Finite Group Databasebeta

← #6497305

Subgroups — #6497305

Order 512.

1485 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35334C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#33155C214yesmaximal, complemented
4#29945C214yesmaximal, complemented
5#27967C212yesmaximal, complemented
6#27953C212yesmaximal, complemented
7#27881C212yesmaximal, complemented
8#27880C212yesmaximal, complemented
9#27621C212yesmaximal, complemented
10#27620C212yesmaximal, complemented
11#27609C212yesmaximal
12#27608C212yesmaximal
13#21986C212yesmaximal, complemented
14#21839C212yesmaximal, complemented
15#20719C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#432C2212yes—
19#2851C2212yes—
20#2851C2212yes—
21#2628C2212yes—
22#2628C2212yes—
23#2581—22no—
24#2580—22no—
25#2564—22no—
26#2564—22no—
27#2355—22no—
28#2354—22no—
29#2352—22no—
30#2352—22no—
31#2316C2211yescharacteristic
32#2316C2211yescharacteristic
33#2314C2212yes—
34#2314C2212yes—
35#2311C2211yescharacteristic
36#2311C2211yescharacteristic
37#2311C2212yes—
38#2285C2212yes—
39#2285C2212yes—
40#2285C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#382C2211yescharacteristic
43#2274C2212yes—
44#2101C2214yes—
45#2101C2214yes—
46#2084C2214yes—
47#2083C2214yes—
48#2077C2214yes—
49#2077C2214yes—
50#2076C2214yes—
51#2075C2214yes—
52#1992C2212yes—
53#1992C2212yes—
54#1992C2212yes—
55#1992C2212yes—
56#1988C2214yes—
57#1987C2214yes—
58#1949C2214yes—
59#1949C2214yes—
60#1921C2214yes—
61#1920C2214yes—
62#1919C2214yes—
63#1918C2214yes—
64#1853C2214yes—
65#1853C2214yes—
66#1852C2214yes—
67#1852C2214yes—
68#1847C2212yes—
69#1847C2212yes—
70#1836C2212yes—
71#1836C2212yes—
72#1825C2214yes—
73#1825C2214yes—
74#1825C2214yes—
75#1825C2214yes—
76#1818C2212yes—
77#1798C2212yes—
78#1798C2212yes—
79#1780C2212yes—
80#1749C2212yes—
81#1749C2211yescharacteristic
82#1749C2211yescharacteristic
83C22×#41C2311yescharacteristic
84#533—28no—
85C2×C4⋊Q8C2312yes—
86C2×#125C2312yes—
87C2×C22⋊Q8—22no—
88C4×#41—28no—
89C2×C4×Q8—22no—
90C2×C4×Q8D412yes—
91C2×C4×D4D412yes—
92C2×C4×D4D412yes—
93C2×C4×D4D412yes—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2312yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#432—28no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119C2×#97C2312yes—
120C2×#97C2312yes—
121#400—22no—
122#400—22no—
123#399—22no—
124#399—22no—
125#395—22no—
126#395—22no—
127#393—22no—
128#393—22no—
129#391—21no—
130#391—21no—
131#391—22no—
132#390—21no—
133#390—21no—
134#390—22no—
135#389—22no—
136#389—22no—
137#389C2312yes—
138#389C2312yes—
139#389C2312yes—
140#389C2312yes—
141#388—24no—
142#388—24no—
143#387—24no—
144#387—24no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#386—21no—
150#386—21no—
151#386—21no—
152#386—21no—
153#386—22no—
154#386—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#384—22no—
160#384C2312yes—
161#384C2312yes—
162#384C2312yes—
163#382—22no—
164#382—22no—
165#382—22no—
166#382C2312yes—
167#382C2312yes—
168#382C2312yes—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380C2314yes—
175#380C2314yes—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180C4×C4⋊C4—24no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4—24no—
187C2×#95—22no—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2312yes—
202#336C2312yes—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—42no—
206C22×Q8—22no—
207C22×Q8C2×D411yescharacteristic
208C22×D4C2×D411yescharacteristic
209C22×D4C2×D411yescharacteristic
210C22×D4C2×D411yescharacteristic
211C23×C4—22no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C22×C8C2414yes—
221C22×C8C2414yes—
222C4⋊Q8—28no—
223#125—28no—
224C22⋊Q8—48no—
225C4×Q8—28no—
226C4×Q8—48no—
227C4×D4—28no—
228C4×D4—28no—
229C4×D4—28no—
230#117—24no—
231#117—24no—
232#117—24no—
233#117—24no—
234#117—24no—
235#117—28no—
236#117—84no—
237#117—84no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—84no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42#4118yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—44no—
295#107—44no—
296#106—44no—
297#106—44no—
298C4⋊C8—28no—
299C22⋊C8—28no—
300C22⋊C8—28no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303#97—24no—
304#97—24no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322C24C22×D411yescharacteristic
323C24C22×D411yescharacteristic
324C24C22×D411yescharacteristic
325#41—416no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—44no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—84no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4#12318yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4114yes—
377C22×C42−1+414yes—
378C22×C4C2511yescharacteristic
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—84no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C42—24no—
423C42—24no—
424C42—22no—
425C42—22no—
426C42—42no—
427C42—42no—
428C42—44no—
429C42—84no—
430C42—84no—
431C42#12318yes—
432C42C2×#4114yes—
433C42C2×#4114yes—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—22no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23C2×#12312yes—
448C23#53612yes—
449C23C23×D411yescharacteristic
450Q8—44no—
451Q8—88no—
452D4—44no—
453D4—44no—
454D4—44no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—164no—
497C2×C4—162no—
498C2×C4—162no—
499C2×C4C2×#12314yes—
500C2×C4C2×#12312yes—
501C2×C4C2×#12314yes—
502C2×C4#53612yes—
503C2×C4C22×#4112yes—
504C2×C4C2×2−1+412yes—
505C2×C4#58512yes—
506C8—44no—
507C8—44no—
508C22—24no—
509C22—24no—
510C22—22no—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—42no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22C22×#12311yescharacteristic
524C22C2×#53611yescharacteristic
525C22#344911yescharacteristic
526C22#358512yes—
527C22#358612yes—
528C4—24no—
529C4—24no—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—41no—
535C4—84no—
536C4—82no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—84no—
541C4#349514yes—
542C2—42no—
543C2—42no—
544C2—81no—
545C2#3531111yescharacteristic
546C2#6051712yes—
547C2#6088612yes—
548C2#6101711yescharacteristic
549C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33155C2semidirect
4. #29945C2semidirect
5. #27967C2semidirect
6. #27953C2semidirect
7. #27881C2semidirect
8. #27880C2semidirect
9. #27621C2semidirect
10. #27620C2semidirect
13. #21986C2semidirect
14. #21839C2semidirect

Maximal subgroups

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

subgroupindex
#353342
#341882
#331552
#299452
#279672
#279532
#278812
#278802
#276212
#276202
#276092
#276082
#219862
#218392
#207192