Finite Group Databasebeta

← C4×#1793 #354824

Subgroups — C4×#1793 #354824

Order 512.

3688 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×2−1+4C211yesmaximal, characteristic
2C2×#1793C212yesmaximal
3C4×#417C212yesmaximal, complemented
4C2×C4×#104C212yesmaximal, complemented
5C2×C4×#101C211yesmaximal, characteristic, complemented
6C4×#391C211yesmaximal, characteristic, complemented
7#10781C214yesmaximal, complemented
8#10750C212yesmaximal, complemented
9C22×2−1+4C2211yescharacteristic
10C4×2−1+4—28no—
11C4×2−1+4—24no—
12C4×2−1+4C418yescomplemented
13C2×#520—22no—
14C2×#520—22no—
15C2×#520C2212yes—
16C2×#519—22no—
17C2×#519C2211yescharacteristic
18C2×C4×#41—22no—
19C2×C4×#41—22no—
20C2×C4×#41C2212yes—
21C22×C4×Q8—22no—
22C22×C4×Q8C2211yescharacteristic
23C22×#104C2214yes—
24C2×#417C2214yes—
25C22×#101C2212yes—
26C2×C4×M16C2212yes—
27C2×#391C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1793C4116yesdirect product factor
31#1763—22no—
32#1762—22no—
33C4×#103—22no—
34C4×#102—22no—
35C4×#104—28no—
36C4×#101—24no—
37C2×#343C2212yes—
38#1736C2214yes—
39C2×#339C2214yes—
40#1436—22no—
41C4⋊#95C2212yes—
42C4×#95—22no—
43C2×2−1+4—24no—
44C2×2−1+4—28no—
45C2×2−1+4—22no—
46C2×2−1+4—24no—
47C2×2−1+4C2×C418yes—
48C2×2−1+4C2×C414yes—
49C22×#41—22no—
50C22×#41—22no—
51C22×#41C2312yes—
52C23×Q8—22no—
53C23×Q8C2311yescharacteristic
54C22×M16C2314yes—
55#520—216no—
56#520—216no—
57#520—24no—
58#520—216no—
59#520—24no—
60#520—216no—
61#520—24no—
62#520C2×C418yes—
63#519—216no—
64#519—22no—
65#519—216no—
66#519—24no—
67#519C2×C414yes—
68C4×#41—216no—
69C4×#41—216no—
70C4×#41—216no—
71C4×#41—216no—
72C4×#41—24no—
73C4×#41—24no—
74C4×#41C2×C418yes—
75C4×#41D418yes—
76C2×C4×Q8—22no—
77C2×C4×Q8—24no—
78C2×C4×Q8—216no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8—216no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—22no—
85C2×C4×Q8—22no—
86C2×C4×Q8—24no—
87C2×C4×Q8C2×C414yes—
88C2×C4×Q8D412yes—
89C2×C4×Q8D412yes—
90C2×C4×Q8D414yes—
91C2×C4×D4—22no—
92C2×C4×D4—24no—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4—24no—
97C2×C4×D4—22no—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117C2312yes—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4D412yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C42—24no—
112C22×C42—22no—
113C22×C42—21no—
114C22×C42—21no—
115C22×C42—21no—
116C22×C42D412yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C2×#104—28no—
120C2×#104—28no—
121C2×#104C2×C4116yescomplemented
122#417—28no—
123#417C2×C4116yescomplemented
124C2×#103—24no—
125C2×#102—24no—
126C2×#101—24no—
127C2×#101—24no—
128C2×#101C2×C418yescomplemented
129C4×M16—24no—
130C4×M16—24no—
131C2×#97—22no—
132C2×C4×C8—22no—
133#391—24no—
134#391C2×C418yescomplemented
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95—21no—
140C43—24no—
141#343—24no—
142#340—24no—
143#340—24no—
144#339—28no—
1452−1+4—216no—
1462−1+4—232no—
1472−1+4—216no—
1482−1+4—232no—
1492−1+4—28no—
1502−1+4—216no—
1512−1+4C42116yescomplemented
152C2×#41—28no—
153C2×#41—28no—
154C2×#41—216no—
155C2×#41—216no—
156C2×#41—28no—
157C2×#41—216no—
158C2×#41—28no—
159C2×#41—216no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—24no—
163C2×#41—24no—
164C2×#41C22×C414yes—
165C2×#41C22×C418yes—
166C2×#41C2×D414yes—
167C2×#41#4118yes—
168C22×Q8—22no—
169C22×Q8—28no—
170C22×Q8—216no—
171C22×Q8—28no—
172C22×Q8—216no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8C22×C412yes—
177C22×Q8C22×C414yes—
178C22×Q8C2×D412yes—
179C22×Q8C2×D412yes—
180C22×Q8#4114yes—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4C2×D412yes—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2411yescharacteristic
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16C22×C418yes—
202C22×C8—24no—
203C4×Q8—216no—
204C4×Q8—28no—
205C4×Q8—28no—
206C4×Q8—232no—
207C4×Q8—28no—
208C4×Q8—216no—
209C4×Q8—28no—
210C4×Q8—28no—
211C4×Q8—28no—
212C4×Q8—28no—
213C4×Q8—28no—
214C4×Q8—24no—
215C4×Q8—28no—
216C4×Q8—28no—
217C4×Q8—24no—
218C4×Q8C22⋊C418yes—
219C4×Q8C22⋊C418yes—
220C4×D4—216no—
221C4×D4—216no—
222C4×D4—232no—
223C4×D4—28no—
224C4×D4—216no—
225C4×D4—28no—
226C4×D4—28no—
227C4×D4—216no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—28no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4C22⋊C418yes—
234C4×D4C22⋊C418yes—
235#117—28no—
236#117—28no—
237#117—232no—
238#117—232no—
239#117—28no—
240#117—22no—
241#117—22no—
242#117—28no—
243#117—28no—
244#117—28no—
245#117C22×C414yes—
246#117C2×D414yes—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4C22⋊C418yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—232no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—28no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42C22⋊C418yes—
295C2×C42C22×C414yes—
296C2×C42C22×C412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D414yes—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301#104—232no—
302#103—216no—
303#102—216no—
304#101—216no—
305#97—44no—
306C4×C8—44no—
307#95—216no—
308#95—48no—
309#95—42no—
310C24—22no—
311C24C22×D411yescharacteristic
312#41—232no—
313#41—264no—
314#41—232no—
315#41—264no—
316#41—232no—
317#41—24no—
318#41—24no—
319#41—28no—
320#41—28no—
321#41—28no—
322#41—28no—
323#41—28no—
324#41—232no—
325#41C2×C4218yes—
326#41#11718yes—
327#41C4×D4116yes—
328C2×Q8—28no—
329C2×Q8—216no—
330C2×Q8—232no—
331C2×Q8—264no—
332C2×Q8—232no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—28no—
341C2×Q8—24no—
342C2×Q8—28no—
343C2×Q8C2×C4214yes—
344C2×Q8C2×C22⋊C414yes—
345C2×Q8#11714yes—
346C2×Q8C4×D418yes—
347C2×Q8C4×D418yes—
348C2×D4—28no—
349C2×D4—216no—
350C2×D4—24no—
351C2×D4—28no—
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—22no—
355C2×D4—24no—
356C2×D4C2×C22⋊C414yes—
357C2×D4C4×D418yes—
358C22×C4—216no—
359C22×C4—232no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—84no—
394C22×C4—88no—
395C22×C4C2×C22⋊C414yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C414yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C4×D418yes—
400C22×C4#12012yes—
401C22×C4#12011yescharacteristic
402C22×C4#12012yes—
403C22×C4#12012yes—
404C22×C4#12011yescharacteristic
405C22×C4C23×C411yescharacteristic
406C22×C4C23×C412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4114yes—
411C22×C4C2×#4112yes—
412M16—28no—
413M16—28no—
414M16—44no—
415M16—44no—
416C2×C8—28no—
417C2×C8—44no—
418C2×C8—44no—
419C4⋊C4—216no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—28no—
427C4⋊C4—216no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—44no—
431C4⋊C4C2×C22⋊C414yes—
432C4⋊C4C2×C22⋊C414yes—
433C22⋊C4—216no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—28no—
437C22⋊C4—28no—
438C42—216no—
439C42—216no—
440C42—24no—
441C42—24no—
442C42—28no—
443C42—28no—
444C42—28no—
445C42—28no—
446C42—28no—
447C42—48no—
448C42—48no—
449C42—44no—
450C42—82no—
451C42—82no—
452C42C2×C22⋊C414yes—
453C42C2×C22⋊C414yes—
454C42#12014yes—
455C42#12014yes—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—24no—
460C23—28no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23C22×C22⋊C411yescharacteristic
465C23C2×C4×D412yes—
466C23C2×#12011yescharacteristic
467C23C2×#12011yescharacteristic
468C23#52512yes—
469Q8—232no—
470Q8—28no—
471Q8—24no—
472Q8—28no—
473Q8—28no—
474Q8—216no—
475Q8C4×C22⋊C418yes—
476D4—232no—
477D4—28no—
478D4—28no—
479D4—216no—
480D4—24no—
481D4—28no—
482D4C4×C22⋊C418yes—
483C2×C4—264no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—216no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—416no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—88no—
519C2×C4—82no—
520C2×C4—816no—
521C2×C4—84no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4C4×C22⋊C418yes—
525C2×C4#37912yes—
526C2×C4#37914yes—
527C2×C4#37912yes—
528C2×C4#45412yes—
529C2×C4#45612yes—
530C2×C4C22×C4211yescharacteristic
531C2×C4C22×C22⋊C412yes—
532C2×C4C2×#11712yes—
533C2×C4C2×#11711yescharacteristic
534C2×C4C2×C4×D414yes—
535C2×C4C2×C4×D414yes—
536C2×C4C2×C4×D412yes—
537C2×C4C2×#12012yes—
538C2×C4#52514yes—
539C2×C4#52514yes—
540C2×C4#52512yes—
541C8—48no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—44no—
552C22—44no—
553C22—48no—
554C22—48no—
555C22C2×C4×C22⋊C411yescharacteristic
556C22C2×#37911yescharacteristic
557C22#228011yescharacteristic
558C22C4×#12012yes—
559C22C4×#12012yes—
560C22C2×#45411yescharacteristic
561C22C2×#45611yescharacteristic
562C22#301812yes—
563C4—22no—
564C4—24no—
565C4—28no—
566C4—28no—
567C4—48no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—84no—
572C4#179314yesdirect product factor
573C4C2×C4×C22⋊C412yes—
574C4#228012yes—
575C4C4×#12014yes—
576C2—22no—
577C2—21no—
578C2—21no—
579C2—48no—
580C2C4×#37911yescharacteristic
581C2C2×#179311yescharacteristic
582C2#2061511yescharacteristic
583C2C4×#45412yes—
584C2C4×#45612yes—
subgroupG/H
centerC22×C4#120
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC22×2−1+4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#417C2semidirect
4. C2×C4×#104C2semidirect
5. C2×C4×#101C2semidirect
6. C4×#391C2semidirect
7. #10781C2semidirect
8. #10750C2semidirect
12. C4×2−1+4C4semidirect
30. #1793C4direct (N × H)
121. C2×#104C2×C4semidirect
123. #417C2×C4semidirect
128. C2×#101C2×C4semidirect
134. #391C2×C4semidirect
151. 2−1+4C42semidirect
572. C4#1793direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×2−1+42
C2×#17932
C4×#4172
C2×C4×#1042
C2×C4×#1012
C4×#3912
#107812
#107502