Finite Group Databasebeta

← #6487771

Subgroups — #6487771

Order 512.

2606 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3253C211yesmaximal, characteristic, complemented
2C22×#423C211yesmaximal, characteristic, complemented
3#36624C212yesmaximal, complemented
4#32936C214yesmaximal, complemented
5#29782C214yesmaximal, complemented
6#24843C214yesmaximal, complemented
7#24836C214yesmaximal, complemented
8#24758C214yesmaximal, complemented
9#24750C214yesmaximal, complemented
10#21964C211yesmaximal, characteristic, complemented
11#21893C212yesmaximal, complemented
12C23×M16C2212yes—
13C2×#532C2212yes—
14C2×#525—22no—
15C23×C42C2211yescharacteristic
16#3253—28no—
17C2×#476C2214yescomplemented
18C2×#469C2214yes—
19C2×#469C2214yes—
20C2×#466C2214yes—
21C2×#459C2214yescomplemented
22C2×#423—28no—
23C2×#423C2214yes—
24C2×#423C2212yes—
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C22×C4⋊C8C2214yes—
28#2857C2212yes—
29#2857C2212yes—
30#2856C2212yes—
31#2856C2212yes—
32#2854—22no—
33#2854C2214yes—
34#2850—22no—
35#2850—22no—
36#2847—22no—
37#2847C2214yes—
38C4×#121—22no—
39C4×#121—22no—
40C4×#121C2214yes—
41C4×#121C2214yes—
42#2297—22no—
43#2285C2212yes—
44#2088C2218yescomplemented
45#2067C2218yescomplemented
46#2065C2218yescomplemented
47#2054C2218yescomplemented
48#1987C2214yescomplemented
49#1982C2214yescomplemented
50#1948C2214yescomplemented
51#1947C2214yescomplemented
52#1895C2218yes—
53#1895C2218yes—
54#1893C2218yes—
55#1809C2214yes—
56#1809C2214yes—
57#1807C2214yes—
58#1802C2214yescomplemented
59#1802C2214yescomplemented
60C22×#41—22no—
61C24×C4D412yes—
62C24×C4C2311yescharacteristic
63C22×D8C2314yes—
64C22×M16—28no—
65C22×M16—44no—
66C22×M16—44no—
67C22×M16C2314yes—
68C22×M16C2312yes—
69C22×M16C2312yes—
70C23×C8C2314yes—
71#532—28no—
72#532—28no—
73C2×C4⋊Q8C2312yes—
74C2×#127C2312yes—
75C2×#124—22no—
76#525—416no—
77C2×#123—24no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—22no—
84C2×#121—22no—
85C2×#121C2314yes—
86C2×#121C2314yes—
87C2×#121C2314yes—
88C2×#120—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—22no—
92C2×C4×D4C2314yes—
93C2×#117—22no—
94C22×C42—24no—
95C22×C42—24no—
96C22×C42D411yescharacteristic
97C22×C42D411yescharacteristic
98C22×C42D412yes—
99C22×C42D411yescharacteristic
100C22×C42D411yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105#476—216no—
106#469—216no—
107#469—48no—
108#466—48no—
109#459—216no—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113#423—28no—
114#423—24no—
115#423—216no—
116#423D418yescomplemented
117C2×C4⋊C8—216no—
118C2×C4⋊C8C2318yes—
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2314yes—
121C2×#102C2318yes—
122C2×#102C2318yes—
123C2×#102C2318yes—
124#391—22no—
125#391—22no—
126#390—22no—
127#390—22no—
128#390—22no—
129#390—22no—
130#390C2314yes—
131#390C2314yes—
132#389—22no—
133#389—22no—
134#389C2314yes—
135#389C2314yes—
136#388—24no—
137#388—24no—
138#388—24no—
139#388—24no—
140#386—24no—
141#386—24no—
142#386—24no—
143#386—24no—
144#386—24no—
145#386—24no—
146#384C2312yes—
147#384C2312yes—
148#382—22no—
149#380—22no—
150#380—22no—
151#380—22no—
152#380C2314yes—
153#380C2314yes—
154#380C2314yes—
155C4×C4⋊C4—22no—
156C4×C4⋊C4C2314yes—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159#340C2318yes—
160#340C2318yes—
161#340C2318yes—
162C25C2×D411yescharacteristic
163C2×#41—48no—
164C2×#41—48no—
165C2×#41—48no—
166C22×Q8—22no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4C2414yes—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—44no—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C2×D8—28no—
196C2×D8—44no—
197C2×D8—44no—
198C2×M16—24no—
199C2×M16—28no—
200C2×M16—28no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—44no—
204C2×M16—42no—
205C2×M16—42no—
206C2×M16—432no—
207C2×M16C2×D418yes—
208C22×C8—28no—
209C22×C8—28no—
210C22×C8—44no—
211C22×C8C2414yes—
212C22×C8C2414yes—
213C4⋊Q8—24no—
214C4⋊Q8—24no—
215#127—24no—
216#127—24no—
217#124—48no—
218#123—416no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221#121—28no—
222#121—28no—
223#121—28no—
224#121—28no—
225#121—28no—
226#121—28no—
227#121—48no—
228#121—48no—
229#121—48no—
230#121—48no—
231#121—48no—
232#120—416no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—416no—
236C4×D4—416no—
237C4×D4—48no—
238#117—48no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C2414yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C42—28no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D414yes—
281C2×C42#4114yes—
282C2×C42C2412yes—
283C2×C42C2412yes—
284#107—44no—
285#106—28no—
286#106—44no—
287C4⋊C8—216no—
288C4⋊C8—28no—
289C4⋊C8C2×D4116yes—
290#102—216no—
291#102—48no—
292#102—48no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—24no—
303C24—24no—
304C24—42no—
305C24C22×D411yescharacteristic
306C24C22×D411yescharacteristic
307C24C22×D411yescharacteristic
308#41—832no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—432no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4#12118yes—
382C22×C4AGL(1,ℤ/8)18yes—
383C22×C4C22×D412yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C2×#4112yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+412yes—
395C22×C42−1+412yes—
396C22×C4C2511yescharacteristic
397D8—88no—
398M16—48no—
399M16—416no—
400M16—48no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—416no—
407C2×C8C22×D418yes—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C42—22no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—48no—
441C42#12114yes—
442C42#12114yes—
443C42C22×D414yes—
444C42C2×#4114yes—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—28no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—48no—
458C23—48no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23C2×#12112yes—
469C23#53712yes—
470C23#53712yes—
471C23C2×AGL(1,ℤ/8)14yes—
472C23C23×D411yescharacteristic
473Q8—88no—
474D4—44no—
475D4—44no—
476D4—44no—
477D4—44no—
478D4—88no—
479D4—816no—
480D4—816no—
481D4—816no—
482D4—88no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—216no—
509C2×C4—216no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—816no—
527C2×C4C2×#12112yes—
528C2×C4C2×#12114yes—
529C2×C4C2×#12114yes—
530C2×C4#53712yes—
531C2×C4#53712yes—
532C2×C4C2×AGL(1,ℤ/8)14yes—
533C2×C4C2×AGL(1,ℤ/8)14yes—
534C2×C4#57514yes—
535C2×C4C23×D411yescharacteristic
536C2×C4C22×#4111yescharacteristic
537C2×C4C2×H(5,𝔽2)11yescharacteristic
538C2×C4C2×H(5,𝔽2)11yescharacteristic
539C2×C4C2×2−1+411yescharacteristic
540C2×C4C2×2−1+411yescharacteristic
541C8—28no—
542C8—44no—
543C22—24no—
544C22—22no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—22no—
549C22—22no—
550C22—24no—
551C22—416no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22#325314yescomplemented
566C22C22×#12111yescharacteristic
567C22C2×#53711yescharacteristic
568C22C2×#53711yescharacteristic
569C22C22×AGL(1,ℤ/8)12yes—
570C22C2×#57512yes—
571C4—24no—
572C4—24no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—42no—
577C4—84no—
578C4—84no—
579C4—84no—
580C4#305518yes—
581C4#348411yescharacteristic
582C4#348611yescharacteristic
583C4#349211yescharacteristic
584C4#349211yescharacteristic
585C2—24no—
586C2—42no—
587C2—42no—
588C2—164no—
589C2#3522311yescharacteristic
590C2C2×#305512yes—
591C2C2×#325311yescharacteristic
592C2#6090711yescharacteristic
593C2#6091612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3253C2semidirect
2. C22×#423C2semidirect
3. #36624C2semidirect
4. #32936C2semidirect
5. #29782C2semidirect
6. #24843C2semidirect
7. #24836C2semidirect
8. #24758C2semidirect
9. #24750C2semidirect
10. #21964C2semidirect
11. #21893C2semidirect
17. C2×#476C22semidirect
21. C2×#459C22semidirect
44. #2088C22semidirect
45. #2067C22semidirect
46. #2065C22semidirect
47. #2054C22semidirect
48. #1987C22semidirect
49. #1982C22semidirect
50. #1948C22semidirect
51. #1947C22semidirect
58. #1802C22semidirect
59. #1802C22semidirect
116. #423D4semidirect
565. C22#3253semidirect

Maximal subgroups

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

subgroupindex
C2×#32532
C22×#4232
#366242
#329362
#297822
#248432
#248362
#247582
#247502
#219642
#218932