Finite Group Databasebeta

← #6498116

Subgroups — #6498116

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#34058C211yesmaximal, characteristic
3#33154C212yesmaximal, complemented
4#32953C212yesmaximal
5#29986C212yesmaximal, complemented
6#29864C212yesmaximal, complemented
7#29586C214yesmaximal, complemented
8#29404C214yesmaximal, complemented
9#29306C212yesmaximal, complemented
10#29115C212yesmaximal
11#27620C212yesmaximal, complemented
12#27591C212yesmaximal
13#22116C212yesmaximal, complemented
14#22110C212yesmaximal, complemented
15#20786C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#474C2212yes—
18C2×#474C2212yes—
19#2868C2211yescharacteristic, complemented
20#2859C2211yescharacteristic, complemented
21#2857C2212yes—
22#2647C2212yescomplemented
23#2629C2212yes—
24#2507—22no—
25#2497—22no—
26#2446—22no—
27#2429—22no—
28#2330C2212yescomplemented
29#2319—22no—
30#2319C2212yes—
31#2319C2212yes—
32#2316C2211yescharacteristic
33#2316C2211yescharacteristic
34#2314C2212yes—
35#2312—22no—
36#2309C2211yescharacteristic
37#2309C2211yescharacteristic
38C4×C4⋊Q8C2212yescomplemented
39#2293—22no—
40#2293C2212yes—
41#2293C2212yes—
42#2280—22no—
43C4×#117C2211yescharacteristic
44C4×#117C2211yescharacteristic
45#2101C2214yescomplemented
46#2092C2214yes—
47#2083C2214yescomplemented
48#2083C2214yescomplemented
49#2071C2214yes—
50#2071C2214yes—
51#2059C2214yescomplemented
52#2025C2214yes—
53#1988C2212yescomplemented
54#1988C2212yescomplemented
55#1948C2212yescomplemented
56#1947C2212yescomplemented
57#1943C2214yescomplemented
58#1942C2214yescomplemented
59#1936C2214yes—
60#1934C2214yes—
61#1926C2214yes—
62#1925C2214yes—
63#1921C2214yescomplemented
64#1921C2214yescomplemented
65#1920C2214yescomplemented
66#1918C2214yescomplemented
67#1862C2214yes—
68#1862C2214yes—
69#1854C2214yes—
70#1852C2214yes—
71#1840C2212yescomplemented
72#1831C2212yescomplemented
73#1825C2214yes—
74#1818C2212yes—
75#1802C2212yes—
76#1802C2212yes—
77#1797C2212yes—
78#1786C2214yes—
79#1767C2211yescharacteristic, complemented
80C4×C4⋊C8C2211yescharacteristic, complemented
81#1750C2212yes—
82#1746C2212yescomplemented
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2312yes—
87C2×#125C2312yes—
88C2×C22⋊Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×D4—28no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×C4×D4C2312yes—
94C2×C4×D4C2312yes—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C42C2311yescharacteristic
104#474—28no—
105#474—28no—
106C2×#107C2314yes—
107C2×#107C2314yes—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2312yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×C22⋊C8C2314yes—
117C2×#97C2312yes—
118C2×C4×C8C2312yes—
119#400—24no—
120#398—24no—
121#398—24no—
122#396—24no—
123#391—22no—
124#391—22no—
125#390—22no—
126#390—22no—
127#390C2312yes—
128#390C2312yes—
129#390C2312yes—
130#390C2312yes—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—24no—
138#389C2314yes—
139#389C2312yes—
140#389C2312yes—
141#389C2312yes—
142#389C2312yes—
143#389C2314yes—
144#388—24no—
145#388—22no—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#387—24no—
154#387—24no—
155#384—22no—
156#384C2312yes—
157#384C2312yes—
158#384C2312yes—
159#382C2312yes—
160#381—22no—
161#381—22no—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—22no—
166#380—22no—
167#380C2314yes—
168#379—22no—
169#379—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—22no—
179C4×C22⋊C4—22no—
180C4×C22⋊C4D412yes—
181C4×C22⋊C4D412yes—
182#376—21no—
183#376C2311yescharacteristic
184#376C2311yescharacteristic
185C43—21no—
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201C25—21no—
202C22×Q8—22no—
203C22×D4—24no—
204C22×D4—21no—
205C22×D4—24no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4C2412yes—
209C23×C4—28no—
210C23×C4—21no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—84no—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—84no—
222#125—42no—
223#125—42no—
224C22⋊Q8—84no—
225C4×Q8—84no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4#4118yes—
232#117—24no—
233#117—24no—
234#117—41no—
235#117—41no—
236#117—41no—
237#117—41no—
238#117—84no—
239#117—84no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D412yes—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295#107—28no—
296#107—28no—
297#107—44no—
298#106—44no—
299C4⋊C8—24no—
300C4⋊C8—24no—
301#102—28no—
302#102—28no—
303#102—44no—
304#102—44no—
305C22⋊C8—28no—
306#97—42no—
307C4×C8—42no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95C2×D412yes—
320#95C2×D412yes—
321#95C2×D412yes—
322#95C2×D412yes—
323C24—24no—
324C24—22no—
325C24—42no—
326C24—42no—
327C24—82no—
328C24—82no—
329C24C22×D411yescharacteristic
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4C2×#4114yes—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—816no—
364C22×C4—82no—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C42−1+414yes—
375C22×C4C2511yescharacteristic
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4C2×#4114yes—
406C22⋊C4—28no—
407C22⋊C4—24no—
408C22⋊C4—28no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—84no—
421C42—22no—
422C42—22no—
423C42—24no—
424C42—24no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—44no—
437C42—82no—
438C42—84no—
439C42C2×#4114yes—
440C23—22no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—24no—
445C23—48no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—88no—
453C23—84no—
454C23—84no—
455C23—88no—
456C23#53612yes—
457C23C23×D411yescharacteristic
458Q8—88no—
459D4—24no—
460D4—24no—
461D4—48no—
462D4—42no—
463D4—42no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—416no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—82no—
503C2×C4—88no—
504C2×C4—84no—
505C2×C4#52512yes—
506C2×C4#52512yes—
507C2×C4#53612yes—
508C2×C4#57614yes—
509C2×C4C22×#4112yes—
510C2×C4C2×2−1+412yes—
511C2×C4#58511yescharacteristic
512C2×C4#58511yescharacteristic
513C8—44no—
514C8—44no—
515C22—24no—
516C22—22no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—816no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22C2×#52511yescharacteristic
529C22C2×#53611yescharacteristic
530C22#344911yescharacteristic
531C22C2×#57612yes—
532C22#358612yes—
533C4—22no—
534C4—22no—
535C4—42no—
536C4—42no—
537C4—41no—
538C4—44no—
539C4—41no—
540C4—42no—
541C4—42no—
542C4—82no—
543C4—84no—
544C4—84no—
545C4—82no—
546C4#312214yes—
547C4#349512yes—
548C4#349512yes—
549C2—42no—
550C2—84no—
551C2#3532511yescharacteristic
552C2C2×#312211yescharacteristic
553C2#6056911yescharacteristic
554C2#6088812yes—
555C2#6098812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
3. #33154C2semidirect
5. #29986C2semidirect
6. #29864C2semidirect
7. #29586C2semidirect
8. #29404C2semidirect
9. #29306C2semidirect
11. #27620C2semidirect
13. #22116C2semidirect
14. #22110C2semidirect
15. #20786C2semidirect
19. #2868C22semidirect
20. #2859C22semidirect
22. #2647C22semidirect
28. #2330C22semidirect
38. C4×C4⋊Q8C22semidirect
45. #2101C22semidirect
47. #2083C22semidirect
48. #2083C22semidirect
51. #2059C22semidirect
53. #1988C22semidirect
54. #1988C22semidirect
55. #1948C22semidirect
56. #1947C22semidirect
57. #1943C22semidirect
58. #1942C22semidirect
63. #1921C22semidirect
64. #1921C22semidirect
65. #1920C22semidirect
66. #1918C22semidirect
71. #1840C22semidirect
72. #1831C22semidirect
79. #1767C22semidirect
80. C4×C4⋊C8C22semidirect
82. #1746C22semidirect

Maximal subgroups

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

subgroupindex
#352842
#340582
#331542
#329532
#299862
#298642
#295862
#294042
#293062
#291152
#276202
#275912
#221162
#221102
#207862