Finite Group Databasebeta

← #192482

Subgroups — #192482

Order 512.

1023 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60234C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#21934C211yesmaximal, characteristic, complemented
4#21774C211yesmaximal, characteristic
5C2×#1501C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8051C211yesmaximal, characteristic, complemented
9#7925C211yesmaximal, characteristic, complemented
10C2×#525C2211yescharacteristic
11#2978—22no—
12#2976C2214yes—
13#2975—21no—
14#2974—22no—
15#2973—21no—
16#2933—22no—
17#2930—22no—
18#2924—21no—
19#2880—21no—
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2280C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1897—22no—
27#1871C2214yes—
28#1855—22no—
29#1855C2212yescomplemented
30#1855C2212yescomplemented
31#1851—22no—
32#1851C2214yescomplemented
33#1826—21no—
34#1810—21no—
35#1810—21no—
36#1797—22no—
37#1789—21no—
38#1501C412yescomplemented
39#1501C412yescomplemented
40#1501C2212yes—
41#1501C2212yes—
42C2×#328C411yescharacteristic, complemented
43C2×#328C411yescharacteristic, complemented
44C2×#327C411yescharacteristic, complemented
45C2×#327C411yescharacteristic, complemented
46#1297C2214yescomplemented
47#1282C2212yescomplemented
48#1282C2212yescomplemented
49C22×#41D411yescharacteristic
50C24×C4C2311yescharacteristic
51C2×#131—22no—
52C22×M16D412yes—
53C22×M16D412yes—
54C22×M16C2312yes—
55C23×C8—21no—
56#525—21no—
57#525—21no—
58#525—21no—
59#525—21no—
60#525C2×C412yes—
61#525D412yes—
62#525Q812yes—
63#525C2312yes—
64C2×#123—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8Q811yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×#121Q811yescharacteristic
69C2×#120—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C22×C22⋊C4—21no—
75C22×C42—21no—
76#436—24no—
77#436—24no—
78#436—24no—
79#435—24no—
80#435—24no—
81#435—24no—
82#435—24no—
83C8×D4—24no—
84#433—24no—
85#432—22no—
86#431—22no—
87C2×#108—22no—
88C2×#108—22no—
89#424—22no—
90C2×C4⋊C8—24no—
91C2×#104—24no—
92#408—22no—
93#408—24no—
94#407—22no—
95#407—21no—
96#407—21no—
97#407—21no—
98#407—21no—
99#407—21no—
100#407—21no—
101#407—21no—
102#407—21no—
103#407—21no—
104#407—21no—
105#407D411yescharacteristic
106#407D411yescharacteristic
107#407C2312yes—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8C2×C411yescharacteristic
117C2×C22⋊C8C2×C411yescharacteristic
118C2×C22⋊C8C2×C411yescharacteristic
119C2×C22⋊C8C2×C411yescharacteristic
120C2×C22⋊C8D412yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2311yescharacteristic
123C2×C22⋊C8C2311yescharacteristic
124#405—22no—
125#381—21no—
126#380—21no—
127#380—21no—
128#379—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4C2×C412yes—
133C4×C22⋊C4C2312yes—
134C4×C22⋊C4C2312yes—
135C2×#95C2×C411yescharacteristic
136C2×#95C2×C411yescharacteristic
137#341—22no—
138#341—22no—
139#339—24no—
140#336—24no—
141#328C2×C412yes—
142#328C2×C412yes—
143#328C2×C412yes—
144#328C2×C412yes—
145#327C2×C412yes—
146#327C2×C412yes—
147#327C2×C412yes—
148#327C2×C412yes—
149C25#4111yescharacteristic
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—82no—
153C2×#41C4⋊C412yes—
154C2×#41C2×D412yes—
155C22×Q8C4⋊C411yescharacteristic
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C4⋊C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—82no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×Q811yescharacteristic
173C23×C4C2×Q811yescharacteristic
174C23×C4#4111yescharacteristic
175C23×C4#4111yescharacteristic
176C23×C4#4111yescharacteristic
177C23×C4C2411yescharacteristic
178#131—84no—
179C2×M16—22no—
180C2×M16—24no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—82no—
186C2×M16—82no—
187C2×M16—82no—
188C2×M16C2×D412yes—
189C2×M16C2×D412yes—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—24no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—82no—
200C22×C8C2×D414yes—
201#123—22no—
202#123—22no—
203#123—22no—
204#123—22no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8C4⋊C412yes—
210C22⋊Q8C4⋊C412yes—
211C22⋊Q8C22×C412yes—
212C22⋊Q8C2×Q812yes—
213#121—21no—
214#121—21no—
215#121—21no—
216#121—21no—
217#121C4⋊C412yes—
218#121C4⋊C412yes—
219#121C22×C412yes—
220#121C2×Q812yes—
221#120—22no—
222#120—22no—
223#120—22no—
224#120—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233#117—21no—
234#117—21no—
235#117—82no—
236#117C4⋊C412yes—
237#117C2×D412yes—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4C4⋊C411yescharacteristic
242C2×C4⋊C4C4⋊C411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—82no—
259#108—82no—
260#108—82no—
261C4⋊C8—24no—
262C4⋊C8—24no—
263C4⋊C8—84no—
264#104—84no—
265C22⋊C8—24no—
266C22⋊C8—24no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—22no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—41no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286C22⋊C8—41no—
287C22⋊C8C22×C411yescharacteristic
288C22⋊C8C22×C411yescharacteristic
289C22⋊C8C22×C411yescharacteristic
290C22⋊C8C22×C411yescharacteristic
291#97—24no—
292C4×C8—24no—
293#95—21no—
294#95—21no—
295#95—22no—
296#95—41no—
297#95C22×C412yes—
298#95C22×C412yes—
299C24—21no—
300C24—42no—
301C24—41no—
302C24—41no—
303C24—81no—
304C24C4×D411yescharacteristic
305C24C4×D411yescharacteristic
306C24C22⋊Q811yescharacteristic
307C24C22⋊Q811yescharacteristic
308C24#12311yescharacteristic
309C24#12311yescharacteristic
310C24C2×#4111yescharacteristic
311#41—44no—
312#41—44no—
313#41—82no—
314#41—82no—
315C2×Q8—21no—
316C2×Q8—81no—
317C2×Q8C2×C4⋊C412yes—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—81no—
326C2×D4—82no—
327C2×D4—81no—
328C2×D4—81no—
329C2×D4—82no—
330C2×D4C2×C4⋊C412yes—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—82no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4C2×C4⋊C411yescharacteristic
361C22×C4C2×C4⋊C411yescharacteristic
362C22×C4C2×C4⋊C411yescharacteristic
363C22×C4C2×C4⋊C411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12011yescharacteristic
372C22×C4#12011yescharacteristic
373C22×C4#12011yescharacteristic
374C22×C4C22⋊Q811yescharacteristic
375C22×C4C22⋊Q811yescharacteristic
376C22×C4C22⋊Q811yescharacteristic
377C22×C4C22⋊Q811yescharacteristic
378C22×C4C22⋊Q811yescharacteristic
379C22×C4C22⋊Q811yescharacteristic
380C22×C4#12311yescharacteristic
381C22×C4#12311yescharacteristic
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q811yescharacteristic
387C22×C4C2×#4111yescharacteristic
388M16—44no—
389M16—42no—
390M16—42no—
391M16—44no—
392M16—42no—
393M16—42no—
394M16—82no—
395M16—82no—
396M16—84no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—42no—
410C2×C8—81no—
411C2×C8—81no—
412C2×C8—81no—
413C2×C8—81no—
414C2×C8—84no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—82no—
422C4⋊C4C2×C4⋊C412yes—
423C4⋊C4C2×C4⋊C412yes—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C42—22no—
441C42—22no—
442C42—44no—
443C42—44no—
444C42—82no—
445C23—22no—
446C23—22no—
447C23—21no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—81no—
472C23—81no—
473C23—84no—
474C23—84no—
475C23#38511yescharacteristic
476C23#38511yescharacteristic
477C23#38511yescharacteristic
478C23#38511yescharacteristic
479C23C2×C4×D411yescharacteristic
480C23C2×C22⋊Q811yescharacteristic
481C23C2×#12311yescharacteristic
482Q8—42no—
483Q8—42no—
484D4—42no—
485D4—42no—
486D4—44no—
487D4—42no—
488D4—42no—
489D4—82no—
490D4—82no—
491D4—82no—
492D4—82no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#38511yescharacteristic
541C2×C4#38511yescharacteristic
542C2×C4#38511yescharacteristic
543C2×C4#38511yescharacteristic
544C2×C4C22×C4⋊C411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×#12011yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—42no—
553C8—42no—
554C8—44no—
555C8—82no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—162no—
589C22#186112yes—
590C22C2×#38511yescharacteristic
591C22#295711yescharacteristic
592C22#295711yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4—162no—
606C4#302612yes—
607C2—21no—
608C2—21no—
609C2—22no—
610C2—41no—
611C2—41no—
612C2—41no—
613C2—41no—
614C2—41no—
615C2—82no—
616C2C2×#186111yescharacteristic
617C2#2088611yescharacteristic
618C2#2088911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #21934C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8051C2semidirect
9. #7925C2semidirect
29. #1855C22semidirect
30. #1855C22semidirect
32. #1851C22semidirect
38. #1501C4semidirect
39. #1501C4semidirect
42. C2×#328C4semidirect
43. C2×#328C4semidirect
44. C2×#327C4semidirect
45. C2×#327C4semidirect
46. #1297C22semidirect
47. #1282C22semidirect
48. #1282C22semidirect

Maximal subgroups

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

subgroupindex
#602342
#219482
#219342
#217742
C2×#15012
#88752
#88742
#80512
#79252