Finite Group Databasebeta

← C2×#29484 #6464243

Subgroups — C2×#29484 #6464243

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2624C211yesmaximal, characteristic, complemented
2C2×#2324C211yesmaximal, characteristic, complemented
3#29484C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic, complemented
5C2×#2034C211yesmaximal, characteristic, complemented
6C2×#2033C211yesmaximal, characteristic, complemented
7C2×#1999C211yesmaximal, characteristic
8C2×#1972C211yesmaximal, characteristic, complemented
9C2×#1944C211yesmaximal, characteristic, complemented
10C2×#1934C211yesmaximal, characteristic, complemented
11C2×#1934C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1920C211yesmaximal, characteristic, complemented
14C2×#1879C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic, complemented
16C2×#1750C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2624—24no—
36#2624C2218yescomplemented
37C2×#399—21no—
38C2×#397—21no—
39C2×#394—21no—
40C2×#394—21no—
41#2324—24no—
42#2324C2218yescomplemented
43C2×#390—21no—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#386—21no—
52C2×#385—21no—
53C2×#380C2211yescharacteristic
54C2×#380C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2077C2218yescomplemented
57#2034C2218yescomplemented
58#2033C2218yescomplemented
59#1999C2218yes—
60#1972C2218yescomplemented
61#1944C2218yescomplemented
62#1934C2218yescomplemented
63#1934C2218yescomplemented
64#1926C2218yescomplemented
65#1920C2218yescomplemented
66#1879C2218yescomplemented
67#1862C2218yescomplemented
68#1750C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C22×D8—24no—
84C22×D8—24no—
85C22×D8—42no—
86C22×D8C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×#121—24no—
92C2×#121—22no—
93C2×#121—22no—
94C2×#121—24no—
95C2×#121—42no—
96C2×#121—42no—
97C2×#121C2314yes—
98C2×#121C2314yes—
99C2×C4×D4—22no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
103C2×#117—42no—
104C2×#117—42no—
105C2×#117D414yes—
106C2×#117C2314yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C2×#107—42no—
135C2×#107C2314yes—
136C2×#106—24no—
137C2×#106—24no—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—24no—
143C2×#102—24no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8—24no—
153C2×C22⋊C8—24no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×#97—42no—
157C2×#97C2314yes—
158C2×C4×C8—42no—
159C2×C4×C8C2314yes—
160#399—28no—
161#397—28no—
162#394—28no—
163#394—28no—
164#390—28no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390C2314yes—
169#390C2314yes—
170#390C2314yes—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#388—28no—
178#386—28no—
179#385—28no—
180#380—22no—
181#380—22no—
182#380C2314yes—
183#380C2314yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2314yes—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25—21no—
202C25C2×D411yescharacteristic
203C22×D4—28no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—44no—
220C22×D4—82no—
221C22×D4—82no—
222C22×D4—82no—
223C22×D4—82no—
224C22×D4#4114yes—
225C22×D4C2412yes—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4—82no—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×D8—28no—
252C2×D8—28no—
253C2×D8—44no—
254C2×D8—416no—
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×C8—22no—
265C22×C8#4114yes—
266C22×C8#4114yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271#121—216no—
272#121—28no—
273#121—28no—
274#121—44no—
275#121—44no—
276#121—48no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—48no—
280#117—48no—
281#117—44no—
282#117—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
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—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4#4114yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
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—42no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D414yes—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—82no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353#107—44no—
354#106—28no—
355#106—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359#102—28no—
360#102—44no—
361#102—44no—
362C22⋊C8—28no—
363C22⋊C8—28no—
364#97—44no—
365C4×C8—44no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
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—
394C24—42no—
395C24C22×D412yes—
396C24C22×D411yescharacteristic
397C2×D4—216no—
398C2×D4—24no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—816no—
429C2×D4—816no—
430C2×D4C2×#4114yes—
431C22×C4—22no—
432C22×C4—24no—
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—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
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—22no—
462C22×C4—48no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
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—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—816no—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4C2511yescharacteristic
500D8—416no—
501C2×C8—28no—
502C2×C8—28no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—44no—
514C2×C8—44no—
515C2×C8C2×#4114yes—
516C2×C8C2×#4114yes—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4C2×#4114yes—
540C22⋊C4—216no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—84no—
560C23—24no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—44no—
569C23—48no—
570C23—44no—
571C23—48no—
572C23—44no—
573C23—48no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23—84no—
602C23#52512yes—
603C23#53212yes—
604C23#53412yes—
605C23#57612yes—
606C23#57612yes—
607C23#57712yes—
608C23#57712yes—
609C23C23×D411yescharacteristic
610D4—24no—
611D4—24no—
612D4—48no—
613D4—42no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—42no—
619D4—816no—
620D4—816no—
621D4—84no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—48no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—816no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4#52512yes—
673C2×C4#53212yes—
674C2×C4#53412yes—
675C2×C4#55114yes—
676C2×C4#55314yes—
677C2×C4#55314yes—
678C2×C4#55914yes—
679C2×C4C22×#4111yescharacteristic
680C2×C4C22×#4111yescharacteristic
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C8—24no—
686C8—24no—
687C8—42no—
688C8—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—44no—
695C22—42no—
696C22—44no—
697C22—42no—
698C22—44no—
699C22—82no—
700C22—82no—
701C22—88no—
702C22—816no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—168no—
713C22#260214yes—
714C22#311814yes—
715C22#311914yes—
716C22#317914yes—
717C22#317914yes—
718C22#319914yes—
719C22#320414yes—
720C22C2×#52511yescharacteristic
721C22C2×#53211yescharacteristic
722C22C2×#53411yescharacteristic
723C22C2×#57611yescharacteristic
724C22C2×#57611yescharacteristic
725C22C2×#57711yescharacteristic
726C22C2×#57711yescharacteristic
727C4—42no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—42no—
732C4—44no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4#260214yes—
739C4C2×#55111yescharacteristic
740C4C2×#55311yescharacteristic
741C4C2×#55311yescharacteristic
742C4C2×#55911yescharacteristic
743C2—82no—
744C2—84no—
745C2—162no—
746C2#2948418yesdirect product factor
747C2C2×#260211yescharacteristic
748C2C2×#311811yescharacteristic
749C2C2×#311911yescharacteristic
750C2C2×#317911yescharacteristic
751C2C2×#317911yescharacteristic
752C2C2×#319911yescharacteristic
753C2C2×#320411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2624C2semidirect
2. C2×#2324C2semidirect
3. #29484C2direct (N × H)
4. C2×#2077C2semidirect
5. C2×#2034C2semidirect
6. C2×#2033C2semidirect
8. C2×#1972C2semidirect
9. C2×#1944C2semidirect
10. C2×#1934C2semidirect
11. C2×#1934C2semidirect
12. C2×#1926C2semidirect
13. C2×#1920C2semidirect
14. C2×#1879C2semidirect
15. C2×#1862C2semidirect
16. C2×#1750C2semidirect
36. #2624C22semidirect
42. #2324C22semidirect
56. #2077C22semidirect
57. #2034C22semidirect
58. #2033C22semidirect
60. #1972C22semidirect
61. #1944C22semidirect
62. #1934C22semidirect
63. #1934C22semidirect
64. #1926C22semidirect
65. #1920C22semidirect
66. #1879C22semidirect
67. #1862C22semidirect
68. #1750C22semidirect
746. C2#29484direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26242
C2×#23242
#294842
C2×#20772
C2×#20342
C2×#20332
C2×#19992
C2×#19722
C2×#19442
C2×#19342
C2×#19262
C2×#19202
C2×#18792
C2×#18622
C2×#17502