Finite Group Databasebeta

← C2×#31153 #7503591

Subgroups — C2×#31153 #7503591

Order 512.

2324 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C22×#463C211yesmaximal, characteristic
3C2×#2448C211yesmaximal, characteristic, complemented
4C2×#2454C211yesmaximal, characteristic, complemented
5#31153C2116yesmaximal, direct product factor
6C2×#2086C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic, complemented
8C2×#2055C211yesmaximal, characteristic, complemented
9C2×#2054C211yesmaximal, characteristic
10C2×#2042C211yesmaximal, characteristic, complemented
11C2×#2025C211yesmaximal, characteristic, complemented
12C2×#1970C211yesmaximal, characteristic
13C2×#1939C211yesmaximal, characteristic, complemented
14C2×#1933C211yesmaximal, characteristic, complemented
15C2×#1880C211yesmaximal, characteristic, complemented
16C2×#1864C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#479—28no—
24C2×#479C2218yescomplemented
25C2×#463—28no—
26C2×#463C2218yes—
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2454—24no—
42#2454C2218yescomplemented
43#2448—24no—
44#2448C2218yescomplemented
45C2×#399—21no—
46C2×#399—21no—
47C2×#397—21no—
48C2×#396—21no—
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#388—21no—
58C2×#388—21no—
59C2×#385—21no—
60C2×#380—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2086C2218yescomplemented
63#2065C2218yescomplemented
64#2055C2218yescomplemented
65#2054C2218yes—
66#2042C2218yescomplemented
67#2025C2218yescomplemented
68#1970C2218yes—
69#1939C2218yescomplemented
70#1933C2218yescomplemented
71#1880C2218yescomplemented
72#1864C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—42no—
86C22×SD16C2314yes—
87C22×D8—24no—
88C22×D8—24no—
89C22×D8—42no—
90C22×D8C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×#125—24no—
96C2×#125—22no—
97C2×#125—42no—
98C2×#125C2314yes—
99C2×#124—24no—
100C2×#124—22no—
101C2×#124—42no—
102C2×#124C2314yes—
103C2×C4×Q8—22no—
104C2×C4×Q8—24no—
105C2×C4×Q8—42no—
106C2×C4×Q8C2314yes—
107C2×C4×D4—24no—
108C2×C4×D4—22no—
109C2×C4×D4—42no—
110C2×C4×D4C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#479—232no—
140#463—232no—
141C2×#107—24no—
142C2×#107C2314yes—
143C2×#106—24no—
144C2×#106C2314yes—
145C2×C4⋊C8—24no—
146C2×C4⋊C8—24no—
147C2×C4⋊C8—42no—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×#103—24no—
152C2×#103—42no—
153C2×#103D418yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102—42no—
160C2×#102—42no—
161C2×#102—42no—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C4×C8—42no—
168C2×C4×C8C2314yes—
169#399—28no—
170#399—28no—
171#397—28no—
172#396—28no—
173#391—22no—
174#391C2314yes—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390C2314yes—
180#390C2314yes—
181#390C2314yes—
182#390C2314yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2314yes—
187#389C2314yes—
188#389C2314yes—
189#388—28no—
190#388—28no—
191#385—28no—
192#380—28no—
193C4×C4⋊C4D414yes—
194C4×C4⋊C4C2314yes—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25—21no—
209C25—21no—
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8C2×D414yes—
217C22×Q8C2412yes—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4#4114yes—
232C22×D4C2412yes—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—82no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—416no—
255C2×D8—28no—
256C2×D8—28no—
257C2×D8—44no—
258C2×D8—416no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8C2×D414yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#125—28no—
276#125—44no—
277#125—48no—
278#124—28no—
279#124—44no—
280#124—48no—
281C4×Q8—28no—
282C4×Q8—44no—
283C4×Q8—48no—
284C4×D4—28no—
285C4×D4—44no—
286C4×D4—48no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4—82no—
342C2×C22⋊C4—82no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4—82no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—24no—
352C2×C42—44no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359#107—28no—
360#106—28no—
361C4⋊C8—28no—
362C4⋊C8—28no—
363C4⋊C8—44no—
364#103—28no—
365#103—44no—
366#103C2×D4116yes—
367#102—28no—
368#102—28no—
369#102—44no—
370#102—44no—
371#102—44no—
372C4×C8—44no—
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—22no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C22×D411yescharacteristic
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8C22×D414yes—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4C2×#4114yes—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
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—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—48no—
462C22×C4—48no—
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—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—84no—
480C22×C4—816no—
481C22×C4#13514yes—
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—
497D8—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—28no—
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—22no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4C22×D414yes—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C42—24no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—48no—
558C23—24no—
559C23—48no—
560C23—48no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23—84no—
584C23—84no—
585C23—84no—
586C23#53712yes—
587C23D4212yes—
588C23#54812yes—
589C23C2×#13512yes—
590C23#57512yes—
591C23#57612yes—
592C23#57712yes—
593C23C23×D411yescharacteristic
594Q8—24no—
595Q8—24no—
596Q8—48no—
597Q8—42no—
598Q8—42no—
599D4—24no—
600D4—24no—
601D4—42no—
602D4—42no—
603D4—48no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—84no—
609C2×C4—22no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—42no—
630C2×C4—48no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—48no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—816no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4#53614yes—
658C2×C4#53712yes—
659C2×C4#53914yes—
660C2×C4D4212yes—
661C2×C4#54614yes—
662C2×C4#54812yes—
663C2×C4#56214yes—
664C2×C4C2×#13514yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C8—24no—
672C8—42no—
673C8—42no—
674C8—42no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—82no—
679C22—82no—
680C22—816no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—168no—
691C22#268114yes—
692C22#325014yes—
693C22#325714yes—
694C22#327514yes—
695C22#328714yes—
696C22#336014yes—
697C22#337014yes—
698C22C2×#53711yescharacteristic
699C22C2×D4211yescharacteristic
700C22C2×#54811yescharacteristic
701C22C22×#13511yescharacteristic
702C22C2×#57511yescharacteristic
703C22C2×#57611yescharacteristic
704C22C2×#57711yescharacteristic
705C4—24no—
706C4—42no—
707C4—42no—
708C4—44no—
709C4—42no—
710C4—42no—
711C4—44no—
712C4—42no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4#268114yes—
718C4C2×#53611yescharacteristic
719C4C2×#53911yescharacteristic
720C4C2×#54611yescharacteristic
721C4C2×#56211yescharacteristic
722C2—84no—
723C2—162no—
724C2#3115318yesdirect product factor
725C2C2×#268111yescharacteristic
726C2C2×#325011yescharacteristic
727C2C2×#325711yescharacteristic
728C2C2×#327511yescharacteristic
729C2C2×#328711yescharacteristic
730C2C2×#336011yescharacteristic
731C2C2×#337011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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×#479C2semidirect
3. C2×#2448C2semidirect
4. C2×#2454C2semidirect
5. #31153C2direct (N × H)
6. C2×#2086C2semidirect
7. C2×#2065C2semidirect
8. C2×#2055C2semidirect
10. C2×#2042C2semidirect
11. C2×#2025C2semidirect
13. C2×#1939C2semidirect
14. C2×#1933C2semidirect
15. C2×#1880C2semidirect
16. C2×#1864C2semidirect
24. C2×#479C22semidirect
42. #2454C22semidirect
44. #2448C22semidirect
62. #2086C22semidirect
63. #2065C22semidirect
64. #2055C22semidirect
66. #2042C22semidirect
67. #2025C22semidirect
69. #1939C22semidirect
70. #1933C22semidirect
71. #1880C22semidirect
72. #1864C22semidirect
724. C2#31153direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4792
C22×#4632
C2×#24482
C2×#24542
#311532
C2×#20862
C2×#20652
C2×#20552
C2×#20542
C2×#20422
C2×#20252
C2×#19702
C2×#19392
C2×#19332
C2×#18802
C2×#18642