Finite Group Databasebeta

← C2×#30780 #7501868

Subgroups — C2×#30780 #7501868

Order 512.

2286 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic, complemented
2C22×#482C211yesmaximal, characteristic
3C2×#2491C211yesmaximal, characteristic
4C2×#2439C211yesmaximal, characteristic, complemented
5#30780C2116yesmaximal, direct product factor
6C2×#2092C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2057C211yesmaximal, characteristic, complemented
10C2×#2034C211yesmaximal, characteristic
11C2×#2023C211yesmaximal, characteristic, complemented
12C2×#1949C211yesmaximal, characteristic, complemented
13C2×#1934C211yesmaximal, characteristic, complemented
14C2×#1923C211yesmaximal, characteristic, complemented
15C2×#1919C211yesmaximal, characteristic, complemented
16C2×#1854C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124—21no—
20C22×#121C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C2×#498—28no—
23C2×#498C2218yescomplemented
24C2×#482—28no—
25C2×#482C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2491—24no—
41#2491C2218yes—
42#2439—24no—
43#2439C2218yescomplemented
44C2×#396—21no—
45C2×#394—21no—
46C2×#393—21no—
47C2×#393—21no—
48C2×#391—21no—
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#387—21no—
55C2×#382C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2092C2218yescomplemented
61#2077C2218yescomplemented
62#2065C2218yescomplemented
63#2057C2218yescomplemented
64#2034C2218yes—
65#2023C2218yescomplemented
66#1949C2218yescomplemented
67#1934C2218yescomplemented
68#1923C2218yescomplemented
69#1919C2218yescomplemented
70#1854C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8—21no—
81C23×D4—21no—
82C23×D4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—42no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2314yes—
96C2×#124—28no—
97C2×#124—48no—
98C2×#121—22no—
99C2×#121—24no—
100C2×#121—42no—
101C2×#121C2314yes—
102C2×#117—22no—
103C2×#117—24no—
104C2×#117—42no—
105C2×#117C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42—21no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#498—232no—
132#482—232no—
133C2×#107—24no—
134C2×#107—42no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106D418yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8—24no—
146C2×C4⋊C8C2314yes—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102—24no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8—24no—
160C2×C4×C8C2314yes—
161#396—28no—
162#394—28no—
163#393—28no—
164#393—28no—
165#391—28no—
166#390—22no—
167#390—22no—
168#390C2314yes—
169#390C2314yes—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#388—28no—
175#387—28no—
176#382—22no—
177#382C2314yes—
178#380—22no—
179#380—22no—
180#380C2314yes—
181#380C2314yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2314yes—
184C4×C22⋊C4D414yes—
185C4×C22⋊C4C2314yes—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D411yescharacteristic
194C2×#95D411yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C25C2×D411yescharacteristic
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—44no—
211C22×Q8—82no—
212C22×Q8—82no—
213C22×D4—24no—
214C22×D4—22no—
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—44no—
222C22×D4—44no—
223C22×D4—82no—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C22×C8—22no—
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×C8C2×D414yes—
257C22×C8#4114yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268#124—432no—
269#121—28no—
270#121—44no—
271#121—48no—
272#117—28no—
273#117—44no—
274#117—48no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
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—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4C2×D414yes—
330C2×C22⋊C4C2×D414yes—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—28no—
344#107—44no—
345#106—28no—
346#106—28no—
347#106—28no—
348#106C2×D4116yes—
349C4⋊C8—28no—
350#102—28no—
351#102—28no—
352#102—28no—
353#102—44no—
354C22⋊C8—28no—
355C22⋊C8—28no—
356C4×C8—28no—
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—
372C24—22no—
373C24—22no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—84no—
388C2×Q8—84no—
389C2×Q8—816no—
390C2×Q8—816no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—84no—
407C2×D4—816no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
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—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
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—42no—
462C22×C4#13514yes—
463C22×C4AGL(1,ℤ/8)14yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C22×C42−1+412yes—
476C22×C4C2511yescharacteristic
477C2×C8—28no—
478C2×C8—28no—
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—44no—
490C2×C8—44no—
491C2×C8C22×D414yes—
492C2×C8C2×#4114yes—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
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—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—48no—
515C4⋊C4—84no—
516C4⋊C4C22×D414yes—
517C4⋊C4C22×D414yes—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—48no—
535C42—48no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23#53612yes—
565C23D4212yes—
566C23#54712yes—
567C23C2×#13512yes—
568C23C2×AGL(1,ℤ/8)12yes—
569C23#57712yes—
570C23#57712yes—
571C23C23×D411yescharacteristic
572Q8—816no—
573Q8—816no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—816no—
579D4—84no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—48no—
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—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4#53612yes—
632C2×C4#53614yes—
633C2×C4#54214yes—
634C2×C4D4212yes—
635C2×C4#54712yes—
636C2×C4#54814yes—
637C2×C4#56314yes—
638C2×C4C2×#13514yes—
639C2×C4C2×AGL(1,ℤ/8)14yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C8—24no—
647C8—24no—
648C8—42no—
649C8—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—88no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—168no—
666C22#267414yes—
667C22#322814yes—
668C22#324414yes—
669C22#326514yes—
670C22#328214yes—
671C22#334014yes—
672C22#335414yes—
673C22C2×#53611yescharacteristic
674C22C2×D4211yescharacteristic
675C22C2×#54711yescharacteristic
676C22C22×#13511yescharacteristic
677C22C22×AGL(1,ℤ/8)11yescharacteristic
678C22C2×#57711yescharacteristic
679C22C2×#57711yescharacteristic
680C4—24no—
681C4—24no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4#267414yes—
693C4C2×#53611yescharacteristic
694C4C2×#54211yescharacteristic
695C4C2×#54811yescharacteristic
696C4C2×#56311yescharacteristic
697C2—82no—
698C2—162no—
699C2#3078018yesdirect product factor
700C2C2×#267411yescharacteristic
701C2C2×#322811yescharacteristic
702C2C2×#324411yescharacteristic
703C2C2×#326511yescharacteristic
704C2C2×#328211yescharacteristic
705C2C2×#334011yescharacteristic
706C2C2×#335411yescharacteristic
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×#498C2semidirect
4. C2×#2439C2semidirect
5. #30780C2direct (N × H)
6. C2×#2092C2semidirect
7. C2×#2077C2semidirect
8. C2×#2065C2semidirect
9. C2×#2057C2semidirect
11. C2×#2023C2semidirect
12. C2×#1949C2semidirect
13. C2×#1934C2semidirect
14. C2×#1923C2semidirect
15. C2×#1919C2semidirect
16. C2×#1854C2semidirect
23. C2×#498C22semidirect
43. #2439C22semidirect
60. #2092C22semidirect
61. #2077C22semidirect
62. #2065C22semidirect
63. #2057C22semidirect
65. #2023C22semidirect
66. #1949C22semidirect
67. #1934C22semidirect
68. #1923C22semidirect
69. #1919C22semidirect
70. #1854C22semidirect
699. C2#30780direct (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×#4822
C2×#24912
C2×#24392
#307802
C2×#20922
C2×#20772
C2×#20652
C2×#20572
C2×#20342
C2×#20232
C2×#19492
C2×#19342
C2×#19232
C2×#19192
C2×#18542