Finite Group Databasebeta

← C2×#8868 #192302

Subgroups — C2×#8868 #192302

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5C2×#1444C211yesmaximal, characteristic
6#8868C218yesmaximal, direct product factor
7C2×#1294C211yesmaximal, characteristic
8C2×#1280C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C23×C4⋊C4—21no—
11C23×C4⋊C4C2211yescharacteristic
12C22×#107—21no—
13C22×#103—21no—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2329—22no—
17#2329—22no—
18#2329—22no—
19#2329C2214yes—
20C2×#391—21no—
21C2×#389—21no—
22C2×#387—21no—
23C2×#387—21no—
24C2×#385—21no—
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4—21no—
27C22×#95—21no—
28C22×#95C2211yescharacteristic
29C22×#95C2211yescharacteristic
30#1872—22no—
31#1872C2214yes—
32#1852—22no—
33#1852C2214yes—
34C2×#340—21no—
35C2×#340—21no—
36C2×#328C418yescomplemented
37C2×#328C2214yes—
38#1444—22no—
39#1444C2214yes—
40#1294C2214yes—
41#1280C2214yes—
42C23×Q8D411yescharacteristic
43C24×C4—21no—
44C24×C4—21no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—21no—
49C23×C8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8C2×C414yes—
55C2×C22⋊Q8D412yes—
56C2×C22⋊Q8Q814yes—
57C2×C22⋊Q8C2312yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—44no—
68C22×C4⋊C4—44no—
69C22×C4⋊C4—44no—
70C22×C4⋊C4—44no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—21no—
78C22×C42—21no—
79C2×#107—24no—
80C2×#107—44no—
81C2×#103—24no—
82C2×#103—44no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8C2×C414yes—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91#391—24no—
92#391—24no—
93#389—24no—
94#389—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#385—24no—
100#385—24no—
101#380—21no—
102#380—21no—
103#380D412yes—
104#380C2312yes—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—24no—
119C2×#95—44no—
120C2×#95C2×C414yes—
121C2×#95D412yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124#340—24no—
125#340—24no—
126#328C2×C4116yescomplemented
127C25C2×D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8C4⋊C414yescomplemented
133C22×Q8C2×D412yes—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—41no—
165C23×C4—41no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4Q1614yes—
169C23×C4C22×C412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×Q812yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C22×C8—24no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8C4⋊C418yescomplemented
197C22⋊Q8C22×C414yes—
198C22⋊Q8C2×Q814yes—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—432no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C4⋊C414yescomplemented
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—82no—
244C2×C42—82no—
245#107—48no—
246#103—48no—
247C22⋊C8—24no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—42no—
252C22⋊C8C22×C414yes—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—48no—
266#95C22×C414yes—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24C4×D412yes—
272C24#12011yescharacteristic
273C24#12011yescharacteristic
274C24C22⋊Q812yes—
275C24C2×Q1612yes—
276C24#13512yes—
277C24AGL(1,ℤ/8)12yes—
278C24#13712yes—
279C24C22×D411yescharacteristic
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—42no—
283C2×Q8—48no—
284C2×Q8—42no—
285C2×Q8—84no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×Q8C2×C4⋊C414yescomplemented
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—216no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—48no—
324C22×C4—42no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—48no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—84no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—84no—
377C22×C4C2×C4⋊C412yes—
378C22×C4C2×C4⋊C412yes—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4C4×D412yes—
382C22×C4#12011yescharacteristic
383C22×C4#12011yescharacteristic
384C22×C4C22⋊Q812yes—
385C22×C4C22⋊Q812yes—
386C22×C4C22⋊Q812yes—
387C22×C4#12312yes—
388C22×C4#12312yes—
389C22×C4C2×Q1614yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×Q811yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—48no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—84no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—28no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—432no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4C2×C4⋊C414yescomplemented
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—28no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—44no—
429C42—84no—
430C42—84no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23#38512yes—
452C23#38512yes—
453C23C4×Q1614yes—
454C23#44014yes—
455C23#44912yes—
456C23#44912yes—
457C23#45112yes—
458C23#45112yes—
459C23#45312yes—
460C23#45612yes—
461C23B(4,𝔽2)12yes—
462C23#45812yes—
463C23#47714yes—
464C23#47914yes—
465C23C2×C4×D411yescharacteristic
466C23C2×#12011yescharacteristic
467C23C2×C22⋊Q811yescharacteristic
468C23C22×Q1611yescharacteristic
469C23C2×#13511yescharacteristic
470C23C2×AGL(1,ℤ/8)11yescharacteristic
471C23C2×#13711yescharacteristic
472Q8—44no—
473Q8—82no—
474Q8—84no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—432no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—88no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—88no—
540C2×C4—84no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#38512yes—
545C2×C4#38512yes—
546C2×C4C22×C4⋊C411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C8—44no—
551C8—82no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—21no—
556C22—21no—
557C22—22no—
558C22—28no—
559C22—28no—
560C22—28no—
561C22—28no—
562C22—21no—
563C22—21no—
564C22—22no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22#160014yes—
575C22#160514yes—
576C22#160914yes—
577C22#161414yes—
578C22#185914yes—
579C22#186314yes—
580C22#186314yes—
581C22#186714yes—
582C22C2×#38511yescharacteristic
583C22C2×C4×Q1611yescharacteristic
584C22C2×#44011yescharacteristic
585C22C2×#45111yescharacteristic
586C22C2×#44911yescharacteristic
587C22C2×#45311yescharacteristic
588C22C2×#45611yescharacteristic
589C22C2×B(4,𝔽2)11yescharacteristic
590C22C2×#45811yescharacteristic
591C22C2×#47711yescharacteristic
592C22C2×#47911yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—22no—
598C4—22no—
599C4—48no—
600C4—44no—
601C4—44no—
602C4—44no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—84no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—24no—
611C2—21no—
612C2#886818yesdirect product factor
613C2C2×#161411yescharacteristic
614C2C2×#160911yescharacteristic
615C2C2×#160511yescharacteristic
616C2C2×#160011yescharacteristic
617C2C2×#185911yescharacteristic
618C2C2×#186311yescharacteristic
619C2C2×#186711yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
6. #8868C2direct (N × H)
36. C2×#328C4semidirect
126. #328C2×C4semidirect
132. C22×Q8C4⋊C4semidirect
196. C22⋊Q8C4⋊C4semidirect
229. C2×C4⋊C4C4⋊C4semidirect
288. C2×Q8C2×C4⋊C4semidirect
422. C4⋊C4C2×C4⋊C4semidirect
612. C2#8868direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23292
C2×#18722
C2×#18522
C22×#3282
C2×#14442
#88682
C2×#12942
C2×#12802