Finite Group Databasebeta

← #6434112

Subgroups — #6434112

Order 512.

1802 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35254C211yesmaximal, characteristic
2#34184C211yesmaximal, characteristic, complemented
3#29917C214yesmaximal, complemented
4#29916C214yesmaximal, complemented
5#29914C214yesmaximal, complemented
6#29912C214yesmaximal, complemented
7#23749C212yesmaximal, complemented
8#23735C212yesmaximal, complemented
9#23659C211yesmaximal, characteristic, complemented
10#23656C211yesmaximal, characteristic, complemented
11#23628C211yesmaximal, characteristic, complemented
12C2×#2073C211yesmaximal, characteristic, complemented
13#20914C212yesmaximal
14#20849C212yesmaximal, complemented
15#20684C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#520C2212yes—
18C2×C4×#41C2211yescharacteristic
19C2×#428C2212yes—
20C2×#427C2212yes—
21C2×#418C2212yes—
22C22×#102C2212yes—
23C2×#408C2212yes—
24#2854C2214yes—
25#2624C2214yes—
26#2448—22no—
27#2445—22no—
28#2418—24no—
29#2351—22no—
30#2349—24no—
31#2349—22no—
32#2317C2212yes—
33#2317C2214yescomplemented
34#2316C2211yescharacteristic
35#2311C2211yescharacteristic
36#2309C2211yescharacteristic
37#2296C2211yescharacteristic
38C2×#390C2211yescharacteristic
39#2285C2211yescharacteristic
40#2285C2212yescomplemented
41C2×#382C2211yescharacteristic
42C2×#382C2212yescomplemented
43#2101C2214yescomplemented
44#2101C2214yescomplemented
45#2100C2214yescomplemented
46#2099C2214yescomplemented
47#2094C2214yes—
48#2093C2214yes—
49#2092C2214yes—
50#2092C2214yes—
51#2085C2212yescomplemented
52#2085C2212yescomplemented
53#2085C2214yescomplemented
54#2085C2214yescomplemented
55#2084C2212yescomplemented
56#2083C2212yescomplemented
57#2073C2214yes—
58#2073C2212yes—
59#2073C2212yes—
60#2073C2212yes—
61#2073C2212yes—
62#2071C2214yes—
63#1875C2214yes—
64#1875C2214yes—
65#1874C2214yes—
66#1874C2214yes—
67#1854C2214yescomplemented
68#1854C2214yescomplemented
69#1853C2214yescomplemented
70#1852C2214yescomplemented
71#1818C2212yescomplemented
72#1818C2212yescomplemented
73#1817C2214yescomplemented
74#1736C2212yes—
75#1736C2212yes—
76#1735C2212yes—
77C2×#340C2212yes—
78#1729C2212yes—
79#1728C2211yescharacteristic
80C2×#336C2211yescharacteristic
81C22×#41C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×M16C2312yes—
84C23×C8C2312yes—
85#535—28no—
86C2×#127—21no—
87C2×#124—21no—
88C2×#123—24no—
89C2×#121C2314yes—
90C2×#120—22no—
91#520—44no—
92#520—44no—
93C4×#41—28no—
94C2×C4×Q8—22no—
95C2×C4×Q8D412yes—
96C2×C4×D4—24no—
97C2×C4×D4—22no—
98C2×C4×D4D414yes—
99C2×C4×D4D412yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2312yes—
103C2×#117C2314yes—
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4—22no—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112#428—44no—
113#427—44no—
114C2×#107C2312yes—
115C2×#107C2312yes—
116C2×#107C2314yes—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×#106C2314yes—
120#418—44no—
121C2×#102—44no—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126#408—28no—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129#399—24no—
130#397—24no—
131#394—24no—
132#394—24no—
133#391—21no—
134#390—21no—
135#390—22no—
136#390—22no—
137#390C2314yes—
138#390C2312yes—
139#390C2312yes—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—24no—
144#389—24no—
145#389C2314yes—
146#389C2312yes—
147#389C2312yes—
148#388—24no—
149#388—24no—
150#386—22no—
151#386—24no—
152#385—22no—
153#385—24no—
154#385—24no—
155#385—24no—
156#384C2314yes—
157#384C2312yes—
158#382—22no—
159#382—24no—
160#382C2314yes—
161#382C2312yes—
162#382C2312yes—
163#382C2311yescharacteristic
164#382C2311yescharacteristic
165#380—22no—
166#380—24no—
167#380—22no—
168#380C2314yes—
169#379—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—22no—
173C2×#95—22no—
174C2×#95—24no—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2312yes—
180#340C2312yes—
181#340C2312yes—
182#340C2312yes—
183#340C2314yes—
184#340C2314yes—
185#336C2312yes—
186#336C2312yes—
187#336C2312yes—
188#336C2314yes—
189#336C2314yes—
190#336C2311yescharacteristic
191#336C2311yescharacteristic
192C25—21no—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—24no—
196C2×#41#4118yes—
197C22×Q8C2×D411yescharacteristic
198C22×D4—24no—
199C22×D4—21no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4C2×D412yes—
205C22×D4C2×D411yescharacteristic
206C22×D4C2412yes—
207C23×C4—21no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4C2×D412yes—
211C23×C4C2×D414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C2×M16—24no—
215C2×M16—24no—
216C2×M16—24no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8C2412yes—
221C22×C8C2412yes—
222C22×C8C2414yes—
223#127—44no—
224#124—44no—
225#123—88no—
226#121—216no—
227#120—48no—
228C4×Q8—28no—
229C4×Q8—48no—
230C4×D4—216no—
231C4×D4—28no—
232C4×D4—416no—
233C4×D4—48no—
234#117—28no—
235#117—24no—
236#117—24no—
237#117—24no—
238#117—24no—
239#117—22no—
240#117—22no—
241#117—44no—
242#117—44no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—84no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D414yes—
279C2×C42—28no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C2×D412yes—
286C2×C42C2×D414yes—
287C2×C42C2412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290#107—42no—
291#107—42no—
292#107—44no—
293#106—42no—
294#106—42no—
295#106—44no—
296#102—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300C22⋊C8—28no—
301C22⋊C8—28no—
302#95—24no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312C24—22no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—82no—
317C24—82no—
318C24C22×D412yes—
319C24C22×D411yescharacteristic
320#41—216no—
321#41—216no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8#12314yes—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—48no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—84no—
340C2×D4—84no—
341C2×D4#12318yes—
342C2×D4#12314yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—48no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—84no—
364C22×C4—82no—
365C22×C4—84no—
366C22×C4C22×D412yes—
367C22×C4C22×D414yes—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4118yes—
372C22×C4C2×#4114yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C42−1+414yes—
376C22×C4C2511yescharacteristic
377M16—48no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—48no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—48no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—84no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C22⋊C4—28no—
411C22⋊C4—24no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—48no—
418C22⋊C4—84no—
419C22⋊C4—84no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C42—28no—
423C42—24no—
424C42—22no—
425C42—24no—
426C42—22no—
427C42—48no—
428C42—42no—
429C42—42no—
430C42—44no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—88no—
443C23—84no—
444C23—164no—
445C23—162no—
446C23—162no—
447C23C2×#12314yes—
448C23C2×#12312yes—
449C23#53512yes—
450C23#53612yes—
451C23C23×D411yescharacteristic
452Q8—28no—
453D4—216no—
454D4—28no—
455D4—44no—
456D4—44no—
457D4—84no—
458D4—168no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4—88no—
491C2×C4—164no—
492C2×C4—162no—
493C2×C4—168no—
494C2×C4C2×#12314yes—
495C2×C4C2×#12312yes—
496C2×C4#53512yes—
497C2×C4#53612yes—
498C2×C4#53918yes—
499C2×C4#53918yes—
500C2×C4C22×#4112yes—
501C2×C4C2×H(5,𝔽2)11yescharacteristic
502C2×C4C2×H(5,𝔽2)11yescharacteristic
503C2×C4C2×2−1+412yes—
504C8—42no—
505C8—42no—
506C8—44no—
507C22—24no—
508C22—24no—
509C22—22no—
510C22—22no—
511C22—24no—
512C22—28no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—84no—
518C22—84no—
519C22—82no—
520C22—82no—
521C22—168no—
522C22#262314yes—
523C22C22×#12311yescharacteristic
524C22C2×#53511yescharacteristic
525C22C2×#53611yescharacteristic
526C22#358412yes—
527C22#358512yes—
528C4—24no—
529C4—44no—
530C4—42no—
531C4—42no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—84no—
536C4—82no—
537C4—82no—
538C4#262314yes—
539C4C2×#53912yes—
540C4C2×#53912yes—
541C2—24no—
542C2—44no—
543C2—162no—
544C2C2×#262311yescharacteristic
545C2#6052312yes—
546C2#6083412yes—
547C2#6087911yescharacteristic
548C2#6088311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
2. #34184C2semidirect
3. #29917C2semidirect
4. #29916C2semidirect
5. #29914C2semidirect
6. #29912C2semidirect
7. #23749C2semidirect
8. #23735C2semidirect
9. #23659C2semidirect
10. #23656C2semidirect
11. #23628C2semidirect
12. C2×#2073C2semidirect
14. #20849C2semidirect
15. #20684C2semidirect
33. #2317C22semidirect
40. #2285C22semidirect
42. C2×#382C22semidirect
43. #2101C22semidirect
44. #2101C22semidirect
45. #2100C22semidirect
46. #2099C22semidirect
51. #2085C22semidirect
52. #2085C22semidirect
53. #2085C22semidirect
54. #2085C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
67. #1854C22semidirect
68. #1854C22semidirect
69. #1853C22semidirect
70. #1852C22semidirect
71. #1818C22semidirect
72. #1818C22semidirect
73. #1817C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#341842
#299172
#299162
#299142
#299122
#237492
#237352
#236592
#236562
#236282
C2×#20732
#209142
#208492
#206842