Finite Group Databasebeta

← #6468647

Subgroups — #6468647

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35331C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#29363C214yesmaximal, complemented
4#29344C212yesmaximal, complemented
5#29343C212yesmaximal
6#29224C212yesmaximal, complemented
7#28753C212yesmaximal, complemented
8#27978C212yesmaximal, complemented
9#25909C212yesmaximal, complemented
10#23831C212yesmaximal, complemented
11#23830C212yesmaximal, complemented
12#21655C212yesmaximal, complemented
13C4×#489C212yesmaximal, complemented
14#21397C214yesmaximal, complemented
15#20465C211yesmaximal, characteristic, complemented
16C2×#542C2212yes—
17C2×#533C2211yescharacteristic
18C2×#489C2212yes—
19C2×#489C2212yes—
20C2×#433C2212yes—
21#2862C2211yescharacteristic
22#2858C2211yescharacteristic
23#2850C2212yes—
24#2640C2212yes—
25#2621C2211yescharacteristic
26#2619C2211yescharacteristic
27#2560—22no—
28#2560—22no—
29#2557—22no—
30#2557—22no—
31#2360—22no—
32#2360—22no—
33#2355—22no—
34Q8×C4⋊C4—22no—
35#2330C2212yes—
36#2330C2212yes—
37#2328C2211yescharacteristic
38#2327C2211yescharacteristic
39#2317C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×#124C2212yes—
43C2×#384C2211yescharacteristic
44#2274C2212yes—
45C4×#117C2211yescharacteristic
46#2101C2214yes—
47#2101C2214yes—
48#2072C2214yes—
49#2071C2214yes—
50#2043C2214yes—
51#2041C2214yes—
52#2026C2214yes—
53#2025C2214yes—
54#1993C2212yes—
55#1993C2212yes—
56#1988C2214yes—
57#1987C2214yes—
58#1983C2214yes—
59#1983C2214yes—
60#1959C2212yes—
61#1959C2212yes—
62#1936C2214yes—
63#1935C2214yes—
64#1934C2214yes—
65#1933C2214yes—
66#1875C2214yes—
67#1874C2214yes—
68#1873C2214yes—
69#1872C2214yes—
70#1836C2212yes—
71#1836C2212yes—
72#1768C2212yes—
73#1767C2212yes—
74#1763C2214yes—
75#1762C2214yes—
76C4×#103C2214yes—
77C4×#102C2214yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83#542—216no—
84#533—28no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#126—22no—
90C2×#125C2312yes—
91C2×#124—22no—
92C2×#124C2312yes—
93C2×#124C2312yes—
94C2×#123—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8C2312yes—
98C2×#121C2312yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8C2312yes—
103C2×C4×D4C2312yes—
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C4⋊C4C2312yes—
109C22×C42C2311yescharacteristic
110#489—28no—
111#489—28no—
112#433—28no—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2312yes—
124C2×#97C2312yes—
125C2×#97C2312yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128C2×C4×C8C2312yes—
129#398—21no—
130#398—21no—
131#397—21no—
132#397—21no—
133#397—22no—
134#397—22no—
135#394—21no—
136#394—21no—
137#393—22no—
138#393—22no—
139#392—21no—
140#392—21no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391C2312yes—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390C2312yes—
150#390C2312yes—
151#389—22no—
152#389—22no—
153#389—24no—
154#389—24no—
155#389C2314yes—
156#389C2314yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#387—22no—
162#387—22no—
163#387—24no—
164#387—24no—
165#386—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#385—22no—
170#385—22no—
171#384—22no—
172#384—22no—
173#384C2312yes—
174#384C2312yes—
175#382—22no—
176#382C2311yescharacteristic
177#382C2311yescharacteristic
178#382C2312yes—
179C4×C4⋊C4—24no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4—24no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4D412yes—
190C4×C22⋊C4D412yes—
191#376C2311yescharacteristic
192#376C2311yescharacteristic
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2314yes—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8C2412yes—
208C22×D4—22no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4C2×D412yes—
215C23×C4C2411yescharacteristic
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—24no—
219C4⋊Q8—28no—
220C4⋊Q8—42no—
221C4⋊Q8#4118yes—
222#127—42no—
223#126—48no—
224#125—28no—
225#124—28no—
226#124—24no—
227#124—24no—
228#124—48no—
229#123—48no—
230C22⋊Q8—216no—
231C22⋊Q8—28no—
232C22⋊Q8—48no—
233#121—28no—
234C4×Q8—216no—
235C4×Q8—28no—
236C4×Q8—84no—
237C4×Q8—84no—
238C4×D4—28no—
239#117—28no—
240#117—42no—
241#117#4118yes—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
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—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—84no—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
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⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C2×D411yescharacteristic
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C22⋊C4C2×D412yes—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C4⋊C8—28no—
297#103—28no—
298#103—28no—
299#103—44no—
300#103—44no—
301#102—28no—
302#102—28no—
303#102—44no—
304#102—44no—
305C22⋊C8—28no—
306#97—24no—
307#97—24no—
308#97—24no—
309#97—42no—
310C4×C8—24no—
311C4×C8—42no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—22no—
325#95—22no—
326#95C2×D412yes—
327#95C2×D412yes—
328#95C2×D412yes—
329#95C2×D412yes—
330C24—22no—
331C24C22×D411yescharacteristic
332C2×Q8—28no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—82no—
341C2×Q8—82no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—44no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—84no—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4114yes—
378C22×C4C2511yescharacteristic
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C4⋊C4—216no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4—84no—
405C4⋊C4#12418yes—
406C4⋊C4C2×#4118yes—
407C22⋊C4—28no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C22⋊C4#12418yes—
422C42—28no—
423C42—24no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—44no—
432C42—84no—
433C42—84no—
434C42C2×#4114yes—
435C42C2×#4114yes—
436C42C2×#4114yes—
437C23—22no—
438C23—22no—
439C23—21no—
440C23—21no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23C2×#12412yes—
449C23C23×D411yescharacteristic
450Q8—416no—
451Q8—44no—
452Q8—44no—
453Q8—164no—
454D4—44no—
455D4—44no—
456D4—162no—
457D4—162no—
458D4—164no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—28no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—164no—
496C2×C4—164no—
497C2×C4#52514yes—
498C2×C4C2×#12414yes—
499C2×C4C2×#12414yes—
500C2×C4C2×#12412yes—
501C2×C4C22×#4112yes—
502C2×C4C22×#4112yes—
503C2×C4#58512yes—
504C8—44no—
505C8—44no—
506C22—22no—
507C22—24no—
508C22—41no—
509C22—41no—
510C22—42no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22C2×#52511yescharacteristic
519C22C22×#12411yescharacteristic
520C22#344911yescharacteristic
521C22#358512yes—
522C22#358612yes—
523C4—22no—
524C4—24no—
525C4—24no—
526C4—24no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—42no—
532C4—48no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4#348014yes—
538C2—42no—
539C2—162no—
540C2#3495511yescharacteristic
541C2#6056512yes—
542C2#6062512yes—
543C2#6101711yescharacteristic
544C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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
2. #34214C2semidirect
3. #29363C2semidirect
4. #29344C2semidirect
6. #29224C2semidirect
7. #28753C2semidirect
8. #27978C2semidirect
9. #25909C2semidirect
10. #23831C2semidirect
11. #23830C2semidirect
12. #21655C2semidirect
13. C4×#489C2semidirect
14. #21397C2semidirect
15. #20465C2semidirect

Maximal subgroups

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

subgroupindex
#353312
#342142
#293632
#293442
#293432
#292242
#287532
#279782
#259092
#238312
#238302
#216552
C4×#4892
#213972
#204652