Finite Group Databasebeta

← C2×#30885 #7502180

Subgroups — C2×#30885 #7502180

Order 512.

2388 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic
2C22×#464C211yesmaximal, characteristic
3C2×#2547C211yesmaximal, characteristic
4C2×#2453C211yesmaximal, characteristic, complemented
5#30885C2116yesmaximal, direct product factor
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2043C211yesmaximal, characteristic, complemented
8C2×#2042C211yesmaximal, characteristic
9C2×#2026C211yesmaximal, characteristic, complemented
10C2×#2016C211yesmaximal, characteristic, complemented
11C2×#2015C211yesmaximal, characteristic
12C2×#1969C211yesmaximal, characteristic, complemented
13C2×#1916C211yesmaximal, characteristic, complemented
14C2×#1878C211yesmaximal, characteristic, complemented
15C2×#1877C211yesmaximal, characteristic
16C2×#1862C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C23×SD16C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C2×#488—28no—
27C2×#488C2218yes—
28C2×#464—28no—
29C2×#464C2218yes—
30C22×#107C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43#2547—24no—
44#2547C2218yes—
45#2453—24no—
46#2453C2218yescomplemented
47C2×#400—21no—
48C2×#398—21no—
49C2×#397—21no—
50C2×#397—21no—
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#389—21no—
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#387—21no—
59C2×#380—21no—
60C2×#380—21no—
61C2×C4×C4⋊C4C2211yescharacteristic
62#2068C2218yescomplemented
63#2043C2218yescomplemented
64#2042C2218yes—
65#2026C2218yescomplemented
66#2016C2218yescomplemented
67#2015C2218yes—
68#1969C2218yescomplemented
69#1916C2218yescomplemented
70#1878C2218yescomplemented
71#1877C2218yes—
72#1862C2218yes—
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4—21no—
84C22×Q16—24no—
85C22×Q16—24no—
86C22×Q16—24no—
87C22×Q16—42no—
88C22×Q16—42no—
89C22×Q16—42no—
90C22×Q16C2314yes—
91C22×Q16C2314yes—
92C22×SD16—24no—
93C22×SD16—24no—
94C22×SD16—42no—
95C22×SD16C2314yes—
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99C23×C8C2311yescharacteristic
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8—42no—
103C2×C4⋊Q8C2314yes—
104C2×#125—24no—
105C2×#125—22no—
106C2×#125—42no—
107C2×#125C2314yes—
108C2×#124—24no—
109C2×#124—24no—
110C2×#124—22no—
111C2×#124—22no—
112C2×#124—42no—
113C2×#124—42no—
114C2×#124C2314yes—
115C2×#124C2314yes—
116C2×C4×Q8—22no—
117C2×C4×Q8—24no—
118C2×C4×Q8—42no—
119C2×C4×Q8C2314yes—
120C2×C4×D4—22no—
121C2×C4×D4—42no—
122C2×C4×D4—42no—
123C2×C4×D4C2314yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146#488—232no—
147#464—232no—
148C2×#107—42no—
149C2×#107C2314yes—
150C2×C4⋊C8—24no—
151C2×C4⋊C8C2314yes—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—24no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103—42no—
158C2×#103—42no—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102—24no—
167C2×#102—42no—
168C2×#102D418yes—
169C2×#102C2314yes—
170C2×#102C2314yes—
171C2×#102C2314yes—
172C2×C4×C8—24no—
173C2×C4×C8C2314yes—
174#400—28no—
175#398—28no—
176#397—28no—
177#397—28no—
178#391—22no—
179#391—22no—
180#391—22no—
181#391—22no—
182#391C2314yes—
183#391C2314yes—
184#391C2314yes—
185#391C2314yes—
186#389—28no—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#387—28no—
192#380—28no—
193#380—28no—
194C4×C4⋊C4D414yes—
195C4×C4⋊C4C2314yes—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25—21no—
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8#4114yes—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4C2×D414yes—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—24no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×Q16—416no—
272C2×Q16—416no—
273C2×SD16—28no—
274C2×SD16—28no—
275C2×SD16—44no—
276C2×SD16—416no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8C2×D414yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—48no—
296#125—28no—
297#125—44no—
298#125—48no—
299#124—28no—
300#124—28no—
301#124—44no—
302#124—44no—
303#124—48no—
304#124—48no—
305C4×Q8—28no—
306C4×Q8—44no—
307C4×Q8—48no—
308C4×D4—44no—
309C4×D4—44no—
310C4×D4—48no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—44no—
354C2×C22⋊C4—44no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—44no—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2412yes—
367C2×C42C2412yes—
368C2×C42C2412yes—
369#107—44no—
370C4⋊C8—28no—
371#103—28no—
372#103—28no—
373#103—28no—
374#103—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#102—28no—
379#102—44no—
380#102C2×D4116yes—
381C4×C8—28no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95C2×D414yes—
393#95C2×D414yes—
394C24—24no—
395C24—22no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24C22×D411yescharacteristic
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8C2×#4114yes—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4C22×D414yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—48no—
503C22×C4—816no—
504C22×C4#13514yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C2×#4112yes—
514C22×C4H(5,𝔽2)12yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C42−1+412yes—
518C22×C4C2511yescharacteristic
519Q16—416no—
520Q16—416no—
521SD16—416no—
522C2×C8—28no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—44no—
535C2×C8—44no—
536C2×C8—44no—
537C2×C8C22×D414yes—
538C4⋊C4—24no—
539C4⋊C4—28no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—44no—
548C4⋊C4—48no—
549C4⋊C4—48no—
550C4⋊C4—48no—
551C4⋊C4—48no—
552C4⋊C4—48no—
553C4⋊C4—48no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—48no—
558C4⋊C4C22×D414yes—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C22⋊C4—48no—
564C42—24no—
565C42—24no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—48no—
574C23—24no—
575C23—48no—
576C23—42no—
577C23—42no—
578C23—48no—
579C23—48no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23#53512yes—
589C23D4212yes—
590C23#54812yes—
591C23C2×#13512yes—
592C23#57512yes—
593C23#57812yes—
594C23#57812yes—
595C23C23×D411yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—48no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—84no—
610Q8—84no—
611D4—24no—
612D4—24no—
613D4—48no—
614D4—42no—
615D4—42no—
616C2×C4—22no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—48no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—816no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4#53512yes—
669C2×C4#53714yes—
670C2×C4#53814yes—
671C2×C4D4212yes—
672C2×C4#54614yes—
673C2×C4#54812yes—
674C2×C4#56214yes—
675C2×C4C2×#13514yes—
676C2×C4C23×D411yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4C2×H(5,𝔽2)11yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4C2×2−1+411yescharacteristic
682C8—24no—
683C8—42no—
684C8—42no—
685C8—42no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—816no—
694C22#267614yes—
695C22#321114yes—
696C22#322314yes—
697C22#326914yes—
698C22#328314yes—
699C22#336414yes—
700C22#337714yes—
701C22C2×#53511yescharacteristic
702C22C2×D4211yescharacteristic
703C22C2×#54811yescharacteristic
704C22C22×#13511yescharacteristic
705C22C2×#57511yescharacteristic
706C22C2×#57811yescharacteristic
707C22C2×#57811yescharacteristic
708C4—24no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—44no—
714C4—42no—
715C4—44no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4#267614yes—
722C4C2×#53711yescharacteristic
723C4C2×#53811yescharacteristic
724C4C2×#54611yescharacteristic
725C4C2×#56211yescharacteristic
726C2—84no—
727C2#3088518yesdirect product factor
728C2C2×#267611yescharacteristic
729C2C2×#321111yescharacteristic
730C2C2×#322311yescharacteristic
731C2C2×#326911yescharacteristic
732C2C2×#328311yescharacteristic
733C2C2×#336411yescharacteristic
734C2C2×#337711yescharacteristic
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
4. C2×#2453C2semidirect
5. #30885C2direct (N × H)
6. C2×#2068C2semidirect
7. C2×#2043C2semidirect
9. C2×#2026C2semidirect
10. C2×#2016C2semidirect
12. C2×#1969C2semidirect
13. C2×#1916C2semidirect
14. C2×#1878C2semidirect
46. #2453C22semidirect
62. #2068C22semidirect
63. #2043C22semidirect
65. #2026C22semidirect
66. #2016C22semidirect
68. #1969C22semidirect
69. #1916C22semidirect
70. #1878C22semidirect
727. C2#30885direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4882
C22×#4642
C2×#25472
C2×#24532
#308852
C2×#20682
C2×#20432
C2×#20422
C2×#20262
C2×#20162
C2×#20152
C2×#19692
C2×#19162
C2×#18782
C2×#18772
C2×#18622