Finite Group Databasebeta

← C2×#22162 #6375846

Subgroups — C2×#22162 #6375846

Order 512.

2360 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#461C211yesmaximal, characteristic
2C2×#2327C211yesmaximal, characteristic, complemented
3C2×#2319C211yesmaximal, characteristic
4#22162C2116yesmaximal, direct product factor
5C2×#1981C211yesmaximal, characteristic, complemented
6C2×#1973C211yesmaximal, characteristic
7C2×#1968C211yesmaximal, characteristic
8C2×#1949C211yesmaximal, characteristic, complemented
9C2×#1935C211yesmaximal, characteristic, complemented
10C2×#1933C211yesmaximal, characteristic, complemented
11C2×#1880C211yesmaximal, characteristic
12C2×#1880C211yesmaximal, characteristic
13C2×#1865C211yesmaximal, characteristic, complemented
14C2×#1863C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×#1768C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#461C4116yescomplemented
24C2×#461C2218yes—
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#97C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39#2327—24no—
40#2327C2218yescomplemented
41#2319—24no—
42#2319C2218yes—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#388—21no—
50C2×#388—21no—
51C2×#387—21no—
52C2×#387—21no—
53C2×#387—21no—
54C2×#387—21no—
55C2×#382C2211yescharacteristic
56C2×#381—21no—
57C2×#380—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×#376C2211yescharacteristic
60#1981C2218yescomplemented
61#1973C2218yes—
62#1968C2218yes—
63#1949C2218yescomplemented
64#1935C2218yescomplemented
65#1933C2218yescomplemented
66#1880C2218yes—
67#1880C2218yes—
68#1865C2218yescomplemented
69#1863C2218yescomplemented
70#1804C2218yes—
71#1768C2218yescomplemented
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×SD16—42no—
83C22×SD16—42no—
84C22×SD16—42no—
85C22×SD16—42no—
86C22×SD16C2×C418yes—
87C22×SD16C2×C418yes—
88C22×SD16C2314yes—
89C22×SD16C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8C2×C418yes—
97C2×C4⋊Q8C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—42no—
101C2×C4×Q8—42no—
102C2×C4×Q8—42no—
103C2×C4×Q8—42no—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2314yes—
106C2×C4×D4—22no—
107C2×C4×D4—24no—
108C2×C4×D4C2×C418yes—
109C2×C4×D4C2314yes—
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⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—21no—
127C22×C42—21no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#461C2×C4164yescomplemented
133C2×#107—42no—
134C2×#107C2314yes—
135C2×#106—42no—
136C2×#106C2314yes—
137C2×C4⋊C8—24no—
138C2×C4⋊C8C2×C418yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103C2×C418yes—
146C2×#103C2×C418yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102—42no—
154C2×#102—42no—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#97—42no—
158C2×#97C2314yes—
159C2×C4×C8—42no—
160C2×C4×C8C2314yes—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#388—28no—
174#388—28no—
175#387—28no—
176#387—28no—
177#387—28no—
178#387—28no—
179#382—22no—
180#382C2314yes—
181#381—28no—
182#380—28no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185#376D414yes—
186#376C2314yes—
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×#95—21no—
196C2×#95D411yescharacteristic
197C2×#95D411yescharacteristic
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8C22×C414yes—
220C22×Q8C22×C414yes—
221C22×Q8C2412yes—
222C22×Q8C2412yes—
223C22×D4—21no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4C22×C414yes—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—28no—
235C23×C4—24no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—82no—
241C23×C4—82no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×SD16—416no—
250C2×SD16—416no—
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16C22×C4116yes—
256C2×SD16C22×C4116yes—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8C22×C414yes—
268C22×C8C22×C414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—216no—
274C4⋊Q8—28no—
275C4⋊Q8C22×C4116yes—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8—48no—
282C4×D4—216no—
283C4×D4—28no—
284C4×D4C22×C4116yes—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4C22×C414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—82no—
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—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42—82no—
337C2×C42C22×C414yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42#4114yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—44no—
346#106—44no—
347C4⋊C8—28no—
348C4⋊C8C22×C4116yes—
349#103—44no—
350#103—44no—
351#103—44no—
352#103—44no—
353#103C22×C4116yes—
354#103C22×C4116yes—
355#102—44no—
356#102—44no—
357#97—44no—
358C4×C8—44no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370#95C2×D414yes—
371#95C2×D414yes—
372C24—24no—
373C24—22no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24C22×D411yescharacteristic
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8C23×C414yes—
406C2×Q8C23×C414yes—
407C2×D4—24no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4C23×C414yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
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—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
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—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—816no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4C4×D414yes—
469C22×C4C4×D414yes—
470C22×C4C23×C412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484SD16—416no—
485SD16—416no—
486C2×C8—28no—
487C2×C8—28no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—44no—
499C2×C8—44no—
500C2×C8C23×C414yes—
501C2×C8C23×C414yes—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4C23×C414yes—
523C22⋊C4—216no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—48no—
535C42—48no—
536C42—84no—
537C42C4×D418yes—
538C42C23×C414yes—
539C42C2×#4114yes—
540C23—24no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23—48no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23C2×C4×D412yes—
555C23#53512yes—
556C23#53712yes—
557C23#57712yes—
558C23#57712yes—
559C23#57712yes—
560C23#57812yes—
561C23C23×D411yescharacteristic
562Q8—24no—
563Q8—24no—
564Q8—24no—
565Q8—24no—
566Q8—48no—
567Q8—48no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572D4—24no—
573D4—24no—
574D4—48no—
575D4—42no—
576D4—42no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
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—816no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4C2×C4×D414yes—
629C2×C4C2×C4×D414yes—
630C2×C4C2×C4×D412yes—
631C2×C4#51814yes—
632C2×C4#52014yes—
633C2×C4#53512yes—
634C2×C4#53712yes—
635C2×C4#54014yes—
636C2×C4#54214yes—
637C2×C4C24×C411yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×2−1+411yescharacteristic
643C8—24no—
644C8—24no—
645C8—42no—
646C8—42no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—82no—
651C22—82no—
652C22—816no—
653C22—82no—
654C22—82no—
655C22#232414yes—
656C22#294714yes—
657C22#294914yes—
658C22#322114yes—
659C22#322514yes—
660C22#325814yes—
661C22#326014yes—
662C22C22×C4×D411yescharacteristic
663C22C2×#53511yescharacteristic
664C22C2×#53711yescharacteristic
665C22C2×#57711yescharacteristic
666C22C2×#57711yescharacteristic
667C22C2×#57711yescharacteristic
668C22C2×#57811yescharacteristic
669C4—24no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#232414yes—
683C4C2×#51811yescharacteristic
684C4C2×#52011yescharacteristic
685C4C2×#54011yescharacteristic
686C4C2×#54211yescharacteristic
687C2—84no—
688C2#2216218yesdirect product factor
689C2C2×#232411yescharacteristic
690C2C2×#294711yescharacteristic
691C2C2×#294911yescharacteristic
692C2C2×#322111yescharacteristic
693C2C2×#322511yescharacteristic
694C2C2×#325811yescharacteristic
695C2C2×#326011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2327C2semidirect
4. #22162C2direct (N × H)
5. C2×#1981C2semidirect
8. C2×#1949C2semidirect
9. C2×#1935C2semidirect
10. C2×#1933C2semidirect
13. C2×#1865C2semidirect
14. C2×#1863C2semidirect
16. C2×#1768C2semidirect
23. C2×#461C4semidirect
40. #2327C22semidirect
60. #1981C22semidirect
63. #1949C22semidirect
64. #1935C22semidirect
65. #1933C22semidirect
68. #1865C22semidirect
69. #1863C22semidirect
71. #1768C22semidirect
132. #461C2×C4semidirect
688. C2#22162direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4612
C2×#23272
C2×#23192
#221622
C2×#19812
C2×#19732
C2×#19682
C2×#19492
C2×#19352
C2×#19332
C2×#18802
C2×#18652
C2×#18632
C2×#18042
C2×#17682