Finite Group Databasebeta

← C2×#26902 #6439628

Subgroups — C2×#26902 #6439628

Order 512.

2558 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C22×C4×SD16C211yesmaximal, characteristic
3C2×#2452C211yesmaximal, characteristic
4C2×#2365C211yesmaximal, characteristic, complemented
5#26902C2116yesmaximal, direct product factor
6C2×#2055C211yesmaximal, characteristic
7C2×#2043C211yesmaximal, characteristic, complemented
8C2×#2024C211yesmaximal, characteristic, complemented
9C2×#2013C211yesmaximal, characteristic
10C2×#1933C211yesmaximal, characteristic, complemented
11C2×#1926C211yesmaximal, characteristic
12C2×#1919C211yesmaximal, characteristic, complemented
13C2×#1916C211yesmaximal, characteristic, complemented
14C2×#1880C211yesmaximal, characteristic
15C2×#1877C211yesmaximal, characteristic
16C2×#1865C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8—21no—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C2×#475—28no—
26C2×#475C2218yescomplemented
27C2×C4×SD16—28no—
28C2×C4×SD16C2218yes—
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41#2452—24no—
42#2452C2218yes—
43C2×#396—21no—
44C2×#393—21no—
45#2365—24no—
46#2365C2218yescomplemented
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389—21no—
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#387—21no—
55C2×#387—21no—
56C2×#385—21no—
57C2×C4×C4⋊C4—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2055C2218yes—
61#2043C2218yescomplemented
62#2024C2218yescomplemented
63#2013C2218yes—
64#1933C2218yescomplemented
65#1926C2218yes—
66#1919C2218yescomplemented
67#1916C2218yescomplemented
68#1880C2218yes—
69#1877C2218yes—
70#1865C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×SD16—24no—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16—24no—
88C22×SD16—24no—
89C22×SD16—42no—
90C22×SD16C2314yes—
91C22×SD16C2314yes—
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8C2314yes—
103C2×C4⋊Q8C2314yes—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—48no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8—42no—
111C2×C4×Q8—42no—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2314yes—
114C2×C4×D4—24no—
115C2×C4×D4—22no—
116C2×C4×D4—42no—
117C2×C4×D4C2314yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C42—21no—
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#475—232no—
144C4×SD16—232no—
145C2×#106—24no—
146C2×#106D418yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8—24no—
150C2×C4⋊C8C2314yes—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—42no—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102—24no—
164C2×#102—24no—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C4×C8—24no—
168C2×C4×C8C2314yes—
169#396—28no—
170#393—28no—
171#391—22no—
172#391—22no—
173#391C2314yes—
174#391C2314yes—
175#390—22no—
176#390—22no—
177#390C2314yes—
178#390C2314yes—
179#389—28no—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#387—28no—
185#387—28no—
186#385—28no—
187C4×C4⋊C4—28no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4D414yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2314yes—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95D411yescharacteristic
200C2×#95D411yescharacteristic
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2314yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209C25—21no—
210C22×Q8—21no—
211C22×Q8—21no—
212C22×Q8—24no—
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×Q8—22no—
222C22×Q8—22no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—82no—
226C22×Q8#4114yes—
227C22×Q8#4114yes—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4#4114yes—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—24no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—232no—
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—28no—
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—44no—
263C2×SD16—416no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8C2×D414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8—216no—
281C4⋊Q8—28no—
282C4⋊Q8—28no—
283C4⋊Q8—28no—
284C4⋊Q8—44no—
285C4⋊Q8—48no—
286C22⋊Q8—432no—
287C4×Q8—28no—
288C4×Q8—28no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×Q8—48no—
292C4×Q8—48no—
293C4×D4—28no—
294C4×D4—44no—
295C4×D4—48no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4SD1618yescomplemented
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—82no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—44no—
354C2×C42—44no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2412yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362#106—28no—
363#106C2×D4116yes—
364C4⋊C8—28no—
365#103—28no—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—28no—
370#103—44no—
371#102—28no—
372#102—28no—
373C4×C8—28no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95C2×D414yes—
382#95C2×D414yes—
383C24—24no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—48no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—48no—
418C2×Q8—84no—
419C2×Q8—816no—
420C2×Q8C2×#4114yes—
421C2×Q8C2×#4114yes—
422C2×D4—28no—
423C2×D4—24no—
424C2×D4—28no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4C2×#4114yes—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—48no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—48no—
480C22×C4—84no—
481C22×C4—816no—
482C22×C4C2×SD1614yes—
483C22×C4C2×SD1614yes—
484C22×C4C2×SD1614yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C22×C4C2511yescharacteristic
499SD16—232no—
500SD16—416no—
501C2×C8—28no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—44no—
514C2×C8—44no—
515C2×C8—44no—
516C2×C8C22×D414yes—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4C2×SD16116yescomplemented
539C4⋊C4C22×D414yes—
540C4⋊C4C22×D414yes—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—48no—
556C42—48no—
557C42—48no—
558C23—24no—
559C23—48no—
560C23—48no—
561C23—42no—
562C23—42no—
563C23—48no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23#52512yes—
573C23D4212yes—
574C23#54812yes—
575C23C22×SD1612yes—
576C23#57512yes—
577C23#57712yes—
578C23#57812yes—
579C23C23×D411yescharacteristic
580Q8—24no—
581Q8—24no—
582Q8—24no—
583Q8—24no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—48no—
589Q8—48no—
590Q8—816no—
591D4—24no—
592D4—24no—
593D4—42no—
594D4—42no—
595D4—48no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—816no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4#52514yes—
647C2×C4#52512yes—
648C2×C4#53914yes—
649C2×C4D4212yes—
650C2×C4#54814yes—
651C2×C4#54812yes—
652C2×C4#55114yes—
653C2×C4C22×SD1614yes—
654C2×C4C22×SD1614yes—
655C2×C4C22×SD1614yes—
656C2×C4C23×D411yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4C2×2−1+411yescharacteristic
662C8—24no—
663C8—42no—
664C8—42no—
665C8—42no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—816no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22#246314yes—
675C22#310314yes—
676C22#312114yes—
677C22#326914yes—
678C22D4×SD1614yes—
679C22#335914yes—
680C22#337614yes—
681C22C2×#52511yescharacteristic
682C22C2×D4211yescharacteristic
683C22C2×#54811yescharacteristic
684C22C23×SD1611yescharacteristic
685C22C2×#57511yescharacteristic
686C22C2×#57711yescharacteristic
687C22C2×#57811yescharacteristic
688C4—24no—
689C4—24no—
690C4—24no—
691C4—44no—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—44no—
696C4—44no—
697C4—42no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#246314yes—
702C4C2×#52511yescharacteristic
703C4C2×#53911yescharacteristic
704C4C2×#54811yescharacteristic
705C4C2×#55111yescharacteristic
706C2—84no—
707C2#2690218yesdirect product factor
708C2C2×#246311yescharacteristic
709C2C2×#310311yescharacteristic
710C2C2×#312111yescharacteristic
711C2C2×#326911yescharacteristic
712C2C2×D4×SD1611yescharacteristic
713C2C2×#335911yescharacteristic
714C2C2×#337611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#475C2semidirect
4. C2×#2365C2semidirect
5. #26902C2direct (N × H)
7. C2×#2043C2semidirect
8. C2×#2024C2semidirect
10. C2×#1933C2semidirect
12. C2×#1919C2semidirect
13. C2×#1916C2semidirect
16. C2×#1865C2semidirect
26. C2×#475C22semidirect
46. #2365C22semidirect
61. #2043C22semidirect
62. #2024C22semidirect
64. #1933C22semidirect
66. #1919C22semidirect
67. #1916C22semidirect
70. #1865C22semidirect
324. C2×C4⋊C4SD16semidirect
538. C4⋊C4C2×SD16semidirect
707. C2#26902direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4752
C22×C4×SD162
C2×#24522
C2×#23652
#269022
C2×#20552
C2×#20432
C2×#20242
C2×#20132
C2×#19332
C2×#19262
C2×#19192
C2×#19162
C2×#18802
C2×#18772
C2×#18652