Finite Group Databasebeta

← #302387

Subgroups — #302387

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#28689C211yesmaximal, characteristic
5#10578C214yesmaximal
6#9898C211yesmaximal, characteristic
7#9870C211yesmaximal, characteristic
8#8895C214yesmaximal
9C4×Q8⋊C8C211yesmaximal, characteristic
10C2×#500—21no—
11C2×#475—21no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14#2862—21no—
15#2582—21no—
16#2571C2212yes—
17#2570—22no—
18#2568—21no—
19#2511—22no—
20#2446—22no—
21#2365—21no—
22#2355—21no—
23#2355C2212yes—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#2083—22no—
28#1990—22no—
29#1990—21no—
30#1990—22no—
31#1988—21no—
32#1948—21no—
33#1948C2212yes—
34#1947C2214yes—
35#1842C2211yescharacteristic
36#1834C2214yes—
37#1803—21no—
38#1803C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42#1296C2214yes—
43#1279C2212yes—
44#1279C2214yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D412yes—
52C2×C4×Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57#500—44no—
58#475—44no—
59C2×#107—24no—
60C2×#106—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8C2312yes—
70#400—24no—
71#400—24no—
72#398—24no—
73#391—22no—
74#391—22no—
75#391—22no—
76#391—22no—
77#391—22no—
78#391—22no—
79#391—21no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—21no—
84#389D412yes—
85#387—24no—
86#387—24no—
87#387—24no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384D411yescharacteristic
94#384D412yes—
95#384C2312yes—
96#384C2312yes—
97#382—22no—
98#382—22no—
99#382—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D411yescharacteristic
109C4×C4⋊C4C2312yes—
110C4×C4⋊C4C2311yescharacteristic
111C43—21no—
112C43C2311yescharacteristic
113#340—22no—
114#340—22no—
115#340—24no—
116#340—28no—
117#336—21no—
118Q8⋊C8Q818yescomplemented
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—41no—
122C22×Q8C2×D411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—84no—
146C4×Q8C2×Q814yes—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D414yes—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2411yescharacteristic
187#107—48no—
188#106—44no—
189C4⋊C8—22no—
190C4⋊C8—24no—
191C4⋊C8—42no—
192C4⋊C8—42no—
193C4⋊C8C2×Q814yes—
194#103—44no—
195C4×C8—22no—
196C4×C8—42no—
197C4×C8—42no—
198C4×C8—42no—
199C4×C8C2×Q814yes—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—21no—
207#95—21no—
208#95—22no—
209#95—21no—
210#95—22no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—82no—
225C2×Q8C4⋊Q812yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4#12011yescharacteristic
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—21no—
251C4⋊C4—24no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—24no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—84no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4C4⋊Q812yes—
283C42—24no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—84no—
295C42#13512yes—
296C42#13512yes—
297C42#13512yes—
298C42AGL(1,ℤ/8)12yes—
299C42AGL(1,ℤ/8)12yes—
300C42#13712yes—
301C42C22×Q812yes—
302C422−1+412yes—
303C23C2×#12011yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—41no—
307Q8—41no—
308Q8—84no—
309Q8—88no—
310Q8—164no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#44912yes—
342C2×C4#44912yes—
343C2×C4#44912yes—
344C2×C4#44912yes—
345C2×C4#45212yes—
346C2×C4#45212yes—
347C2×C4#45412yes—
348C2×C4#45512yes—
349C2×C4C2×C4⋊Q811yescharacteristic
350C2×C4#53611yescharacteristic
351C2×C4D4×Q812yes—
352C2×C4D4×Q812yes—
353C2×C4C2×#13511yescharacteristic
354C2×C4C2×#13511yescharacteristic
355C2×C4C2×#13511yescharacteristic
356C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
357C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
358C2×C4C2×#13711yescharacteristic
359C2×C4#57812yes—
360C8—44no—
361C8—44no—
362C8—82no—
363C22#243411yescharacteristic
364C22C2×#44911yescharacteristic
365C22C2×#44911yescharacteristic
366C22C2×#45211yescharacteristic
367C22#301111yescharacteristic
368C22C2×#45511yescharacteristic
369C22C2×#45411yescharacteristic
370C4—22no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—84no—
376C4—82no—
377C4—81no—
378C4—82no—
379C4—81no—
380C4—162no—
381C4#162814yes—
382C4#164014yes—
383C4#316011yescharacteristic
384C4#316511yescharacteristic
385C4#322811yescharacteristic
386C4#323111yescharacteristic
387C4#338312yes—
388C4#338612yes—
389C2C2×#164011yescharacteristic
390C2C2×#162811yescharacteristic
391C2#1863611yescharacteristic
392C2#1864511yescharacteristic
393C2#2581211yescharacteristic
394C2#2582911yescharacteristic
395C2#2584011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
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
118. Q8⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
#332892
#331962
#286892
#105782
#98982
#98702
#88952
C4×Q8⋊C82