Finite Group Databasebeta

← C2×#26679 #6446921

Subgroups — C2×#26679 #6446921

Order 512.

2142 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2423C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3#26679C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2068C211yesmaximal, characteristic, complemented
6C2×#2036C211yesmaximal, characteristic
7C2×#2001C211yesmaximal, characteristic
8C2×#1935C211yesmaximal, characteristic, complemented
9C2×#1924C211yesmaximal, characteristic
10C2×#1918C211yesmaximal, characteristic, complemented
11C2×#1878C211yesmaximal, characteristic, complemented
12C2×#1878C211yesmaximal, characteristic, complemented
13C2×#1872C211yesmaximal, characteristic
14C2×#1823C211yesmaximal, characteristic
15C2×#1803C211yesmaximal, characteristic, complemented
16C2×#1763C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2423—24no—
35#2423C2218yes—
36C2×#399—21no—
37C2×#398—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×#384C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4C2211yescharacteristic
54#2099C2218yescomplemented
55#2068C2218yescomplemented
56#2036C2218yes—
57#2001C2218yes—
58#1935C2218yescomplemented
59#1924C2218yes—
60#1918C2218yescomplemented
61#1878C2218yescomplemented
62#1878C2218yescomplemented
63#1872C2218yes—
64#1823C2218yes—
65#1803C2218yescomplemented
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—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—42no—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C2×#107—24no—
135C2×#107C2314yes—
136C2×C4⋊C8—42no—
137C2×C4⋊C8C2314yes—
138C2×#103—24no—
139C2×#103—24no—
140C2×#103—24no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×C22⋊C8—42no—
151C2×C22⋊C8—42no—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2314yes—
154C2×#97—42no—
155C2×#97C2314yes—
156#399—28no—
157#398—28no—
158#391—22no—
159#391—22no—
160#391C2314yes—
161#391C2314yes—
162#389—22no—
163#389—22no—
164#389—28no—
165#389C2314yes—
166#389C2314yes—
167#388—28no—
168#387—28no—
169#387—28no—
170#385—28no—
171#384—22no—
172#384C2314yes—
173#380—22no—
174#380C2314yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4C2314yes—
177C4×C22⋊C4—28no—
178C4×C22⋊C4D414yes—
179C4×C22⋊C4C2314yes—
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—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4#4114yes—
313C2×C4⋊C4#4114yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4C2×D414yes—
329C2×C22⋊C4C2×D414yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42—82no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—28no—
345C4⋊C8—44no—
346#103—28no—
347#103—28no—
348#103—28no—
349#103—44no—
350#103—44no—
351#103—44no—
352C22⋊C8—44no—
353C22⋊C8—44no—
354#97—44no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24#13714yes—
372C24C22×D412yes—
373C24C22×D411yescharacteristic
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8C2×#4114yes—
399C2×Q8C2×#4114yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
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—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
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—22no—
435C22×C4—48no—
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—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—84no—
462C22×C4—816no—
463C22×C4#13714yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4C2511yescharacteristic
477Q16—416no—
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—22no—
489C2×C8—22no—
490C2×C8—44no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8—44no—
494C4⋊C4—216no—
495C4⋊C4—24no—
496C4⋊C4—24no—
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—22no—
506C4⋊C4—22no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—48no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4C2×#4114yes—
520C4⋊C4C2×#4114yes—
521C22⋊C4—216no—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—48no—
525C22⋊C4—84no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—48no—
533C42—48no—
534C42—84no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23#52512yes—
549C23#54612yes—
550C23#54612yes—
551C23C2×#13714yes—
552C23C2×#13712yes—
553C23C2×#13712yes—
554C23#57612yes—
555C23#57812yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—24no—
559Q8—24no—
560Q8—24no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—48no—
566Q8—48no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
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—48no—
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—42no—
619C2×C4—816no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4#52512yes—
625C2×C4#52914yes—
626C2×C4#54014yes—
627C2×C4#54612yes—
628C2×C4#54612yes—
629C2×C4#55114yes—
630C2×C4#55314yes—
631C2×C4C2×#13714yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—416no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22#245714yes—
651C22#311314yes—
652C22#312414yes—
653C22#331214yes—
654C22#331214yes—
655C22#331414yes—
656C22#333314yes—
657C22C2×#52511yescharacteristic
658C22C2×#54611yescharacteristic
659C22C2×#54611yescharacteristic
660C22C22×#13711yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57611yescharacteristic
663C22C2×#57811yescharacteristic
664C4—24no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#245714yes—
678C4C2×#52911yescharacteristic
679C4C2×#54011yescharacteristic
680C4C2×#55111yescharacteristic
681C4C2×#55311yescharacteristic
682C2—44no—
683C2#2667918yesdirect product factor
684C2C2×#245711yescharacteristic
685C2C2×#311311yescharacteristic
686C2C2×#312411yescharacteristic
687C2C2×#331411yescharacteristic
688C2C2×#331211yescharacteristic
689C2C2×#331211yescharacteristic
690C2C2×#333311yescharacteristic
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. #26679C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2068C2semidirect
8. C2×#1935C2semidirect
10. C2×#1918C2semidirect
11. C2×#1878C2semidirect
12. C2×#1878C2semidirect
15. C2×#1803C2semidirect
16. C2×#1763C2semidirect
54. #2099C22semidirect
55. #2068C22semidirect
58. #1935C22semidirect
60. #1918C22semidirect
61. #1878C22semidirect
62. #1878C22semidirect
65. #1803C22semidirect
66. #1763C22semidirect
683. C2#26679direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24232
C2×#23602
#266792
C2×#20992
C2×#20682
C2×#20362
C2×#20012
C2×#19352
C2×#19242
C2×#19182
C2×#18782
C2×#18722
C2×#18232
C2×#18032
C2×#17632