Finite Group Databasebeta

← C2×#30954 #7502411

Subgroups — C2×#30954 #7502411

Order 512.

2378 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic, complemented
2C22×#476C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic, complemented
4C2×#2452C211yesmaximal, characteristic
5#30954C2116yesmaximal, direct product factor
6C2×#2094C211yesmaximal, characteristic
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2084C211yesmaximal, characteristic, complemented
9C2×#2058C211yesmaximal, characteristic, complemented
10C2×#2058C211yesmaximal, characteristic, complemented
11C2×#2023C211yesmaximal, characteristic
12C2×#1934C211yesmaximal, characteristic
13C2×#1926C211yesmaximal, characteristic
14C2×#1919C211yesmaximal, characteristic, complemented
15C2×#1916C211yesmaximal, characteristic, complemented
16C2×#1864C211yesmaximal, characteristic
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8—21no—
22C22×C4×D4C2211yescharacteristic
23C2×#498—28no—
24C2×#498C2218yescomplemented
25C2×#476—28no—
26C2×#476C2218yes—
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39#2551—24no—
40#2551C2218yescomplemented
41#2452—24no—
42#2452C2218yes—
43C2×#396—21no—
44C2×#395—21no—
45C2×#395—21no—
46C2×#393—21no—
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389—21no—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2094C2218yes—
59#2088C2218yescomplemented
60#2084C2218yescomplemented
61#2058C2218yescomplemented
62#2058C2218yescomplemented
63#2023C2218yes—
64#1934C2218yes—
65#1926C2218yes—
66#1919C2218yescomplemented
67#1916C2218yescomplemented
68#1864C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8—21no—
81C23×Q8—21no—
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8—48no—
99C2×C4⋊Q8C2314yes—
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2314yes—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—48no—
104C2×C4×D4—24no—
105C2×C4×D4—22no—
106C2×C4×D4—42no—
107C2×C4×D4C2314yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
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⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#498—232no—
136#476—232no—
137C2×#106—24no—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106—24no—
141C2×#106—42no—
142C2×#106D418yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8—24no—
150C2×C4⋊C8C2314yes—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102—42no—
154C2×#102Q818yescomplemented
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C4×C8—24no—
160C2×C4×C8C2314yes—
161#396—28no—
162#395—28no—
163#395—28no—
164#393—28no—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#389—28no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#389C2314yes—
178#385—28no—
179#384—22no—
180#384C2314yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D414yes—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2314yes—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95D411yescharacteristic
191C2×#95D411yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202C25—21no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—44no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8—82no—
213C22×Q8—82no—
214C22×Q8—82no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2×Q814yes—
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—82no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8C2×D414yes—
252C22×C8C2×Q814yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259C4⋊Q8—28no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263C4⋊Q8—48no—
264C4⋊Q8—48no—
265C4⋊Q8—48no—
266C4⋊Q8—432no—
267C22⋊Q8—432no—
268C4×D4—28no—
269C4×D4—44no—
270C4×D4—48no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
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—24no—
287C2×C4⋊C4—24no—
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—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4SD1618yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×Q814yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—82no—
331C2×C42—22no—
332C2×C42—22no—
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×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#106—28no—
346#106—28no—
347#106—28no—
348#106—28no—
349#106—44no—
350#106C2×D4116yes—
351C4⋊C8—28no—
352#102—28no—
353#102—28no—
354#102—44no—
355#102C2×Q8116yescomplemented
356C4×C8—28no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95C2×D414yes—
363#95C2×D414yes—
364C24—24no—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24C22×D411yescharacteristic
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—84no—
380C2×Q8—84no—
381C2×Q8—84no—
382C2×Q8—816no—
383C2×Q8—816no—
384C2×Q8—816no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4C22×Q814yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—48no—
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—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—84no—
443C22×C4—816no—
444C22×C4C2×SD1614yes—
445C22×C4C2×SD1614yes—
446C22×C4C2×SD1614yes—
447C22×C4AGL(1,ℤ/8)14yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×Q812yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
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×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—44no—
475C2×C8—44no—
476C2×C8C22×D414yes—
477C2×C8C22×Q814yes—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—28no—
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—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—84no—
506C4⋊C4C2×SD16116yes—
507C4⋊C4C22×D414yes—
508C4⋊C4C22×D414yes—
509C4⋊C4C22×Q814yes—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—48no—
525C23—24no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23#53612yes—
540C23D4212yes—
541C23D4×Q812yes—
542C23C22×SD1612yes—
543C23C2×AGL(1,ℤ/8)12yes—
544C23#57712yes—
545C23#57712yes—
546C23C23×D411yescharacteristic
547Q8—816no—
548Q8—816no—
549Q8—816no—
550D4—24no—
551D4—24no—
552D4—48no—
553D4—42no—
554D4—42no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
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—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—816no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4#53612yes—
605C2×C4#53614yes—
606C2×C4#54414yes—
607C2×C4D4212yes—
608C2×C4D4×Q812yes—
609C2×C4D4×Q814yes—
610C2×C4#56314yes—
611C2×C4C22×SD1614yes—
612C2×C4C22×SD1614yes—
613C2×C4C22×SD1614yes—
614C2×C4C2×AGL(1,ℤ/8)14yes—
615C2×C4C23×D411yescharacteristic
616C2×C4C23×Q811yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C8—24no—
622C8—24no—
623C8—42no—
624C8—42no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—816no—
633C22#267714yes—
634C22#322714yes—
635C22#324414yes—
636C22#326514yes—
637C22D4×SD1614yes—
638C22Q8×SD1614yes—
639C22#338414yes—
640C22C2×#53611yescharacteristic
641C22C2×D4211yescharacteristic
642C22C2×D4×Q811yescharacteristic
643C22C23×SD1611yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C2×#57711yescharacteristic
646C22C2×#57711yescharacteristic
647C4—24no—
648C4—24no—
649C4—24no—
650C4—24no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#267714yes—
661C4C2×#53611yescharacteristic
662C4C2×#54411yescharacteristic
663C4C2×D4×Q811yescharacteristic
664C4C2×#56311yescharacteristic
665C2—84no—
666C2#3095418yesdirect product factor
667C2C2×#267711yescharacteristic
668C2C2×#322711yescharacteristic
669C2C2×#324411yescharacteristic
670C2C2×#326511yescharacteristic
671C2C2×D4×SD1611yescharacteristic
672C2C2×Q8×SD1611yescharacteristic
673C2C2×#338411yescharacteristic
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×#498C2semidirect
3. C2×#2551C2semidirect
5. #30954C2direct (N × H)
7. C2×#2088C2semidirect
8. C2×#2084C2semidirect
9. C2×#2058C2semidirect
10. C2×#2058C2semidirect
14. C2×#1919C2semidirect
15. C2×#1916C2semidirect
24. C2×#498C22semidirect
40. #2551C22semidirect
59. #2088C22semidirect
60. #2084C22semidirect
61. #2058C22semidirect
62. #2058C22semidirect
66. #1919C22semidirect
67. #1916C22semidirect
154. C2×#102Q8semidirect
355. #102C2×Q8semidirect
666. C2#30954direct (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×#4762
C2×#25512
C2×#24522
#309542
C2×#20942
C2×#20882
C2×#20842
C2×#20582
C2×#20232
C2×#19342
C2×#19262
C2×#19192
C2×#19162
C2×#18642