Finite Group Databasebeta

← C2×#31159 #7503594

Subgroups — C2×#31159 #7503594

Order 512.

2378 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C22×#461C211yesmaximal, characteristic
3C2×#2446C211yesmaximal, characteristic, complemented
4C2×#2452C211yesmaximal, characteristic
5#31159C2116yesmaximal, direct product factor
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2066C211yesmaximal, characteristic, complemented
8C2×#2055C211yesmaximal, characteristic
9C2×#2055C211yesmaximal, characteristic
10C2×#2043C211yesmaximal, characteristic, complemented
11C2×#2024C211yesmaximal, characteristic, complemented
12C2×#1968C211yesmaximal, characteristic
13C2×#1939C211yesmaximal, characteristic, complemented
14C2×#1933C211yesmaximal, characteristic, complemented
15C2×#1880C211yesmaximal, characteristic
16C2×#1864C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8—21no—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C2×#475—28no—
25C2×#475C2218yescomplemented
26C2×#461—28no—
27C2×#461C2218yes—
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2452—24no—
43#2452C2218yes—
44#2446—24no—
45#2446C2218yescomplemented
46C2×#400—21no—
47C2×#398—21no—
48C2×#396—21no—
49C2×#393—21no—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#387—21no—
59C2×#387—21no—
60C2×#385—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2088C2218yescomplemented
63#2066C2218yescomplemented
64#2055C2218yes—
65#2055C2218yes—
66#2043C2218yescomplemented
67#2024C2218yescomplemented
68#1968C2218yes—
69#1939C2218yescomplemented
70#1933C2218yescomplemented
71#1880C2218yes—
72#1864C2218yes—
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16—24no—
88C22×SD16—42no—
89C22×SD16—42no—
90C22×SD16C2314yes—
91C22×SD16C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8C2314yes—
103C2×C4⋊Q8C2314yes—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—48no—
106C2×C4×Q8—22no—
107C2×C4×Q8—24no—
108C2×C4×Q8—42no—
109C2×C4×Q8C2314yes—
110C2×C4×D4—24no—
111C2×C4×D4—22no—
112C2×C4×D4—42no—
113C2×C4×D4C2314yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#475—232no—
138#461—232no—
139C2×#106—24no—
140C2×#106—24no—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8—24no—
144C2×C4⋊C8—24no—
145C2×C4⋊C8—42no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103—42no—
152C2×#103—42no—
153C2×#103—42no—
154C2×#103D418yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102—24no—
162C2×#102—42no—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×C4×C8—42no—
166C2×C4×C8C2314yes—
167#400—28no—
168#398—28no—
169#396—28no—
170#393—28no—
171#391—22no—
172#391—22no—
173#391—22no—
174#391C2314yes—
175#391C2314yes—
176#391C2314yes—
177#390—22no—
178#390—22no—
179#390C2314yes—
180#390C2314yes—
181#389—22no—
182#389—22no—
183#389—22no—
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187#387—28no—
188#387—28no—
189#385—28no—
190C4×C4⋊C4D414yes—
191C4×C4⋊C4C2314yes—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209C25—21no—
210C22×Q8—21no—
211C22×Q8—21no—
212C22×Q8—24no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—82no—
227C22×Q8C2×D414yes—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4#4114yes—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—24no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—28no—
260C2×SD16—44no—
261C2×SD16—44no—
262C2×SD16—416no—
263C2×SD16—416no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8C2×D414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8—28no—
281C4⋊Q8—28no—
282C4⋊Q8—44no—
283C4⋊Q8—48no—
284C4⋊Q8—44no—
285C4⋊Q8—48no—
286C22⋊Q8—432no—
287C4×Q8—28no—
288C4×Q8—44no—
289C4×Q8—48no—
290C4×D4—28no—
291C4×D4—44no—
292C4×D4—48no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4SD1618yescomplemented
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—82no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—44no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355#106—28no—
356#106—28no—
357C4⋊C8—28no—
358C4⋊C8—28no—
359C4⋊C8—44no—
360#103—28no—
361#103—28no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103C2×D4116yes—
366#102—28no—
367#102—44no—
368C4×C8—44no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95C2×D414yes—
379#95C2×D414yes—
380C24—24no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—48no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—48no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—84no—
417C2×Q8—816no—
418C2×Q8C22×D414yes—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4C2×#4114yes—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—48no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—48no—
477C22×C4—84no—
478C22×C4—816no—
479C22×C4C2×SD1614yes—
480C22×C4C2×SD1614yes—
481C22×C4C2×SD1614yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C42−1+412yes—
495C22×C4C2511yescharacteristic
496SD16—416no—
497SD16—416no—
498C2×C8—28no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—44no—
511C2×C8—44no—
512C2×C8—44no—
513C2×C8C22×D414yes—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4C2×SD16116yescomplemented
537C4⋊C4C22×D414yes—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—84no—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—48no—
553C23—24no—
554C23—48no—
555C23—48no—
556C23—42no—
557C23—48no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23#53712yes—
568C23D4212yes—
569C23#54812yes—
570C23C22×SD1612yes—
571C23#57512yes—
572C23#57712yes—
573C23#57812yes—
574C23C23×D411yescharacteristic
575Q8—24no—
576Q8—24no—
577Q8—48no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—42no—
584Q8—816no—
585Q8—84no—
586D4—24no—
587D4—24no—
588D4—42no—
589D4—48no—
590D4—42no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—816no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4#53614yes—
642C2×C4#53712yes—
643C2×C4#53914yes—
644C2×C4D4212yes—
645C2×C4#54614yes—
646C2×C4#54812yes—
647C2×C4#56214yes—
648C2×C4C22×SD1614yes—
649C2×C4C22×SD1614yes—
650C2×C4C22×SD1614yes—
651C2×C4C23×D411yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C8—24no—
658C8—42no—
659C8—42no—
660C8—42no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—82no—
665C22—82no—
666C22—816no—
667C22—82no—
668C22—82no—
669C22#268114yes—
670C22#324614yes—
671C22#325814yes—
672C22#327714yes—
673C22D4×SD1614yes—
674C22#335914yes—
675C22#337614yes—
676C22C2×#53711yescharacteristic
677C22C2×D4211yescharacteristic
678C22C2×#54811yescharacteristic
679C22C23×SD1611yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57711yescharacteristic
682C22C2×#57811yescharacteristic
683C4—24no—
684C4—24no—
685C4—24no—
686C4—42no—
687C4—42no—
688C4—44no—
689C4—44no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#268114yes—
697C4C2×#53611yescharacteristic
698C4C2×#53911yescharacteristic
699C4C2×#54611yescharacteristic
700C4C2×#56211yescharacteristic
701C2—84no—
702C2#3115918yesdirect product factor
703C2C2×#268111yescharacteristic
704C2C2×#324611yescharacteristic
705C2C2×#325811yescharacteristic
706C2C2×#327711yescharacteristic
707C2C2×D4×SD1611yescharacteristic
708C2C2×#335911yescharacteristic
709C2C2×#337611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#475C2semidirect
3. C2×#2446C2semidirect
5. #31159C2direct (N × H)
6. C2×#2088C2semidirect
7. C2×#2066C2semidirect
10. C2×#2043C2semidirect
11. C2×#2024C2semidirect
13. C2×#1939C2semidirect
14. C2×#1933C2semidirect
25. C2×#475C22semidirect
45. #2446C22semidirect
62. #2088C22semidirect
63. #2066C22semidirect
66. #2043C22semidirect
67. #2024C22semidirect
69. #1939C22semidirect
70. #1933C22semidirect
322. C2×C4⋊C4SD16semidirect
536. C4⋊C4C2×SD16semidirect
702. C2#31159direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4752
C22×#4612
C2×#24462
C2×#24522
#311592
C2×#20882
C2×#20662
C2×#20552
C2×#20432
C2×#20242
C2×#19682
C2×#19392
C2×#19332
C2×#18802
C2×#18642