Finite Group Databasebeta

← C2×#30962 #7502427

Subgroups — C2×#30962 #7502427

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C22×#474C211yesmaximal, characteristic
3C2×#2551C211yesmaximal, characteristic, complemented
4C2×#2454C211yesmaximal, characteristic
5#30962C2116yesmaximal, direct product factor
6C2×#2094C211yesmaximal, characteristic
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2083C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#2058C211yesmaximal, characteristic, complemented
11C2×#2023C211yesmaximal, characteristic
12C2×#1936C211yesmaximal, characteristic
13C2×#1926C211yesmaximal, characteristic
14C2×#1921C211yesmaximal, characteristic, complemented
15C2×#1917C211yesmaximal, characteristic, complemented
16C2×#1862C211yesmaximal, characteristic
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#501—28no—
23C2×#501C2218yescomplemented
24C2×#474—28no—
25C2×#474C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2551—24no—
39#2551C2218yescomplemented
40#2454—24no—
41#2454C2218yes—
42C2×#399—21no—
43C2×#396—21no—
44C2×#395—21no—
45C2×#395—21no—
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#385—21no—
54C2×#384C2211yescharacteristic
55C2×#380—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2094C2218yes—
59#2088C2218yescomplemented
60#2083C2218yescomplemented
61#2059C2218yescomplemented
62#2058C2218yescomplemented
63#2023C2218yes—
64#1936C2218yes—
65#1926C2218yes—
66#1921C2218yescomplemented
67#1917C2218yescomplemented
68#1862C2218yes—
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×D4C2311yescharacteristic
82C24×C4—21no—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—28no—
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⋊Q8—48no—
95C2×C4⋊Q8C2314yes—
96C2×C4⋊Q8C2314yes—
97C2×#125—24no—
98C2×#125—22no—
99C2×#125—42no—
100C2×#125C2314yes—
101C2×C4×D4—24no—
102C2×C4×D4—22no—
103C2×C4×D4—42no—
104C2×C4×D4C2314yes—
105C22×C4⋊C4—21no—
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—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#501—232no—
134#474—232no—
135C2×#107—24no—
136C2×#107—24no—
137C2×#107D418yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106—24no—
142C2×#106—24no—
143C2×#106—42no—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8—24no—
148C2×C4⋊C8C2314yes—
149C2×#102—24no—
150C2×#102—24no—
151C2×#102—42no—
152C2×#102Q818yescomplemented
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#97—24no—
158C2×#97C2314yes—
159#399—28no—
160#396—28no—
161#395—28no—
162#395—28no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#389—28no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#385—28no—
177#384—22no—
178#384C2314yes—
179#380—28no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4D414yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2314yes—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95D411yescharacteristic
190C2×#95D411yescharacteristic
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8—82no—
209C22×Q8—82no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4C2×Q814yes—
217C22×D4C2412yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—82no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2411yescharacteristic
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8C2×D414yes—
247C22×C8C2×Q814yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—48no—
257C4⋊Q8—48no—
258C4⋊Q8—432no—
259#125—28no—
260#125—44no—
261#125—48no—
262C4×D4—28no—
263C4×D4—44no—
264C4×D4—48no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
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—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×Q814yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4—82no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2412yes—
341#107—28no—
342#107—28no—
343#107C2×D4116yes—
344#106—28no—
345#106—28no—
346#106—44no—
347C4⋊C8—28no—
348#102—28no—
349#102—28no—
350#102—44no—
351#102C2×Q8116yescomplemented
352#97—28no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—24no—
361C24—22no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24C22×D411yescharacteristic
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—84no—
374C2×Q8—84no—
375C2×Q8—816no—
376C2×Q8—816no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—24no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4C22×Q814yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
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—22no—
416C22×C4—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
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—48no—
436C22×C4—84no—
437C22×C4—816no—
438C22×C4AGL(1,ℤ/8)14yes—
439C22×C4#13714yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×Q812yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+412yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C22×C4C2511yescharacteristic
454C2×C8—28no—
455C2×C8—28no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—44no—
467C2×C8—44no—
468C2×C8C22×D414yes—
469C2×C8C22×Q814yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4C22×D414yes—
501C4⋊C4C22×D414yes—
502C4⋊C4C22×Q814yes—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—48no—
518C23—24no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23#53612yes—
533C23D4212yes—
534C23D4×Q812yes—
535C23C2×AGL(1,ℤ/8)12yes—
536C23C2×#13712yes—
537C23#57612yes—
538C23#57712yes—
539C23C23×D411yescharacteristic
540Q8—816no—
541Q8—816no—
542D4—24no—
543D4—24no—
544D4—48no—
545D4—42no—
546D4—42no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—44no—
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—816no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4#53612yes—
599C2×C4#53614yes—
600C2×C4#54414yes—
601C2×C4D4212yes—
602C2×C4D4×Q812yes—
603C2×C4D4×Q814yes—
604C2×C4#56314yes—
605C2×C4C2×AGL(1,ℤ/8)14yes—
606C2×C4C2×#13714yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C23×Q811yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C8—24no—
614C8—24no—
615C8—42no—
616C8—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—816no—
625C22#267714yes—
626C22#323914yes—
627C22#324114yes—
628C22#326414yes—
629C22#327814yes—
630C22#338414yes—
631C22#338514yes—
632C22C2×#53611yescharacteristic
633C22C2×D4211yescharacteristic
634C22C2×D4×Q811yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C22×#13711yescharacteristic
637C22C2×#57611yescharacteristic
638C22C2×#57711yescharacteristic
639C4—24no—
640C4—24no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4#267714yes—
653C4C2×#53611yescharacteristic
654C4C2×#54411yescharacteristic
655C4C2×D4×Q811yescharacteristic
656C4C2×#56311yescharacteristic
657C2—84no—
658C2#3096218yesdirect product factor
659C2C2×#267711yescharacteristic
660C2C2×#323911yescharacteristic
661C2C2×#324111yescharacteristic
662C2C2×#326411yescharacteristic
663C2C2×#327811yescharacteristic
664C2C2×#338411yescharacteristic
665C2C2×#338511yescharacteristic
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×#501C2semidirect
3. C2×#2551C2semidirect
5. #30962C2direct (N × H)
7. C2×#2088C2semidirect
8. C2×#2083C2semidirect
9. C2×#2059C2semidirect
10. C2×#2058C2semidirect
14. C2×#1921C2semidirect
15. C2×#1917C2semidirect
23. C2×#501C22semidirect
39. #2551C22semidirect
59. #2088C22semidirect
60. #2083C22semidirect
61. #2059C22semidirect
62. #2058C22semidirect
66. #1921C22semidirect
67. #1917C22semidirect
152. C2×#102Q8semidirect
351. #102C2×Q8semidirect
658. C2#30962direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4742
C2×#25512
C2×#24542
#309622
C2×#20942
C2×#20882
C2×#20832
C2×#20592
C2×#20582
C2×#20232
C2×#19362
C2×#19262
C2×#19212
C2×#19172
C2×#18622