Finite Group Databasebeta

← C2×#26345 #6439151

Subgroups — C2×#26345 #6439151

Order 512.

2500 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic, complemented
2C22×#427C211yesmaximal, characteristic, complemented
3C2×#2441C211yesmaximal, characteristic, complemented
4C2×#2363C211yesmaximal, characteristic
5#26345C2116yesmaximal, direct product factor
6C2×#2094C211yesmaximal, characteristic, complemented
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#1947C211yesmaximal, characteristic, complemented
11C2×#1919C211yesmaximal, characteristic, complemented
12C2×#1893C211yesmaximal, characteristic
13C2×#1875C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic, complemented
16C2×#1850C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#478—28no—
24C2×#478C2218yescomplemented
25C2×#427—28no—
26C2×#427C2218yescomplemented
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40#2441—24no—
41#2441C2218yescomplemented
42C2×#399—21no—
43C2×#396—21no—
44#2363—24no—
45#2363C2218yes—
46C2×#390—21no—
47C2×#390C2211yescharacteristic
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#387—21no—
54C2×#386—21no—
55C2×#384C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4—21no—
59C2×C4×C22⋊C4C2211yescharacteristic
60#2094C2218yescomplemented
61#2088C2218yescomplemented
62#2067C2218yescomplemented
63#2059C2218yescomplemented
64#1947C2218yescomplemented
65#1919C2218yescomplemented
66#1893C2218yes—
67#1875C2218yes—
68#1862C2218yescomplemented
69#1854C2218yescomplemented
70#1850C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C23×C8—24no—
84C23×C8—24no—
85C23×C8—24no—
86C23×C8C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×#125—24no—
92C2×#125—22no—
93C2×#125—42no—
94C2×#125C2314yes—
95C2×#121—22no—
96C2×#121—24no—
97C2×#121Q818yes—
98C2×#121C2314yes—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117—42no—
108C2×#117—42no—
109C2×#117C2314yes—
110C2×#117C2314yes—
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⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#478—232no—
140#427—232no—
141C2×#107—24no—
142C2×#107—24no—
143C2×#107—24no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106—24no—
148C2×#106—42no—
149C2×#106D418yescomplemented
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8—24no—
154C2×C4⋊C8—24no—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2314yes—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102—24no—
160C2×#102—42no—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×C22⋊C8—24no—
166C2×C22⋊C8C2314yes—
167#399—28no—
168#396—28no—
169#390—28no—
170#390—22no—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—28no—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#388—28no—
180#387—28no—
181#386—28no—
182#384—22no—
183#384C2314yes—
184#380—22no—
185#380C2314yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—28no—
189C4×C22⋊C4D414yes—
190C4×C22⋊C4C2314yes—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95D411yescharacteristic
197C2×#95D411yescharacteristic
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C25C2×D411yescharacteristic
208C22×D4—28no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—82no—
217C22×D4—82no—
218C22×D4C2×Q814yes—
219C22×D4C2412yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
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×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×Q814yes—
242C23×C4#4114yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C22×C8—232no—
246C22×C8—22no—
247C22×C8—28no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—28no—
252C22×C8—28no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8C2×D414yes—
260C22×C8#4114yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265#125—28no—
266#125—44no—
267#125—48no—
268#121—216no—
269#121—28no—
270#121C2×Q8116yes—
271C4×D4—28no—
272C4×D4—44no—
273C4×D4—48no—
274#117—28no—
275#117—28no—
276#117—44no—
277#117—44no—
278#117—48no—
279#117—48no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
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—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
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⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×Q814yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42—82no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354#107—28no—
355#107—28no—
356#107—28no—
357#106—28no—
358#106—44no—
359#106C2×D4116yescomplemented
360C4⋊C8—28no—
361C4⋊C8—28no—
362#102—28no—
363#102—28no—
364#102—28no—
365#102—44no—
366C22⋊C8—28no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—24no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24C22⋊Q814yes—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391C2×D4—216no—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—816no—
405C2×D4—84no—
406C2×D4—84no—
407C2×D4C22×Q814yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
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—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
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—42no—
461C22×C4—84no—
462C22×C4—816no—
463C22×C4C22⋊Q814yes—
464C22×C4C22⋊Q818yes—
465C22×C4#13514yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×Q814yes—
474C22×C4C22×Q812yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481C2×C8—232no—
482C2×C8—28no—
483C2×C8—28no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—44no—
495C2×C8—44no—
496C2×C8C22×D414yes—
497C2×C8C2×#4114yes—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C22×D414yes—
524C4⋊C4C22×D414yes—
525C4⋊C4C22×Q814yes—
526C22⋊C4—216no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C42—24no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—48no—
542C42—48no—
543C42—84no—
544C23—28no—
545C23—28no—
546C23—22no—
547C23—22no—
548C23—28no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23C2×C22⋊Q814yes—
572C23C2×C22⋊Q812yes—
573C23D4212yes—
574C23#54712yes—
575C23C2×#13512yes—
576C23#57512yes—
577C23#57612yes—
578C23#57712yes—
579C23C23×D411yescharacteristic
580D4—24no—
581D4—24no—
582D4—42no—
583D4—42no—
584D4—48no—
585D4—816no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—216no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
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—42no—
632C2×C4—42no—
633C2×C4—816no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4C2×C22⋊Q814yes—
639C2×C4C2×C22⋊Q812yes—
640C2×C4#52514yes—
641C2×C4#53914yes—
642C2×C4D4212yes—
643C2×C4#54712yes—
644C2×C4D4×Q814yes—
645C2×C4#55414yes—
646C2×C4C2×#13514yes—
647C2×C4C23×D411yescharacteristic
648C2×C4C23×Q811yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C8—24no—
654C8—24no—
655C8—42no—
656C8—42no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—416no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—816no—
673C22#244914yes—
674C22#307614yes—
675C22#308114yes—
676C22#326814yes—
677C22#328414yes—
678C22#333914yes—
679C22#334814yes—
680C22C22×C22⋊Q811yescharacteristic
681C22C2×D4211yescharacteristic
682C22C2×#54711yescharacteristic
683C22C22×#13511yescharacteristic
684C22C2×#57511yescharacteristic
685C22C2×#57611yescharacteristic
686C22C2×#57711yescharacteristic
687C4—24no—
688C4—24no—
689C4—42no—
690C4—44no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—44no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#244914yes—
700C4C2×#52511yescharacteristic
701C4C2×#53911yescharacteristic
702C4C2×D4×Q811yescharacteristic
703C4C2×#55411yescharacteristic
704C2—44no—
705C2—84no—
706C2#2634518yesdirect product factor
707C2C2×#244911yescharacteristic
708C2C2×#307611yescharacteristic
709C2C2×#308111yescharacteristic
710C2C2×#326811yescharacteristic
711C2C2×#328411yescharacteristic
712C2C2×#333911yescharacteristic
713C2C2×#334811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
1. C22×#478C2semidirect
2. C22×#427C2semidirect
3. C2×#2441C2semidirect
5. #26345C2direct (N × H)
6. C2×#2094C2semidirect
7. C2×#2088C2semidirect
8. C2×#2067C2semidirect
9. C2×#2059C2semidirect
10. C2×#1947C2semidirect
11. C2×#1919C2semidirect
14. C2×#1862C2semidirect
15. C2×#1854C2semidirect
16. C2×#1850C2semidirect
24. C2×#478C22semidirect
26. C2×#427C22semidirect
41. #2441C22semidirect
60. #2094C22semidirect
61. #2088C22semidirect
62. #2067C22semidirect
63. #2059C22semidirect
64. #1947C22semidirect
65. #1919C22semidirect
68. #1862C22semidirect
69. #1854C22semidirect
70. #1850C22semidirect
149. C2×#106D4semidirect
359. #106C2×D4semidirect
706. C2#26345direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4782
C22×#4272
C2×#24412
C2×#23632
#263452
C2×#20942
C2×#20882
C2×#20672
C2×#20592
C2×#19472
C2×#19192
C2×#18932
C2×#18752
C2×#18622
C2×#18542
C2×#18502