Finite Group Databasebeta

← #501509

Subgroups — #501509

Order 512.

716 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2207C211yesmaximal, characteristic, complemented
2C2×#2163C211yesmaximal, characteristic, complemented
3#33171C211yesmaximal, characteristic
4C4×#467C211yesmaximal, characteristic, complemented
5#13380C212yesmaximal, complemented
6#13305C212yesmaximal, complemented
7#13252C211yesmaximal, characteristic, complemented
8#13238C212yesmaximal, complemented
9#13176C211yesmaximal, characteristic, complemented
10#13149C212yesmaximal, complemented
11#12812C211yesmaximal, characteristic, complemented
12C22×SD32C2212yes—
13C2×#493C2211yescharacteristic
14C2×#467C2211yescharacteristic
15C2×#466—21no—
16C2×#466C2212yes—
17C2×C4×Q16C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2856—21no—
20C4×C22⋊Q8—21no—
21#2207—24no—
22#2163—24no—
23C2×#366C2212yes—
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2212yes—
27C2×#357C2212yes—
28C2×#357C2212yes—
29C2×#357C2212yes—
30C2×C22⋊C16C2212yes—
31C2×#346C2212yes—
32#1970—21no—
33#1970C2212yes—
34#1969C2211yescharacteristic
35#1957—21no—
36#1942C2211yescharacteristic
37#1933—21no—
38#1892—21no—
39#1845C2211yescharacteristic
40#1808C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#103—21no—
43#1380C2212yes—
44#1380C2212yes—
45C22×Q16C2311yescharacteristic
46C22×D8—21no—
47C22×D8C2312yes—
48C23×C8D411yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#127—21no—
51C2×C22⋊Q8—21no—
52C2×#121—22no—
53C2×#121—41no—
54C2×#121—41no—
55C2×C4×Q8—21no—
56C22×C42D411yescharacteristic
57C2×SD32—24no—
58C2×SD32—24no—
59C2×SD32—42no—
60C22×C16C2312yes—
61C22×C16C2312yes—
62#493—22no—
63#493—22no—
64#467—22no—
65#467—22no—
66#466—42no—
67#466—42no—
68#466—44no—
69C4×Q16—22no—
70C4×Q16—22no—
71C2×#107—21no—
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2312yes—
75C2×#107C2311yescharacteristic
76C2×#103—21no—
77C2×#103—21no—
78C2×#103—21no—
79C2×#102—22no—
80C2×#102—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#391—21no—
88#390—22no—
89#390—41no—
90#390—41no—
91#389—21no—
92#387—41no—
93#385—41no—
94#384—21no—
95#380—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—41no—
98C4×C22⋊C4—41no—
99#366—42no—
100C4⋊C16—22no—
101C4⋊C16D414yescomplemented
102#358—42no—
103#357—24no—
104#357—24no—
105#357—42no—
106C22⋊C16—24no—
107#346—24no—
108#340—21no—
109#340—21no—
110#340—21no—
111C22×Q8—21no—
112C22×D4—22no—
113C22×D4—42no—
114C22×D4—41no—
115C22×D4—41no—
116C23×C4D811yescharacteristic
117C23×C4D811yescharacteristic
118C23×C4C2×D411yescharacteristic
119C2×Q16—21no—
120C2×Q16—21no—
121C2×Q16—21no—
122C2×Q16—21no—
123C2×Q16—21no—
124C2×Q16#4112yes—
125C2×D8—22no—
126C2×D8—22no—
127C2×D8—22no—
128C2×D8—22no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—42no—
132C2×D8—42no—
133C2×D8—42no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—41no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147#127—42no—
148#127—42no—
149C22⋊Q8—24no—
150C22⋊Q8—42no—
151#121—44no—
152#121—44no—
153#121—84no—
154#121—84no—
155C4×Q8—42no—
156C4×Q8—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42D811yescharacteristic
176C2×C42D811yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179SD32—48no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16C2×D414yes—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—22no—
191#107—21no—
192#107—22no—
193#107—42no—
194#107#4112yes—
195#103—24no—
196#103—42no—
197#103—42no—
198#102—44no—
199#102—44no—
200C4×C8—22no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—22no—
204C4×C8C2×D412yes—
205C4×C8#4112yes—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211C24—42no—
212C24C2×D811yescharacteristic
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—82no—
226C2×D4—82no—
227C2×D4—84no—
228C2×D4—84no—
229C2×D4—84no—
230C2×D4—82no—
231C2×D4—82no—
232C2×D4—82no—
233C2×D4—82no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C22×D411yescharacteristic
257Q16—22no—
258Q16—22no—
259Q16—41no—
260Q16—41no—
261D8—42no—
262D8—42no—
263D8—42no—
264D8—42no—
265D8—84no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—44no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8#12112yes—
279C2×C8AGL(1,ℤ/8)12yes—
280C2×C8C22×D411yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C8H(5,𝔽2)11yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C22⋊C4—44no—
300C22⋊C4—84no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C22⋊C4—82no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—82no—
309C42—82no—
310C42#12112yes—
311C42#13512yes—
312C16—24no—
313C16—42no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—82no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—82no—
327C23#50912yes—
328C23C22×D811yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—81no—
332Q8—81no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—82no—
337D4—164no—
338D4—168no—
339D4—164no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—44no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4#46612yes—
373C2×C4#46612yes—
374C2×C4#50912yes—
375C2×C4#51012yes—
376C2×C4C2×#12111yescharacteristic
377C2×C4#52511yescharacteristic
378C2×C4#53511yescharacteristic
379C2×C4C2×#13511yescharacteristic
380C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
381C2×C4#57511yescharacteristic
382C8—22no—
383C8—22no—
384C8—41no—
385C8#54611yescharacteristic
386C8#55011yescharacteristic
387C8#55311yescharacteristic
388C8#55511yescharacteristic
389C22—22no—
390C22—22no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—162no—
397C22—162no—
398C22—162no—
399C22—162no—
400C22—162no—
401C22—162no—
402C22—162no—
403C22C2×#46611yescharacteristic
404C22#310011yescharacteristic
405C22#320511yescharacteristic
406C22C2×#50911yescharacteristic
407C22C2×#50911yescharacteristic
408C22C2×#51011yescharacteristic
409C22#341411yescharacteristic
410C4—22no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—82no—
416C4—161no—
417C4—161no—
418C4—162no—
419C4#222014yes—
420C4#241411yescharacteristic
421C4#305511yescharacteristic
422C4#310611yescharacteristic
423C4#321211yescharacteristic
424C2—42no—
425C2—322no—
426C2#2465011yescharacteristic
427C2C2×#222011yescharacteristic
428C2#3360511yescharacteristic
429C2#3367211yescharacteristic
430C2#3367511yescharacteristic
431C2#3378211yescharacteristic
432C2#3378311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#33171C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2207C2semidirect
2. C2×#2163C2semidirect
4. C4×#467C2semidirect
5. #13380C2semidirect
6. #13305C2semidirect
7. #13252C2semidirect
8. #13238C2semidirect
9. #13176C2semidirect
10. #13149C2semidirect
11. #12812C2semidirect
101. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22072
C2×#21632
#331712
C4×#4672
#133802
#133052
#132522
#132382
#131762
#131492
#128122