Finite Group Databasebeta

← C2×#26480 #6444646

Subgroups — C2×#26480 #6444646

Order 512.

2466 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic, complemented
2C22×C4×SD16C211yesmaximal, characteristic
3C2×#2451C211yesmaximal, characteristic, complemented
4C2×#2452C211yesmaximal, characteristic
5#26480C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2024C211yesmaximal, characteristic, complemented
9C2×#2023C211yesmaximal, characteristic
10C2×#1968C211yesmaximal, characteristic
11C2×#1968C211yesmaximal, characteristic
12C2×#1941C211yesmaximal, characteristic, complemented
13C2×#1941C211yesmaximal, characteristic, complemented
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic
16C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#498—28no—
24C2×#498C2218yescomplemented
25C2×C4×SD16—28no—
26C2×C4×SD16C2218yes—
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2452—24no—
43#2452C2218yes—
44#2451—24no—
45#2451C2218yescomplemented
46C2×#398—21no—
47C2×#396—21no—
48C2×#395—21no—
49C2×#393—21no—
50C2×#391—21no—
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#390C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#389C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#387—21no—
60C2×#385—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2066C2218yescomplemented
63#2065C2218yes—
64#2024C2218yescomplemented
65#2023C2218yes—
66#1968C2218yes—
67#1968C2218yes—
68#1941C2218yescomplemented
69#1941C2218yescomplemented
70#1865C2218yescomplemented
71#1864C2218yes—
72C8×C4⋊C4C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8—21no—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16Q818yescomplemented
87C22×SD16C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8C2314yes—
99C2×C4⋊Q8C2314yes—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—48no—
102C2×C4×Q8—22no—
103C2×C4×Q8—24no—
104C2×C4×Q8—42no—
105C2×C4×Q8C2314yes—
106C2×C4×D4—24no—
107C2×C4×D4—22no—
108C2×C4×D4—42no—
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×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#498—232no—
135C4×SD16—232no—
136C2×#106—24no—
137C2×#106—24no—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106D418yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8—24no—
147C2×C4⋊C8C2314yes—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103—42no—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102—24no—
155C2×#102—24no—
156C2×#102—42no—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C4×C8—24no—
161C2×C4×C8—24no—
162C2×C4×C8—24no—
163C2×C4×C8C2314yes—
164C2×C4×C8C2314yes—
165C2×C4×C8C2314yes—
166#398—28no—
167#396—28no—
168#395—28no—
169#393—28no—
170#391—22no—
171#391—22no—
172#391—28no—
173#391C2314yes—
174#391C2314yes—
175#390—22no—
176#390—22no—
177#390C2314yes—
178#390C2314yes—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187#387—28no—
188#385—28no—
189C4×C4⋊C4D414yes—
190C4×C4⋊C4C2314yes—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
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—
208C22×Q8—21no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8—44no—
220C22×Q8—82no—
221C22×Q8—82no—
222C22×Q8C2×Q814yes—
223C22×Q8C2412yes—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4C2×Q814yes—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—232no—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16C2×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×C8C2×D414yes—
263C22×C8C2×Q814yes—
264C22×C8#4114yes—
265C22×C8#4114yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—28no—
271C4⋊Q8—28no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—48no—
275C4⋊Q8—48no—
276C22⋊Q8—432no—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×D4—28no—
281C4×D4—44no—
282C4×D4—48no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
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—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
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—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4SD1618yescomplemented
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—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#106—28no—
350#106—28no—
351#106—28no—
352#106—28no—
353#106C2×D4116yes—
354C4⋊C8—28no—
355#103—28no—
356#103—28no—
357#103—44no—
358#102—28no—
359#102—28no—
360#102—44no—
361C4×C8—28no—
362C4×C8—28no—
363C4×C8—28no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—24no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24C22×D411yescharacteristic
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8—816no—
402C2×Q8—816no—
403C2×Q8C22×Q814yes—
404C2×D4—28no—
405C2×D4—24no—
406C2×D4—28no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4C22×Q814yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
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—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
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—48no—
461C22×C4—84no—
462C22×C4—816no—
463C22×C4C2×SD1614yes—
464C22×C4C2×SD1614yes—
465C22×C4C2×SD1614yes—
466C22×C4#13514yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×Q812yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481SD16—232no—
482C2×C8—28no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8C22⋊Q8116yescomplemented
494C2×C8C22×D414yes—
495C2×C8C22×Q814yes—
496C2×C8C2×#4114yes—
497C2×C8C2×#4114yes—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4C2×SD16116yescomplemented
523C4⋊C4C22×D414yes—
524C4⋊C4C22×D414yes—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—84no—
530C42—24no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—48no—
540C23—24no—
541C23—48no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23#53212yes—
555C23D4212yes—
556C23D4×Q812yes—
557C23C22×SD1612yes—
558C23C2×#13512yes—
559C23#57712yes—
560C23#57712yes—
561C23C23×D411yescharacteristic
562Q8—24no—
563Q8—24no—
564Q8—42no—
565Q8—42no—
566Q8—48no—
567Q8—816no—
568Q8—816no—
569D4—24no—
570D4—24no—
571D4—42no—
572D4—42no—
573D4—48no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—816no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4C2×C22⋊Q814yes—
624C2×C4#53212yes—
625C2×C4#54214yes—
626C2×C4D4212yes—
627C2×C4#54814yes—
628C2×C4D4×Q812yes—
629C2×C4#55414yes—
630C2×C4C22×SD1614yes—
631C2×C4C22×SD1614yes—
632C2×C4C22×SD1614yes—
633C2×C4C2×#13514yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C23×Q811yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C8—24no—
641C8—42no—
642C8—42no—
643C8C2×C22⋊Q818yescomplemented
644C22—44no—
645C22—44no—
646C22—44no—
647C22—82no—
648C22—816no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22#245214yes—
653C22#316714yes—
654C22#317314yes—
655C22D4×SD1614yes—
656C22#328214yes—
657C22Q8×SD1614yes—
658C22#338114yes—
659C22C2×#53211yescharacteristic
660C22C2×D4211yescharacteristic
661C22C2×D4×Q811yescharacteristic
662C22C23×SD1611yescharacteristic
663C22C22×#13511yescharacteristic
664C22C2×#57711yescharacteristic
665C22C2×#57711yescharacteristic
666C4—24no—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#245214yes—
680C4C22×C22⋊Q811yescharacteristic
681C4C2×#54211yescharacteristic
682C4C2×#54811yescharacteristic
683C4C2×#55411yescharacteristic
684C2—84no—
685C2#2648018yesdirect product factor
686C2C2×#245211yescharacteristic
687C2C2×#316711yescharacteristic
688C2C2×#317311yescharacteristic
689C2C2×D4×SD1611yescharacteristic
690C2C2×#328211yescharacteristic
691C2C2×Q8×SD1611yescharacteristic
692C2C2×#338111yescharacteristic
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×#498C2semidirect
3. C2×#2451C2semidirect
5. #26480C2direct (N × H)
6. C2×#2066C2semidirect
8. C2×#2024C2semidirect
12. C2×#1941C2semidirect
13. C2×#1941C2semidirect
14. C2×#1865C2semidirect
16. C2×C8×C4⋊C4C2semidirect
24. C2×#498C22semidirect
45. #2451C22semidirect
62. #2066C22semidirect
64. #2024C22semidirect
68. #1941C22semidirect
69. #1941C22semidirect
70. #1865C22semidirect
72. C8×C4⋊C4C22semidirect
86. C22×SD16Q8semidirect
253. C2×SD16C2×Q8semidirect
314. C2×C4⋊C4SD16semidirect
493. C2×C8C22⋊Q8semidirect
522. C4⋊C4C2×SD16semidirect
643. C8C2×C22⋊Q8semidirect
685. C2#26480direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4982
C22×C4×SD162
C2×#24512
C2×#24522
#264802
C2×#20662
C2×#20652
C2×#20242
C2×#20232
C2×#19682
C2×#19412
C2×#18652
C2×#18642
C2×C8×C4⋊C42