Finite Group Databasebeta

← #7470773

Subgroups — #7470773

Order 512.

1489 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46004C211yesmaximal, characteristic, complemented
2#39489C211yesmaximal, characteristic
3#32645C212yesmaximal, complemented
4#31743C214yesmaximal, complemented
5#31272C212yesmaximal, complemented
6#30986C214yesmaximal, complemented
7#30205C212yesmaximal, complemented
8#29554C212yesmaximal, complemented
9#29003C214yesmaximal, complemented
10#27283C214yesmaximal, complemented
11#25945C212yesmaximal, complemented
12#23977C212yesmaximal
13#23272C211yesmaximal, characteristic, complemented
14C2×#555C2211yescharacteristic
15C2×#554C2211yescharacteristic
16C2×#491C2212yes—
17C2×#461C2212yes—
18C2×#448C2212yes—
19C2×#419C2212yes—
20#2868C2211yescharacteristic
21#2862C2212yes—
22#2844—21no—
23#2811—21no—
24#2804—21no—
25#2763—24no—
26#2674—24no—
27#2620—21no—
28#2568C2212yes—
29#2567C2211yescharacteristic
30#2563C2211yescharacteristic
31#2501—21no—
32#2499—21no—
33#2492C2214yes—
34#2441C2214yes—
35#2434C2214yes—
36#2422C2214yes—
37#2409C2212yes—
38#2379—21no—
39#2317—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2077C2214yes—
43#2072C2214yes—
44#2072C2214yes—
45#2066C2214yes—
46#2058C2212yes—
47#2043C2214yes—
48#2043C2212yes—
49#2035C2214yes—
50#2026C2214yes—
51#2024C2214yes—
52#2024C2214yes—
53#2023C2212yes—
54#2013C2212yes—
55#2000C2214yes—
56#1990C2214yes—
57#1989C2212yes—
58#1985C2212yes—
59#1968C2214yes—
60#1956C2214yes—
61#1935C2214yes—
62#1933C2214yes—
63#1927C2214yes—
64#1923C2214yes—
65#1920C2214yes—
66#1895C2212yes—
67#1894C2214yes—
68#1877C2214yes—
69#1877C2214yes—
70#1873C2214yes—
71#1870C2212yes—
72#1825C2212yes—
73#1787C2214yes—
74#1762C2212yes—
75#1753C2214yes—
76#1744C2212yes—
77C23×Q8C2312yes—
78C22×SD16C2314yes—
79C22×M16C2312yes—
80#555—42no—
81#555—42no—
82#554—42no—
83#554—42no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×#125C2312yes—
91C2×#124—22no—
92C2×#124D411yescharacteristic
93C2×#123—24no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8C2314yes—
97C2×#121—21no—
98C2×#121C2312yes—
99C2×C4×Q8C2311yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×C4×D4C2312yes—
102C2×C4×D4C2311yescharacteristic
103C2×#117—21no—
104C2×#117—21no—
105C2×#117C2312yes—
106C22×C4⋊C4—21no—
107#491—44no—
108#461—44no—
109#448—44no—
110C2×#106C2314yes—
111C2×C4⋊C8C2312yes—
112#419—44no—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2312yes—
123C2×#97C2312yes—
124C2×C4×C8C2314yes—
125#400—22no—
126#400—22no—
127#400—24no—
128#399—24no—
129#397—24no—
130#396—24no—
131#395—22no—
132#395—22no—
133#395—24no—
134#395—24no—
135#394—24no—
136#393—22no—
137#393—22no—
138#393—24no—
139#391—21no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—24no—
144#391D411yescharacteristic
145#391C2314yes—
146#391C2314yes—
147#391C2314yes—
148#390—21no—
149#390D411yescharacteristic
150#390C2312yes—
151#389—24no—
152#389C2314yes—
153#389C2312yes—
154#388—24no—
155#388—22no—
156#388D414yes—
157#387—24no—
158#386—22no—
159#385—24no—
160#385—22no—
161#385—24no—
162#385—24no—
163#384—21no—
164#384—21no—
165#384—21no—
166#384C2314yes—
167#382—21no—
168#382—22no—
169#382—22no—
170#382—22no—
171#382D411yescharacteristic
172#380C2314yes—
173#380C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—21no—
177C4×C22⋊C4C2314yes—
178C43C2311yescharacteristic
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2312yes—
186#336C2312yes—
187#336C2314yes—
188#336C2314yes—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—44no—
194C22×Q8—42no—
195C22×Q8—44no—
196C22×Q8—42no—
197C22×Q8C2×D411yescharacteristic
198C22×Q8C2414yes—
199C22×D4—22no—
200C22×D4C2×D411yescharacteristic
201C22×D4C2411yescharacteristic
202C23×C4—21no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4C2412yes—
206C2×SD16—28no—
207C2×SD16—44no—
208C2×SD16—44no—
209C2×M16—44no—
210C2×M16—42no—
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—41no—
214C4⋊Q8—41no—
215C4⋊Q8—41no—
216C4⋊Q8—41no—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—84no—
222#125—24no—
223#125—42no—
224#124—24no—
225#124—48no—
226#123—88no—
227C22⋊Q8—48no—
228C22⋊Q8—44no—
229C22⋊Q8—44no—
230C22⋊Q8—44no—
231#121—24no—
232#121—42no—
233#121—44no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—24no—
239C4×D4—42no—
240C4×D4—41no—
241C4×D4—41no—
242#117—24no—
243#117—42no—
244#117—44no—
245#117—44no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4C2×D412yes—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2412yes—
289C2×C42C2412yes—
290#106—44no—
291C4⋊C8—42no—
292#103—28no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299#102—44no—
300C22⋊C8—44no—
301C22⋊C8—42no—
302#97—42no—
303C4×C8—44no—
304#95—22no—
305#95—21no—
306#95—22no—
307#95—24no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95C2×D414yes—
317#95C2×D414yes—
318C24—21no—
319C24C22×D412yes—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—44no—
332C2×Q8—48no—
333C2×Q8—82no—
334C2×Q8—84no—
335C2×Q8—82no—
336C2×Q8—82no—
337C2×Q8—82no—
338C2×Q8—82no—
339C2×Q8—84no—
340C2×Q8—88no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—24no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—84no—
348C2×D4H(5,𝔽2)12yes—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—84no—
364C22×C4—82no—
365C22×C4—84no—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C2511yescharacteristic
374SD16—88no—
375M16—84no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—84no—
407C4⋊C4H(5,𝔽2)12yes—
408C22⋊C4—24no—
409C22⋊C4—28no—
410C22⋊C4—28no—
411C22⋊C4—42no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C22⋊C4—84no—
419C22⋊C4—84no—
420C42—24no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42—24no—
426C42—41no—
427C42—44no—
428C42—42no—
429C42—82no—
430C42#13714yes—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23C23×D411yescharacteristic
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—44no—
446Q8—82no—
447Q8—88no—
448Q8—164no—
449Q8—162no—
450Q8—162no—
451D4—24no—
452D4—42no—
453D4—82no—
454D4—164no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—48no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—88no—
491C2×C4—164no—
492C2×C4C2×#13712yes—
493C2×C4C2×#13714yes—
494C2×C4#57714yes—
495C2×C4C2×H(5,𝔽2)11yescharacteristic
496C2×C4C2×H(5,𝔽2)11yescharacteristic
497C2×C4#58514yes—
498C8—44no—
499C8—44no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22—44no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—82no—
509C22—82no—
510C22#348411yescharacteristic
511C22#350012yes—
512C22C22×#13711yescharacteristic
513C22C2×#57712yes—
514C22#358611yescharacteristic
515C4—24no—
516C4—24no—
517C4—22no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—41no—
523C4—84no—
524C4—82no—
525C4—84no—
526C4—82no—
527C4—82no—
528C4#354012yes—
529C4#354512yes—
530C2—82no—
531C2—82no—
532C2#5174911yescharacteristic
533C2#6158611yescharacteristic
534C2#6166211yescharacteristic
535C2#6254012yes—
536C2#6260312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46004C2semidirect
3. #32645C2semidirect
4. #31743C2semidirect
5. #31272C2semidirect
6. #30986C2semidirect
7. #30205C2semidirect
8. #29554C2semidirect
9. #29003C2semidirect
10. #27283C2semidirect
11. #25945C2semidirect
13. #23272C2semidirect

Maximal subgroups

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

subgroupindex
#460042
#394892
#326452
#317432
#312722
#309862
#302052
#295542
#290032
#272832
#259452
#239772
#232722