Finite Group Databasebeta

← C2×#26309 #6447219

Subgroups — C2×#26309 #6447219

Order 512.

1996 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
1C2×#2562C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#26309C2116yesmaximal, direct product factor
4C2×#2077C211yesmaximal, characteristic, complemented
5C2×#2077C211yesmaximal, characteristic, complemented
6C2×#2077C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic, complemented
8C2×#1992C211yesmaximal, characteristic, complemented
9C2×#1989C211yesmaximal, characteristic, complemented
10C2×#1955C211yesmaximal, characteristic
11C2×#1948C211yesmaximal, characteristic, complemented
12C2×#1854C211yesmaximal, characteristic, complemented
13C2×#1854C211yesmaximal, characteristic, complemented
14C2×#1825C211yesmaximal, characteristic, complemented
15C2×#1825C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2562—24no—
32#2562C2218yes—
33C2×#400—21no—
34C2×#394—21no—
35#2354—24no—
36#2354C2218yes—
37C2×#390—21no—
38C2×#388—21no—
39C2×#388—21no—
40C2×#388—21no—
41C2×#388—21no—
42C2×#384C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2077C2218yescomplemented
53#2077C2218yescomplemented
54#2077C2218yescomplemented
55#2077C2218yescomplemented
56#1992C2218yescomplemented
57#1989C2218yescomplemented
58#1955C2218yes—
59#1948C2218yescomplemented
60#1854C2218yescomplemented
61#1854C2218yescomplemented
62#1825C2218yescomplemented
63#1825C2218yescomplemented
64#1776C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C23×D4D411yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#125—22no—
85C2×#125—24no—
86C2×#125—24no—
87C2×#125C2314yes—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4D414yes—
91C2×C4×D4C2314yes—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—42no—
97C2×#117—42no—
98C2×#117C2314yes—
99C2×#117C2314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C2×#107—24no—
129C2×#107C2314yes—
130C2×#106—24no—
131C2×#106—42no—
132C2×#106—42no—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8—24no—
137C2×C4⋊C8C2314yes—
138C2×C22⋊C8—24no—
139C2×C22⋊C8—24no—
140C2×C22⋊C8—42no—
141C2×C22⋊C8—42no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97—24no—
147C2×#97C2314yes—
148#400—28no—
149#394—28no—
150#390—28no—
151#388—28no—
152#388—28no—
153#388—28no—
154#388—28no—
155#384—22no—
156#384C2314yes—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380C2314yes—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167#380C2314yes—
168#380C2314yes—
169C4×C4⋊C4—28no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194C25C2×D411yescharacteristic
195C25C2×D411yescharacteristic
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4C2×D414yes—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C2×D412yes—
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8#4114yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245#125—28no—
246#125—28no—
247#125—48no—
248C4×D4—216no—
249C4×D4—28no—
250C4×D4—28no—
251#117—28no—
252#117—28no—
253#117—44no—
254#117—44no—
255#117—48no—
256#117—48no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4#4114yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4—82no—
317C2×C22⋊C4C2×D414yes—
318C2×C22⋊C4C2×D414yes—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—82no—
327C2×C42C2×D412yes—
328C2×C42#4114yes—
329C2×C42#4114yes—
330C2×C42C2412yes—
331#107—28no—
332#106—28no—
333#106—44no—
334#106—44no—
335C4⋊C8—28no—
336C22⋊C8—28no—
337C22⋊C8—28no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340#97—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C22×D411yescharacteristic
366C2×D4—28no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—416no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
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#12314yes—
439C22×C4#12314yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+412yes—
451C22×C4C2511yescharacteristic
452C2×C8—28no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—44no—
465C2×C8—44no—
466C2×C8—44no—
467C2×C8C2×#4114yes—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—84no—
496C4⋊C4C2×#4114yes—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C42—48no—
508C42—48no—
509C42—48no—
510C42—84no—
511C42#12318yes—
512C42C2×#4114yes—
513C42C2×#4114yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23C2×#12312yes—
543C23#54612yes—
544C23#54712yes—
545C23#57512yes—
546C23#57512yes—
547C23#57712yes—
548C23#57712yes—
549C23C23×D411yescharacteristic
550D4—416no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
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—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4C2×#12314yes—
608C2×C4C2×#12314yes—
609C2×C4C2×#12312yes—
610C2×C4#52914yes—
611C2×C4#54214yes—
612C2×C4#54612yes—
613C2×C4#54712yes—
614C2×C4#55614yes—
615C2×C4#55914yes—
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C8—24no—
623C8—42no—
624C8—42no—
625C8—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—88no—
641C22—88no—
642C22#244714yes—
643C22#309314yes—
644C22#309914yes—
645C22#330714yes—
646C22#331714yes—
647C22#334814yes—
648C22#335014yes—
649C22C22×#12311yescharacteristic
650C22C2×#54611yescharacteristic
651C22C2×#54711yescharacteristic
652C22C2×#57511yescharacteristic
653C22C2×#57511yescharacteristic
654C22C2×#57711yescharacteristic
655C22C2×#57711yescharacteristic
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#244714yes—
669C4C2×#52911yescharacteristic
670C4C2×#54211yescharacteristic
671C4C2×#55611yescharacteristic
672C4C2×#55911yescharacteristic
673C2—82no—
674C2—82no—
675C2#2630918yesdirect product factor
676C2C2×#244711yescharacteristic
677C2C2×#309311yescharacteristic
678C2C2×#309911yescharacteristic
679C2C2×#330711yescharacteristic
680C2C2×#331711yescharacteristic
681C2C2×#334811yescharacteristic
682C2C2×#335011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26309C2direct (N × H)
4. C2×#2077C2semidirect
5. C2×#2077C2semidirect
6. C2×#2077C2semidirect
7. C2×#2077C2semidirect
8. C2×#1992C2semidirect
9. C2×#1989C2semidirect
11. C2×#1948C2semidirect
12. C2×#1854C2semidirect
13. C2×#1854C2semidirect
14. C2×#1825C2semidirect
15. C2×#1825C2semidirect
16. C2×#1776C2semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2077C22semidirect
55. #2077C22semidirect
56. #1992C22semidirect
57. #1989C22semidirect
59. #1948C22semidirect
60. #1854C22semidirect
61. #1854C22semidirect
62. #1825C22semidirect
63. #1825C22semidirect
64. #1776C22semidirect
675. C2#26309direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#23542
#263092
C2×#20772
C2×#19922
C2×#19892
C2×#19552
C2×#19482
C2×#18542
C2×#18252
C2×#17762