Finite Group Databasebeta

← #242833

Subgroups — #242833

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21890C212yesmaximal
2#21375C211yesmaximal, characteristic
3#20797C211yesmaximal, characteristic, complemented
4#10620C212yesmaximal, complemented
5#10619C212yesmaximal, complemented
6#10618C211yesmaximal, characteristic, complemented
7#10617C211yesmaximal, characteristic, complemented
8#10616C211yesmaximal, characteristic, complemented
9#10615C211yesmaximal, characteristic, complemented
10#8042C212yesmaximal, complemented
11#7874C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#427—21no—
15C2×#427—21no—
16C2×#423C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2285C2212yes—
19#1989—22no—
20#1984C2211yescharacteristic
21#1984C2211yescharacteristic
22#1983—22no—
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1895—22no—
26#1894—22no—
27#1869—21no—
28#1869—21no—
29#1846—21no—
30#1846—21no—
31#1845C2211yescharacteristic
32#1845C2211yescharacteristic
33#1844C2212yes—
34#1803C2212yes—
35#1803C2212yes—
36#1802C2212yes—
37#1802C2212yes—
38#1750—21no—
39#1750—21no—
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2211yescharacteristic
45#1277C2211yescharacteristic
46#1277C2212yes—
47C22×M16—22no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50#529—24no—
51#529—24no—
52C2×#126—21no—
53C2×#125D411yescharacteristic
54C2×#124D411yescharacteristic
55C2×#123—21no—
56C2×C22⋊Q8D411yescharacteristic
57C2×#121D411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4—21no—
60C2×C4×D4C2311yescharacteristic
61C2×#117—21no—
62C2×#117—22no—
63C22×C42C2311yescharacteristic
64#432—24no—
65#427—44no—
66#427—44no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#106—22no—
72C2×#106—22no—
73#423—24no—
74#423—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×#103—22no—
81C2×#103—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×#97—21no—
90C2×C4×C8—24no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#384C2312yes—
96#384C2312yes—
97#382—22no—
98#380—22no—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110C22×Q8C2×D411yescharacteristic
111C22×D4—21no—
112C22×D4C2×D411yescharacteristic
113C23×C4—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C2×M16—44no—
118C2×M16—82no—
119C2×M16—82no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134#126—44no—
135#125—24no—
136#124—24no—
137#123—44no—
138C22⋊Q8—24no—
139#121—24no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—42no—
145C4×D4—42no—
146#117—42no—
147#117—42no—
148#117—44no—
149#117—44no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—42no—
168C2×C42—21no—
169C2×C42—42no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4114yes—
173C2×C42#4114yes—
174C2×C42#4111yescharacteristic
175C2×C42C2411yescharacteristic
176#107—42no—
177#107—42no—
178#107—42no—
179#107—42no—
180#106—44no—
181#106—44no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—44no—
189#103—82no—
190#103—82no—
191#102—82no—
192#102—82no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198#97—42no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203#95—42no—
204#95—42no—
205C24—21no—
206C24#12111yescharacteristic
207C2×Q8—22no—
208C2×Q8—21no—
209C2×Q8—22no—
210C2×Q8—21no—
211C2×D4—22no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—21no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12312yes—
242C22×C4#12314yes—
243C22×C4#12412yes—
244C22×C4#12614yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C2×#4111yescharacteristic
248M16—88no—
249C2×C8—28no—
250C2×C8—28no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8AGL(1,ℤ/8)12yes—
272C2×C8AGL(1,ℤ/8)12yes—
273C2×C8#13712yes—
274C2×C8#13712yes—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—84no—
297C42—41no—
298C42—41no—
299C42—84no—
300C42#12312yes—
301C42#12314yes—
302C42#12412yes—
303C42#12614yes—
304C42C2×#4112yes—
305C42C2×#4112yes—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23#39712yes—
320C23C2×#12111yescharacteristic
321Q8—42no—
322Q8—42no—
323D4—42no—
324D4—42no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—44no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#39712yes—
361C2×C4#39712yes—
362C2×C4#39712yes—
363C2×C4C2×#12311yescharacteristic
364C2×C4C2×#12312yes—
365C2×C4C2×#12411yescharacteristic
366C2×C4C2×#12612yes—
367C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
369C2×C4C2×#13711yescharacteristic
370C2×C4C2×#13711yescharacteristic
371C2×C4#57611yescharacteristic
372C2×C4#57712yes—
373C2×C4#57811yescharacteristic
374C8—22no—
375C8—22no—
376C8—22no—
377C8—22no—
378C8—82no—
379C8—82no—
380C8—81no—
381C8—81no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22C2×#39711yescharacteristic
397C22#303811yescharacteristic
398C22#303911yescharacteristic
399C22#306011yescharacteristic
400C22#306111yescharacteristic
401C22#306711yescharacteristic
402C22#306711yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—81no—
408C4—81no—
409C4—81no—
410C4—162no—
411C4—162no—
412C4#309211yescharacteristic
413C4#309211yescharacteristic
414C4#309612yes—
415C4#309712yes—
416C4#314111yescharacteristic
417C4#314111yescharacteristic
418C2—42no—
419C2—161no—
420C2#1935511yescharacteristic
421C2#1935611yescharacteristic
422C2#1936011yescharacteristic
423C2#1936111yescharacteristic
424C2#2363011yescharacteristic
425C2#2364111yescharacteristic
426C2#2364611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20797C2semidirect
4. #10620C2semidirect
5. #10619C2semidirect
6. #10618C2semidirect
7. #10617C2semidirect
8. #10616C2semidirect
9. #10615C2semidirect
10. #8042C2semidirect
11. #7874C2semidirect

Maximal subgroups

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

subgroupindex
#218902
#213752
#207972
#106202
#106192
#106182
#106172
#106162
#106152
#80422
#78742