Finite Group Databasebeta

← #6458881

Subgroups — #6458881

Order 512.

1850 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35254C211yesmaximal, characteristic
2#34189C211yesmaximal, characteristic, complemented
3#27871C214yesmaximal, complemented
4#27868C214yesmaximal, complemented
5#27865C214yesmaximal, complemented
6#27859C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic, complemented
9#23659C211yesmaximal, characteristic, complemented
10C2×#2073C211yesmaximal, characteristic, complemented
11#21063C212yesmaximal, complemented
12#21036C212yesmaximal, complemented
13#20975C212yesmaximal
14#20664C212yesmaximal, complemented
15#20336C211yesmaximal, characteristic, complemented
16C22×#131C2212yes—
17C2×#535C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#418C2212yes—
22C22×#102C2212yes—
23#2854C2214yes—
24#2624C2214yes—
25#2448—22no—
26#2445—22no—
27#2418—24no—
28#2357—22no—
29#2356—22no—
30#2353—24no—
31#2316C2211yescharacteristic
32#2314C2212yescomplemented
33#2314C2212yescomplemented
34#2311C2212yes—
35#2311C2211yescharacteristic
36#2309C2211yescharacteristic
37#2296C2211yescharacteristic
38C2×#390C2211yescharacteristic
39#2285C2211yescharacteristic
40#2283C2214yescomplemented
41C2×#382C2211yescharacteristic
42#2274C2212yes—
43#2101C2212yescomplemented
44#2101C2212yescomplemented
45#2100C2212yescomplemented
46#2099C2212yescomplemented
47#2094C2212yes—
48#2093C2212yes—
49#2092C2214yes—
50#2085C2214yescomplemented
51#2085C2214yescomplemented
52#2073C2212yes—
53#2073C2212yes—
54#2073C2214yes—
55#1926C2214yes—
56#1926C2214yes—
57#1925C2214yes—
58#1925C2214yes—
59#1921C2214yescomplemented
60#1920C2214yescomplemented
61#1919C2214yescomplemented
62#1918C2214yescomplemented
63#1895C2214yes—
64#1895C2214yes—
65#1893C2214yes—
66#1893C2214yes—
67#1809C2214yescomplemented
68#1809C2214yescomplemented
69#1808C2214yescomplemented
70#1807C2214yescomplemented
71#1736C2212yes—
72#1735C2212yes—
73#1735C2212yes—
74C2×#340C2212yes—
75#1729C2212yes—
76#1729C2214yescomplemented
77#1728C2211yescharacteristic
78#1728C2212yescomplemented
79C2×#336C2211yescharacteristic
80C2×#336C2212yescomplemented
81C22×#41C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C2×#131—28no—
84C2×#131—28no—
85C2×#131—44no—
86C22×M16C2314yes—
87C22×M16C2312yes—
88C23×C8C2312yes—
89C23×C8C2314yes—
90#535—28no—
91C2×#127—21no—
92C2×#124—21no—
93C2×#123—24no—
94C2×#121C2314yes—
95C2×#120—22no—
96C4×#41—28no—
97C2×C4×Q8D412yes—
98C2×C4×D4D414yes—
99C2×C4×D4D412yes—
100C2×#117C2311yescharacteristic
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—22no—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110#428—28no—
111#428—44no—
112C2×#107C2314yes—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116#418—28no—
117C2×#102—44no—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122#399—24no—
123#397—24no—
124#394—24no—
125#394—24no—
126#391—22no—
127#391—21no—
128#390—22no—
129#390—21no—
130#390—22no—
131#390—22no—
132#390C2314yes—
133#390C2312yes—
134#390C2312yes—
135#389—22no—
136#389C2314yes—
137#389C2312yes—
138#389C2312yes—
139#388—24no—
140#388—24no—
141#388—24no—
142#388—24no—
143#387—24no—
144#387—24no—
145#386—24no—
146#386—22no—
147#386—24no—
148#385—22no—
149#384C2311yescharacteristic
150#384C2311yescharacteristic
151#383—24no—
152#382—22no—
153#382C2312yes—
154#382C2312yes—
155#382C2312yes—
156#381—24no—
157#380—22no—
158#380—22no—
159#380C2314yes—
160#380C2314yes—
161#380C2314yes—
162#379—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2314yes—
169C4×C22⋊C4—24no—
170C4×C22⋊C4—24no—
171#376—24no—
172C2×#95—22no—
173#340C2312yes—
174#340C2314yes—
175#340C2312yes—
176#340C2314yes—
177#340C2312yes—
178#340C2312yes—
179#340C2314yes—
180#340C2314yes—
181#340C2312yes—
182#340C2312yes—
183#340C2312yes—
184#340C2312yes—
185#336C2312yes—
186#336C2311yescharacteristic
187#336C2311yescharacteristic
188#336C2312yes—
189#336C2312yes—
190#336C2314yes—
191#336C2314yes—
192C25—21no—
193C2×#41—24no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41#4118yes—
197C22×Q8C2×D411yescharacteristic
198C22×D4—24no—
199C22×D4—21no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C2×D412yes—
205C22×D4C2×D411yescharacteristic
206C22×D4C2412yes—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4C2×D412yes—
210C23×C4C2×D414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213#131—432no—
214C2×M16—24no—
215C2×M16—24no—
216C2×M16—28no—
217C2×M16—28no—
218C2×M16—44no—
219C2×M16#4118yes—
220C22×C8—28no—
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—28no—
225C22×C8—44no—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2414yes—
229#127—44no—
230#124—44no—
231#123—88no—
232#121—216no—
233#120—48no—
234C4×Q8—28no—
235C4×D4—216no—
236C4×D4—28no—
237#117—28no—
238#117—24no—
239#117—24no—
240#117—24no—
241#117—24no—
242#117—22no—
243#117—22no—
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—22no—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—84no—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D414yes—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—21no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42C2×D414yes—
284C2×C42C2×D412yes—
285C2×C42C2412yes—
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288#107—28no—
289#107—44no—
290#106—28no—
291#106—44no—
292#102—28no—
293#102—28no—
294#102—44no—
295#102—44no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308C24—22no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—82no—
313C24—82no—
314C24C22×D412yes—
315C24C22×D411yescharacteristic
316#41—216no—
317#41—216no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8#12314yes—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—48no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4#12318yes—
338C2×D4#12314yes—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C22×D412yes—
360C22×C4C22×D414yes—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4118yes—
365C22×C4C2×#4114yes—
366C22×C4C2×#4114yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C42−1+412yes—
369C22×C4C2511yescharacteristic
370M16—216no—
371M16—416no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—416no—
381C2×C8—48no—
382C2×C8C2×#4118yes—
383C4⋊C4—28no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—84no—
402C22⋊C4—28no—
403C22⋊C4—24no—
404C22⋊C4—48no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—48no—
409C22⋊C4—84no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C42—28no—
416C42—24no—
417C42—22no—
418C42—24no—
419C42—22no—
420C42—42no—
421C42—44no—
422C42—42no—
423C42—84no—
424C42—84no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—22no—
430C23—22no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—88no—
437C23—84no—
438C23—162no—
439C23—162no—
440C23—164no—
441C23C2×#12314yes—
442C23C2×#12312yes—
443C23#53212yes—
444C23#53712yes—
445C23C23×D411yescharacteristic
446Q8—28no—
447D4—216no—
448D4—28no—
449D4—44no—
450D4—44no—
451D4—84no—
452D4—168no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—84no—
484C2×C4—88no—
485C2×C4—162no—
486C2×C4—164no—
487C2×C4C2×#12314yes—
488C2×C4C2×#12312yes—
489C2×C4#53212yes—
490C2×C4#53712yes—
491C2×C4#55118yes—
492C2×C4#55218yes—
493C2×C4C22×#4112yes—
494C2×C4C22×#4112yes—
495C2×C4C2×H(5,𝔽2)11yescharacteristic
496C2×C4C2×2−1+411yescharacteristic
497C8—28no—
498C8—42no—
499C8—42no—
500C22—22no—
501C22—22no—
502C22—24no—
503C22—24no—
504C22—28no—
505C22—24no—
506C22—42no—
507C22—42no—
508C22—44no—
509C22—82no—
510C22—82no—
511C22—84no—
512C22—82no—
513C22—84no—
514C22—168no—
515C22#249814yes—
516C22C22×#12311yescharacteristic
517C22C2×#53211yescharacteristic
518C22C2×#53711yescharacteristic
519C22#358412yes—
520C22#358512yes—
521C4—24no—
522C4—44no—
523C4—42no—
524C4—42no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—84no—
531C4#249814yes—
532C4C2×#55112yes—
533C4C2×#55212yes—
534C2—24no—
535C2—44no—
536C2—162no—
537C2C2×#249811yescharacteristic
538C2#6052312yes—
539C2#6072411yescharacteristic
540C2#6072511yescharacteristic
541C2#6092612yes—
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. #34189C2semidirect
3. #27871C2semidirect
4. #27868C2semidirect
5. #27865C2semidirect
6. #27859C2semidirect
7. #23744C2semidirect
8. #23731C2semidirect
9. #23659C2semidirect
10. C2×#2073C2semidirect
11. #21063C2semidirect
12. #21036C2semidirect
14. #20664C2semidirect
15. #20336C2semidirect
32. #2314C22semidirect
33. #2314C22semidirect
40. #2283C22semidirect
43. #2101C22semidirect
44. #2101C22semidirect
45. #2100C22semidirect
46. #2099C22semidirect
50. #2085C22semidirect
51. #2085C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
61. #1919C22semidirect
62. #1918C22semidirect
67. #1809C22semidirect
68. #1809C22semidirect
69. #1808C22semidirect
70. #1807C22semidirect
76. #1729C22semidirect
78. #1728C22semidirect
80. C2×#336C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#341892
#278712
#278682
#278652
#278592
#237442
#237312
#236592
C2×#20732
#210632
#210362
#209752
#206642
#203362