Finite Group Databasebeta

← C2×#31241 #7503869

Subgroups — C2×#31241 #7503869

Order 512.

2460 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic, complemented
2C22×#476C211yesmaximal, characteristic, complemented
3C2×#2454C211yesmaximal, characteristic, complemented
4C2×#2452C211yesmaximal, characteristic, complemented
5#31241C2116yesmaximal, direct product factor
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2065C211yesmaximal, characteristic, complemented
10C2×#2059C211yesmaximal, characteristic, complemented
11C2×#2057C211yesmaximal, characteristic, complemented
12C2×#1970C211yesmaximal, characteristic
13C2×#1941C211yesmaximal, characteristic, complemented
14C2×#1939C211yesmaximal, characteristic, complemented
15C2×#1879C211yesmaximal, characteristic
16C2×#1879C211yesmaximal, characteristic
17C23×D8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#478—28no—
24C2×#478C2218yescomplemented
25C2×#476—28no—
26C2×#476C2218yescomplemented
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2454—24no—
43#2454C2218yescomplemented
44#2452—24no—
45#2452C2218yescomplemented
46C2×#399—21no—
47C2×#396—21no—
48C2×#396—21no—
49C2×#393—21no—
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#385—21no—
59C2×#385—21no—
60C2×#380—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2067C2218yescomplemented
63#2065C2218yescomplemented
64#2065C2218yescomplemented
65#2065C2218yescomplemented
66#2059C2218yescomplemented
67#2057C2218yescomplemented
68#1970C2218yes—
69#1941C2218yescomplemented
70#1939C2218yescomplemented
71#1879C2218yes—
72#1879C2218yes—
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C22×D8—42no—
85C22×D8—42no—
86C22×D8Q818yescomplemented
87C22×D8C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—42no—
95C2×C4⋊Q8C2314yes—
96C2×#125—24no—
97C2×#125—22no—
98C2×#125—42no—
99C2×#125C2314yes—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—48no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—42no—
107C2×C4×D4—42no—
108C2×C4×D4C2314yes—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
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×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#478—232no—
139#476—232no—
140C2×#107—24no—
141C2×#107—42no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106—42no—
146C2×#106D418yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8—24no—
151C2×C4⋊C8—24no—
152C2×C4⋊C8—42no—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2314yes—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102—42no—
161C2×#102—42no—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×C4×C8—42no—
169C2×C4×C8C2314yes—
170#399—28no—
171#396—28no—
172#396—28no—
173#393—28no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#390C2314yes—
181#390C2314yes—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389C2314yes—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#385—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×#95D411yescharacteristic
200C2×#95D411yescharacteristic
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207C25—21no—
208C25—21no—
209C22×Q8—24no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8—82no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4C2×Q814yes—
226C22×D4C2×Q814yes—
227C22×D4C2412yes—
228C22×D4C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×D8—416no—
251C2×D8—44no—
252C2×D8—44no—
253C2×D8C2×Q8116yescomplemented
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8C2×D414yes—
265C22×C8C2×Q814yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—28no—
271C4⋊Q8—44no—
272C4⋊Q8—48no—
273#125—28no—
274#125—44no—
275#125—48no—
276C22⋊Q8—432no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—48no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
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—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
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—24no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
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×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357#107—28no—
358#107—44no—
359#106—28no—
360#106—44no—
361#106C2×D4116yes—
362C4⋊C8—28no—
363C4⋊C8—28no—
364C4⋊C8—44no—
365#102—28no—
366#102—28no—
367#102—28no—
368#102—28no—
369#102—44no—
370#102—44no—
371C4×C8—44no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95C2×D414yes—
377#95C2×D414yes—
378C24—24no—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22×D411yescharacteristic
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—84no—
398C2×Q8—816no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4C22×Q814yes—
424C2×D4C22×Q814yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
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—48no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—816no—
472C22×C4—816no—
473C22×C4#13514yes—
474C22×C4AGL(1,ℤ/8)14yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×Q812yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+412yes—
487C22×C42−1+412yes—
488C22×C4C2511yescharacteristic
489D8—416no—
490C2×C8—28no—
491C2×C8—28no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8C22×D414yes—
505C2×C8C22×Q814yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—48no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4C22×D414yes—
532C4⋊C4C22×D414yes—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C23—24no—
553C23—24no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—42no—
559C23—42no—
560C23—48no—
561C23—48no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—84no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23#53712yes—
581C23D4212yes—
582C23D4×Q812yes—
583C23C2×#13512yes—
584C23C2×AGL(1,ℤ/8)12yes—
585C23#57612yes—
586C23#57712yes—
587C23C23×D411yescharacteristic
588Q8—816no—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—24no—
593D4—42no—
594D4—42no—
595D4—48no—
596D4—48no—
597D4—42no—
598D4—42no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—48no—
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—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—816no—
642C2×C4—816no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4#53714yes—
648C2×C4#53712yes—
649C2×C4#54314yes—
650C2×C4D4212yes—
651C2×C4D4×Q812yes—
652C2×C4D4×Q814yes—
653C2×C4#56314yes—
654C2×C4C2×#13514yes—
655C2×C4C2×AGL(1,ℤ/8)14yes—
656C2×C4C23×D411yescharacteristic
657C2×C4C23×Q811yescharacteristic
658C2×C4C2×H(5,𝔽2)11yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C2×C4C2×2−1+411yescharacteristic
662C8—24no—
663C8—24no—
664C8—42no—
665C8—42no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—816no—
673C22—816no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22#268314yes—
683C22#324714yes—
684C22#325714yes—
685C22#327414yes—
686C22#328214yes—
687C22#338014yes—
688C22#338414yes—
689C22C2×#53711yescharacteristic
690C22C2×D4211yescharacteristic
691C22C2×D4×Q811yescharacteristic
692C22C22×#13511yescharacteristic
693C22C22×AGL(1,ℤ/8)11yescharacteristic
694C22C2×#57611yescharacteristic
695C22C2×#57711yescharacteristic
696C4—24no—
697C4—24no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4#268314yes—
709C4C2×#53711yescharacteristic
710C4C2×#54311yescharacteristic
711C4C2×D4×Q811yescharacteristic
712C4C2×#56311yescharacteristic
713C2—84no—
714C2—84no—
715C2#3124118yesdirect product factor
716C2C2×#268311yescharacteristic
717C2C2×#324711yescharacteristic
718C2C2×#325711yescharacteristic
719C2C2×#327411yescharacteristic
720C2C2×#328211yescharacteristic
721C2C2×#338011yescharacteristic
722C2C2×#338411yescharacteristic
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×#478C2semidirect
2. C22×#476C2semidirect
3. C2×#2454C2semidirect
4. C2×#2452C2semidirect
5. #31241C2direct (N × H)
6. C2×#2067C2semidirect
7. C2×#2065C2semidirect
8. C2×#2065C2semidirect
9. C2×#2065C2semidirect
10. C2×#2059C2semidirect
11. C2×#2057C2semidirect
13. C2×#1941C2semidirect
14. C2×#1939C2semidirect
24. C2×#478C22semidirect
26. C2×#476C22semidirect
43. #2454C22semidirect
45. #2452C22semidirect
62. #2067C22semidirect
63. #2065C22semidirect
64. #2065C22semidirect
65. #2065C22semidirect
66. #2059C22semidirect
67. #2057C22semidirect
69. #1941C22semidirect
70. #1939C22semidirect
86. C22×D8Q8semidirect
253. C2×D8C2×Q8semidirect
715. C2#31241direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4782
C22×#4762
C2×#24542
C2×#24522
#312412
C2×#20672
C2×#20652
C2×#20592
C2×#20572
C2×#19702
C2×#19412
C2×#19392
C2×#18792