Finite Group Databasebeta

← #443522

Subgroups — #443522

Order 512.

1290 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 258 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#35640C211yesmaximal, characteristic
2C2×#2660C211yesmaximal, characteristic
3C2×#2060C212yesmaximal
4#23552C212yesmaximal
5#11383C211yesmaximal, characteristic
6#11339C214yesmaximal
7#10891C212yesmaximal
8#10879C212yesmaximal
9C2×#544C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C2×#428C2214yes—
13#2660—28no—
14#2583—22no—
15#2552—24no—
16#2550—24no—
17#2502—22no—
18C2×#398—22no—
19C2×#395—22no—
20#2317C2212yes—
21#2296C2212yes—
22C2×#384C2212yes—
23C2×#382C2212yes—
24#2274C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#2060—28no—
27#2058—24no—
28#2057—24no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31#1736C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1728C2212yes—
35C23×Q8—21no—
36C22×M16C2314yes—
37#544—24no—
38#544—42no—
39C2×C4⋊Q8—24no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—42no—
43C2×#125—24no—
44C2×#125—22no—
45C2×#125—24no—
46C2×#125—42no—
47C2×C22⋊Q8—22no—
48C2×#117C2312yes—
49C2×#117C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59#428—28no—
60#428—44no—
61C2×#107—24no—
62C2×#106—24no—
63#398—24no—
64#398—216no—
65#398—28no—
66#395—24no—
67#395—216no—
68#391—24no—
69#389—28no—
70#389—24no—
71#389D414yes—
72#389D414yes—
73#385—24no—
74#385—22no—
75#385—22no—
76#384—24no—
77#384—22no—
78#384—24no—
79#384D418yes—
80#382—22no—
81#382—24no—
82#382—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85C4×C22⋊C4—22no—
86#376—24no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—21no—
90#343—24no—
91#343—42no—
92#340—28no—
93#339—28no—
94#339Q8116yescomplemented
95#336—24no—
96C22×Q8—22no—
97C22×Q8—24no—
98C22×Q8—42no—
99C22×Q8—42no—
100C22×Q8—84no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2411yescharacteristic
108C2×M16—24no—
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16C2×Q818yes—
114C22×C8—24no—
115C4⋊Q8—416no—
116C4⋊Q8—48no—
117#125—416no—
118#125—48no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121#117—24no—
122#117—24no—
123#117—24no—
124#117C2×Q818yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—216no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—88no—
149C2×C4⋊C4C2×D414yes—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D414yes—
153C2×C4⋊C4C2×D414yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—22no—
162C2×C42—24no—
163C2×C42C2×D414yes—
164C2×C42C2×D414yes—
165C2×C42C2×Q818yes—
166#107—48no—
167#106—48no—
168#95—24no—
169#95—24no—
170#95—216no—
171#95—24no—
172#95—28no—
173#95—24no—
174#95—24no—
175C24C22×D411yescharacteristic
176C2×Q8—48no—
177C2×Q8—44no—
178C2×Q8—88no—
179C2×Q8—88no—
180C2×Q8—84no—
181C2×Q8—88no—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—28no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—48no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4#12012yes—
207C22×C4#12012yes—
208C22×C4#12014yes—
209C22×C4C22×D412yes—
210C22×C4C22×D412yes—
211C22×C4C22×Q814yes—
212C22×C4H(5,𝔽2)12yes—
213C22×C42−1+414yes—
214M16—28no—
215M16—28no—
216M16—44no—
217M16—44no—
218C2×C8—28no—
219C2×C8—44no—
220C2×C8—44no—
221C4⋊C4—24no—
222C4⋊C4—28no—
223C4⋊C4—24no—
224C4⋊C4—28no—
225C4⋊C4—24no—
226C4⋊C4—48no—
227C4⋊C4—48no—
228C4⋊C4—48no—
229C4⋊C4—44no—
230C4⋊C4—48no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—88no—
236C4⋊C4—816no—
237C4⋊C4C4⋊Q818yes—
238C22⋊C4—28no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C42—24no—
242C42—28no—
243C42—28no—
244C42—28no—
245C42—44no—
246C42—84no—
247C42C4⋊Q818yes—
248C23—22no—
249C23—22no—
250C23C2×#12011yescharacteristic
251C23C2×#12011yescharacteristic
252C23#53411yescharacteristic
253C23D4×Q814yes—
254Q8—816no—
255Q8—164no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—28no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—416no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—48no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—82no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4—82no—
287C2×C4C2×C4⋊Q812yes—
288C2×C4C2×C4⋊Q812yes—
289C2×C4#53612yes—
290C2×C4#53612yes—
291C2×C4#54414yes—
292C2×C4D4214yes—
293C2×C4D4×Q814yes—
294C2×C4D4×Q814yes—
295C2×C4Q8212yes—
296C2×C4#56312yes—
297C8—48no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—28no—
303C22#240311yescharacteristic
304C22#243412yes—
305C22#243412yes—
306C22#273412yes—
307C22#302112yes—
308C22#302212yes—
309C4—48no—
310C4—42no—
311C4—44no—
312C4—42no—
313C4—48no—
314C4—84no—
315C4—82no—
316C4—82no—
317C4—82no—
318C4#255114yes—
319C4#267714yes—
320C2—22no—
321C2—22no—
322C2#1576211yescharacteristic
323C2#2391311yescharacteristic
324C2#2391611yescharacteristic
325C2#2583314yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
94. #339Q8semidirect

Maximal subgroups

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

subgroupindex
#356402
C2×#26602
C2×#20602
#235522
#113832
#113392
#108912
#108792