Finite Group Databasebeta

← C2×#26072 #6443439

Subgroups — C2×#26072 #6443439

Order 512.

2402 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic, complemented
2C22×#428C211yesmaximal, characteristic
3C2×#2439C211yesmaximal, characteristic
4C2×#2440C211yesmaximal, characteristic
5#26072C2116yesmaximal, direct product factor
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2058C211yesmaximal, characteristic, complemented
9C2×#2057C211yesmaximal, characteristic, complemented
10C2×#1944C211yesmaximal, characteristic, complemented
11C2×#1942C211yesmaximal, characteristic, complemented
12C2×#1854C211yesmaximal, characteristic
13C2×#1853C211yesmaximal, characteristic
14C2×#1847C211yesmaximal, characteristic, complemented
15C2×#1809C211yesmaximal, characteristic
16C2×#1753C211yesmaximal, characteristic
17C23×M16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C2×#498—28no—
24C2×#498C2218yescomplemented
25C2×#428—28no—
26C2×#428C2218yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C22×#106C2211yescharacteristic
34C22×#106C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×#97C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2440—24no—
41#2440C2218yes—
42#2439—24no—
43#2439C2218yes—
44C2×#398—21no—
45C2×#396—21no—
46C2×#395—21no—
47C2×#393—21no—
48C2×#391—21no—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#387—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×#380C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2085C2218yescomplemented
61#2084C2218yescomplemented
62#2058C2218yescomplemented
63#2057C2218yescomplemented
64#1944C2218yescomplemented
65#1942C2218yescomplemented
66#1854C2218yes—
67#1853C2218yes—
68#1847C2218yescomplemented
69#1809C2218yes—
70#1753C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8—21no—
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C22×M16—24no—
84C22×M16—24no—
85C22×M16Q818yescomplemented
86C22×M16C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8C2314yes—
98C2×C4⋊Q8C2314yes—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—48no—
101C2×#117—22no—
102C2×#117—24no—
103C2×#117—42no—
104C2×#117C2314yes—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
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⋊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×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#498—232no—
135#428—232no—
136C2×#107—24no—
137C2×#107—24no—
138C2×#107—24no—
139C2×#107—42no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106—24no—
146C2×#106—24no—
147C2×#106—24no—
148C2×#106—42no—
149C2×#106D418yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155C2×#106C2314yes—
156C2×C22⋊C8Q818yescomplemented
157C2×C22⋊C8C2314yes—
158C2×#97—24no—
159C2×#97C2314yes—
160C2×C4×C8—24no—
161C2×C4×C8C2314yes—
162#398—28no—
163#396—28no—
164#395—28no—
165#393—28no—
166#391—28no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#389C2314yes—
175#387—28no—
176#385—28no—
177#384—22no—
178#384C2314yes—
179#382—22no—
180#382C2314yes—
181#380—22no—
182#380—22no—
183#380C2314yes—
184#380C2314yes—
185C4×C22⋊C4D414yes—
186C4×C22⋊C4C2314yes—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95D411yescharacteristic
196C2×#95D411yescharacteristic
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2314yes—
205C25C2×D411yescharacteristic
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8—82no—
213C22×Q8—82no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—24no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—82no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×Q814yes—
237C23×C4C2×Q814yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C2×M16—232no—
241C2×M16—28no—
242C2×M16—28no—
243C2×M16C2×Q8116yescomplemented
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8C2×D414yes—
251C22×C8C2×Q814yes—
252C22×C8C2×Q814yes—
253C22×C8C2×Q814yes—
254C22×C8C2×Q814yes—
255C22×C8#4114yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—48no—
265C4⋊Q8—48no—
266C22⋊Q8—432no—
267#117—28no—
268#117—44no—
269#117—48no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
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—28no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
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—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4C2×D414yes—
332C2×C22⋊C4C2×D414yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—28no—
344#107—28no—
345#107—28no—
346#107—44no—
347#106—28no—
348#106—28no—
349#106—28no—
350#106—28no—
351#106—44no—
352#106C2×D4116yes—
353C22⋊C8C2×Q8116yescomplemented
354#97—28no—
355C4×C8—28no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95C2×D414yes—
365#95C2×D414yes—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24C4⋊Q814yes—
373C24C22×D412yes—
374C24C22×D411yescharacteristic
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—84no—
380C2×Q8—84no—
381C2×Q8—816no—
382C2×Q8—816no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
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—48no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—84no—
440C22×C4—816no—
441C22×C4C4⋊Q814yes—
442C22×C4C4⋊Q818yes—
443C22×C4AGL(1,ℤ/8)14yes—
444C22×C4#13714yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×Q814yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q812yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C4C2511yescharacteristic
460M16—232no—
461C2×C8—28no—
462C2×C8—28no—
463C2×C8—28no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8C22⋊Q8116yescomplemented
471C2×C8C22×D414yes—
472C2×C8C22×Q814yes—
473C2×C8C22×Q814yes—
474C2×C8C22×Q814yes—
475C2×C8C22×Q814yes—
476C2×C8C2×#4114yes—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4C22×D414yes—
509C4⋊C4C22×D414yes—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—48no—
522C23—28no—
523C23—28no—
524C23—22no—
525C23—28no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23C2×C4⋊Q814yes—
536C23C2×C4⋊Q812yes—
537C23D4212yes—
538C23#54712yes—
539C23C2×AGL(1,ℤ/8)12yes—
540C23C2×#13712yes—
541C23#57712yes—
542C23#57712yes—
543C23C23×D411yescharacteristic
544Q8—816no—
545Q8—816no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—216no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—816no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4C2×C22⋊Q814yes—
602C2×C4C2×C4⋊Q814yes—
603C2×C4C2×C4⋊Q812yes—
604C2×C4#54414yes—
605C2×C4D4212yes—
606C2×C4#54712yes—
607C2×C4D4×Q814yes—
608C2×C4#55414yes—
609C2×C4C2×AGL(1,ℤ/8)14yes—
610C2×C4C2×#13714yes—
611C2×C4C23×D411yescharacteristic
612C2×C4C23×Q811yescharacteristic
613C2×C4C23×Q811yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C8—24no—
618C8—24no—
619C8—24no—
620C8C2×C22⋊Q818yescomplemented
621C22—24no—
622C22—24no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—416no—
629C22#244014yes—
630C22#316314yes—
631C22#316514yes—
632C22#326514yes—
633C22#326614yes—
634C22#335414yes—
635C22#335614yes—
636C22C22×C4⋊Q811yescharacteristic
637C22C2×D4211yescharacteristic
638C22C2×#54711yescharacteristic
639C22C22×AGL(1,ℤ/8)11yescharacteristic
640C22C22×#13711yescharacteristic
641C22C2×#57711yescharacteristic
642C22C2×#57711yescharacteristic
643C4—24no—
644C4—24no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#244014yes—
657C4C22×C22⋊Q811yescharacteristic
658C4C2×#54411yescharacteristic
659C4C2×D4×Q811yescharacteristic
660C4C2×#55411yescharacteristic
661C2—44no—
662C2#2607218yesdirect product factor
663C2C2×#244011yescharacteristic
664C2C2×#316311yescharacteristic
665C2C2×#316511yescharacteristic
666C2C2×#326511yescharacteristic
667C2C2×#326611yescharacteristic
668C2C2×#335411yescharacteristic
669C2C2×#335611yescharacteristic
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
1. C22×#498C2semidirect
5. #26072C2direct (N × H)
6. C2×#2084C2semidirect
7. C2×#2085C2semidirect
8. C2×#2058C2semidirect
9. C2×#2057C2semidirect
10. C2×#1944C2semidirect
11. C2×#1942C2semidirect
14. C2×#1847C2semidirect
24. C2×#498C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
62. #2058C22semidirect
63. #2057C22semidirect
64. #1944C22semidirect
65. #1942C22semidirect
68. #1847C22semidirect
85. C22×M16Q8semidirect
156. C2×C22⋊C8Q8semidirect
243. C2×M16C2×Q8semidirect
353. C22⋊C8C2×Q8semidirect
470. C2×C8C22⋊Q8semidirect
620. C8C2×C22⋊Q8semidirect
662. C2#26072direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4982
C22×#4282
C2×#24392
C2×#24402
#260722
C2×#20842
C2×#20852
C2×#20582
C2×#20572
C2×#19442
C2×#19422
C2×#18542
C2×#18532
C2×#18472
C2×#18092
C2×#17532