Finite Group Databasebeta

← C4×#1874 #354209

Subgroups — C4×#1874 #354209

Order 512.

2850 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1874C212yesmaximal
4C2×C4×#102C212yesmaximal, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
6#10931C212yesmaximal, complemented
7#10884C212yesmaximal, complemented
8#10730C212yesmaximal, complemented
9C4×#340C212yesmaximal, complemented
10C22×#121C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C42—21no—
15C23×C42C2211yescharacteristic
16C22×#102C2214yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2212yes—
19C4×#121—24no—
20C4×#121C418yescomplemented
21C2×#390C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#386—21no—
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—28no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4—21no—
33C22×#95—21no—
34#1874C4116yesdirect product factor
35C4×#102—28no—
36C4×C22⋊C8—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#336C2212yes—
41C23×D4—21no—
42C23×D4C2311yescharacteristic
43C24×C4—22no—
44C24×C4—21no—
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2314yes—
49C2×#121—22no—
50C2×#121—24no—
51C2×#121C2×C418yes—
52C2×#121C2×C414yes—
53C2×C4×D4—28no—
54C2×C4×D4—22no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—44no—
58C2×C4×D4—44no—
59C2×C4×D4C2×C414yescomplemented
60C22×C4⋊C4—22no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—28no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2×C418yescomplemented
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2314yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—24no—
76C22×C42—24no—
77C22×C42—28no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42—44no—
82C22×C42—44no—
83C22×C42C2×C414yes—
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D414yes—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#102—28no—
90C2×#102—28no—
91C2×#102C2×C4116yescomplemented
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8C2×C418yescomplemented
95C2×C4×C8—24no—
96C2×C4×C8—24no—
97C2×C4×C8—24no—
98#390—22no—
99#390C2×C414yescomplemented
100#389—22no—
101#389C2×C414yescomplemented
102#388—28no—
103#386—28no—
104#380—22no—
105#380C2×C414yes—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—28no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C414yescomplemented
112C4×C4⋊C4D418yescomplemented
113C4×C22⋊C4—28no—
114C2×#95—28no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120#340—28no—
121#340—28no—
122#340C2×C4116yescomplemented
123#336—24no—
124C25—21no—
125C25C2×D411yescharacteristic
126C22×D4—24no—
127C22×D4—28no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4—44no—
136C22×D4C22×C412yes—
137C22×D4C22×C414yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—28no—
146C23×C4—28no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—24no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C22⋊C414yes—
168C23×C4C22⋊C414yes—
169C23×C4D814yes—
170C23×C4SD1614yes—
171C23×C4C22×C412yes—
172C23×C4C22×C414yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4#4114yes—
180C23×C4C2411yescharacteristic
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8C22×C418yes—
188#121—28no—
189#121—216no—
190#121C42116yescomplemented
191C4×D4—216no—
192C4×D4—432no—
193C4×D4—48no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—232no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—232no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C418yes—
217C2×C4⋊C4C2×D418yes—
218C2×C4⋊C4#4118yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—216no—
239C2×C42—216no—
240C2×C42—28no—
241C2×C42—24no—
242C2×C42—28no—
243C2×C42—44no—
244C2×C42—432no—
245C2×C42—42no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—42no—
249C2×C42—44no—
250C2×C42C22⋊C414yes—
251C2×C42C22⋊C414yes—
252C2×C42C22×C412yes—
253C2×C42C22×C412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D414yes—
256C2×C42C2×D412yes—
257C2×C42#4114yes—
258#102—232no—
259C22⋊C8—216no—
260C4×C8—48no—
261#95—28no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—28no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—24no—
274C24—41no—
275C24—41no—
276C24—42no—
277C24—41no—
278C24—41no—
279C24—42no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24C2×C22⋊C412yes—
284C24C4×D414yes—
285C24C2×D812yes—
286C24C2×SD1612yes—
287C24#13514yes—
288C24C22×D411yescharacteristic
289C2×D4—216no—
290C2×D4—22no—
291C2×D4—28no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—28no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—416no—
299C2×D4—416no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4C2×C4214yescomplemented
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—232no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—21no—
333C22×C4—28no—
334C22×C4—21no—
335C22×C4—28no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—44no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—416no—
346C22×C4—416no—
347C22×C4—416no—
348C22×C4—416no—
349C22×C4—416no—
350C22×C4—44no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—416no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4#102116yescomplemented
376C22×C4C2×C4214yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4#11714yes—
383C22×C4C4×D418yes—
384C22×C4C4×D414yes—
385C22×C4#12012yes—
386C22×C4#12012yes—
387C22×C4#12312yes—
388C22×C4#12312yes—
389C22×C4C2×D812yes—
390C22×C4C2×D812yes—
391C22×C4C2×SD1612yes—
392C22×C4C2×SD1612yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4#13712yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C23×C412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4114yes—
402C2×C8—28no—
403C2×C8—28no—
404C2×C8—28no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C4⋊C4—232no—
410C4⋊C4—22no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4C2×C4214yescomplemented
421C4⋊C4C4×D4116yescomplemented
422C22⋊C4—216no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C42—24no—
428C42—24no—
429C42—28no—
430C42—28no—
431C42—416no—
432C42—48no—
433C42—416no—
434C42C4×D418yes—
435C42#12118yes—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—81no—
468C23—81no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—81no—
475C23—81no—
476C23—82no—
477C23C4×C22⋊C414yes—
478C23C2×#10214yes—
479C23#41614yes—
480C23C4×D818yes—
481C23C4×SD1618yes—
482C23C22×C22⋊C411yescharacteristic
483C23C2×C4×D412yes—
484C23C2×#12011yescharacteristic
485C23C2×#12311yescharacteristic
486C23#52512yes—
487C23C22×D811yescharacteristic
488C23C22×SD1611yescharacteristic
489C23C2×#13512yes—
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×#13711yescharacteristic
492C23#57512yes—
493D4—28no—
494D4—432no—
495D4—44no—
496D4—48no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—232no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—216no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—216no—
520C2×C4—416no—
521C2×C4—464no—
522C2×C4—42no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—816no—
547C2×C4C4×C22⋊C414yes—
548C2×C4#38614yes—
549C2×C4#38614yes—
550C2×C4C2×#10214yes—
551C2×C4C2×#10214yes—
552C2×C4#41714yes—
553C2×C4#44714yes—
554C2×C4#44814yes—
555C2×C4#48014yes—
556C2×C4#48114yes—
557C2×C4C22×C4211yescharacteristic
558C2×C4C2×#11711yescharacteristic
559C2×C4C2×C4×D412yes—
560C2×C4C2×C4×D414yes—
561C2×C4C2×C4×D412yes—
562C2×C4C4×#4112yes—
563C2×C4C4×#4114yes—
564C2×C4C2×#12112yes—
565C2×C4#52514yes—
566C2×C4#52914yes—
567C2×C4#53212yes—
568C8—48no—
569C22—28no—
570C22—28no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—28no—
578C22—28no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—84no—
583C22—84no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22—84no—
590C22C4×#10218yescomplemented
591C22C2×C4×C22⋊C411yescharacteristic
592C22C2×#38611yescharacteristic
593C22#229711yescharacteristic
594C22C4×#12012yes—
595C22C4×#12312yes—
596C22C22×#10211yescharacteristic
597C22C2×#41611yescharacteristic
598C22C2×#41711yescharacteristic
599C22#289611yescharacteristic
600C22C2×C4×D812yes—
601C22C2×C4×SD1612yes—
602C22C4×AGL(1,ℤ/8)12yes—
603C22C4×#13712yes—
604C22C2×#44711yescharacteristic
605C22C2×#44811yescharacteristic
606C22#300212yes—
607C22C2×#48011yescharacteristic
608C22C2×#48111yescharacteristic
609C22#309112yes—
610C22#310012yes—
611C22#310312yes—
612C22#310512yes—
613C22#310812yes—
614C4—24no—
615C4—24no—
616C4—24no—
617C4—42no—
618C4—44no—
619C4—416no—
620C4—82no—
621C4—82no—
622C4#187418yesdirect product factor
623C4C42×D412yes—
624C4#229012yes—
625C4C4×#12114yes—
626C2—28no—
627C2—84no—
628C2C4×#38611yescharacteristic
629C2C2×C4×#10211yescharacteristic
630C2C4×#41711yescharacteristic
631C2C2×#187411yescharacteristic
632C2#2090811yescharacteristic
633C2#2138611yescharacteristic
634C2#2138811yescharacteristic
635C2C4×#44712yes—
636C2C4×#44812yes—
637C2C4×#48012yes—
638C2C4×#48112yes—
subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC24C22×D4
prime radicalC22×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4×C4⋊C4C2semidirect
4. C2×C4×#102C2semidirect
5. C2×C4×C22⋊C8C2semidirect
6. #10931C2semidirect
7. #10884C2semidirect
8. #10730C2semidirect
9. C4×#340C2semidirect
20. C4×#121C4semidirect
34. #1874C4direct (N × H)
59. C2×C4×D4C2×C4semidirect
65. C22×C4⋊C4C2×C4semidirect
91. C2×#102C2×C4semidirect
94. C2×C22⋊C8C2×C4semidirect
99. #390C2×C4semidirect
101. #389C2×C4semidirect
111. C4×C4⋊C4C2×C4semidirect
112. C4×C4⋊C4D4semidirect
122. #340C2×C4semidirect
190. #121C42semidirect
304. C2×D4C2×C42semidirect
375. C22×C4#102semidirect
420. C4⋊C4C2×C42semidirect
421. C4⋊C4C4×D4semidirect
590. C22C4×#102semidirect
622. C4#1874direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C22×C4×C4⋊C42
C2×#18742
C2×C4×#1022
C2×C4×C22⋊C82
#109312
#108842
#107302
C4×#3402