Finite Group Databasebeta

← C2×#8556 #188075

Subgroups — C2×#8556 #188075

Order 512.

2802 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
1C23×#126C211yesmaximal, characteristic
2C2×#1821C212yesmaximal, complemented
3C2×#1463C212yesmaximal, complemented
4C2×#1462C212yesmaximal
5#8556C218yesmaximal, direct product factor
6C22×#126—24no—
7C22×#126—24no—
8C22×#126—24no—
9C22×#126—24no—
10C22×#126—24no—
11C22×#126—24no—
12C22×#126C414yes—
13C22×#126C2214yes—
14C23×C4⋊C4C411yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C411yescharacteristic
18C23×C22⋊C4C411yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C42C411yescharacteristic
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
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2311yescharacteristic
38C22×M16—24no—
39C2×#126—232no—
40C2×#126—232no—
41C2×#126—232no—
42C2×#126—232no—
43C2×#126—216no—
44C2×#126—216no—
45C2×#126—216no—
46C2×#126—216no—
47C2×#126—216no—
48C2×#126—216no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4D414yes—
67C22×C4⋊C4D414yes—
68C22×C4⋊C4D414yes—
69C22×C4⋊C4D414yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C22⋊C4C2×C414yes—
89C22×C22⋊C4C2×C414yes—
90C22×C22⋊C4C2×C412yes—
91C22×C22⋊C4D414yes—
92C22×C22⋊C4D414yes—
93C22×C22⋊C4C2312yes—
94C22×C42—22no—
95C22×C42—22no—
96C22×C42—22no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42—22no—
100C22×C42C2×C414yes—
101#407—24no—
102#407—24no—
103#407—24no—
104#407C2318yes—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107#343—216no—
108#324—216no—
109#323—216no—
110C25—21no—
111C25—21no—
112C25—21no—
113C25—21no—
114C25—44no—
115C25—44no—
116C25—44no—
117C25C22⋊C412yes—
118C25C22⋊C412yes—
119C25C22×C411yescharacteristic
120C25C22×C412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22⋊C412yes—
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2411yescharacteristic
170C2×M16—28no—
171C2×M16—44no—
172C2×M16—416no—
173C22×C8—44no—
174#126—2128no—
175#126—2128no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C22⋊C418yes—
200C2×C4⋊C4C22⋊C418yes—
201C2×C4⋊C4C2×D418yes—
202C2×C4⋊C4C2×D418yes—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4C22⋊C418yes—
226C2×C22⋊C4C2×D418yes—
227C2×C42—28no—
228C2×C42—28no—
229C2×C42—28no—
230C2×C42—28no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—28no—
234C22⋊C8—28no—
235C22⋊C8—28no—
236C22⋊C8—44no—
237C22⋊C8—44no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—28no—
248C24—44no—
249C24—44no—
250C24—44no—
251C24—44no—
252C24—44no—
253C24—44no—
254C24—44no—
255C24—44no—
256C24—44no—
257C24—44no—
258C24—44no—
259C24—44no—
260C24—44no—
261C24—44no—
262C24—44no—
263C24—44no—
264C24—44no—
265C24—44no—
266C24—44no—
267C24—44no—
268C24—832no—
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C411yescharacteristic
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C414yes—
276C24C23×C411yescharacteristic
277C22×C4—28no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—28no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4C2×C22⋊C414yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C414yes—
353C22×C4C2×C22⋊C414yes—
354C22×C4C2×C22⋊C414yes—
355C22×C4C2×C22⋊C414yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4#12014yes—
359C22×C4#12014yes—
360C22×C4#12014yes—
361C22×C4#12014yes—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365M16—416no—
366M16—88no—
367C2×C8—48no—
368C2×C8—88no—
369C4⋊C4—232no—
370C4⋊C4—216no—
371C4⋊C4—216no—
372C4⋊C4—216no—
373C4⋊C4—216no—
374C22⋊C4—232no—
375C22⋊C4—232no—
376C22⋊C4—216no—
377C22⋊C4—216no—
378C42—232no—
379C23—232no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—28no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—816no—
421C23—816no—
422C23—816no—
423C23—816no—
424C23—816no—
425C23—816no—
426C23—816no—
427C23#37911yescharacteristic
428C23#37911yescharacteristic
429C23#37911yescharacteristic
430C23#37911yescharacteristic
431C23#41014yes—
432C23#41314yes—
433C23#41314yes—
434C23C22×C22⋊C411yescharacteristic
435C23C22×C22⋊C411yescharacteristic
436C23C22×C22⋊C411yescharacteristic
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4#37914yes—
488C2×C4C2×#12014yes—
489C8—88no—
490C22—28no—
491C22—28no—
492C22—28no—
493C22—28no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—28no—
499C22—28no—
500C22—28no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—88no—
513C22—88no—
514C22—88no—
515C22—88no—
516C22—88no—
517C22—88no—
518C22—88no—
519C22#178212yes—
520C22#178212yes—
521C22#178512yes—
522C22#178512yes—
523C22#178512yes—
524C22#178512yes—
525C22C2×#37911yescharacteristic
526C22C2×#41012yes—
527C22C2×#41312yes—
528C22C2×#41312yes—
529C4—24no—
530C4—24no—
531C4—44no—
532C4—44no—
533C4—44no—
534C4—44no—
535C4—44no—
536C2—22no—
537C2—22no—
538C2—22no—
539C2—22no—
540C2—22no—
541C2—22no—
542C2—84no—
543C2#855614yesdirect product factor
544C2C2×#178211yescharacteristic
545C2C2×#178511yescharacteristic
546C2C2×#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×#1821C2semidirect
3. C2×#1463C2semidirect
5. #8556C2direct (N × H)
23. #1821C22semidirect
26. #1463C22semidirect
543. C2#8556direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#1262
C2×#18212
C2×#14632
C2×#14622
#85562