Finite Group Databasebeta

← C2×#26574 #6439294

Subgroups — C2×#26574 #6439294

Order 512.

2370 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C22×#441C211yesmaximal, characteristic, complemented
3C2×#2423C211yesmaximal, characteristic
4C2×#2360C211yesmaximal, characteristic
5#26574C2116yesmaximal, direct product factor
6C2×#2096C211yesmaximal, characteristic, complemented
7C2×#2056C211yesmaximal, characteristic, complemented
8C2×#2036C211yesmaximal, characteristic
9C2×#2001C211yesmaximal, characteristic
10C2×#1948C211yesmaximal, characteristic, complemented
11C2×#1943C211yesmaximal, characteristic, complemented
12C2×#1928C211yesmaximal, characteristic, complemented
13C2×#1924C211yesmaximal, characteristic
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1865C211yesmaximal, characteristic, complemented
16C2×#1853C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C2×#483—28no—
24C2×#483C2218yes—
25C2×#441—28no—
26C2×#441C2218yescomplemented
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40C22×C22⋊C8C2211yescharacteristic
41C22×#97C2211yescharacteristic
42#2423—24no—
43#2423C2218yes—
44C2×#399—21no—
45C2×#398—21no—
46#2360—24no—
47#2360C2218yes—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#389—21no—
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#387—21no—
55C2×#387—21no—
56C2×#385—21no—
57C2×#384C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C22⋊C4—21no—
61C2×C4×C22⋊C4C2211yescharacteristic
62#2096C2218yescomplemented
63#2056C2218yescomplemented
64#2036C2218yes—
65#2001C2218yes—
66#1948C2218yescomplemented
67#1943C2218yescomplemented
68#1928C2218yescomplemented
69#1924C2218yes—
70#1865C2218yescomplemented
71#1865C2218yescomplemented
72#1853C2218yes—
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C22×Q16—24no—
84C22×Q16—24no—
85C22×Q16—24no—
86C22×Q16C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—42no—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—42no—
104C2×C4×Q8—42no—
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2314yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—22no—
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⋊C4—21no—
118C22×C4⋊C4—42no—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42—21no—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#483—232no—
137#441—232no—
138C2×#107—24no—
139C2×#107C2314yes—
140C2×#106—24no—
141C2×#106—24no—
142C2×#106—24no—
143C2×#106—42no—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8—24no—
149C2×C4⋊C8C2314yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—42no—
155C2×#103D418yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×C22⋊C8—24no—
163C2×C22⋊C8—24no—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×#97—24no—
167C2×#97C2314yes—
168#399—28no—
169#398—28no—
170#391—22no—
171#391—22no—
172#391C2314yes—
173#391C2314yes—
174#389—22no—
175#389—22no—
176#389—28no—
177#389C2314yes—
178#389C2314yes—
179#388—28no—
180#387—28no—
181#387—28no—
182#385—28no—
183#384—22no—
184#384C2314yes—
185#380—22no—
186#380C2314yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4C2314yes—
189C4×C22⋊C4—28no—
190C4×C22⋊C4D414yes—
191C4×C22⋊C4C2314yes—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25C2×D411yescharacteristic
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8C2×D414yes—
222C22×Q8#4114yes—
223C22×Q8C2412yes—
224C22×Q8C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
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×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×Q16—232no—
252C2×Q16—28no—
253C2×Q16—28no—
254C2×Q16—28no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8C2×D414yes—
266C22×C8#4114yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22⋊Q8—216no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—44no—
276C22⋊Q8—48no—
277C4×Q8—28no—
278C4×Q8—28no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8—48no—
282C4×Q8—48no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
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—22no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4#4114yes—
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—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4C2×D414yes—
337C2×C22⋊C4C2×D414yes—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—44no—
346C2×C42—44no—
347C2×C42—82no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352#107—28no—
353#106—28no—
354#106—28no—
355#106—28no—
356#106—44no—
357C4⋊C8—28no—
358#103—28no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—44no—
363#103C2×D4116yes—
364C22⋊C8—28no—
365C22⋊C8—28no—
366#97—28no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95C2×D414yes—
376#95C2×D414yes—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24#13714yes—
384C24C22×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8C22×D414yes—
411C2×Q8C2×#4114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
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—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
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—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—48no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—84no—
474C22×C4—816no—
475C22×C4#13714yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+412yes—
488C22×C4C2511yescharacteristic
489Q16—232no—
490C2×C8—28no—
491C2×C8—28no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8C22×D414yes—
505C2×C8C2×#4114yes—
506C4⋊C4—216no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—48no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4C22×D414yes—
532C4⋊C4C2×#4114yes—
533C22⋊C4—216no—
534C22⋊C4—28no—
535C22⋊C4—28no—
536C22⋊C4—48no—
537C22⋊C4—84no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—48no—
545C42—48no—
546C42—84no—
547C23—28no—
548C23—28no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—28no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23#52512yes—
561C23D4212yes—
562C23#54712yes—
563C23C2×#13714yes—
564C23C2×#13712yes—
565C23C2×#13712yes—
566C23#57712yes—
567C23#57712yes—
568C23C23×D411yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—24no—
572Q8—24no—
573Q8—42no—
574Q8—42no—
575Q8—48no—
576Q8—48no—
577Q8—42no—
578Q8—42no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—816no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4#52514yes—
637C2×C4#52512yes—
638C2×C4#53914yes—
639C2×C4D4212yes—
640C2×C4#54714yes—
641C2×C4#54712yes—
642C2×C4#55914yes—
643C2×C4C2×#13714yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×2−1+411yescharacteristic
650C8—24no—
651C8—24no—
652C8—42no—
653C8—42no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—416no—
662C22#245514yes—
663C22#311314yes—
664C22#312514yes—
665C22#326614yes—
666C22#327814yes—
667C22#334414yes—
668C22#334414yes—
669C22C2×#52511yescharacteristic
670C22C2×D4211yescharacteristic
671C22C2×#54711yescharacteristic
672C22C22×#13711yescharacteristic
673C22C22×#13711yescharacteristic
674C22C2×#57711yescharacteristic
675C22C2×#57711yescharacteristic
676C4—24no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#245514yes—
690C4C2×#52511yescharacteristic
691C4C2×#53911yescharacteristic
692C4C2×#54711yescharacteristic
693C4C2×#55911yescharacteristic
694C2—44no—
695C2#2657418yesdirect product factor
696C2C2×#245511yescharacteristic
697C2C2×#311311yescharacteristic
698C2C2×#312511yescharacteristic
699C2C2×#326611yescharacteristic
700C2C2×#327811yescharacteristic
701C2C2×#334411yescharacteristic
702C2C2×#334411yescharacteristic
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
2. C22×#441C2semidirect
5. #26574C2direct (N × H)
6. C2×#2096C2semidirect
7. C2×#2056C2semidirect
10. C2×#1948C2semidirect
11. C2×#1943C2semidirect
12. C2×#1928C2semidirect
14. C2×#1865C2semidirect
15. C2×#1865C2semidirect
26. C2×#441C22semidirect
62. #2096C22semidirect
63. #2056C22semidirect
66. #1948C22semidirect
67. #1943C22semidirect
68. #1928C22semidirect
70. #1865C22semidirect
71. #1865C22semidirect
695. C2#26574direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4832
C22×#4412
C2×#24232
C2×#23602
#265742
C2×#20962
C2×#20562
C2×#20362
C2×#20012
C2×#19482
C2×#19432
C2×#19282
C2×#19242
C2×#18652
C2×#18532