Finite Group Databasebeta

← C2×#26678 #6446927

Subgroups — C2×#26678 #6446927

Order 512.

2120 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3#26678C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2066C211yesmaximal, characteristic, complemented
6C2×#2035C211yesmaximal, characteristic
7C2×#2001C211yesmaximal, characteristic
8C2×#1935C211yesmaximal, characteristic, complemented
9C2×#1924C211yesmaximal, characteristic
10C2×#1920C211yesmaximal, characteristic, complemented
11C2×#1878C211yesmaximal, characteristic, complemented
12C2×#1878C211yesmaximal, characteristic, complemented
13C2×#1872C211yesmaximal, characteristic
14C2×#1824C211yesmaximal, characteristic
15C2×#1805C211yesmaximal, characteristic, complemented
16C2×#1763C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2421—24no—
35#2421C2218yes—
36C2×#400—21no—
37C2×#397—21no—
38#2360—24no—
39#2360C2218yes—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389—21no—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#387—21no—
47C2×#387—21no—
48C2×#385—21no—
49C2×#382C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4C2211yescharacteristic
54#2101C2218yescomplemented
55#2066C2218yescomplemented
56#2035C2218yes—
57#2001C2218yes—
58#1935C2218yescomplemented
59#1924C2218yes—
60#1920C2218yescomplemented
61#1878C2218yescomplemented
62#1878C2218yescomplemented
63#1872C2218yes—
64#1824C2218yes—
65#1805C2218yescomplemented
66#1763C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C22×Q16—24no—
82C22×Q16—24no—
83C22×Q16—42no—
84C22×Q16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—42no—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—42no—
102C2×C4×Q8—42no—
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—42no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C2×#106—24no—
133C2×#106C2314yes—
134C2×C4⋊C8—42no—
135C2×C4⋊C8C2314yes—
136C2×#103—24no—
137C2×#103—24no—
138C2×#103—24no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C22⋊C8—42no—
149C2×C22⋊C8—42no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97—42no—
153C2×#97C2314yes—
154#400—28no—
155#397—28no—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#389—22no—
161#389—22no—
162#389—28no—
163#389C2314yes—
164#389C2314yes—
165#388—28no—
166#387—28no—
167#387—28no—
168#385—28no—
169#382—22no—
170#382C2314yes—
171#380—22no—
172#380C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—28no—
176C4×C22⋊C4D414yes—
177C4×C22⋊C4C2314yes—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8#4114yes—
214C22×Q8#4114yes—
215C22×Q8C2412yes—
216C22×Q8C2412yes—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—82no—
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—44no—
246C2×Q16—416no—
247C22×C8—22no—
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×C8—22no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22⋊Q8—216no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—44no—
268C22⋊Q8—48no—
269C4×Q8—28no—
270C4×Q8—28no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×Q8—48no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—48no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4#4114yes—
309C2×C4⋊C4#4114yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D414yes—
325C2×C22⋊C4C2×D414yes—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42—82no—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340#106—28no—
341C4⋊C8—44no—
342#103—28no—
343#103—28no—
344#103—28no—
345#103—44no—
346#103—44no—
347#103—44no—
348C22⋊C8—44no—
349C22⋊C8—44no—
350#97—44no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95C2×D414yes—
362#95C2×D414yes—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24#13714yes—
370C24C22×D412yes—
371C24C22×D411yescharacteristic
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8C2×#4114yes—
397C2×Q8C2×#4114yes—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—48no—
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—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
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—84no—
461C22×C4—816no—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4C2511yescharacteristic
475Q16—416no—
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—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—44no—
489C2×C8—44no—
490C2×C8—44no—
491C2×C8—44no—
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—24no—
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—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—84no—
515C4⋊C4C2×#4114yes—
516C4⋊C4C2×#4114yes—
517C22⋊C4—216no—
518C22⋊C4—28no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—48no—
529C42—48no—
530C42—84no—
531C23—28no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—28no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23#52512yes—
545C23#54612yes—
546C23#54612yes—
547C23C2×#13714yes—
548C23C2×#13712yes—
549C23#57512yes—
550C23#57712yes—
551C23#57812yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—24no—
555Q8—24no—
556Q8—24no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—48no—
562Q8—48no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—48no—
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—816no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4#52512yes—
622C2×C4#52914yes—
623C2×C4#54014yes—
624C2×C4#54612yes—
625C2×C4#54612yes—
626C2×C4#55114yes—
627C2×C4#55314yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C8—42no—
635C8—42no—
636C8—42no—
637C8—42no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—416no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22#245714yes—
647C22#311714yes—
648C22#312014yes—
649C22#331114yes—
650C22#331214yes—
651C22#331614yes—
652C22#332714yes—
653C22C2×#52511yescharacteristic
654C22C2×#54611yescharacteristic
655C22C2×#54611yescharacteristic
656C22C22×#13711yescharacteristic
657C22C2×#57511yescharacteristic
658C22C2×#57711yescharacteristic
659C22C2×#57811yescharacteristic
660C4—44no—
661C4—42no—
662C4—44no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#245714yes—
674C4C2×#52911yescharacteristic
675C4C2×#54011yescharacteristic
676C4C2×#55111yescharacteristic
677C4C2×#55311yescharacteristic
678C2—44no—
679C2#2667818yesdirect product factor
680C2C2×#245711yescharacteristic
681C2C2×#311711yescharacteristic
682C2C2×#312011yescharacteristic
683C2C2×#331611yescharacteristic
684C2C2×#331211yescharacteristic
685C2C2×#331111yescharacteristic
686C2C2×#332711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26678C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2066C2semidirect
8. C2×#1935C2semidirect
10. C2×#1920C2semidirect
11. C2×#1878C2semidirect
12. C2×#1878C2semidirect
15. C2×#1805C2semidirect
16. C2×#1763C2semidirect
54. #2101C22semidirect
55. #2066C22semidirect
58. #1935C22semidirect
60. #1920C22semidirect
61. #1878C22semidirect
62. #1878C22semidirect
65. #1805C22semidirect
66. #1763C22semidirect
679. C2#26678direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24212
C2×#23602
#266782
C2×#21012
C2×#20662
C2×#20352
C2×#20012
C2×#19352
C2×#19242
C2×#19202
C2×#18782
C2×#18722
C2×#18242
C2×#18052
C2×#17632