Finite Group Databasebeta

← C2×#26058 #6443372

Subgroups — C2×#26058 #6443372

Order 512.

2560 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C22×#474C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic, complemented
4C2×#2349C211yesmaximal, characteristic
5#26058C2116yesmaximal, direct product factor
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2058C211yesmaximal, characteristic, complemented
9C2×#2057C211yesmaximal, characteristic, complemented
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1940C211yesmaximal, characteristic, complemented
12C2×#1864C211yesmaximal, characteristic
13C2×#1862C211yesmaximal, characteristic
14C2×#1847C211yesmaximal, characteristic, complemented
15C2×#1802C211yesmaximal, characteristic
16C2×#1762C211yesmaximal, characteristic
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#476—28no—
24C2×#476C2218yes—
25C2×#474—28no—
26C2×#474C2218yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#97C2211yescharacteristic
41#2551—24no—
42#2551C2218yescomplemented
43C2×#395—21no—
44C2×#395—21no—
45#2349—24no—
46#2349C2218yes—
47C2×#389—21no—
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#385—21no—
53C2×#385—21no—
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#380—21no—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C22×#95—21no—
60#2088C2218yescomplemented
61#2087C2218yescomplemented
62#2058C2218yescomplemented
63#2057C2218yescomplemented
64#1949C2218yescomplemented
65#1940C2218yescomplemented
66#1864C2218yes—
67#1862C2218yes—
68#1847C2218yescomplemented
69#1802C2218yes—
70#1762C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8—21no—
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C24×C4—21no—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—42no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8—48no—
97C2×C4⋊Q8C2314yes—
98C2×C4⋊Q8C2314yes—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—24no—
105C2×C4×D4—42no—
106C2×C4×D4—42no—
107C2×C4×D4Q818yescomplemented
108C2×C4×D4C2314yes—
109C2×C4×D4C2314yes—
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⋊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×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#476—232no—
136#474—232no—
137C2×#107—24no—
138C2×#107—24no—
139C2×#107—42no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106—24no—
144C2×#106—24no—
145C2×#106—42no—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8—24no—
150C2×C4⋊C8—24no—
151C2×C4⋊C8—24no—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155C2×#102—24no—
156C2×#102—24no—
157C2×#102D418yes—
158C2×#102Q818yescomplemented
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#97—24no—
164C2×#97C2314yes—
165#395—28no—
166#395—28no—
167#389—28no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#385—28no—
177#385—28no—
178#384—22no—
179#384C2314yes—
180#382—22no—
181#382C2314yes—
182#380—28no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4D414yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—28no—
195C2×#95D411yescharacteristic
196C2×#95D411yescharacteristic
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2314yes—
205C25—21no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8—82no—
213C22×Q8—82no—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4C2×D414yes—
219C22×D4C2×Q814yes—
220C22×D4C2×Q814yes—
221C22×D4C2412yes—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—28no—
226C23×C4—24no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—82no—
238C23×C4—82no—
239C23×C4—82no—
240C23×C4—82no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8C2×D414yes—
259C22×C8C2×Q814yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—28no—
265C4⋊Q8—28no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—48no—
270C4⋊Q8—432no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—216no—
274C4×D4—28no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4C2×Q8116yescomplemented
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
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—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—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⋊C4—82no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×Q814yes—
320C2×C4⋊C4C2×Q814yes—
321C2×C4⋊C4C2412yes—
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—44no—
333C2×C22⋊C4—44no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—44no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×Q814yes—
344C2×C42#4114yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348#107—28no—
349#107—28no—
350#107—44no—
351#106—28no—
352#106—28no—
353#106—44no—
354C4⋊C8—28no—
355C4⋊C8—28no—
356C4⋊C8—28no—
357#102—28no—
358#102—28no—
359#102C2×D4116yes—
360#102C2×Q8116yescomplemented
361#97—28no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95C2×D414yes—
370#95C2×D414yes—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—22no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24C22×D411yescharacteristic
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—84no—
385C2×Q8—84no—
386C2×Q8—816no—
387C2×Q8—816no—
388C2×D4—24no—
389C2×D4—28no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4C4⋊Q8116yescomplemented
398C2×D4C22×D414yes—
399C2×D4C22×Q814yes—
400C2×D4C22×Q814yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—216no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—816no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—816no—
454C22×C4—816no—
455C22×C4C22⋊Q814yes—
456C22×C4C22⋊Q814yes—
457C22×C4AGL(1,ℤ/8)14yes—
458C22×C4#13714yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+412yes—
472C22×C4C2511yescharacteristic
473C2×C8—28no—
474C2×C8—28no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—44no—
486C2×C8—44no—
487C2×C8C22×D414yes—
488C2×C8C22×Q814yes—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—48no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4C22×D414yes—
518C4⋊C4C22×Q814yes—
519C4⋊C4C22×Q814yes—
520C22⋊C4—216no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—48no—
530C42C22⋊Q818yes—
531C42C22×Q814yes—
532C42C2×#4114yes—
533C23—24no—
534C23—24no—
535C23—216no—
536C23—24no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23C2×C22⋊Q812yes—
548C23D4212yes—
549C23D4×Q812yes—
550C23C2×AGL(1,ℤ/8)12yes—
551C23C2×#13712yes—
552C23#57612yes—
553C23#57712yes—
554C23C23×D411yescharacteristic
555Q8—816no—
556Q8—816no—
557D4—24no—
558D4—216no—
559D4—24no—
560D4—24no—
561D4C2×C4⋊Q818yescomplemented
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—816no—
607C2×C4—816no—
608C2×C4—816no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4C2×C22⋊Q814yes—
614C2×C4C2×C22⋊Q814yes—
615C2×C4C2×C22⋊Q812yes—
616C2×C4C2×C4⋊Q814yes—
617C2×C4#54414yes—
618C2×C4D4212yes—
619C2×C4#54714yes—
620C2×C4D4×Q812yes—
621C2×C4#55414yes—
622C2×C4C2×AGL(1,ℤ/8)14yes—
623C2×C4C2×#13714yes—
624C2×C4C23×D411yescharacteristic
625C2×C4C23×Q811yescharacteristic
626C2×C4C23×Q811yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C8—24no—
631C8—24no—
632C8—42no—
633C8—42no—
634C22—28no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—432no—
642C22#244014yes—
643C22#307914yes—
644C22#308114yes—
645C22#326414yes—
646C22#326614yes—
647C22#338414yes—
648C22#338514yes—
649C22C22×C22⋊Q811yescharacteristic
650C22C2×D4211yescharacteristic
651C22C2×D4×Q811yescharacteristic
652C22C22×AGL(1,ℤ/8)11yescharacteristic
653C22C22×#13711yescharacteristic
654C22C2×#57611yescharacteristic
655C22C2×#57711yescharacteristic
656C4—24no—
657C4—24no—
658C4—24no—
659C4—24no—
660C4—44no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—44no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4#244014yes—
670C4C22×C4⋊Q811yescharacteristic
671C4C2×#54411yescharacteristic
672C4C2×#54711yescharacteristic
673C4C2×#55411yescharacteristic
674C2—48no—
675C2#2605818yesdirect product factor
676C2C2×#244011yescharacteristic
677C2C2×#307911yescharacteristic
678C2C2×#308111yescharacteristic
679C2C2×#326411yescharacteristic
680C2C2×#326611yescharacteristic
681C2C2×#338411yescharacteristic
682C2C2×#338511yescharacteristic
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
3. C2×#2551C2semidirect
5. #26058C2direct (N × H)
6. C2×#2087C2semidirect
7. C2×#2088C2semidirect
8. C2×#2058C2semidirect
9. C2×#2057C2semidirect
10. C2×#1949C2semidirect
11. C2×#1940C2semidirect
14. C2×#1847C2semidirect
42. #2551C22semidirect
60. #2088C22semidirect
61. #2087C22semidirect
62. #2058C22semidirect
63. #2057C22semidirect
64. #1949C22semidirect
65. #1940C22semidirect
68. #1847C22semidirect
107. C2×C4×D4Q8semidirect
158. C2×#102Q8semidirect
279. C4×D4C2×Q8semidirect
360. #102C2×Q8semidirect
397. C2×D4C4⋊Q8semidirect
561. D4C2×C4⋊Q8semidirect
675. C2#26058direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4762
C22×#4742
C2×#25512
C2×#23492
#260582
C2×#20872
C2×#20882
C2×#20582
C2×#20572
C2×#19492
C2×#19402
C2×#18642
C2×#18622
C2×#18472
C2×#18022
C2×#17622