no—
324C4#301012yes—
325C4#305514yes—
326C4#312612yes—
327C2—42no—
328C2—322no—
329C2#1943412yes—
330C2#1943712yes—
331C2#3291211yescharacteristic
332C2#3293611yescharacteristic
333C2#3294511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#29806C2
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. C2×#2983C2semidirect
2. #32903C2semidirect
4. C2×#1688C2semidirect
5. #9230C2semidirect
6. #9134C2semidirect
7. #8657C2semidirect
8. #8598C2semidirect
59. #446D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#329032
#298062
C2×#16882
#92302
#91342
#86572
#85982