Finite Group Databasebeta

← #192480

Subgroups — #192480

Order 512.

1013 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60234C211yesmaximal, characteristic
2#21949C211yesmaximal, characteristic, complemented
3#21935C211yesmaximal, characteristic, complemented
4#21773C211yesmaximal, characteristic
5#17559C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8051C211yesmaximal, characteristic, complemented
9#7929C211yesmaximal, 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
23C2×#407C2211yescharacteristic
24#2280C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1896—22no—
27#1871C2214yes—
28#1855—22no—
29#1855C2212yescomplemented
30#1855C2212yescomplemented
31#1851—22no—
32#1851C2214yescomplemented
33#1826—21no—
34#1811—21no—
35#1811—21no—
36#1798—22no—
37#1789—21no—
38#1513C414yescomplemented
39#1511C412yescomplemented
40#1510C412yescomplemented
41#1501C2212yes—
42#1501C2212yes—
43#1297C2214yescomplemented
44#1282C2212yescomplemented
45#1282C2212yescomplemented
46C22×#41D411yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#131—22no—
49C22×M16—21no—
50C22×M16D412yes—
51C22×M16D412yes—
52C22×M16C2312yes—
53#525—21no—
54#525—21no—
55#525—21no—
56#525—21no—
57#525C2×C412yes—
58#525D412yes—
59#525Q812yes—
60#525C2312yes—
61C2×#123—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8Q811yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×#121Q811yescharacteristic
66C2×#120—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C42—21no—
73#436—24no—
74#436—24no—
75#436—24no—
76#435—24no—
77#435—24no—
78#435—24no—
79#435—24no—
80C8×D4—24no—
81#433—24no—
82#432—22no—
83#431—22no—
84C2×#108—22no—
85C2×#108—22no—
86#424—22no—
87C2×C4⋊C8—24no—
88C2×#104—24no—
89#408—22no—
90#408—24no—
91#407—22no—
92#407—21no—
93#407—21no—
94#407—21no—
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#407C2×C412yes—
105#407C2×C412yes—
106#407D411yescharacteristic
107#407D411yescharacteristic
108#407C2312yes—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8D412yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2311yescharacteristic
118C2×C22⋊C8C2311yescharacteristic
119#405—22no—
120#381C2×C412yes—
121#380—21no—
122#380C2×C412yes—
123#379—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4C2312yes—
129C4×C22⋊C4C2312yes—
130C2×#95—21no—
131#341—22no—
132#341—22no—
133#339—24no—
134#336—24no—
135#328C2×C414yescomplemented
136#328C2×C414yescomplemented
137#327C2×C414yescomplemented
138#327C2×C414yescomplemented
139C25#4111yescharacteristic
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—82no—
143C2×#41C4⋊C412yes—
144C2×#41C2×D412yes—
145C22×Q8C4⋊C411yescharacteristic
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C4⋊C411yescharacteristic
150C23×C4—21no—
151C23×C4—41no—
152C23×C4—82no—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×Q811yescharacteristic
162C23×C4C2×Q811yescharacteristic
163C23×C4#4111yescharacteristic
164C23×C4#4111yescharacteristic
165C23×C4#4111yescharacteristic
166C23×C4C2411yescharacteristic
167#131—84no—
168C2×M16—22no—
169C2×M16—24no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—82no—
177C2×M16—82no—
178C2×M16—82no—
179C2×M16—82no—
180C2×M16C2×D412yes—
181C2×M16C2×D412yes—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—24no—
186C22×C8—22no—
187C22×C8C2×D414yes—
188#123—22no—
189#123—22no—
190#123—22no—
191#123—22no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8C4⋊C412yes—
197C22⋊Q8C4⋊C412yes—
198C22⋊Q8C22×C412yes—
199C22⋊Q8C2×Q812yes—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204#121C4⋊C412yes—
205#121C4⋊C412yes—
206#121C22×C412yes—
207#121C2×Q812yes—
208#120—22no—
209#120—22no—
210#120—22no—
211#120—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220#117—21no—
221#117—21no—
222#117—82no—
223#117C4⋊C412yes—
224#117C2×D412yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4C4⋊C411yescharacteristic
229C2×C4⋊C4C4⋊C411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—41no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—41no—
245C2×C42—82no—
246#108—82no—
247#108—82no—
248C4⋊C8—24no—
249C4⋊C8—24no—
250C4⋊C8—84no—
251#104—84no—
252C22⋊C8—24no—
253C22⋊C8—24no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—21no—
257C22⋊C8—22no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—41no—
269C22⋊C8—41no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272C22⋊C8C22×C412yes—
273C22⋊C8C22×C412yes—
274#97—24no—
275C4×C8—24no—
276#95—22no—
277#95—22no—
278#95—41no—
279#95C22×C414yes—
280C24—21no—
281C24—42no—
282C24—81no—
283C24—81no—
284C24C4×D411yescharacteristic
285C24C4×D411yescharacteristic
286C24C22⋊Q811yescharacteristic
287C24C22⋊Q811yescharacteristic
288C24#12311yescharacteristic
289C24#12311yescharacteristic
290C24C2×#4111yescharacteristic
291#41—44no—
292#41—44no—
293#41—82no—
294#41—82no—
295C2×Q8—21no—
296C2×Q8—81no—
297C2×Q8C2×C4⋊C412yes—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—81no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4C2×C4⋊C412yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—82no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4C2×C4⋊C411yescharacteristic
336C22×C4C2×C4⋊C411yescharacteristic
337C22×C4C2×C4⋊C411yescharacteristic
338C22×C4C2×C4⋊C411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4C22⋊Q811yescharacteristic
350C22×C4C22⋊Q811yescharacteristic
351C22×C4C22⋊Q811yescharacteristic
352C22×C4C22⋊Q811yescharacteristic
353C22×C4C22⋊Q811yescharacteristic
354C22×C4C22⋊Q811yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4#12311yescharacteristic
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4111yescharacteristic
363M16—44no—
364M16—42no—
365M16—42no—
366M16—44no—
367M16—42no—
368M16—42no—
369M16—82no—
370M16—82no—
371M16—82no—
372M16—82no—
373M16—84no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—22no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—44no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—42no—
385C2×C8—84no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—82no—
393C4⋊C4C2×C4⋊C412yes—
394C4⋊C4C2×C4⋊C412yes—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—24no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C42—22no—
412C42—22no—
413C42—44no—
414C42—44no—
415C42—82no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—81no—
440C23—81no—
441C23—88no—
442C23#38511yescharacteristic
443C23#38511yescharacteristic
444C23#38511yescharacteristic
445C23#38511yescharacteristic
446C23C2×C4×D411yescharacteristic
447C23C2×C22⋊Q811yescharacteristic
448C23C2×#12311yescharacteristic
449Q8—42no—
450Q8—42no—
451D4—42no—
452D4—42no—
453D4—44no—
454D4—44no—
455D4—82no—
456D4—82no—
457D4—82no—
458D4—82no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—88no—
502C2×C4#38511yescharacteristic
503C2×C4#38511yescharacteristic
504C2×C4#38511yescharacteristic
505C2×C4#38511yescharacteristic
506C2×C4C22×C4⋊C411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×#12011yescharacteristic
509C2×C4C2×C22⋊Q811yescharacteristic
510C8—42no—
511C8—42no—
512C8—42no—
513C8—42no—
514C8—44no—
515C8—82no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—162no—
547C22C2×#38511yescharacteristic
548C22#295612yes—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—84no—
560C4—84no—
561C4#302612yes—
562C2—22no—
563C2—41no—
564C2—41no—
565C2—41no—
566C2—41no—
567C2—41no—
568C2—41no—
569C2—82no—
570C2#2086711yescharacteristic
571C2#2088511yescharacteristic
572C2#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. #21949C2semidirect
3. #21935C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8051C2semidirect
9. #7929C2semidirect
29. #1855C22semidirect
30. #1855C22semidirect
32. #1851C22semidirect
38. #1513C4semidirect
39. #1511C4semidirect
40. #1510C4semidirect
43. #1297C22semidirect
44. #1282C22semidirect
45. #1282C22semidirect
135. #328C2×C4semidirect
136. #328C2×C4semidirect
137. #327C2×C4semidirect
138. #327C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602342
#219492
#219352
#217732
#175592
#88752
#88742
#80512
#79292