Finite Group Databasebeta

← C4×#1867 #354456

Subgroups — C4×#1867 #354456

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#518C211yesmaximal, characteristic
2C2×#1867C212yesmaximal
3C2×C4×#99C211yesmaximal, characteristic
4C4×#385C212yesmaximal, complemented
5C4×#386C211yesmaximal, characteristic
6#10918C214yesmaximal, complemented
7#10745C212yesmaximal
8C4×#342C212yesmaximal, complemented
9C2×#518C2212yes—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C22×#99C2211yescharacteristic
13C22×#99C2211yescharacteristic
14C2×#386C2212yes—
15C2×#385C2214yes—
16#2275—24no—
17#2274—21no—
18#2273—21no—
19#2273C2212yes—
20C42×D4—22no—
21C42×D4—22no—
22C4×#117—21no—
23C2×C4×C4⋊C4—22no—
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1867C4116yesdirect product factor
30C4×#99C418yescomplemented
31C2×#342C2214yes—
32C2×#342C2214yes—
33C2×#342C2212yes—
34#1437—21no—
35#1436—24no—
36#1436—21no—
37C22⋊#95C2212yes—
38C22⋊#95C2214yes—
39C4×#95—21no—
40C4×#95—22no—
41C23×D4C2311yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45#518—24no—
46#518—28no—
47#518C2×C418yes—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4C2×C414yes—
54C2×C4×D4Q814yes—
55C2×#117—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4D411yescharacteristic
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2312yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C22⋊C4C2314yes—
65C22×C22⋊C4C2312yes—
66C22×C42—21no—
67C22×C42—22no—
68C22×C42—22no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—44no—
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D412yes—
78C2×#99—24no—
79C2×#99C2×C414yes—
80C2×#99C2×C418yes—
81C2×#99C2×C414yes—
82#386—24no—
83#386C2×C418yes—
84#385—28no—
85#385C2×C4116yescomplemented
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—48no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—44no—
107C4×C22⋊C4C2×C418yescomplemented
108#376—22no—
109C2×#95—22no—
110C2×#95—24no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—24no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—24no—
117C2×#95—21no—
118C2×#95D414yes—
119C43—22no—
120#342—28no—
121#342C2×C4116yescomplemented
122#342C2×C418yescomplemented
123C25C2×D411yescharacteristic
124C25C2411yescharacteristic
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—24no—
128C22×D4C22×C412yes—
129C22×D4C22×C414yes—
130C22×D4C2×Q812yes—
131C22×D4#4114yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4C4⋊C414yescomplemented
151C23×C4C22×C414yes—
152C23×C4C22×C412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D414yes—
159C23×C4C2×D414yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×Q812yes—
163C23×C4#4114yes—
164C4×D4—216no—
165C4×D4—216no—
166C4×D4—28no—
167C4×D4—28no—
168C4×D4—216no—
169C4×D4—216no—
170C4×D4—28no—
171C4×D4—28no—
172#117—28no—
173#117—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—216no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4C4⋊C414yescomplemented
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4C4⋊C414yescomplemented
205C2×C22⋊C4C22×C414yes—
206C2×C22⋊C4C22×C414yes—
207C2×C22⋊C4C22×C412yes—
208C2×C22⋊C4C22×C418yes—
209C2×C22⋊C4C22×C414yes—
210C2×C22⋊C4C22×C418yes—
211C2×C22⋊C4C2×Q812yes—
212C2×C22⋊C4#4118yes—
213C2×C42—24no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—28no—
219C2×C42—28no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—48no—
231C2×C42—44no—
232C2×C42—42no—
233C2×C42—41no—
234C2×C42—42no—
235C2×C42—41no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—82no—
242C2×C42—82no—
243C2×C42C4⋊C414yescomplemented
244#99—28no—
245#99C42116yescomplemented
246#95—24no—
247#95—22no—
248#95—22no—
249#95—28no—
250#95—216no—
251#95—28no—
252#95—24no—
253#95—24no—
254#95—44no—
255#95—48no—
256#95—44no—
257#95—48no—
258#95—42no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—41no—
263C24—41no—
264C24—42no—
265C24—41no—
266C24—42no—
267C24—41no—
268C24C2×C4⋊C412yes—
269C24C4×D414yes—
270C24C23×C411yescharacteristic
271C24C23×C412yes—
272C24C22×D411yescharacteristic
273C24C22×D412yes—
274C24C22×Q811yescharacteristic
275C24C2×#4112yes—
276C24C2×#4112yes—
277C24C2×#4114yes—
278C2×D4—24no—
279C2×D4—28no—
280C2×D4—24no—
281C2×D4—28no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—28no—
285C2×D4—24no—
286C2×D4C2×C4214yes—
287C2×D4#11714yes—
288C2×D4C4×Q818yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—28no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—44no—
319C22×C4—48no—
320C22×C4—44no—
321C22×C4—48no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—48no—
327C22×C4—416no—
328C22×C4—48no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—84no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—82no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4C2×C4⋊C412yes—
366C22×C4C2×C4⋊C412yes—
367C22×C4C2×C4⋊C412yes—
368C22×C4C2×C4⋊C412yes—
369C22×C4C2×C4⋊C412yes—
370C22×C4C4×D414yes—
371C22×C4C4×D414yes—
372C22×C4C4×D418yes—
373C22×C4C4×D418yes—
374C22×C4C4×D414yes—
375C22×C4#12011yescharacteristic
376C22×C4#12012yes—
377C22×C4#12011yescharacteristic
378C22×C4C22⋊Q814yes—
379C22×C4C22⋊Q814yes—
380C22×C4#12312yes—
381C22×C4#12312yes—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—432no—
386C22⋊C4—216no—
387C22⋊C4—28no—
388C22⋊C4—24no—
389C22⋊C4—28no—
390C22⋊C4—24no—
391C22⋊C4—28no—
392C22⋊C4—24no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—28no—
399C22⋊C4—416no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4C2×C4218yescomplemented
403C42—216no—
404C42—24no—
405C42—24no—
406C42—416no—
407C42—48no—
408C42—44no—
409C42—44no—
410C42—44no—
411C42—44no—
412C42—82no—
413C42—82no—
414C42—84no—
415C42—84no—
416C42—82no—
417C42—82no—
418C23—22no—
419C23—21no—
420C23—22no—
421C23—21no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23—42no—
432C23—44no—
433C23—42no—
434C23—41no—
435C23—42no—
436C23—41no—
437C23—42no—
438C23—44no—
439C23—42no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—81no—
448C23—81no—
449C23—82no—
450C23—82no—
451C23C4×C4⋊C414yescomplemented
452C23C22×C4211yescharacteristic
453C23C22×C4⋊C411yescharacteristic
454C23C2×#11711yescharacteristic
455C23C2×C4×D412yes—
456C23C2×C4×D412yes—
457C23C2×C4×D414yes—
458C23C2×C4×Q812yes—
459C23C4×#4112yes—
460C23C4×#4114yes—
461C23C2×#12011yescharacteristic
462C23C2×C22⋊Q812yes—
463C23C2×#12311yescharacteristic
464C23#52514yes—
465C23#52512yes—
466C23#52914yes—
467C23#53312yes—
468D4—28no—
469D4—28no—
470D4—216no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—216no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—84no—
526C2×C4—81no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—81no—
530C2×C4—88no—
531C2×C4—88no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4—82no—
538C2×C4C4×C4⋊C414yescomplemented
539C2×C4#38512yes—
540C2×C4#38514yes—
541C2×C4#38512yes—
542C2×C4B(4,𝔽2)12yes—
543C2×C4#45812yes—
544C22—22no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—22no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—24no—
553C22—22no—
554C22—24no—
555C22—22no—
556C22—24no—
557C22—22no—
558C22—24no—
559C22—22no—
560C22—24no—
561C22—41no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—44no—
566C22—42no—
567C22—41no—
568C22—44no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—84no—
574C22—82no—
575C22C2×C4×C4⋊C411yescharacteristic
576C22C42×D412yes—
577C22C2×#38511yescharacteristic
578C22#229012yes—
579C22#229611yescharacteristic
580C22C4×#12012yes—
581C22C4×#12312yes—
582C22C4×C22⋊Q814yes—
583C22C2×B(4,𝔽2)11yescharacteristic
584C22C2×#45811yescharacteristic
585C22#302512yes—
586C4—22no—
587C4—24no—
588C4—24no—
589C4—48no—
590C4—42no—
591C4—42no—
592C4—48no—
593C4—48no—
594C4—48no—
595C4—84no—
596C4—84no—
597C4#186714yesdirect product factor
598C2—22no—
599C2—22no—
600C2—22no—
601C2—21no—
602C2—21no—
603C2—24no—
604C2—44no—
605C2C4×#38511yescharacteristic
606C2C2×#186711yescharacteristic
607C2#2088811yescharacteristic
608C2C4×B(4,𝔽2)12yes—
609C2C4×#45812yes—
subgroupG/H
centerC22×C4#120
commutatorC23C22×C42
Frattini, cosocleC25C24
socleC23C2×#120
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C4×#385C2semidirect
6. #10918C2semidirect
8. C4×#342C2semidirect
29. #1867C4direct (N × H)
30. C4×#99C4semidirect
85. #385C2×C4semidirect
107. C4×C22⋊C4C2×C4semidirect
121. #342C2×C4semidirect
122. #342C2×C4semidirect
150. C23×C4C4⋊C4semidirect
181. C2×C4⋊C4C4⋊C4semidirect
204. C2×C22⋊C4C4⋊C4semidirect
243. C2×C42C4⋊C4semidirect
245. #99C42semidirect
402. C22⋊C4C2×C42semidirect
451. C23C4×C4⋊C4semidirect
538. C2×C4C4×C4⋊C4semidirect
597. C4#1867direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#5182
C2×#18672
C2×C4×#992
C4×#3852
C4×#3862
#109182
#107452
C4×#3422