Finite Group Databasebeta

← C2×#8555 #188073

Subgroups — C2×#8555 #188073

Order 512.

3950 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#1821C212yesmaximal, complemented
3C2×#1463C212yesmaximal, complemented
4C2×#1462C212yesmaximal
5#8555C218yesmaximal, direct product factor
6C24×Q8C411yescharacteristic
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⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C411yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C2×#407C2214yes—
22#1821—24no—
23#1821C2218yescomplemented
24C2×#343—24no—
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—22no—
37C23×Q8—22no—
38C23×Q8—48no—
39C23×Q8C2×C414yes—
40C24×C4—22no—
41C24×C4—22no—
42C24×C4—22no—
43C24×C4—22no—
44C24×C4—22no—
45C24×C4—22no—
46C24×C4—48no—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C414yes—
49C24×C4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—24no—
55C2×C22⋊Q8—232no—
56C2×C22⋊Q8—232no—
57C2×C22⋊Q8—232no—
58C2×C22⋊Q8—232no—
59C2×C22⋊Q8—216no—
60C2×C22⋊Q8—216no—
61C2×C22⋊Q8—216no—
62C2×C22⋊Q8—216no—
63C2×C22⋊Q8—216no—
64C2×C22⋊Q8—216no—
65C22×C4⋊C4—22no—
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⋊C4C2×C414yes—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C412yes—
82C22×C4⋊C4D414yes—
83C22×C4⋊C4D414yes—
84C22×C4⋊C4D414yes—
85C22×C4⋊C4D414yes—
86C22×C4⋊C4C2312yes—
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—
103#407—24no—
104#407—24no—
105#407—24no—
106#407C2318yes—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—24no—
109#343—216no—
110#324—216no—
111#323—216no—
112C25—28no—
113C25—21no—
114C25—21no—
115C25—21no—
116C25—21no—
117C25—44no—
118C25—44no—
119C25C22⋊C412yes—
120C25C22⋊C412yes—
121C25C22×C411yescharacteristic
122C25C22×C412yes—
123C22×Q8—28no—
124C22×Q8—28no—
125C22×Q8—28no—
126C22×Q8—28no—
127C22×Q8—28no—
128C22×Q8—28no—
129C22×Q8—28no—
130C22×Q8—416no—
131C22×Q8—416no—
132C22×Q8—416no—
133C22×Q8—416no—
134C22×Q8—416no—
135C22×Q8—48no—
136C22×Q8—416no—
137C22×Q8—48no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—28no—
143C23×C4—28no—
144C23×C4—28no—
145C23×C4—28no—
146C23×C4—28no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
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—28no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—44no—
167C23×C4—416no—
168C23×C4—416no—
169C23×C4—416no—
170C23×C4—416no—
171C23×C4—48no—
172C23×C4—416no—
173C23×C4—416no—
174C23×C4—48no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C22⋊C412yes—
180C23×C4C22⋊C412yes—
181C23×C4C22⋊C412yes—
182C23×C4C22⋊C412yes—
183C23×C4C22⋊C412yes—
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×C411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2411yescharacteristic
202C2×M16—28no—
203C2×M16—44no—
204C2×M16—416no—
205C22×C8—44no—
206C22⋊Q8—2128no—
207C22⋊Q8—2128no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
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—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4C22⋊C418yes—
232C2×C4⋊C4C22⋊C418yes—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×D418yes—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4C22⋊C418yes—
251C2×C22⋊C4C2×D418yes—
252C22⋊C8—28no—
253C22⋊C8—28no—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—28no—
260C24—28no—
261C24—28no—
262C24—28no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—28no—
270C24—24no—
271C24—28no—
272C24—44no—
273C24—44no—
274C24—44no—
275C24—44no—
276C24—44no—
277C24—44no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24—464no—
282C24—44no—
283C24—44no—
284C24—44no—
285C24—44no—
286C24—44no—
287C24#10118yes—
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C24C2×C22⋊C411yescharacteristic
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C414yes—
295C24C23×C411yescharacteristic
296C2×Q8—232no—
297C2×Q8—432no—
298C2×Q8—432no—
299C2×Q8—432no—
300C2×Q8—432no—
301C2×Q8—416no—
302C2×Q8—432no—
303C2×Q8—432no—
304C2×Q8—416no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—232no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
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—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—432no—
352C22×C4—432no—
353C22×C4—432no—
354C22×C4—432no—
355C22×C4—416no—
356C22×C4—432no—
357C22×C4—432no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—416no—
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#10018yes—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C414yes—
391C22×C4C2×C22⋊C414yes—
392C22×C4C2×C22⋊C414yes—
393C22×C4C2×C22⋊C414yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4#12014yes—
397C22×C4#12014yes—
398C22×C4#12014yes—
399C22×C4#12014yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403M16—416no—
404M16—88no—
405C2×C8—48no—
406C2×C8—88no—
407C4⋊C4—232no—
408C4⋊C4—216no—
409C4⋊C4—216no—
410C4⋊C4—216no—
411C4⋊C4—216no—
412C22⋊C4—232no—
413C22⋊C4—216no—
414C22⋊C4—216no—
415C23—232no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—28no—
435C23—28no—
436C23—28no—
437C23—28no—
438C23—24no—
439C23—24no—
440C23—28no—
441C23—28no—
442C23—44no—
443C23—416no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—432no—
449C23—432no—
450C23—432no—
451C23—432no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—432no—
456C23—432no—
457C23—416no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23#37911yescharacteristic
465C23#37911yescharacteristic
466C23#37911yescharacteristic
467C23#37911yescharacteristic
468C23#41014yes—
469C23C2×#10014yes—
470C23C2×#10114yes—
471C23C2×#10118yes—
472C23C22×C22⋊C411yescharacteristic
473C23C22×C22⋊C411yescharacteristic
474C23C22×C22⋊C411yescharacteristic
475Q8—464no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
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—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—464no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
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#37914yes—
528C2×C4C2×#10018yes—
529C2×C4C2×#12014yes—
530C8—88no—
531C22—28no—
532C22—28no—
533C22—28no—
534C22—28no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—28no—
540C22—28no—
541C22—28no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—28no—
551C22—48no—
552C22—416no—
553C22—416no—
554C22—48no—
555C22—416no—
556C22—416no—
557C22—416no—
558C22—44no—
559C22—416no—
560C22—44no—
561C22#178212yes—
562C22#178212yes—
563C22#178312yes—
564C22#178312yes—
565C22#178412yes—
566C22#178412yes—
567C22C2×#37911yescharacteristic
568C22C2×#41012yes—
569C22C22×#10012yes—
570C22C22×#10112yes—
571C4—24no—
572C4—24no—
573C4—28no—
574C4—44no—
575C4—44no—
576C4—44no—
577C4—44no—
578C2—22no—
579C2—22no—
580C2—22no—
581C2—22no—
582C2—22no—
583C2—22no—
584C2—48no—
585C2#855514yesdirect product factor
586C2C2×#178211yescharacteristic
587C2C2×#178311yescharacteristic
588C2C2×#178411yescharacteristic
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×#1821C2semidirect
3. C2×#1463C2semidirect
5. #8555C2direct (N × H)
23. #1821C22semidirect
26. #1463C22semidirect
585. C2#8555direct (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×#18212
C2×#14632
C2×#14622
#85552