Finite Group Databasebeta

← #413238

Subgroups — #413238

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2C2×#2060C214yesmaximal
3#21171C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#14692C211yesmaximal, characteristic
6#11340C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10839C211yesmaximal, characteristic
9C4×#339C211yesmaximal, characteristic
10C2×#544C2212yes—
11C2×#428C2212yes—
12C2×#428C2214yes—
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2868—21no—
16#2859—21no—
17#2660—22no—
18#2552—22no—
19#2551—22no—
20#2440—24no—
21#2317C2212yes—
22#2285C2211yescharacteristic
23C2×#384—24no—
24C2×#384C2211yescharacteristic
25C2×#384C2214yes—
26C2×#384C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#2060—216no—
30#1948—22no—
31#1947—22no—
32#1847—22no—
33#1845—21no—
34#1844—21no—
35C2×#343C2214yes—
36#1736C2212yes—
37C2×#339C2214yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42#544—28no—
43#544—44no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×#125—22no—
48C2×#125—22no—
49C2×C22⋊Q8—24no—
50C2×#117C2312yes—
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C22×C4⋊C4—28no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D414yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42—22no—
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61#428—28no—
62#428—28no—
63#428—44no—
64C2×#107—24no—
65C2×#107—22no—
66C2×#106—24no—
67C2×#106—22no—
68#423—22no—
69#423Q814yescomplemented
70C2×C4⋊C8—22no—
71C4×M16—22no—
72C4×M16Q814yescomplemented
73C2×#97—21no—
74C2×C4×C8—21no—
75#398—28no—
76#395—28no—
77#389—24no—
78#389—24no—
79#389—24no—
80#385—24no—
81#384—24no—
82#384—22no—
83#384—21no—
84#384—21no—
85#384—22no—
86#384—28no—
87#384—21no—
88#384—21no—
89#384—22no—
90#384—44no—
91#384—416no—
92#384D411yescharacteristic
93#384D411yescharacteristic
94#382—22no—
95#382D412yes—
96#380—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D412yes—
99C4×C4⋊C4D412yes—
100C4×C22⋊C4—22no—
101#376—21no—
102C43—21no—
103C43—22no—
104C43Q814yescomplemented
105#343—28no—
106#340—24no—
107#339—28no—
108#339D418yes—
109#339Q818yescomplemented
110C22×Q8—24no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4C2×D412yes—
114C23×C4C2×D414yes—
115C23×C4C2411yescharacteristic
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16C2×D414yes—
123C2×M16C2×Q814yes—
124C2×M16C2×Q814yes—
125C2×M16C2×Q814yes—
126C22×C8—22no—
127C22×C8—22no—
128C4⋊Q8—48no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—88no—
132#125—48no—
133#125—44no—
134#125—44no—
135C22⋊Q8—416no—
136#117—22no—
137#117—22no—
138#117—24no—
139#117—24no—
140#117C2×D414yes—
141#117C2×Q814yes—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—416no—
160C2×C4⋊C4—432no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C42—28no—
173C2×C42—28no—
174C2×C42—28no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42C2×D414yes—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D412yes—
187C2×C42C2×Q814yes—
188C2×C42C2×Q814yes—
189C2×C42C2×Q812yes—
190C2×C42C2×Q812yes—
191C2×C42C2×Q812yes—
192C2×C42#4112yes—
193#107—48no—
194#107—44no—
195#106—48no—
196#106—44no—
197C4⋊C8—28no—
198#97—24no—
199C4×C8—24no—
200#95—28no—
201#95—22no—
202#95—22no—
203#95—24no—
204C24C22×D411yescharacteristic
205C2×Q8—48no—
206C2×Q8—88no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—48no—
222C22×C4—48no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4#12012yes—
226C22×C4#12012yes—
227C22×C4C22⋊Q814yes—
228C22×C4C4⋊Q814yes—
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D412yes—
232C22×C4C22×Q811yescharacteristic
233C22×C4C22×Q811yescharacteristic
234C22×C4C22×Q811yescharacteristic
235C22×C4C22×Q812yes—
236C22×C4C2×#4111yescharacteristic
237C22×C4H(5,𝔽2)12yes—
238C22×C42−1+412yes—
239M16—24no—
240M16—24no—
241M16—24no—
242M16—28no—
243M16—44no—
244M16C4⋊Q818yescomplemented
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—48no—
257C4⋊C4—48no—
258C4⋊C4—48no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—48no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—84no—
269C4⋊C4—832no—
270C4⋊C4—84no—
271C4⋊C4—88no—
272C4⋊C4—84no—
273C4⋊C4#12712yes—
274C4⋊C4#12712yes—
275C4⋊C4C4⋊Q814yes—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—44no—
279C42—232no—
280C42—24no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—24no—
285C42—42no—
286C42—42no—
287C42—44no—
288C42—44no—
289C42C22⋊Q812yes—
290C42C22⋊Q812yes—
291C42C22⋊Q814yes—
292C42C4⋊Q814yes—
293C42C4⋊Q812yes—
294C42C4⋊Q812yes—
295C42C4⋊Q818yescomplemented
296C23—22no—
297C23—22no—
298C23C2×#12011yescharacteristic
299C23C2×C22⋊Q811yescharacteristic
300C23C2×C4⋊Q811yescharacteristic
301C23D4212yes—
302C23D4×Q812yes—
303Q8—168no—
304C2×C4—216no—
305C2×C4—28no—
306C2×C4—28no—
307C2×C4—216no—
308C2×C4—28no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—88no—
327C2×C4—88no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4#39514yes—
331C2×C4#39514yes—
332C2×C4#45314yes—
333C2×C4#45614yes—
334C2×C4C2×C22⋊Q811yescharacteristic
335C2×C4C2×C22⋊Q811yescharacteristic
336C2×C4C2×#12711yescharacteristic
337C2×C4C2×C4⋊Q811yescharacteristic
338C2×C4C2×C4⋊Q811yescharacteristic
339C2×C4C2×C4⋊Q812yes—
340C2×C4C2×C4⋊Q812yes—
341C2×C4C2×C4⋊Q811yescharacteristic
342C2×C4#53511yescharacteristic
343C2×C4#53611yescharacteristic
344C2×C4#54412yes—
345C2×C4#54412yes—
346C2×C4D4212yes—
347C2×C4#54712yes—
348C2×C4D4×Q812yes—
349C2×C4D4×Q812yes—
350C2×C4#55412yes—
351C2×C4Q8212yes—
352C8—28no—
353C8—44no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22C2×#39511yescharacteristic
360C22#243311yescharacteristic
361C22#243411yescharacteristic
362C22#244012yes—
363C22#255112yes—
364C22C2×#45312yes—
365C22C2×#45612yes—
366C4—216no—
367C4—24no—
368C4—42no—
369C4—42no—
370C4—44no—
371C4—82no—
372C4—84no—
373C4—82no—
374C4—162no—
375C4#206018yescomplemented
376C4#244012yes—
377C4#254211yescharacteristic
378C4#255112yes—
379C4#256711yescharacteristic
380C4#266011yescharacteristic
381C4#286811yescharacteristic
382C2—22no—
383C2—22no—
384C2#1514311yescharacteristic
385C2C2×#206012yes—
386C2#2574611yescharacteristic
387C2#2574711yescharacteristic
388C2#2583012yes—
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
69. #423Q8semidirect
72. C4×M16Q8semidirect
104. C43Q8semidirect
109. #339Q8semidirect
244. M16C4⋊Q8semidirect
295. C42C4⋊Q8semidirect
375. C4#2060semidirect

Maximal subgroups

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

subgroupindex
#352752
C2×#20602
#211712
#208002
#146922
#113402
#108572
#108392
C4×#3392