Finite Group Databasebeta

← #186630

Subgroups — #186630

Order 512.

2422 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic, complemented
3#20732C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic
7C2×#1479C211yesmaximal, characteristic, complemented
8#8449C218yesmaximal, complemented
9C2×#556—24no—
10C2×#556—22no—
11C2×#556—24no—
12C2×#556—22no—
13C2×#556—24no—
14C2×#556—22no—
15C2×#556C418yescomplemented
16C2×#556C2214yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×#126—22no—
19C22×#124—21no—
20C22×C22⋊Q8—21no—
21C22×C4×Q8—21no—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×#117—21no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419—21no—
29#2285C2211yescharacteristic
30#1990C2214yescomplemented
31#1987—22no—
32#1824—22no—
33#1823—21no—
34#1805—22no—
35#1803C2214yescomplemented
36#1802C2214yes—
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1549—24no—
41C2×#332C2214yescomplemented
42#1486—24no—
43#1479—24no—
44C2×Q8⋊C8C2214yescomplemented
45C2×D4⋊C8C2214yes—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C22×M16—21no—
51#556—216no—
52#556—216no—
53#556—216no—
54#556—216no—
55#556—28no—
56#556—216no—
57#556—28no—
58#556—216no—
59#556—28no—
60#556C2×C4116yes—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8C2×C414yes—
66C2×C4⋊Q8D414yes—
67C2×C4⋊Q8D412yes—
68C2×C4⋊Q8C2312yes—
69C2×#126—28no—
70C2×#126—216no—
71C2×#126—216no—
72C2×#126—28no—
73C2×#124—28no—
74C2×#124—24no—
75C2×#124—28no—
76C2×#124—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—28no—
79C2×C22⋊Q8—28no—
80C2×C22⋊Q8—24no—
81C2×C4×Q8—24no—
82C2×C4×Q8—28no—
83C2×C4×Q8—28no—
84C2×C4×Q8—24no—
85C2×C4×Q8—22no—
86C2×C4×Q8—21no—
87C2×C4×Q8—22no—
88C2×C4×Q8—21no—
89C2×C4×Q8C2×C414yes—
90C2×C4×Q8D414yes—
91C2×C4×Q8D412yes—
92C2×C4×Q8C2312yes—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4C2×C414yes—
98C2×C4×D4D414yes—
99C2×C4×D4D412yes—
100C2×C4×D4C2312yes—
101C2×#117—24no—
102C2×#117—28no—
103C2×#117—28no—
104C2×#117—24no—
105C2×#117—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D412yes—
111C22×C4⋊C4D411yescharacteristic
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4—21no—
115C22×C42—22no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118#431—22no—
119#431—22no—
120#431—22no—
121#431—22no—
122#423—22no—
123#423—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128#419—44no—
129C2×#103—24no—
130C2×#102—24no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8C2312yes—
135#384C2312yes—
136#382—21no—
137#380—21no—
138#380—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141#340—24no—
142#340—24no—
143#340—28no—
144#332—28no—
145Q8⋊C8—28no—
146D4⋊C8—28no—
147C25—21no—
148C22×Q8—24no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—48no—
159C22×Q8C22⋊C414yes—
160C22×Q8C2×D412yes—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—44no—
164C22×D4C22⋊C414yes—
165C22×D4C2×D412yes—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8C22⋊C418yes—
195C4⋊Q8C22×C414yes—
196C4⋊Q8C2×D414yes—
197#126—232no—
198#126—264no—
199#126—232no—
200#124—216no—
201#124—232no—
202#124—216no—
203C22⋊Q8—216no—
204C22⋊Q8—232no—
205C22⋊Q8—216no—
206C4×Q8—216no—
207C4×Q8—232no—
208C4×Q8—216no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8C22⋊C418yes—
217C4×Q8C22×C414yes—
218C4×Q8C2×D414yes—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4C22⋊C418yes—
227C4×D4C22×C414yes—
228C4×D4C2×D414yes—
229#117—216no—
230#117—232no—
231#117—216no—
232#117—42no—
233#117—42no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C22⋊C414yes—
254C2×C4⋊C4C22⋊C414yes—
255C2×C4⋊C4C22⋊C418yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D414yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—41no—
268C2×C42—28no—
269C2×C42—216no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—41no—
275C2×C42C22×C412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2411yescharacteristic
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—42no—
288C4⋊C8—42no—
289#103—48no—
290#102—48no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C4×C8—22no—
294C4×C8—22no—
295C4×C8—42no—
296#95—41no—
297#95—41no—
298C24—22no—
299C24—24no—
300C24—44no—
301C24—44no—
302C24—42no—
303C24—42no—
304C24#12011yescharacteristic
305C2×Q8—24no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—28no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—28no—
315C2×Q8—28no—
316C2×Q8—48no—
317C2×Q8—416no—
318C2×Q8—416no—
319C2×Q8—48no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—42no—
323C2×Q8—44no—
324C2×Q8C2×C22⋊C414yes—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—48no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4C2×C22⋊C414yes—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—48no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4#12012yes—
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12012yes—
370C22×C4#12012yes—
371C22×C4#12011yescharacteristic
372C22×C4#13512yes—
373C22×C4#13512yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4#13712yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379M16—84no—
380C2×C8—44no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—44no—
387C4⋊C4—232no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—28no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4C2×C22⋊C414yes—
405C4⋊C4C2×C22⋊C414yes—
406C4⋊C4C2×C22⋊C418yes—
407C22⋊C4—232no—
408C22⋊C4—232no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—28no—
412C22⋊C4—82no—
413C42—232no—
414C42—24no—
415C42—24no—
416C42—28no—
417C42—82no—
418C42C2×C22⋊C412yes—
419C42C2×C22⋊C412yes—
420C42C2×C22⋊C412yes—
421C42#12012yes—
422C42#12012yes—
423C42#12012yes—
424C42C23×C411yescharacteristic
425C42C22×D411yescharacteristic
426C42C22×D411yescharacteristic
427C42C22×D411yescharacteristic
428C23—22no—
429C23—22no—
430C23—24no—
431C23—48no—
432C23—48no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23—48no—
438C23—88no—
439C23—88no—
440C23#37912yes—
441C23#44912yes—
442C23#45212yes—
443C23C2×#12011yescharacteristic
444Q8—24no—
445Q8—24no—
446Q8—416no—
447Q8—48no—
448Q8—42no—
449Q8—42no—
450Q8—48no—
451Q8—48no—
452D4—24no—
453D4—24no—
454D4—48no—
455D4—44no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—416no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—48no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—416no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#37912yes—
486C2×C4#37912yes—
487C2×C4#37912yes—
488C2×C4#41614yes—
489C2×C4#41714yes—
490C2×C4#44914yes—
491C2×C4#44912yes—
492C2×C4#45214yes—
493C2×C4#45212yes—
494C2×C4C22×C22⋊C411yescharacteristic
495C2×C4C22×C22⋊C411yescharacteristic
496C2×C4C22×C22⋊C411yescharacteristic
497C2×C4C2×#12011yescharacteristic
498C2×C4C2×#12011yescharacteristic
499C2×C4C2×#12011yescharacteristic
500C2×C4C2×#13511yescharacteristic
501C2×C4C2×#13511yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4C2×#13711yescharacteristic
504C2×C4#57514yes—
505C8—82no—
506C8—82no—
507C8—82no—
508C22—24no—
509C22—24no—
510C22—22no—
511C22—22no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—816no—
516C22—84no—
517C22—84no—
518C22#178814yes—
519C22C2×#37911yescharacteristic
520C22C2×#44911yescharacteristic
521C22C2×#45211yescharacteristic
522C22#300512yes—
523C22#302012yes—
524C4—22no—
525C4—21no—
526C4—21no—
527C4—22no—
528C4—44no—
529C4—44no—
530C4—48no—
531C4—44no—
532C4—162no—
533C4#178814yes—
534C4C2×#41611yescharacteristic
535C4C2×#41711yescharacteristic
536C4#289612yes—
537C4C2×#44911yescharacteristic
538C4C2×#45211yescharacteristic
539C4#300512yes—
540C2—24no—
541C2—84no—
542C2#1876212yes—
543C2#1876312yes—
544C2C2×#178811yescharacteristic
545C2#2060011yescharacteristic
546C2#2061711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21834C2semidirect
4. #20647C2semidirect
5. C2×#1549C2semidirect
7. C2×#1479C2semidirect
8. #8449C2semidirect
15. C2×#556C4semidirect
30. #1990C22semidirect
35. #1803C22semidirect
41. C2×#332C22semidirect
44. C2×Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
#218342
#207322
#206472
C2×#15492
C2×#14862
C2×#14792
#84492