Finite Group Databasebeta

← #196930

Subgroups — #196930

Order 512.

1859 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35236C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic, complemented
3#21788C211yesmaximal, characteristic, complemented
4C2×#1509C211yesmaximal, characteristic, complemented
5C2×#1500C211yesmaximal, characteristic, complemented
6#9150C212yesmaximal, complemented
7#9147C212yesmaximal, complemented
8#9146C212yesmaximal, complemented
9#9143C212yesmaximal, complemented
10#8022C211yesmaximal, characteristic, complemented
11#8014C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2848C2212yes—
18#2847C2212yes—
19#2636—22no—
20#2562C412yescomplemented
21#2540C412yescomplemented
22#2538C412yescomplemented
23#2537C412yescomplemented
24#2487—21no—
25#2411—22no—
26#2408—21no—
27#2324—22no—
28#2285C2211yescharacteristic
29#2275—24no—
30#2275—24no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—22no—
34C2×C4×C22⋊C4C2212yes—
35C2×#376C2211yescharacteristic
36C22×#95C2212yes—
37#1955—22no—
38#1954—22no—
39#1779C2212yes—
40#1779C2212yes—
41#1509—24no—
42#1500—24no—
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2212yes—
46C2×#327C2211yescharacteristic
47C2×#327C2211yescharacteristic
48C2×#327C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×D4D411yescharacteristic
54C24×C4—22no—
55C24×C4C2312yes—
56#529—22no—
57#529—22no—
58#529—22no—
59#529—22no—
60C2×#126—21no—
61C2×#125C2×C411yescharacteristic, complemented
62C2×#124C2×C411yescharacteristic, complemented
63C2×#123—21no—
64C2×C22⋊Q8C2×C414yescomplemented
65C2×C22⋊Q8C2311yescharacteristic
66C2×#121—22no—
67C2×#121C2×C414yescomplemented
68C2×#121C2311yescharacteristic
69C2×#120—22no—
70C2×C4×Q8C2×C411yescharacteristic, complemented
71C2×C4×D4—216no—
72C2×C4×D4—28no—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4C2×C412yes—
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2×C411yescharacteristic, complemented
81C2×C4×D4D412yes—
82C2×C4×D4D412yes—
83C2×#117—21no—
84C2×#117—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C22⋊C4—22no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42—22no—
92C22×C42C2311yescharacteristic
93#432—22no—
94#432—22no—
95#431—22no—
96#431—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8C2311yescharacteristic
102C2×C22⋊C8C2311yescharacteristic
103C2×C22⋊C8C2312yes—
104C2×#97—21no—
105C2×C4×C8—21no—
106#400—22no—
107#399—22no—
108#396—24no—
109#394—22no—
110#394—24no—
111#392—22no—
112#390—22no—
113#390—24no—
114#389—22no—
115#388—24no—
116#386—24no—
117#386—22no—
118#385—24no—
119#385—22no—
120#384C2×C412yescomplemented
121#382—21no—
122#380—21no—
123#380—22no—
124#380C2×C414yescomplemented
125#380C2×C414yescomplemented
126#380C2312yes—
127#379—22no—
128C4×C4⋊C4—28no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2×C412yescomplemented
132C4×C4⋊C4D414yescomplemented
133C4×C22⋊C4—216no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—24no—
136#376—22no—
137#376D414yescomplemented
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145#328—22no—
146#328—22no—
147#328—24no—
148#327—22no—
149#327—22no—
150#327—24no—
151C25C2×D412yes—
152C22×Q8C22×C411yescharacteristic
153C22×D4—28no—
154C22×D4—21no—
155C22×D4—24no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4C22⋊C412yes—
159C22×D4C22⋊C412yes—
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—216no—
165C23×C4—28no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—84no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4—84no—
175C23×C4C22×C414yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D414yes—
180C23×C4C2411yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184#126—42no—
185#126—42no—
186#125—22no—
187#125—22no—
188#124—22no—
189#124—22no—
190#123—42no—
191#123—42no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—84no—
197#121—21no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121—48no—
202#121—84no—
203#120—82no—
204#120—82no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×D4—264no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—42no—
219C4×D4—42no—
220#117—42no—
221#117—42no—
222#117—81no—
223#117—81no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4C22⋊C412yes—
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C414yes—
240C2×C22⋊C4—216no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4C22⋊C414yes—
255C2×C22⋊C4C22⋊C414yes—
256C2×C42—28no—
257C2×C42—24no—
258C2×C42—21no—
259C2×C42—28no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—81no—
263C2×C42—81no—
264C2×C42—84no—
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4112yes—
272C2×C42#4112yes—
273C4⋊C8—42no—
274C4⋊C8—42no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—22no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—22no—
281C22⋊C8—84no—
282C22⋊C8—84no—
283#97—42no—
284C4×C8—42no—
285#95—28no—
286#95—22no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—21no—
291#95—22no—
292#95—21no—
293#95—22no—
294#95C2×D412yes—
295#95C2×D414yes—
296#95C2×D412yes—
297C24—22no—
298C24—22no—
299C24—28no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24—44no—
306C24C2×C22⋊C414yes—
307C24C22×D411yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—42no—
311C2×D4—216no—
312C2×D4—216no—
313C2×D4—28no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—48no—
327C2×D4—84no—
328C2×D4—84no—
329C2×D4#10418yes—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—232no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—88no—
374C22×C4—816no—
375C22×C4—816no—
376C22×C4—88no—
377C22×C4C2×C22⋊C414yes—
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4#12012yes—
383C22×C4#12012yes—
384C22×C4#12312yes—
385C22×C4#12312yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—44no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—42no—
398C4⋊C4—216no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—44no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—82no—
413C4⋊C4—84no—
414C22⋊C4—232no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—48no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C42—216no—
431C42—42no—
432C42—84no—
433C42—82no—
434C42—88no—
435C42C4×D412yes—
436C42C4×D412yes—
437C42C4×D412yes—
438C42C4×D412yes—
439C42#12112yes—
440C42#12112yes—
441C42#12112yes—
442C42#12112yes—
443C23—22no—
444C23—22no—
445C23—28no—
446C23—28no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—48no—
454C23—48no—
455C23—48no—
456C23—48no—
457C23—44no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—41no—
468C23C2×#10418yes—
469C23C22×C22⋊C411yescharacteristic
470C23C2×#12011yescharacteristic
471C23C2×#12311yescharacteristic
472C23#57611yescharacteristic
473C23#57711yescharacteristic
474C23#57711yescharacteristic
475C23#57811yescharacteristic
476Q8—81no—
477Q8—81no—
478D4—216no—
479D4—216no—
480D4—81no—
481D4—81no—
482D4—81no—
483D4—81no—
484D4—81no—
485D4—81no—
486D4—168no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—832no—
539C2×C4—84no—
540C2×C4#38612yes—
541C2×C4#38612yes—
542C2×C4#38612yes—
543C2×C4#38612yes—
544C2×C4C2×#10414yes—
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C8—81no—
550C8—81no—
551C8—82no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—28no—
561C22—28no—
562C22—28no—
563C22—48no—
564C22—48no—
565C22—48no—
566C22—48no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22C2×#38611yescharacteristic
577C22#288312yes—
578C22#289711yescharacteristic
579C22#289811yescharacteristic
580C22C22×#10412yes—
581C22#301211yescharacteristic
582C22#301311yescharacteristic
583C22#309811yescharacteristic
584C22#309911yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—44no—
593C4—48no—
594C4—44no—
595C4—44no—
596C4—84no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4#189614yes—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—48no—
606C2—161no—
607C2#1754712yes—
608C2#1754812yes—
609C2#2091511yescharacteristic
610C2C2×#189611yescharacteristic
611C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35236C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2275C2semidirect
3. #21788C2semidirect
4. C2×#1509C2semidirect
5. C2×#1500C2semidirect
6. #9150C2semidirect
7. #9147C2semidirect
8. #9146C2semidirect
9. #9143C2semidirect
10. #8022C2semidirect
11. #8014C2semidirect
20. #2562C4semidirect
21. #2540C4semidirect
22. #2538C4semidirect
23. #2537C4semidirect
61. C2×#125C2×C4semidirect
62. C2×#124C2×C4semidirect
64. C2×C22⋊Q8C2×C4semidirect
67. C2×#121C2×C4semidirect
70. C2×C4×Q8C2×C4semidirect
80. C2×C4×D4C2×C4semidirect
120. #384C2×C4semidirect
124. #380C2×C4semidirect
125. #380C2×C4semidirect
131. C4×C4⋊C4C2×C4semidirect
132. C4×C4⋊C4D4semidirect
137. #376D4semidirect

Maximal subgroups

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

subgroupindex
#352362
C2×#22752
#217882
C2×#15092
C2×#15002
#91502
#91472
#91462
#91432
#80222
#80142