Finite Group Databasebeta

← C2×#8550 #188069

Subgroups — C2×#8550 #188069

Order 512.

4038 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#1784C212yesmaximal, complemented
3C2×#1463C212yesmaximal, complemented
4C2×#1462C212yesmaximal
5#8550C218yesmaximal, direct product factor
6C24×Q8C2211yescharacteristic
7C25×C4C411yescharacteristic
8C22×C22⋊Q8—24no—
9C22×C22⋊Q8—24no—
10C22×C22⋊Q8—24no—
11C22×C22⋊Q8—24no—
12C22×C22⋊Q8—24no—
13C22×C22⋊Q8—24no—
14C22×C22⋊Q8C414yes—
15C22×C22⋊Q8C2214yes—
16C23×C4⋊C4C411yescharacteristic
17C23×C4⋊C4C411yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C411yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C22×#101—24no—
22C2×#407C2214yes—
23#1784—24no—
24#1784C2218yescomplemented
25#1463—24no—
26#1463C2218yescomplemented
27#1462—24no—
28#1462C2218yes—
29C2×#324—24no—
30C2×#323—24no—
31C26C2×C411yescharacteristic
32C23×Q8—22no—
33C23×Q8—22no—
34C23×Q8—22no—
35C23×Q8—22no—
36C23×Q8—48no—
37C23×Q8C2×C412yes—
38C23×Q8D414yes—
39C23×Q8D414yes—
40C23×Q8C2312yes—
41C24×C4—22no—
42C24×C4—22no—
43C24×C4—22no—
44C24×C4—22no—
45C24×C4—22no—
46C24×C4—22no—
47C24×C4—48no—
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C414yes—
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16—24no—
56C2×C22⋊Q8—232no—
57C2×C22⋊Q8—232no—
58C2×C22⋊Q8—232no—
59C2×C22⋊Q8—232no—
60C2×C22⋊Q8—216no—
61C2×C22⋊Q8—216no—
62C2×C22⋊Q8—216no—
63C2×C22⋊Q8—216no—
64C2×C22⋊Q8—216no—
65C2×C22⋊Q8—216no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4C2×C414yes—
83C22×C4⋊C4C2×C414yes—
84C22×C4⋊C4C2×C412yes—
85C22×C4⋊C4D414yes—
86C22×C4⋊C4D414yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—22no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—22no—
98C22×C22⋊C4C2×C414yes—
99C22×C22⋊C4C2×C412yes—
100C22×C22⋊C4D414yes—
101C22×C22⋊C4D414yes—
102C22×C22⋊C4C2312yes—
103C2×#101—216no—
104C2×#101—416no—
105#407—24no—
106#407—24no—
107#407—24no—
108#407C2318yes—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—24no—
111#324—216no—
112#323—216no—
113C25—28no—
114C25—21no—
115C25—21no—
116C25—21no—
117C25—21no—
118C25—44no—
119C25—44no—
120C25C22⋊C412yes—
121C25C22⋊C412yes—
122C25C22×C411yescharacteristic
123C25C22×C412yes—
124C22×Q8—216no—
125C22×Q8—24no—
126C22×Q8—24no—
127C22×Q8—28no—
128C22×Q8—28no—
129C22×Q8—28no—
130C22×Q8—28no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8—48no—
134C22×Q8—48no—
135C22×Q8—416no—
136C22×Q8—48no—
137C22×Q8—416no—
138C22×Q8—416no—
139C22×Q8—416no—
140C22×Q8—416no—
141C22×Q8C22⋊C418yes—
142C22×Q8C2×D418yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—28no—
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—28no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—416no—
175C23×C4—416no—
176C23×C4—416no—
177C23×C4—416no—
178C23×C4—48no—
179C23×C4—416no—
180C23×C4—416no—
181C23×C4—48no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4C22⋊C412yes—
185C23×C4C22⋊C412yes—
186C23×C4C22⋊C412yes—
187C23×C4C22⋊C412yes—
188C23×C4C22⋊C412yes—
189C23×C4C22⋊C412yes—
190C23×C4C22⋊C412yes—
191C23×C4C22⋊C412yes—
192C23×C4C22⋊C412yes—
193C23×C4C22⋊C412yes—
194C23×C4C22×C412yes—
195C23×C4C22×C412yes—
196C23×C4C22×C412yes—
197C23×C4C22×C412yes—
198C23×C4C22×C412yes—
199C23×C4C22×C411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2×D412yes—
205C23×C4C2×D412yes—
206C23×C4C2411yescharacteristic
207C2×M16—28no—
208C2×M16—44no—
209C2×M16—416no—
210C22×C8—44no—
211C22⋊Q8—2128no—
212C22⋊Q8—2128no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4C22⋊C418yes—
236C2×C4⋊C4C2×D418yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4C22⋊C418yes—
253C2×C22⋊C4C2×D418yes—
254#101—432no—
255C22⋊C8—28no—
256C22⋊C8—28no—
257C22⋊C8—44no—
258C22⋊C8—44no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—28no—
263C24—28no—
264C24—28no—
265C24—28no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—28no—
274C24—28no—
275C24—44no—
276C24—44no—
277C24—44no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24—44no—
282C24—44no—
283C24—44no—
284C24—464no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24#10118yescomplemented
291C24C2×C22⋊C411yescharacteristic
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C414yes—
298C24C23×C411yescharacteristic
299C2×Q8—216no—
300C2×Q8—216no—
301C2×Q8—216no—
302C2×Q8—216no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—432no—
306C2×Q8—432no—
307C2×Q8—416no—
308C2×Q8—416no—
309C2×Q8—432no—
310C2×Q8—432no—
311C2×Q8—416no—
312C2×Q8—416no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—24no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—232no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—432no—
366C22×C4—432no—
367C22×C4—432no—
368C22×C4—432no—
369C22×C4—416no—
370C22×C4—432no—
371C22×C4—432no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—416no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4#9918yes—
396C22×C4C2×C22⋊C414yes—
397C22×C4C2×C22⋊C414yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C414yes—
400C22×C4C2×C22⋊C414yes—
401C22×C4C2×C22⋊C414yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C412yes—
404C22×C4#12014yes—
405C22×C4#12014yes—
406C22×C4#12014yes—
407C22×C4#12014yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411M16—416no—
412M16—88no—
413C2×C8—48no—
414C2×C8—88no—
415C4⋊C4—232no—
416C4⋊C4—232no—
417C4⋊C4—216no—
418C4⋊C4—216no—
419C22⋊C4—232no—
420C22⋊C4—216no—
421C22⋊C4—216no—
422C23—232no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—28no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—44no—
450C23—416no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—432no—
456C23—432no—
457C23—432no—
458C23—432no—
459C23—432no—
460C23—416no—
461C23—432no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23#37911yescharacteristic
472C23#37911yescharacteristic
473C23#37911yescharacteristic
474C23#37911yescharacteristic
475C23C2×#9914yes—
476C23C2×#10114yes—
477C23C2×#10118yescomplemented
478C23#41314yes—
479C23C22×C22⋊C411yescharacteristic
480C23C22×C22⋊C411yescharacteristic
481C23C22×C22⋊C411yescharacteristic
482Q8—432no—
483Q8—432no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—464no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4#37914yes—
536C2×C4C2×#9918yes—
537C2×C4C2×#12014yes—
538C8—88no—
539C22—28no—
540C22—28no—
541C22—28no—
542C22—28no—
543C22—28no—
544C22—28no—
545C22—28no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—28no—
559C22—48no—
560C22—416no—
561C22—48no—
562C22—416no—
563C22—416no—
564C22—416no—
565C22—416no—
566C22—416no—
567C22—44no—
568C22—44no—
569C22#178112yes—
570C22#178112yes—
571C22#178412yes—
572C22#178412yes—
573C22#178512yes—
574C22#178512yes—
575C22C2×#37911yescharacteristic
576C22C22×#9912yes—
577C22C22×#10112yes—
578C22C2×#41312yes—
579C4—24no—
580C4—28no—
581C4—24no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—44no—
586C2—22no—
587C2—22no—
588C2—22no—
589C2—22no—
590C2—22no—
591C2—22no—
592C2—48no—
593C2#855014yesdirect product factor
594C2C2×#178111yescharacteristic
595C2C2×#178411yescharacteristic
596C2C2×#178511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1784C2semidirect
3. C2×#1463C2semidirect
5. #8550C2direct (N × H)
24. #1784C22semidirect
26. #1463C22semidirect
290. C24#101semidirect
477. C23C2×#101semidirect
593. C2#8550direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#17842
C2×#14632
C2×#14622
#85502