Finite Group Databasebeta

← #316308

Subgroups — #316308

Order 512.

823 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27645C211yesmaximal, characteristic
2#24209C211yesmaximal, characteristic
3#21487C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5#10598C212yesmaximal, complemented
6#10477C212yesmaximal, complemented
7#9573C212yesmaximal, complemented
8#9225C212yesmaximal, complemented
9#8724C211yesmaximal, characteristic, complemented
10#8595C211yesmaximal, characteristic, complemented
11#8340C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#481—21no—
14C2×#461C2212yes—
15C2×#448—21no—
16C2×#436C2211yescharacteristic
17C2×#435—21no—
18C22×C4×C8C2211yescharacteristic
19C4×#121C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2084—21no—
22#2071—21no—
23#2043—21no—
24#2013—21no—
25#1990—21no—
26#1990—21no—
27#1990C2212yes—
28#1990C2212yes—
29#1981C2212yes—
30#1954C2211yescharacteristic
31#1948C2212yes—
32#1939C2211yescharacteristic
33#1923—21no—
34#1869—21no—
35#1824—21no—
36#1802C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1787—21no—
41#1770C2211yescharacteristic
42#1753—21no—
43C2×#340—21no—
44C2×#340—21no—
45C2×#334C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8—21no—
52C22×SD16—22no—
53C22×M16—21no—
54C23×C8D411yescharacteristic
55C23×C8D411yescharacteristic
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8C2312yes—
62C2×#121D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2311yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C42C2311yescharacteristic
71#481—44no—
72#461—24no—
73#461—42no—
74#448—44no—
75#436—42no—
76#435—44no—
77C2×#106—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2312yes—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#102—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×#97—21no—
94C2×C4×C8—42no—
95C2×C4×C8D411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#390D411yescharacteristic
100#389D411yescharacteristic
101#388—21no—
102#386—21no—
103#384—21no—
104#384—21no—
105#384—21no—
106#384C2312yes—
107#380D411yescharacteristic
108C4×C4⋊C4C2311yescharacteristic
109C4×C22⋊C4—21no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#336—21no—
121#334—42no—
122#332—24no—
123#332—42no—
124C22×Q8—22no—
125C22×Q8—44no—
126C22×Q8—41no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—82no—
130C22×D4—21no—
131C22×D4C2×D411yescharacteristic
132C23×C4—21no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4111yescharacteristic
138C2×SD16—44no—
139C2×SD16—44no—
140C2×SD16—82no—
141C2×M16—42no—
142C2×M16—42no—
143C2×M16—42no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—41no—
153C22×C8SD1614yescomplemented
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164C4⋊Q8—42no—
165C4⋊Q8#4114yes—
166#121—22no—
167#121—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—42no—
171C4×D4—42no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206#106—44no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—42no—
214#103—44no—
215#103—44no—
216#103—82no—
217#102—44no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224#97—42no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—41no—
228C4×C8—41no—
229#95—21no—
230#95—21no—
231#95—21no—
232C24—21no—
233C24#12111yescharacteristic
234C2×Q8—24no—
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—88no—
242C2×Q8—88no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—22no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4C2×SD1612yes—
282C22×C4AGL(1,ℤ/8)12yes—
283C22×C4AGL(1,ℤ/8)12yes—
284C22×C4#13712yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C2×#4111yescharacteristic
288SD16—88no—
289M16—84no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8C2×SD1612yes—
315C2×C8C2×SD1612yes—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C42—21no—
341C42—21no—
342C42—22no—
343C42—22no—
344C42—41no—
345C42—41no—
346C42C2×#4112yes—
347C42H(5,𝔽2)12yes—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23#46112yes—
360C23#46912yes—
361C23#47312yes—
362C23C2×#12111yescharacteristic
363Q8—44no—
364Q8—44no—
365Q8—88no—
366Q8—82no—
367Q8—82no—
368D4—42no—
369D4—42no—
370D4—84no—
371C2×C4—28no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—88no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—164no—
407C2×C4#46112yes—
408C2×C4#46912yes—
409C2×C4#47112yes—
410C2×C4#52511yescharacteristic
411C2×C4#53411yescharacteristic
412C2×C4#54612yes—
413C2×C4#55112yes—
414C2×C4C22×SD1611yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
417C2×C4C2×#13711yescharacteristic
418C2×C4#57511yescharacteristic
419C2×C4#57511yescharacteristic
420C2×C4#57712yes—
421C8—22no—
422C8—22no—
423C8—41no—
424C8—41no—
425C8—81no—
426C8—81no—
427C8—82no—
428C8—82no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22#169314yes—
443C22#253411yescharacteristic
444C22C2×#46111yescharacteristic
445C22#304011yescharacteristic
446C22C2×#46911yescharacteristic
447C22#305711yescharacteristic
448C22C2×#47111yescharacteristic
449C22C2×#47311yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—81no—
455C4—84no—
456C4—81no—
457C4—81no—
458C4—162no—
459C4#310311yescharacteristic
460C4#311711yescharacteristic
461C4#318711yescharacteristic
462C4#319611yescharacteristic
463C4#330612yes—
464C4#331012yes—
465C2—24no—
466C2—161no—
467C2C2×#169311yescharacteristic
468C2#1919111yescharacteristic
469C2#1930111yescharacteristic
470C2#1936311yescharacteristic
471C2#2810711yescharacteristic
472C2#2811511yescharacteristic
473C2#2813511yescharacteristic
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
4. C2×#1990C2semidirect
5. #10598C2semidirect
6. #10477C2semidirect
7. #9573C2semidirect
8. #9225C2semidirect
9. #8724C2semidirect
10. #8595C2semidirect
11. #8340C2semidirect
153. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#276452
#242092
#214872
C2×#19902
#105982
#104772
#95732
#92252
#87242
#85952
#83402