Finite Group Databasebeta

← C4×#3486 #7625656

Subgroups — C4×#3486 #7625656

Order 512.

11070 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3486C211yesmaximal, characteristic
2C2×C4×2−1+4C211yesmaximal, characteristic, complemented
3C23×C4×Q8C211yesmaximal, characteristic, complemented
4#34294C218yesmaximal, complemented
5#34291C218yesmaximal, complemented
6#34266C216yesmaximal, complemented
7#34257C211yesmaximal, characteristic, complemented
8#34249C211yesmaximal, characteristic, complemented
9#34200C216yesmaximal, complemented
10#34173C211yesmaximal, characteristic, complemented
11C4×D4×Q8C218yesmaximal, complemented
12C4×#548C218yesmaximal, complemented
13C4×#536C216yesmaximal, complemented
14C4×#525C216yesmaximal, complemented
15C4×#519C211yesmaximal, characteristic, complemented
16C22×2−1+4C2211yescharacteristic
17C24×Q8C2211yescharacteristic
18#3486C4132yesdirect product factor
19C2×D4×Q8C2218yes—
20C2×#548C2218yes—
21C2×#536C2216yes—
22C2×#525C2216yes—
23C4×2−1+4—216no—
24C2×#520C2216yes—
25C2×#520C2214yescomplemented
26C2×#519C2211yescharacteristic
27C2×#519C2211yescharacteristic
28C2×#519C2211yescharacteristic
29C2×#519C2214yescomplemented
30C2×C4×#41C2216yes—
31C2×C4×#41C2214yescomplemented
32C22×C4×Q8—216no—
33C22×C4×Q8C2213yes—
34C22×C4×Q8C2218yes—
35C22×C4×Q8C2211yescharacteristic
36C22×C4×Q8C2213yes—
37C22×C4×Q8C2218yes—
38C22×C4×Q8C2211yescharacteristic
39C22×C4×Q8C2214yescomplemented
40C23×C4⋊C4C2213yes—
41C23×C42C2213yes—
42#2367C22124yescomplemented
43#2365C22124yescomplemented
44#2362C22124yescomplemented
45#2361C22124yes—
46#2360C22124yes—
47#2360C22124yescomplemented
48#2352C22124yes—
49#2351C2218yescomplemented
50Q8×C4⋊C4C2218yescomplemented
51D4×C4⋊C4C22124yes—
52#2341C2218yescomplemented
53Q8×C22⋊C4C2218yes—
54Q8×C22⋊C4C2218yes—
55Q8×C22⋊C4C2218yescomplemented
56#2327C22112yescomplemented
57#2326C22112yescomplemented
58#2323C22124yescomplemented
59#2322C22124yes—
60#2317C2216yes—
61#2311C2216yes—
62#2310C2216yes—
63C4×C4⋊Q8C22112yescomplemented
64C4×#124C22112yescomplemented
65C4×C22⋊Q8C22112yes—
66C4×C22⋊Q8C22112yes—
67C4×C22⋊Q8C22124yescomplemented
68C4×#123C22112yes—
69C4×#123C22112yes—
70C4×#121C22112yes—
71C4×#121C22112yes—
72C4×#120C22112yes—
73C4×#120C2214yes—
74#2297C2216yes—
75#2296C2216yes—
76#2290C22124yes—
77#2280C2216yes—
78#2276C2218yescomplemented
79#2274C2216yes—
80C42×Q8C2218yescomplemented
81C42×D4C22124yes—
82C4×#117C2216yes—
83C2×2−1+4—28no—
84C2×2−1+4—28no—
85C2×2−1+4C2×C4116yescomplemented
86C22×#41C2316yes—
87C22×#41C2314yes—
88C23×Q8—28no—
89C23×Q8—28no—
90C23×Q8C2×C4116yescomplemented
91C23×Q8C2313yes—
92C23×Q8C2318yes—
93C23×Q8C2314yes—
94C23×Q8C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2313yes—
97C24×C4C2313yes—
98D4×Q8C2×C41128yescomplemented
99#548C2×C41128yescomplemented
100#536C2×C4196yescomplemented
101C2×C4⋊Q8C23112yes—
102C2×#124C23112yes—
103#525C2×C4196yescomplemented
104C2×#123C23112yes—
105C2×C22⋊Q8C23112yes—
106C2×C22⋊Q8C23124yes—
107C2×#121C23112yes—
108C2×#120C2314yes—
109#520—232no—
110#520—248no—
111#519—28no—
112#519—28no—
113#519—232no—
114#519C2×C4116yescomplemented
115C4×#41—232no—
116C4×#41—248no—
117C2×C4×Q8—28no—
118C2×C4×Q8—264no—
119C2×C4×Q8—264no—
120C2×C4×Q8—232no—
121C2×C4×Q8—224no—
122C2×C4×Q8—224no—
123C2×C4×Q8D4116yescomplemented
124C2×C4×Q8C2316yes—
125C2×C4×Q8C23112yes—
126C2×C4×Q8C23112yes—
127C2×C4×Q8C2316yes—
128C2×C4×Q8C2318yes—
129C2×C4×Q8C2314yes—
130C2×C4×Q8C2314yes—
131C2×C4×Q8C2318yes—
132C2×C4×Q8C23112yes—
133C2×C4×Q8C23124yes—
134C2×C4×Q8C2318yes—
135C2×C4×Q8C2314yes—
136C2×C4×Q8C2314yes—
137C2×C4×D4C23112yes—
138C2×C4×D4C2316yes—
139C2×C4×D4C23112yes—
140C2×C4×D4C2314yes—
141C2×C4×D4C2316yes—
142C2×C4×D4C2318yes—
143C2×C4×D4C23124yes—
144C2×C4×D4C23124yes—
145C2×#117C2316yes—
146C2×#117C2316yes—
147C2×#117C23112yes—
148C2×#117C23112yes—
149C2×#117C2316yes—
150C22×C4⋊C4—224no—
151C22×C4⋊C4C2313yes—
152C22×C4⋊C4C2316yes—
153C22×C4⋊C4C23112yes—
154C22×C4⋊C4C23124yes—
155C22×C4⋊C4C2313yes—
156C22×C42—224no—
157C22×C42C23112yes—
158C22×C42C23112yes—
159C22×C42C2313yes—
160C22×C42C2313yes—
161C22×C42C2313yes—
162C22×C42C23112yes—
163C22×C42C2313yes—
164#391C23112yes—
165#391C23112yes—
166#391C23124yes—
167#391C23112yes—
168#391C23112yes—
169#390C23112yes—
170#390C23112yes—
171#389C23148yes—
172#389C23124yes—
173#389C23112yes—
174#389C23112yes—
175#388C23148yes—
176#388C23124yes—
177#388C23124yes—
178#387C23148yes—
179#387C23124yes—
180#387C23124yes—
181#386C23112yes—
182#386C23124yes—
183#386C2318yes—
184#386C2314yes—
185#385C23112yes—
186#385C23148yes—
187#385C23124yes—
188#385C2318yes—
189#385C2314yes—
190#384C23112yes—
191#382C23112yes—
192#381C23112yes—
193#380C23124yes—
194#380C23112yes—
195#380C23112yes—
196#379C2314yes—
197C4×C4⋊C4C23124yes—
198C4×C4⋊C4C23112yes—
199C4×C4⋊C4C23124yes—
200C4×C4⋊C4C23124yes—
201C4×C4⋊C4C23124yes—
202C4×C4⋊C4C23124yes—
203C4×C4⋊C4C23112yes—
204C4×C22⋊C4C23112yes—
205C4×C22⋊C4C23124yes—
206C4×C22⋊C4C23124yes—
207C4×C22⋊C4C23124yes—
208C4×C22⋊C4C23124yes—
209C4×C22⋊C4C2314yes—
210C4×C22⋊C4C2318yes—
211C4×C22⋊C4C23124yes—
212#376C23112yes—
213C43C23112yes—
214C25C2411yescharacteristic
2152−1+4—2128no—
216C2×#41—216no—
217C2×#41—216no—
218C2×#41—224no—
219C2×#41—224no—
220C2×#41C22×C4148yes—
221C2×#41C22×C4132yescomplemented
222C22×Q8—2128no—
223C22×Q8—232no—
224C22×Q8—216no—
225C22×Q8—232no—
226C22×Q8—216no—
227C22×Q8—212no—
228C22×Q8—212no—
229C22×Q8C22×C4124yes—
230C22×Q8C22×C4164yes—
231C22×Q8C22×C418yes—
232C22×Q8C22×C4132yescomplemented
233C22×Q8C2×D418yes—
234C22×Q8#4118yes—
235C22×Q8C24112yes—
236C22×Q8C2416yes—
237C22×Q8C2414yes—
238C22×D4C2418yes—
239C22×D4C2414yes—
240C22×D4C2416yes—
241C23×C4—212no—
242C23×C4—24no—
243C23×C4—212no—
244C23×C4—28no—
245C23×C4—212no—
246C23×C4—412no—
247C23×C4C22×C4124yes—
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C24112yes—
252C23×C4C24112yes—
253C23×C4C2413yes—
254C23×C4C24112yes—
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C23×C4C2413yes—
258C23×C4C24112yes—
259C23×C4C2413yes—
260C23×C4C2413yes—
261C4⋊Q8C22×C4196yescomplemented
262#124C22×C4196yescomplemented
263#123C22×C4196yes—
264C22⋊Q8C22×C4196yes—
265C22⋊Q8C22×C41192yescomplemented
266#121C22×C4196yes—
267#120C22×C4132yes—
268C4×Q8—232no—
269C4×Q8—216no—
270C4×Q8—216no—
271C4×Q8—232no—
272C4×Q8—296no—
273C4×Q8—248no—
274C4×Q8—248no—
275C4×Q8—248no—
276C4×Q8—224no—
277C4×Q8—224no—
278C4×Q8C22×C4164yescomplemented
279C4×Q8C2×D4132yes—
280C4×Q8C2×D4132yes—
281C4×D4—216no—
282C4×D4—248no—
283C4×D4—232no—
284C4×D4—248no—
285C4×D4—224no—
286C4×D4—224no—
287C4×D4—448no—
288C4×D4C22×C41192yes—
289#117—224no—
290#117—224no—
291#117—248no—
292#117—248no—
293#117C22×C4148yes—
294C2×C4⋊C4—224no—
295C2×C4⋊C4—212no—
296C2×C4⋊C4—296no—
297C2×C4⋊C4—248no—
298C2×C4⋊C4C2×D4124yes—
299C2×C4⋊C4C24112yes—
300C2×C4⋊C4C24124yes—
301C2×C4⋊C4C2416yes—
302C2×C4⋊C4C24112yes—
303C2×C4⋊C4C2416yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C24112yes—
306C2×C4⋊C4C2418yes—
307C2×C4⋊C4C24124yes—
308C2×C4⋊C4C24112yes—
309C2×C4⋊C4C24124yes—
310C2×C4⋊C4C24112yes—
311C2×C22⋊C4C2414yes—
312C2×C22⋊C4C24112yes—
313C2×C22⋊C4C2414yes—
314C2×C22⋊C4C2414yes—
315C2×C22⋊C4C2418yes—
316C2×C22⋊C4C2418yes—
317C2×C22⋊C4C2418yes—
318C2×C22⋊C4C24124yes—
319C2×C22⋊C4C2414yes—
320C2×C22⋊C4C2418yes—
321C2×C22⋊C4C24124yes—
322C2×C42—212no—
323C2×C42—212no—
324C2×C42—212no—
325C2×C42—248no—
326C2×C42—248no—
327C2×C42—248no—
328C2×C42C2×D4124yes—
329C2×C42C24112yes—
330C2×C42C24112yes—
331C2×C42C24112yes—
332C2×C42C24112yes—
333C2×C42C24112yes—
334C2×C42C2416yes—
335C2×C42C2414yes—
336C2×C42C2418yes—
337C2×C42C2416yes—
338C2×C42C24124yes—
339C2×C42C24112yes—
340C2×C42C2416yes—
341C2×C42C2416yes—
342#95C2418yes—
343#95C24124yes—
344#95C2418yes—
345#95C2418yes—
346#95C24124yes—
347#95C24124yes—
348#95C24124yes—
349#95C24124yes—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24C23×C414yes—
355C24C2514yes—
356C24C2511yescharacteristic
357C24C2512yes—
358#41—2128no—
359#41—2192no—
360C2×Q8—2256no—
361C2×Q8—2128no—
362C2×Q8—296no—
363C2×Q8—224no—
364C2×Q8—212no—
365C2×Q8—224no—
366C2×Q8—212no—
367C2×Q8C4×D4164yescomplemented
368C2×Q8C23×C4124yes—
369C2×Q8C23×C4148yes—
370C2×Q8C23×C4116yes—
371C2×Q8C22×D4116yes—
372C2×Q8C2×#41116yes—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—212no—
378C2×D4—212no—
379C2×D4C23×C4132yes—
380C2×D4C23×C4116yes—
381C2×D4C23×C4124yes—
382C22×C4—232no—
383C22×C4—22no—
384C22×C4—26no—
385C22×C4—26no—
386C22×C4—24no—
387C22×C4—296no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—224no—
391C22×C4—224no—
392C22×C4—224no—
393C22×C4—224no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—26no—
399C22×C4—224no—
400C22×C4—28no—
401C22×C4—224no—
402C22×C4—216no—
403C22×C4—224no—
404C22×C4—46no—
405C22×C4—412no—
406C22×C4—46no—
407C22×C4—424no—
408C22×C4—448no—
409C22×C4C23×C4148yes—
410C22×C4C23×C4148yes—
411C22×C4C23×C4112yes—
412C22×C4C23×C4112yes—
413C22×C4C22×D414yes—
414C22×C4C22×D4112yes—
415C22×C4C22×D4112yes—
416C22×C4C2×#41112yes—
417C22×C42−1+418yescomplemented
418C22×C4C2518yes—
419C22×C4C2514yes—
420C22×C4C2514yes—
421C22×C4C2516yes—
422C22×C4C2518yes—
423C22×C4C2516yes—
424C22×C4C2514yes—
425C22×C4C2511yescharacteristic
426C22×C4C2516yes—
427C22×C4C2512yes—
428C22×C4C2516yes—
429C22×C4C2511yescharacteristic
430C4⋊C4—248no—
431C4⋊C4—224no—
432C4⋊C4—224no—
433C4⋊C4—212no—
434C4⋊C4—28no—
435C4⋊C4—212no—
436C4⋊C4—224no—
437C4⋊C4—216no—
438C4⋊C4—424no—
439C4⋊C4C23×C4148yes—
440C4⋊C4C23×C4196yes—
441C4⋊C4C22×D4148yes—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C22⋊C4—216no—
445C22⋊C4—216no—
446C22⋊C4—224no—
447C22⋊C4—28no—
448C22⋊C4—216no—
449C22⋊C4—424no—
450C22⋊C4C23×C4116yes—
451C22⋊C4C23×C4132yes—
452C22⋊C4C23×C4196yes—
453C42—28no—
454C42—224no—
455C42—224no—
456C42—212no—
457C42—212no—
458C42—224no—
459C42—224no—
460C42—224no—
461C42—216no—
462C42—424no—
463C42C23×C4148yes—
464C42C22×D4124yes—
465C42C22×D4124yes—
466C23—28no—
467C23—28no—
468C23—216no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—28no—
476C23—216no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23C24×C418yes—
482C23C24×C412yes—
483C23C24×C414yes—
484C23C23×D412yes—
485C23C22×#4112yes—
486C23C2×2−1+414yes—
487C23#58514yes—
488C23C2611yescharacteristic
489Q8—296no—
490Q8—248no—
491Q8C2×C4×D4164yes—
492D4—232no—
493D4—264no—
494D4—248no—
495C2×C4—216no—
496C2×C4—28no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—26no—
500C2×C4—24no—
501C2×C4—26no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—212no—
507C2×C4—224no—
508C2×C4—296no—
509C2×C4—296no—
510C2×C4—28no—
511C2×C4—216no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—26no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—212no—
521C2×C4—412no—
522C2×C4—448no—
523C2×C4—412no—
524C2×C4—416no—
525C2×C4—824no—
526C2×C4C2×C4×D4148yes—
527C2×C4C24×C4116yes—
528C2×C4C24×C4112yes—
529C2×C4C24×C418yes—
530C2×C4C24×C4112yes—
531C2×C4C23×D412yes—
532C2×C4C23×D4112yes—
533C2×C4C23×D4112yes—
534C2×C4C23×D412yes—
535C2×C4C22×#41112yes—
536C2×C4C2×2−1+414yes—
537C2×C4C2×2−1+414yes—
538C22—232no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—28no—
550C22—216no—
551C22—44no—
552C22—48no—
553C22—44no—
554C22C22×C4×D414yes—
555C22C4×2−1+418yescomplemented
556C22C25×C411yescharacteristic
557C22C24×D411yescharacteristic
558C22C23×#4111yescharacteristic
559C22C22×2−1+412yes—
560C22C2×#58512yes—
561C4—22no—
562C4—24no—
563C4—212no—
564C4—28no—
565C4—212no—
566C4—216no—
567C4—44no—
568C4—48no—
569C4—46no—
570C4—44no—
571C4C22×C4×D4124yes—
572C4#348614yesdirect product factor
573C2—24no—
574C2—28no—
575C2—48no—
576C2C23×C4×D411yescharacteristic
577C2C2×C4×2−1+412yes—
578C2C2×#348611yescharacteristic
579C2#6278812yes—
580C2#6281111yescharacteristic
subgroupG/H
centerC22×C4C25
commutatorC22C25×C4
Frattini, socle, cosocleC23C26
prime radicalC2×#3486C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×2−1+4C2semidirect
3. C23×C4×Q8C2semidirect
4. #34294C2semidirect
5. #34291C2semidirect
6. #34266C2semidirect
7. #34257C2semidirect
8. #34249C2semidirect
9. #34200C2semidirect
10. #34173C2semidirect
11. C4×D4×Q8C2semidirect
12. C4×#548C2semidirect
13. C4×#536C2semidirect
14. C4×#525C2semidirect
15. C4×#519C2semidirect
18. #3486C4direct (N × H)
25. C2×#520C22semidirect
29. C2×#519C22semidirect
31. C2×C4×#41C22semidirect
39. C22×C4×Q8C22semidirect
42. #2367C22semidirect
43. #2365C22semidirect
44. #2362C22semidirect
47. #2360C22semidirect
49. #2351C22semidirect
50. Q8×C4⋊C4C22semidirect
52. #2341C22semidirect
55. Q8×C22⋊C4C22semidirect
56. #2327C22semidirect
57. #2326C22semidirect
58. #2323C22semidirect
63. C4×C4⋊Q8C22semidirect
64. C4×#124C22semidirect
67. C4×C22⋊Q8C22semidirect
78. #2276C22semidirect
80. C42×Q8C22semidirect
85. C2×2−1+4C2×C4semidirect
90. C23×Q8C2×C4semidirect
98. D4×Q8C2×C4semidirect
99. #548C2×C4semidirect
100. #536C2×C4semidirect
103. #525C2×C4semidirect
114. #519C2×C4semidirect
123. C2×C4×Q8D4semidirect
221. C2×#41C22×C4semidirect
232. C22×Q8C22×C4semidirect
261. C4⋊Q8C22×C4semidirect
262. #124C22×C4semidirect
265. C22⋊Q8C22×C4semidirect
278. C4×Q8C22×C4semidirect
367. C2×Q8C4×D4semidirect
417. C22×C42−1+4semidirect
555. C22C4×2−1+4semidirect
572. C4#3486direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#34862
C2×C4×2−1+42
C23×C4×Q82
#342942
#342912
#342662
#342572
#342492
#342002
#341732
C4×D4×Q82
C4×#5482
C4×#5362
C4×#5252
C4×#5192