Finite Group Databasebeta

← C2×#8854 #192059

Subgroups — C2×#8854 #192059

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2315C211yesmaximal, characteristic
2C2×#1853C212yesmaximal, complemented
3C22×#323C211yesmaximal, characteristic
4C2×#1439C211yesmaximal, characteristic
5#8854C218yesmaximal, direct product factor
6C2×#1294C212yesmaximal, complemented
7C23×C4⋊C4C2211yescharacteristic
8C23×C22⋊C4—21no—
9C23×C22⋊C4—21no—
10C23×C22⋊C4C2211yescharacteristic
11C22×#106—22no—
12C22×C22⋊C8C2212yes—
13#2315—24no—
14#2315—22no—
15#2315C2214yes—
16C2×#385—22no—
17C2×#385—22no—
18C2×#380—21no—
19C2×#380C2212yes—
20C22×#95—21no—
21C22×#95C2211yescharacteristic
22#1853—24no—
23#1853C2218yescomplemented
24C2×#340—22no—
25C2×#323C418yescomplemented
26C2×#323C2214yes—
27#1439—22no—
28#1439C2214yes—
29#1294C2218yescomplemented
30C26D411yescharacteristic
31C24×C4—21no—
32C24×C4—21no—
33C24×C4—21no—
34C24×C4—21no—
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—44no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4C2312yes—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—24no—
50C22×C22⋊C4—24no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—24no—
55C22×C22⋊C4—44no—
56C22×C22⋊C4—44no—
57C22×C22⋊C4—44no—
58C22×C22⋊C4—44no—
59C22×C22⋊C4—44no—
60C22×C22⋊C4C2×C414yes—
61C22×C22⋊C4D412yes—
62C22×C22⋊C4Q814yes—
63C22×C22⋊C4C2312yes—
64C2×#106—28no—
65C2×#106—48no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8C2×C418yescomplemented
69C2×C22⋊C8C2314yes—
70#385—28no—
71#385—28no—
72#385—28no—
73#385—28no—
74#380—24no—
75#380—22no—
76#380—22no—
77#380—24no—
78#380D414yes—
79#380C2314yes—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—24no—
83C2×#95—21no—
84C2×#95—24no—
85C2×#95—44no—
86C2×#95C2312yes—
87#340—28no—
88#323C2×C4116yescomplemented
89C25—21no—
90C25—21no—
91C25—22no—
92C25—22no—
93C25—22no—
94C25—42no—
95C25—42no—
96C25—42no—
97C25—42no—
98C25C4⋊C414yes—
99C25SD1614yes—
100C25SD1614yes—
101C25C2×D411yescharacteristic
102C25C2×D412yes—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—41no—
127C23×C4—41no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—82no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4—82no—
137C23×C4C4⋊C414yes—
138C23×C4C22×C412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×Q812yes—
144C23×C4#4114yes—
145C23×C4C2411yescharacteristic
146C22×C8—28no—
147C22×C8—24no—
148C22×C8—44no—
149C22×C8—44no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D414yes—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—48no—
181C2×C22⋊C4—48no—
182C2×C22⋊C4—416no—
183C2×C22⋊C4—432no—
184C2×C22⋊C4—48no—
185C2×C22⋊C4—48no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4C4⋊C418yes—
189C2×C22⋊C4C22×C414yes—
190C2×C22⋊C4C2×Q814yes—
191#106—416no—
192C22⋊C8—28no—
193C22⋊C8—24no—
194C22⋊C8C22×C418yescomplemented
195#95—28no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—48no—
200#95—42no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—24no—
207C24—24no—
208C24—24no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—24no—
213C24—28no—
214C24—24no—
215C24—24no—
216C24—28no—
217C24—28no—
218C24—48no—
219C24—42no—
220C24—42no—
221C24—48no—
222C24—48no—
223C24—48no—
224C24—48no—
225C24—48no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—84no—
233C24—84no—
234C24—84no—
235C24—84no—
236C24—84no—
237C24—84no—
238C24—81no—
239C24—81no—
240C24—81no—
241C24—81no—
242C24#106116yescomplemented
243C24C2×C4⋊C412yes—
244C24C2×C4⋊C414yes—
245C24#12011yescharacteristic
246C24#12011yescharacteristic
247C24#12312yes—
248C24C2×SD1612yes—
249C24C2×SD1612yes—
250C24C2×SD1614yes—
251C24C2×SD1614yes—
252C24AGL(1,ℤ/8)12yes—
253C24#13712yes—
254C24C22×D411yescharacteristic
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—44no—
272C22×C4—48no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—48no—
283C22×C4—42no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—48no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—84no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4—82no—
319C22×C4—816no—
320C22×C4—816no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—84no—
325C22×C4C2×C4⋊C414yes—
326C22×C4C2×C4⋊C412yes—
327C22×C4C4×D414yes—
328C22×C4C4×D414yes—
329C22×C4#12012yes—
330C22×C4C22⋊Q814yes—
331C22×C4C22⋊Q814yes—
332C22×C4#12312yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C22×Q811yescharacteristic
336C22×C4C2×#4112yes—
337C2×C8—28no—
338C2×C8—44no—
339C2×C8—416no—
340C2×C8—44no—
341C4⋊C4—28no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C22⋊C4—28no—
351C22⋊C4—28no—
352C22⋊C4—216no—
353C22⋊C4—416no—
354C22⋊C4—432no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—48no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—216no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—216no—
385C23—24no—
386C23—44no—
387C23—432no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—48no—
394C23—48no—
395C23—48no—
396C23—48no—
397C23—48no—
398C23—48no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—82no—
404C23—84no—
405C23—816no—
406C23—816no—
407C23—816no—
408C23—82no—
409C23—84no—
410C23—82no—
411C23—84no—
412C23—82no—
413C23—84no—
414C23—82no—
415C23—84no—
416C23—82no—
417C23—84no—
418C23#38512yes—
419C23#38512yes—
420C23C2×#10614yes—
421C23C2×#10618yescomplemented
422C23#42814yes—
423C23#44812yes—
424C23#44812yes—
425C23#45012yes—
426C23#45012yes—
427C23#45312yes—
428C23#45512yes—
429C23B(4,𝔽2)12yes—
430C23#45812yes—
431C23#48114yes—
432C23#48314yes—
433C23C22×C4⋊C411yescharacteristic
434C23C2×#12011yescharacteristic
435C23C2×#12311yescharacteristic
436C23C22×SD1611yescharacteristic
437C23C22×SD1611yescharacteristic
438C23C2×AGL(1,ℤ/8)11yescharacteristic
439C23C2×#13711yescharacteristic
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—48no—
455C2×C4—42no—
456C2×C4—42no—
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—82no—
471C2×C4—84no—
472C2×C4—816no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—88no—
479C2×C4—816no—
480C2×C4—816no—
481C2×C4—82no—
482C2×C4—88no—
483C2×C4—88no—
484C2×C4—88no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#38514yes—
488C2×C4C2×C4×D412yes—
489C2×C4C2×C22⋊Q812yes—
490C8—48no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—28no—
495C22—28no—
496C22—28no—
497C22—28no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—28no—
514C22—48no—
515C22—44no—
516C22—432no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—84no—
523C22—84no—
524C22—816no—
525C22—84no—
526C22#159614yes—
527C22#160114yes—
528C22#160814yes—
529C22#161314yes—
530C22#185314yes—
531C22#185314yes—
532C22#185614yes—
533C22#186714yes—
534C22C2×#38511yescharacteristic
535C22C22×#10611yescharacteristic
536C22C2×#42811yescharacteristic
537C22C2×#45011yescharacteristic
538C22C2×#44811yescharacteristic
539C22C2×#45311yescharacteristic
540C22C2×#45511yescharacteristic
541C22C2×B(4,𝔽2)11yescharacteristic
542C22C2×#45811yescharacteristic
543C22C2×#48311yescharacteristic
544C22C2×#48111yescharacteristic
545C4—22no—
546C4—22no—
547C4—24no—
548C4—44no—
549C4—84no—
550C4—84no—
551C4—84no—
552C4—84no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—24no—
557C2—21no—
558C2—48no—
559C2#885418yesdirect product factor
560C2C2×#160111yescharacteristic
561C2C2×#159611yescharacteristic
562C2C2×#161311yescharacteristic
563C2C2×#160811yescharacteristic
564C2C2×#185311yescharacteristic
565C2C2×#185611yescharacteristic
566C2C2×#186711yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1853C2semidirect
5. #8854C2direct (N × H)
6. C2×#1294C2semidirect
23. #1853C22semidirect
25. C2×#323C4semidirect
29. #1294C22semidirect
68. C2×C22⋊C8C2×C4semidirect
88. #323C2×C4semidirect
194. C22⋊C8C22×C4semidirect
242. C24#106semidirect
421. C23C2×#106semidirect
559. C2#8854direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23152
C2×#18532
C22×#3232
C2×#14392
#88542
C2×#12942