Finite Group Databasebeta

← C2×#26394 #6439059

Subgroups — C2×#26394 #6439059

Order 512.

2618 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic, complemented
2C22×#442C211yesmaximal, characteristic
3C2×#2454C211yesmaximal, characteristic, complemented
4C2×#2349C211yesmaximal, characteristic, complemented
5#26394C2116yesmaximal, direct product factor
6C2×#2092C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#1943C211yesmaximal, characteristic, complemented
11C2×#1917C211yesmaximal, characteristic, complemented
12C2×#1879C211yesmaximal, characteristic
13C2×#1879C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic, complemented
16C2×#1862C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#478—28no—
24C2×#478C2218yescomplemented
25C2×#442—28no—
26C2×#442C2218yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2454—24no—
41#2454C2218yescomplemented
42C2×#399—21no—
43C2×#396—21no—
44#2349—24no—
45#2349C2218yescomplemented
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#385—21no—
53C2×#385—21no—
54C2×#385—21no—
55C2×#382C2211yescharacteristic
56C2×#380—21no—
57C2×#380—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C22×#95—21no—
60#2092C2218yescomplemented
61#2085C2218yescomplemented
62#2067C2218yescomplemented
63#2059C2218yescomplemented
64#1943C2218yescomplemented
65#1917C2218yescomplemented
66#1879C2218yes—
67#1879C2218yes—
68#1862C2218yescomplemented
69#1862C2218yescomplemented
70#1862C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4—21no—
83C24×C4—21no—
84C24×C4—21no—
85C22×D8—24no—
86C22×D8—24no—
87C22×D8Q818yes—
88C22×D8C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×#125—24no—
94C2×#125—22no—
95C2×#125—42no—
96C2×#125C2314yes—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—42no—
107C2×C4×D4—42no—
108C2×C4×D4—42no—
109C2×C4×D4C2314yes—
110C2×C4×D4C2314yes—
111C2×C4×D4C2314yes—
112C2×C4×D4C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
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#442—232no—
140C2×#107—24no—
141C2×#107—24no—
142C2×#107—42no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106—24no—
147C2×#106—24no—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8—24no—
151C2×C4⋊C8C2314yes—
152C2×#102—24no—
153C2×#102—24no—
154C2×#102—24no—
155C2×#102—24no—
156C2×#102—42no—
157C2×#102D418yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#97—24no—
165C2×#97C2314yes—
166#399—28no—
167#396—28no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#389C2314yes—
180#385—28no—
181#385—28no—
182#385—28no—
183#382—22no—
184#382C2314yes—
185#380—28no—
186#380—28no—
187C4×C4⋊C4D414yes—
188C4×C4⋊C4C2314yes—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—28no—
196C2×#95D411yescharacteristic
197C2×#95D411yescharacteristic
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2314yes—
203#336C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C25—21no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4C2×D414yes—
219C22×D4C2×Q814yes—
220C22×D4C2×Q814yes—
221C22×D4#4114yes—
222C22×D4C2412yes—
223C22×D4C2412yes—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4—82no—
243C23×C4—82no—
244C23×C4—82no—
245C23×C4—82no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×D8—232no—
254C2×D8—28no—
255C2×D8—28no—
256C2×D8C2×Q8116yes—
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—22no—
266C22×C8—22no—
267C22×C8C2×D414yes—
268C22×C8C2×Q814yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273#125—28no—
274#125—44no—
275#125—48no—
276C4×D4—216no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—48no—
287C4×D4—48no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
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—22no—
299C2×C4⋊C4—22no—
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—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4#4114yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
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—24no—
336C2×C22⋊C4—24no—
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×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2412yes—
356#107—28no—
357#107—28no—
358#107—44no—
359#106—28no—
360#106—28no—
361C4⋊C8—28no—
362#102—28no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—44no—
367#102C2×D4116yes—
368#97—28no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95C2×D414yes—
376#95C2×D414yes—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—22no—
381C24—24no—
382C24—22no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24C22×D411yescharacteristic
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—28no—
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—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4C22⋊Q8116yes—
416C2×D4C22×D414yes—
417C2×D4C22×Q814yes—
418C2×D4C22×Q814yes—
419C2×D4C2×#4114yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
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—22no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
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—42no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—816no—
475C22×C4—816no—
476C22×C4—816no—
477C22×C4—816no—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×Q812yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492D8—232no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—44no—
506C2×C8—44no—
507C2×C8C22×D414yes—
508C2×C8C22×Q814yes—
509C4⋊C4—24no—
510C4⋊C4—24no—
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—22no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4C22×D414yes—
534C4⋊C4C2×#4114yes—
535C22⋊C4—216no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
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—24no—
555C23—216no—
556C23—24no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—48no—
568C23—48no—
569C23—42no—
570C23—42no—
571C23—48no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23#52512yes—
581C23D4212yes—
582C23D4×Q812yes—
583C23#57512yes—
584C23#57512yes—
585C23#57612yes—
586C23#57612yes—
587C23C23×D411yescharacteristic
588D4—24no—
589D4—24no—
590D4—24no—
591D4—24no—
592D4—24no—
593D4—216no—
594D4—42no—
595D4—42no—
596D4—48no—
597D4C2×C22⋊Q818yes—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—48no—
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—42no—
642C2×C4—816no—
643C2×C4—816no—
644C2×C4—816no—
645C2×C4—816no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4C2×C22⋊Q814yes—
651C2×C4#52512yes—
652C2×C4#53914yes—
653C2×C4D4212yes—
654C2×C4#54714yes—
655C2×C4D4×Q812yes—
656C2×C4#55414yes—
657C2×C4C23×D411yescharacteristic
658C2×C4C23×Q811yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×2−1+411yescharacteristic
663C8—24no—
664C8—24no—
665C8—42no—
666C8—42no—
667C22—28no—
668C22—432no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—816no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22#244914yes—
684C22#311114yes—
685C22#312214yes—
686C22#326814yes—
687C22#327314yes—
688C22#338914yes—
689C22#338914yes—
690C22C2×#52511yescharacteristic
691C22C2×D4211yescharacteristic
692C22C2×D4×Q811yescharacteristic
693C22C2×#57511yescharacteristic
694C22C2×#57511yescharacteristic
695C22C2×#57611yescharacteristic
696C22C2×#57611yescharacteristic
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4#244914yes—
710C4C22×C22⋊Q811yescharacteristic
711C4C2×#53911yescharacteristic
712C4C2×#54711yescharacteristic
713C4C2×#55411yescharacteristic
714C2—48no—
715C2—84no—
716C2#2639418yesdirect product factor
717C2C2×#244911yescharacteristic
718C2C2×#311111yescharacteristic
719C2C2×#312211yescharacteristic
720C2C2×#326811yescharacteristic
721C2C2×#327311yescharacteristic
722C2C2×#338911yescharacteristic
723C2C2×#338911yescharacteristic
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
3. C2×#2454C2semidirect
4. C2×#2349C2semidirect
5. #26394C2direct (N × H)
6. C2×#2092C2semidirect
7. C2×#2085C2semidirect
8. C2×#2067C2semidirect
9. C2×#2059C2semidirect
10. C2×#1943C2semidirect
11. C2×#1917C2semidirect
14. C2×#1862C2semidirect
15. C2×#1862C2semidirect
16. C2×#1862C2semidirect
24. C2×#478C22semidirect
41. #2454C22semidirect
45. #2349C22semidirect
60. #2092C22semidirect
61. #2085C22semidirect
62. #2067C22semidirect
63. #2059C22semidirect
64. #1943C22semidirect
65. #1917C22semidirect
68. #1862C22semidirect
69. #1862C22semidirect
70. #1862C22semidirect
716. C2#26394direct (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×#4422
C2×#24542
C2×#23492
#263942
C2×#20922
C2×#20852
C2×#20672
C2×#20592
C2×#19432
C2×#19172
C2×#18792
C2×#18622