Finite Group Databasebeta

← #314724

Subgroups — #314724

Order 512.

639 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27646C212yesmaximal
2#21487C211yesmaximal, characteristic
3#20817C211yesmaximal, characteristic, complemented
4#10656C212yesmaximal, complemented
5#10622C212yesmaximal, complemented
6#9621C211yesmaximal, characteristic, complemented
7#9620C211yesmaximal, characteristic, complemented
8#8726C212yesmaximal, complemented
9#8652C211yesmaximal, characteristic, complemented
10#8652C211yesmaximal, characteristic, complemented
11#8595C211yesmaximal, characteristic, complemented
12C2×#482—22no—
13C2×#436C2211yescharacteristic
14C2×#435—21no—
15C22×C4×C8C2211yescharacteristic
16C4×#121C2211yescharacteristic
17#2285C2212yes—
18#2085—22no—
19#2073—22no—
20#1954C2211yescharacteristic
21#1948C2212yes—
22#1947C2212yes—
23#1939C2211yescharacteristic
24#1923—21no—
25#1869—21no—
26#1845C2211yescharacteristic
27#1845C2211yescharacteristic
28#1834C2212yes—
29#1825—22no—
30#1808—21no—
31#1808—21no—
32#1802C2212yes—
33#1802C2212yes—
34#1797C2211yescharacteristic
35#1797C2212yes—
36C4×#107—21no—
37C4×#107—21no—
38#1770C2211yescharacteristic
39#1753—21no—
40C2×#334C2211yescharacteristic
41C2×#334C2211yescharacteristic
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1277C2212yes—
47C22×M16—21no—
48C23×C8D411yescharacteristic
49C23×C8D411yescharacteristic
50C2×#121D411yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4C2311yescharacteristic
53C2×#117—22no—
54C22×C42C2311yescharacteristic
55#482—48no—
56#436—22no—
57#436—22no—
58#435—44no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—22no—
64C2×#106—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×#102—22no—
71C2×#102—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×#97—21no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8D411yescharacteristic
81C2×C4×C8D411yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#390D411yescharacteristic
85#389D411yescharacteristic
86#388—21no—
87#386—21no—
88#384C2312yes—
89#384C2312yes—
90#382—22no—
91#380—22no—
92#380—22no—
93#380D411yescharacteristic
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2311yescharacteristic
97C4×C22⋊C4—21no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#336—21no—
105#336—21no—
106#336—22no—
107#336—21no—
108#334—22no—
109#334—22no—
110C22×D4—21no—
111C22×D4C2×D411yescharacteristic
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4#4111yescharacteristic
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133#121—24no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—42no—
139C4×D4—42no—
140#117—44no—
141#117—44no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—42no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—42no—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42C2411yescharacteristic
168#107—44no—
169#107—42no—
170#107—42no—
171#107—42no—
172#107—42no—
173#106—44no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—42no—
183#102—44no—
184#102—44no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188C22⋊C8—44no—
189C22⋊C8—42no—
190#97—42no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8#4112yes—
196C4×C8#4112yes—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—42no—
201#95—42no—
202C24—21no—
203C24#12111yescharacteristic
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4AGL(1,ℤ/8)14yes—
239C22×C4C22×D411yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C2×#4111yescharacteristic
242M16—84no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8#12312yes—
266C2×C8#12312yes—
267C2×C8#13512yes—
268C2×C8AGL(1,ℤ/8)12yes—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—84no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—41no—
298C42—41no—
299C42—84no—
300C42C2×#4111yescharacteristic
301C42C2×#4111yescharacteristic
302C422−1+411yescharacteristic
303C422−1+411yescharacteristic
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#46912yes—
318C23C2×#12111yescharacteristic
319D4—42no—
320D4—42no—
321D4—84no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
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—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4#46912yes—
360C2×C4C2×#12311yescharacteristic
361C2×C4#53611yescharacteristic
362C2×C4#54711yescharacteristic
363C2×C4#54711yescharacteristic
364C2×C4#55211yescharacteristic
365C2×C4#55211yescharacteristic
366C2×C4C2×#13511yescharacteristic
367C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C2×AGL(1,ℤ/8)12yes—
369C2×C4#57511yescharacteristic
370C2×C4#57511yescharacteristic
371C2×C4#57812yes—
372C8—22no—
373C8—41no—
374C8—41no—
375C8—81no—
376C8—82no—
377C8—82no—
378C8—81no—
379C8#48214yescomplemented
380C22—22no—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22#248611yescharacteristic
395C22#303511yescharacteristic
396C22#303611yescharacteristic
397C22C2×#46911yescharacteristic
398C22#305911yescharacteristic
399C22#306711yescharacteristic
400C22#306711yescharacteristic
401C4—22no—
402C4—21no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—162no—
410C4—162no—
411C4C2×#48211yescharacteristic
412C4#309311yescharacteristic
413C4#323311yescharacteristic
414C4#323511yescharacteristic
415C4#334312yes—
416C4#334312yes—
417C2—42no—
418C2—161no—
419C2#1915711yescharacteristic
420C2#1922611yescharacteristic
421C2#1928011yescharacteristic
422C2#1933611yescharacteristic
423C2#2764611yescharacteristic
424C2#2766711yescharacteristic
425C2#2767411yescharacteristic
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. #20817C2semidirect
4. #10656C2semidirect
5. #10622C2semidirect
6. #9621C2semidirect
7. #9620C2semidirect
8. #8726C2semidirect
9. #8652C2semidirect
10. #8652C2semidirect
11. #8595C2semidirect
379. C8#482semidirect

Maximal subgroups

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

subgroupindex
#276462
#214872
#208172
#106562
#106222
#96212
#96202
#87262
#86522
#85952