Finite Group Databasebeta

← #320664

Subgroups — #320664

Order 512.

757 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 170 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#33230C211yesmaximal, characteristic
2#29218C212yesmaximal, complemented
3#28655C212yesmaximal
4#21691C211yesmaximal, characteristic, complemented
5#10686C211yesmaximal, characteristic, complemented
6#10677C211yesmaximal, characteristic
7#10309C211yesmaximal, characteristic, complemented
8#10306C211yesmaximal, characteristic
9#9838C212yesmaximal
10#9834C212yesmaximal, complemented
11#8263C211yesmaximal, characteristic, complemented
12C2×#500C2212yes—
13C2×#494C2211yescharacteristic
14C2×#489—22no—
15C2×#486C2212yes—
16C2×#441—22no—
17#2868C2211yescharacteristic
18#2858C2211yescharacteristic
19C4×C4⋊Q8C2212yes—
20#2100—24no—
21#2067—24no—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1987—24no—
26#1971—22no—
27#1959—22no—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2212yes—
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1956C2212yes—
34#1925—24no—
35#1844C2211yescharacteristic
36#1834C2212yes—
37#1834C2212yes—
38C4×C4⋊C8C2212yes—
39C4×#97C2211yescharacteristic
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×Q16—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2312yes—
48C2×#127C2311yescharacteristic
49C2×#124—21no—
50C2×#124—21no—
51C2×C4×Q8—22no—
52#500—24no—
53#500—42no—
54#494—24no—
55#489—48no—
56#486—24no—
57#486—42no—
58#441—48no—
59C2×#107—24no—
60C2×#106—22no—
61C2×C4⋊C8D414yes—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—24no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#97D412yes—
70C2×#97D412yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—22no—
76#390—22no—
77#390—22no—
78#389—22no—
79#389—22no—
80#384—22no—
81#384—21no—
82#384C2312yes—
83#384C2311yescharacteristic
84#384C2311yescharacteristic
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C43C2311yescharacteristic
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#336D414yes—
94#322—42no—
95C22×Q8—22no—
96C22×Q8—21no—
97C22×Q8—41no—
98C22×D4—21no—
99C22×D4—22no—
100C2×Q16—28no—
101C2×Q16—44no—
102C2×Q16—44no—
103C22×C8C2×D414yes—
104C22×C8C2×D414yes—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—21no—
111C4⋊Q8—22no—
112C4⋊Q8—21no—
113C4⋊Q8—48no—
114C4⋊Q8—44no—
115C4⋊Q8#4114yes—
116#127—22no—
117#127—21no—
118#127—21no—
119#124—44no—
120#124—44no—
121C4×Q8—48no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—42no—
138C2×C42—42no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D412yes—
141C2×C42C2×D412yes—
142C2×C42C2×D411yescharacteristic
143C2×C42C2411yescharacteristic
144#107—48no—
145#106—44no—
146C4⋊C8—44no—
147#103—48no—
148#103—44no—
149#103—44no—
150#102—48no—
151#102—48no—
152#102—44no—
153#102—44no—
154#97—24no—
155#97—24no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—21no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—21no—
163#95—42no—
164#95—42no—
165C24—41no—
166C2×Q8—24no—
167C2×Q8—42no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—82no—
175C2×Q8—84no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—44no—
183C2×D4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4#12711yescharacteristic
190C22×C4#12711yescharacteristic
191C22×C4#12712yes—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D412yes—
194Q16—416no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8—24no—
206C2×C8AGL(1,ℤ/8)14yes—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—84no—
221C4⋊C4—82no—
222C4⋊C4—84no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C22⋊C4—84no—
226C42—22no—
227C42—24no—
228C42—24no—
229C42—84no—
230C42AGL(1,ℤ/8)14yes—
231C42C2×#4112yes—
232C42H(5,𝔽2)11yescharacteristic
233C422−1+411yescharacteristic
234C23—81no—
235C23—81no—
236C23—82no—
237C23—82no—
238C23—81no—
239C23C2×#12711yescharacteristic
240Q8—48no—
241Q8—84no—
242Q8—82no—
243Q8—82no—
244Q8—164no—
245D4—82no—
246D4—84no—
247D4—82no—
248D4—84no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—44no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4#49614yes—
269C2×C4#52511yescharacteristic
270C2×C4#53711yescharacteristic
271C2×C4#54612yes—
272C2×C4#54812yes—
273C2×C4C2×AGL(1,ℤ/8)12yes—
274C2×C4C2×AGL(1,ℤ/8)12yes—
275C2×C4#57514yes—
276C2×C4#57814yes—
277C8—24no—
278C8—42no—
279C8—44no—
280C8—44no—
281C22—162no—
282C22—161no—
283C22—161no—
284C22—162no—
285C22—161no—
286C22#254011yescharacteristic
287C22C2×#49611yescharacteristic
288C22#315111yescharacteristic
289C22#315312yes—
290C22#315612yes—
291C4—24no—
292C4—41no—
293C4—41no—
294C4—84no—
295C4—161no—
296C4—161no—
297C4—162no—
298C4—161no—
299C4#310912yes—
300C4#325312yes—
301C4#331314yes—
302C4#337714yes—
303C2—321no—
304C2#1967412yes—
305C2#1973111yescharacteristic
306C2#1974911yescharacteristic
307C2#2838111yescharacteristic
308C2#2842412yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29218C2semidirect
4. #21691C2semidirect
5. #10686C2semidirect
7. #10309C2semidirect
10. #9834C2semidirect
11. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#332302
#292182
#286552
#216912
#106862
#106772
#103092
#103062
#98382
#98342
#82632