Finite Group Databasebeta

← #308269

Subgroups — #308269

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26527C211yesmaximal, characteristic, complemented
2#26087C211yesmaximal, characteristic
3#22962C211yesmaximal, characteristic
4#22713C211yesmaximal, characteristic
5#10680C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10594C211yesmaximal, characteristic, complemented
8#10577C211yesmaximal, characteristic, complemented
9#9615C211yesmaximal, characteristic, complemented
10#9536C211yesmaximal, characteristic, complemented
11#9227C211yesmaximal, characteristic
12#8984C211yesmaximal, characteristic
13#8925C211yesmaximal, characteristic, complemented
14#8886C211yesmaximal, characteristic
15#8847C211yesmaximal, characteristic
16C2×#474C2211yescharacteristic
17C2×#436C2211yescharacteristic
18#2571C2211yescharacteristic
19#2365C2211yescharacteristic
20#2354C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22#2097—21no—
23#2092—21no—
24#2067—21no—
25#2066—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1981C2211yescharacteristic
29#1956C2211yescharacteristic
30#1949—21no—
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1942—21no—
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1921—21no—
40#1918—21no—
41#1863—21no—
42#1862—21no—
43#1854—21no—
44#1852—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1809—21no—
49#1805—21no—
50#1802C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×#102—21no—
54C4×#102C2211yescharacteristic
55#1736—21no—
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4×Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C22×C4⋊C4—21no—
74#474—22no—
75#474Q814yes—
76#436—22no—
77#436—22no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107D411yescharacteristic
84C2×#107D411yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—41no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102D411yescharacteristic
102C2×#102D411yescharacteristic
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#400—41no—
109#391—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#388—41no—
116#387—41no—
117#385—41no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122#382—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—41no—
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149#332—22no—
150D4⋊C8—22no—
151C22×Q8—21no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C23×C4—41no—
155C2×M16—24no—
156C2×M16—81no—
157C2×M16—81no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C2×Q812yes—
171C4×Q8—42no—
172C4×Q8—81no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—42no—
176C4×D4—81no—
177C4×D4C2×Q812yes—
178C4×D4#4112yes—
179#117—42no—
180#117—81no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—41no—
212C2×C42—41no—
213C2×C42C2411yescharacteristic
214#107—24no—
215#107—22no—
216#107—41no—
217#107—42no—
218#107—81no—
219#107—81no—
220#107—81no—
221#106—81no—
222#106—81no—
223#106—81no—
224#106—82no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8C2×Q812yes—
234C4⋊C8#4112yes—
235#103—42no—
236#103—81no—
237#103—81no—
238#103—81no—
239#102—24no—
240#102—22no—
241#102—41no—
242#102—81no—
243C22⋊C8—24no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—81no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4C22⋊Q812yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—161no—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4H(5,𝔽2)11yescharacteristic
297M16—84no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8#13712yes—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—161no—
359C4⋊C4C22⋊Q812yes—
360C4⋊C4#13512yes—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—161no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—82no—
373C42—82no—
374C42C22×Q811yescharacteristic
375C42C2×#4111yescharacteristic
376C42C2×#4111yescharacteristic
377C422−1+411yescharacteristic
378C23—22no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23D4211yescharacteristic
386Q8—42no—
387Q8—42no—
388Q8—81no—
389Q8—81no—
390D4—22no—
391D4—22no—
392D4—41no—
393D4—41no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—164no—
434C2×C4C2×C22⋊Q811yescharacteristic
435C2×C4#52511yescharacteristic
436C2×C4#53911yescharacteristic
437C2×C4#54711yescharacteristic
438C2×C4D4×Q811yescharacteristic
439C2×C4#55411yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C2×C4#57511yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C8—22no—
449C8—42no—
450C8—42no—
451C8—41no—
452C8—41no—
453C8—41no—
454C8—42no—
455C8—81no—
456C8—81no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22#244911yescharacteristic
465C22#326711yescharacteristic
466C22#326811yescharacteristic
467C22#327811yescharacteristic
468C22#328011yescharacteristic
469C22#328811yescharacteristic
470C22#329311yescharacteristic
471C4—21no—
472C4—21no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4#307811yescharacteristic
485C4#308411yescharacteristic
486C4#310211yescharacteristic
487C4#312211yescharacteristic
488C4#334411yescharacteristic
489C4#335111yescharacteristic
490C4#338011yescharacteristic
491C4#338911yescharacteristic
492C2—82no—
493C2#1977511yescharacteristic
494C2#1990711yescharacteristic
495C2#2014411yescharacteristic
496C2#2026511yescharacteristic
497C2#2636011yescharacteristic
498C2#2637011yescharacteristic
499C2#2640611yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26527C2semidirect
5. #10680C2semidirect
6. #10657C2semidirect
7. #10594C2semidirect
8. #10577C2semidirect
9. #9615C2semidirect
10. #9536C2semidirect
13. #8925C2semidirect

Maximal subgroups

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

subgroupindex
#265272
#260872
#229622
#227132
#106802
#106572
#105942
#105772
#96152
#95362
#92272
#89842
#89252
#88862
#88472