Finite Group Databasebeta

← C2×#82859 #10964244

Subgroups — C2×#82859 #10964244

Order 768.

4222 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic, complemented
2C22×#935C211yesmaximal, characteristic, complemented
3C22×#939C211yesmaximal, characteristic
4C22×#929C211yesmaximal, characteristic, complemented
5C23×Dic3⋊C4C211yesmaximal, characteristic, complemented
6C22×#886C211yesmaximal, characteristic, complemented
7#82859C2116yesmaximal, direct product factor
8C2×#4993C212yesmaximal, complemented
9C2×#4975C212yesmaximal, complemented
10C2×#4949C212yesmaximal, complemented
11C2×#4957C212yesmaximal, complemented
12C2×Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
13C2×#2501—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C23×#67C2211yescharacteristic
17C23×#67C2211yescharacteristic
18C24×Dic3C2211yescharacteristic
19C24×Dic3C2211yescharacteristic
20C2×C3×#123—28no—
21C2×C3×#123C2218yescomplemented
22C22×C3×C4⋊C4C2212yes—
23C22×C3×C22⋊C4C2212yes—
24C22×C3×C22⋊C4C2211yescharacteristic
25C22×#217C2212yes—
26C22×#217C2211yescharacteristic
27C22×#217C2211yescharacteristic
28C2×#939—28no—
29C2×#939C2218yes—
30C2×#935—28no—
31C2×#935C2218yescomplemented
32C2×#929—28no—
33C2×#929C2218yescomplemented
34C22×(C2×D3)⋊C4C2211yescharacteristic
35C22×(C2×D3)⋊C4C2212yes—
36C22×(C2×D3)⋊C4C2212yes—
37C22×C4⋊Dic3C2212yes—
38C22×Dic3⋊C4—28no—
39C22×Dic3⋊C4C2212yes—
40C22×Dic3⋊C4C2212yes—
41C22×Dic3⋊C4C2212yes—
42C22×Dic3⋊C4C2212yes—
43C22×Dic3⋊C4C2211yescharacteristic
44C22×Dic3⋊C4C2218yescomplemented
45C22×C4×Dic3C2212yes—
46C2×#886—28no—
47C2×#886C2218yescomplemented
48C2×C3⋊#95C2212yes—
49#4993C22116yescomplemented
50#4975C22116yescomplemented
51#4957C22116yescomplemented
52#4949C22116yescomplemented
53Dic3×C22⋊C4C2218yescomplemented
54C22×#123—31no—
55C22×#123—31no—
56C22×#123—31no—
57C22×#123D311yescharacteristic, complemented
58C22×#121—31no—
59C23×C4⋊C4—31no—
60#2501—316no—
61C2×#399—32no—
62C2×#399—32no—
63C2×#390—32no—
64C2×#384—32no—
65C2×C4×C22⋊C4—31no—
66C25×C3C2311yescharacteristic
67C25×C3C2311yescharacteristic
68C24×D3C2311yescharacteristic
69C22×C3×D4—24no—
70C22×C3×D4—24no—
71C22×C3×D4—24no—
72C22×C3×D4C2314yes—
73C23×C3×C4—24no—
74C23×C3×C4—24no—
75C23×C3×C4—24no—
76C23×C3×C4C2312yes—
77C23×C3×C4C2314yes—
78C23×C3×C4C2311yescharacteristic
79C23×C3×C4C2312yes—
80C22×#67—24no—
81C22×#67—24no—
82C22×#67—24no—
83C22×#67—24no—
84C22×#67—28no—
85C22×#67C2314yes—
86C22×#67C2314yes—
87C23×Dic3—28no—
88C23×Dic3—24no—
89C23×Dic3—24no—
90C23×Dic3—24no—
91C23×Dic3D418yes—
92C23×Dic3C2314yes—
93C23×Dic3C2312yes—
94C23×Dic3C2311yescharacteristic
95C23×Dic3C2314yes—
96C23×Dic3C2311yescharacteristic
97C23×Dic3C2312yes—
98C23×Dic3C2311yescharacteristic
99C3×#123—232no—
100C2×C3×C4⋊C4—28no—
101C2×C3×C4⋊C4C2318yes—
102C2×C3×C22⋊C4—28no—
103C2×C3×C22⋊C4D418yescomplemented
104C2×C3×C22⋊C4C2318yes—
105C2×C3×C22⋊C4C2314yes—
106C2×#217—28no—
107C2×#217—24no—
108C2×#217D418yescomplemented
109C2×#217C2314yes—
110C2×#217C2318yes—
111C2×#217C2314yes—
112#939—232no—
113#935—232no—
114#929—232no—
115C2×(C2×D3)⋊C4—28no—
116C2×(C2×D3)⋊C4—28no—
117C2×(C2×D3)⋊C4—24no—
118C2×(C2×D3)⋊C4C2318yes—
119C2×(C2×D3)⋊C4C2318yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×C4⋊Dic3—28no—
122C2×C4⋊Dic3C2318yes—
123C2×Dic3⋊C4—28no—
124C2×Dic3⋊C4—28no—
125C2×Dic3⋊C4—232no—
126C2×Dic3⋊C4—28no—
127C2×Dic3⋊C4—28no—
128C2×Dic3⋊C4—24no—
129C2×Dic3⋊C4C2318yes—
130C2×Dic3⋊C4C2318yes—
131C2×Dic3⋊C4C2318yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2318yes—
134C2×C4×Dic3—28no—
135C2×C4×Dic3C2318yes—
136#886—232no—
137C3⋊#95C2318yes—
138C23×D4—31no—
139C23×D4—31no—
140C23×D4C2×D311yescharacteristic
141C24×C4—31no—
142C24×C4—31no—
143C24×C4C2×D311yescharacteristic
144C2×#123—28no—
145C2×#123—38no—
146C2×#123—38no—
147C2×#123—38no—
148C2×#123—68no—
149C2×#123—68no—
150C2×#123—68no—
151C2×#123C2×D318yescomplemented
152C2×#121—38no—
153C2×#121—68no—
154C22×C4⋊C4—32no—
155C22×C4⋊C4—32no—
156C22×C4⋊C4—32no—
157C22×C4⋊C4—32no—
158C22×C4⋊C4—32no—
159C22×C4⋊C4—38no—
160C22×C4⋊C4—31no—
161C22×C4⋊C4—68no—
162C22×C4⋊C4C2×D312yes—
163C22×C22⋊C4—31no—
164C22×C22⋊C4—31no—
165C22×C22⋊C4—32no—
166C22×C22⋊C4—31no—
167C22×C22⋊C4—32no—
168C22×C22⋊C4—32no—
169C22×C22⋊C4C2×D312yes—
170C22×C22⋊C4C2×D311yescharacteristic
171C22×C42—32no—
172#399—316no—
173#399—316no—
174#390—316no—
175#384—316no—
176C4×C22⋊C4—38no—
177C2×#95—32no—
178C24×C3—24no—
179C24×C3—24no—
180C24×C3—22no—
181C24×C3—22no—
182C24×C3—22no—
183C24×C3—24no—
184C24×C3—24no—
185C24×C3C2×D414yes—
186C24×C3C2×D414yes—
187C24×C3C2×D414yes—
188C24×C3C2412yes—
189C24×C3C2412yes—
190C24×C3C2411yescharacteristic
191C23×D3—22no—
192C23×D3—22no—
193C23×D3—22no—
194C23×D3—24no—
195C23×D3—24no—
196C23×D3C2412yes—
197C2×C3×D4—28no—
198C2×C3×D4—28no—
199C2×C3×D4—28no—
200C2×C3×D4—416no—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—28no—
204C22×C3×C4—24no—
205C22×C3×C4—22no—
206C22×C3×C4—24no—
207C22×C3×C4—22no—
208C22×C3×C4—28no—
209C22×C3×C4—24no—
210C22×C3×C4—22no—
211C22×C3×C4—28no—
212C22×C3×C4—416no—
213C22×C3×C4C2×D418yes—
214C22×C3×C4C2414yes—
215C22×C3×C4C2412yes—
216C22×C3×C4C2414yes—
217C2×#67—28no—
218C2×#67—28no—
219C2×#67—28no—
220C2×#67—28no—
221C2×#67—216no—
222C2×#67—416no—
223C2×#67—416no—
224C22×Dic3—24no—
225C22×Dic3—24no—
226C22×Dic3—24no—
227C22×Dic3—232no—
228C22×Dic3—216no—
229C22×Dic3—28no—
230C22×Dic3—28no—
231C22×Dic3—24no—
232C22×Dic3—22no—
233C22×Dic3—24no—
234C22×Dic3—22no—
235C22×Dic3—22no—
236C22×Dic3—24no—
237C22×Dic3—28no—
238C22×Dic3—416no—
239C22×Dic3C2×D418yes—
240C22×Dic3C2×D414yes—
241C22×Dic3C2×D4116yes—
242C22×Dic3C2×D414yes—
243C22×Dic3#4118yes—
244C22×Dic3C2412yes—
245C22×Dic3C2414yes—
246C22×Dic3C2412yes—
247C22×Dic3C2414yes—
248C22×Dic3C2412yes—
249C3×C4⋊C4—216no—
250C3×C22⋊C4—216no—
251C3×C22⋊C4C2×D4116yescomplemented
252#217—216no—
253#217—28no—
254#217C2×D4116yescomplemented
255(C2×D3)⋊C4—216no—
256(C2×D3)⋊C4—216no—
257(C2×D3)⋊C4—28no—
258C4⋊Dic3—216no—
259Dic3⋊C4—216no—
260Dic3⋊C4—216no—
261Dic3⋊C4—216no—
262Dic3⋊C4—216no—
263Dic3⋊C4—28no—
264C4×Dic3—216no—
265C25—31no—
266C25C22×D311yescharacteristic
267C25C22×D311yescharacteristic
268C22×D4—24no—
269C22×D4—24no—
270C22×D4—24no—
271C22×D4—34no—
272C22×D4—34no—
273C22×D4—64no—
274C22×D4—68no—
275C22×D4—64no—
276C22×D4—64no—
277C22×D4—64no—
278C22×D4C22×D314yes—
279C23×C4—24no—
280C23×C4—24no—
281C23×C4—24no—
282C23×C4—31no—
283C23×C4—38no—
284C23×C4—34no—
285C23×C4—32no—
286C23×C4—31no—
287C23×C4—34no—
288C23×C4—32no—
289C23×C4—31no—
290C23×C4—68no—
291C23×C4—64no—
292C23×C4—64no—
293C23×C4—64no—
294C23×C4C22×D314yes—
295C23×C4C22×D312yes—
296C23×C4C22×D311yescharacteristic
297C23×C4C22×D312yes—
298#123—232no—
299#123—632no—
300#123—632no—
301#123—632no—
302#121—632no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—38no—
305C2×C4⋊C4—38no—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—38no—
308C2×C4⋊C4—38no—
309C2×C4⋊C4—38no—
310C2×C4⋊C4—68no—
311C2×C4⋊C4—68no—
312C2×C4⋊C4—632no—
313C2×C4⋊C4—64no—
314C2×C4⋊C4—68no—
315C2×C4⋊C4—68no—
316C2×C4⋊C4—68no—
317C2×C4⋊C4C22×D318yes—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—38no—
320C2×C22⋊C4—34no—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—38no—
323C2×C22⋊C4—34no—
324C2×C22⋊C4—38no—
325C2×C22⋊C4—38no—
326C2×C22⋊C4—64no—
327C2×C22⋊C4—68no—
328C2×C22⋊C4—64no—
329C2×C22⋊C4—68no—
330C2×C22⋊C4—68no—
331C2×C22⋊C4#6718yescomplemented
332C2×C22⋊C4C22×D318yes—
333C2×C22⋊C4C22×D314yes—
334C2×C42—38no—
335C2×C42—68no—
336#95—38no—
337C23×C3—28no—
338C23×C3—28no—
339C23×C3—216no—
340C23×C3—216no—
341C23×C3—24no—
342C23×C3—24no—
343C23×C3—22no—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—24no—
347C23×C3—24no—
348C23×C3—44no—
349C23×C3—44no—
350C23×C3—44no—
351C23×C3—48no—
352C23×C3—48no—
353C23×C3#127116yes—
354C23×C3C22×D414yes—
355C23×C3C22×D414yes—
356C23×C3C22×D412yes—
357C23×C3C22×D412yes—
358C23×C3C22×D414yes—
359C23×C3C22×D412yes—
360C23×C3C2×#4114yes—
361C23×C32−1+414yes—
362C23×C3C2511yescharacteristic
363C22×D3—22no—
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—24no—
367C22×D3—24no—
368C22×D3—44no—
369C22×D3—44no—
370C22×D3—44no—
371C22×D3—48no—
372C22×D3—48no—
373C3×D4—416no—
374C2×C3×C4—216no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—24no—
378C2×C3×C4—22no—
379C2×C3×C4—24no—
380C2×C3×C4—22no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—416no—
384C2×C3×C4—48no—
385C2×C3×C4—44no—
386C2×C3×C4C22×D418yes—
387#67—416no—
388#67—416no—
389C2×Dic3—28no—
390C2×Dic3—216no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—232no—
395C2×Dic3—24no—
396C2×Dic3—22no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—24no—
401C2×Dic3—416no—
402C2×Dic3—48no—
403C2×Dic3—44no—
404C2×Dic3#123116yescomplemented
405C2×Dic3C22×D418yes—
406C2×Dic3C22×D414yes—
407C2×Dic3C22×D414yes—
408C2×Dic3C2×#4118yes—
409C24—24no—
410C24—24no—
411C24—22no—
412C24—22no—
413C24—24no—
414C24—24no—
415C24—22no—
416C24—32no—
417C24—62no—
418C24—62no—
419C24—64no—
420C24—64no—
421C24—62no—
422C24D3×D414yes—
423C24D3×D414yes—
424C24C2×#6714yes—
425C24C23×D312yes—
426C24C23×D312yes—
427C24C23×D311yescharacteristic
428C2×D4—28no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—416no—
432C2×D4—68no—
433C2×D4—616no—
434C2×D4—68no—
435C2×D4—68no—
436C2×D4—68no—
437C2×D4—1216no—
438C2×D4—1216no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—28no—
450C22×C4—34no—
451C22×C4—38no—
452C22×C4—32no—
453C22×C4—38no—
454C22×C4—34no—
455C22×C4—316no—
456C22×C4—32no—
457C22×C4—34no—
458C22×C4—34no—
459C22×C4—32no—
460C22×C4—416no—
461C22×C4—632no—
462C22×C4—616no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—64no—
466C22×C4—64no—
467C22×C4—62no—
468C22×C4—68no—
469C22×C4—64no—
470C22×C4—62no—
471C22×C4—68no—
472C22×C4—64no—
473C22×C4—62no—
474C22×C4—68no—
475C22×C4—1216no—
476C22×C4C2×#6718yes—
477C22×C4C23×D314yes—
478C22×C4C23×D312yes—
479C22×C4C23×D314yes—
480C4⋊C4—216no—
481C4⋊C4—616no—
482C4⋊C4—616no—
483C4⋊C4—68no—
484C4⋊C4—616no—
485C4⋊C4—616no—
486C4⋊C4—616no—
487C22⋊C4—216no—
488C22⋊C4—316no—
489C22⋊C4—68no—
490C22⋊C4—616no—
491C22⋊C4—68no—
492C22⋊C4—616no—
493C22⋊C4—616no—
494C22⋊C4C2×#67116yescomplemented
495C42—616no—
496C22×C3—24no—
497C22×C3—24no—
498C22×C3—28no—
499C22×C3—28no—
500C22×C3—232no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—44no—
505C22×C3—44no—
506C22×C3—88no—
507C22×C3C2×#12314yes—
508C22×C3C2×#12718yes—
509C22×C3C2×#12714yes—
510C22×C3#53614yes—
511C22×C3#54718yes—
512C22×C3#54718yes—
513C22×C3C23×D411yescharacteristic
514C22×C3C23×D411yescharacteristic
515C22×C3C23×D411yescharacteristic
516C22×C3C22×#4112yes—
517C22×C3C2×2−1+412yes—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—44no—
522C2×D3—44no—
523C2×D3—88no—
524C3×C4—28no—
525C3×C4—44no—
526C3×C4—42no—
527Dic3—24no—
528Dic3—28no—
529Dic3—44no—
530Dic3—42no—
531Dic3C2×#12318yescomplemented
532C23—28no—
533C23—28no—
534C23—216no—
535C23—24no—
536C23—24no—
537C23—216no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—44no—
544C23—44no—
545C23—48no—
546C23—44no—
547C23—48no—
548C23—62no—
549C23—62no—
550C23—64no—
551C23—64no—
552C23—62no—
553C23—124no—
554C23—124no—
555C23—128no—
556C23—124no—
557C23—128no—
558C23#940116yes—
559C23C2×D3×D414yes—
560C23C2×D3×D414yes—
561C23C2×D3×D412yes—
562C23C2×D3×D412yes—
563C23D3×#4114yes—
564C23#101014yes—
565C23C22×#6714yes—
566C23C22×#6712yes—
567C23C24×D311yescharacteristic
568D4—416no—
569D4—1216no—
570D4—1216no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—216no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—316no—
581C2×C4—34no—
582C2×C4—38no—
583C2×C4—38no—
584C2×C4—34no—
585C2×C4—416no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—632no—
589C2×C4—68no—
590C2×C4—616no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—64no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—62no—
600C2×C4—1216no—
601C2×C4—128no—
602C2×C4—124no—
603C2×C4C22×#6718yes—
604C2×C3—28no—
605C2×C3—82no—
606C2×C3#250118yescomplemented
607C2×C3C22×#12311yescharacteristic
608C2×C3C22×#12711yescharacteristic
609C2×C3C2×#53611yescharacteristic
610C2×C3C2×#54712yes—
611C2×C3C2×#54712yes—
612D3—82no—
613C22—232no—
614C22—28no—
615C22—24no—
616C22—24no—
617C22—28no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—88no—
624C22—122no—
625C22—124no—
626C22—122no—
627C22—122no—
628C22—124no—
629C22—248no—
630C22#564814yes—
631C22D3×#12314yes—
632C22#565518yes—
633C22C2×#94018yes—
634C22C2×#94014yes—
635C22#582418yes—
636C22C22×D3×D411yescharacteristic
637C22C22×D3×D411yescharacteristic
638C22C2×D3×#4112yes—
639C22C2×#101012yes—
640C22C23×#6711yescharacteristic
641C4—28no—
642C4—38no—
643C4—44no—
644C4—42no—
645C4—64no—
646C4—68no—
647C4—124no—
648C4—122no—
649C3C2×#250111yescharacteristic, complemented, Sylow
650C2—28no—
651C2—82no—
652C2—242no—
653C2#8285918yesdirect product factor
654C2C2×#564811yescharacteristic
655C2C2×D3×#12311yescharacteristic
656C2C2×#565512yes—
657C2C22×#94011yescharacteristic
658C2C2×#582412yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×#123C2
socleC24×C3C24
prime radicalC22×#939C2
hypercenterC22×#123D3
hypocenterC3C2×#2501
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C22×C3×#123C2semidirect
2. C22×#935C2semidirect
4. C22×#929C2semidirect
5. C23×Dic3⋊C4C2semidirect
6. C22×#886C2semidirect
7. #82859C2direct (N × H)
8. C2×#4993C2semidirect
9. C2×#4975C2semidirect
10. C2×#4949C2semidirect
11. C2×#4957C2semidirect
12. C2×Dic3×C22⋊C4C2semidirect
21. C2×C3×#123C22semidirect
31. C2×#935C22semidirect
33. C2×#929C22semidirect
44. C22×Dic3⋊C4C22semidirect
47. C2×#886C22semidirect
49. #4993C22semidirect
50. #4975C22semidirect
51. #4957C22semidirect
52. #4949C22semidirect
53. Dic3×C22⋊C4C22semidirect
57. C22×#123D3semidirect
103. C2×C3×C22⋊C4D4semidirect
108. C2×#217D4semidirect
151. C2×#123C2×D3semidirect
251. C3×C22⋊C4C2×D4semidirect
254. #217C2×D4semidirect
331. C2×C22⋊C4#67semidirect
404. C2×Dic3#123semidirect
494. C22⋊C4C2×#67semidirect
531. Dic3C2×#123semidirect
606. C2×C3#2501semidirect
649. C3C2×#2501semidirect
653. C2#82859direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C22×#9352
C22×#9392
C22×#9292
C23×Dic3⋊C42
C22×#8862
#828592
C2×#49932
C2×#49752
C2×#49492
C2×#49572
C2×Dic3×C22⋊C42
C2×#25013