Finite Group Databasebeta

← C2×#83530 #11670436

Subgroups — C2×#83530 #11670436

Order 768.

2858 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C22×#950C212yesmaximal, complemented
3C2×#5553C211yesmaximal, characteristic, complemented
4C2×#5238C212yesmaximal, complemented
5#83530C2116yesmaximal, direct product factor
6C2×#5090C211yesmaximal, characteristic, complemented
7C2×#5085C211yesmaximal, characteristic, complemented
8C2×#5047C212yesmaximal, complemented
9C2×#5048C212yesmaximal, complemented
10C2×C4×#214C211yesmaximal, characteristic, complemented
11C2×#5015C211yesmaximal, characteristic, complemented
12C2×D4×C3⋊C8C211yesmaximal, characteristic, complemented
13C2×#3284—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C22×#235C2212yes—
16C3×#547—24no—
17C3×#547C2218yescomplemented
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C2×#950—216no—
26C2×#950C22116yescomplemented
27C22×#216C2212yes—
28C22×#215C2212yes—
29C22×#215C2212yes—
30C2×#940—21no—
31C22×#214C2211yescharacteristic
32C22×#213C2212yes—
33C2×#930C2212yes—
34C23×C3⋊C8C2212yes—
35#5553—24no—
36#5553C2218yescomplemented
37C2×#893—21no—
38C2×#884—22no—
39C2×#874C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41#5238C22116yescomplemented
42C2×#835C2212yes—
43C2×#832C2211yescharacteristic
44C2×#830C2212yes—
45#5090C2218yescomplemented
46#5085C2218yescomplemented
47#5048C22116yescomplemented
48#5047C22116yescomplemented
49C4×#214C2218yescomplemented
50#5015C2218yescomplemented
51D4×C3⋊C8C2218yescomplemented
52C2×#810C2211yescharacteristic
53C2×#809C2212yes—
54C2×#808C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57C22×#135—32no—
58C2×#550—31no—
59C2×#547D311yescharacteristic, complemented
60#3284—316no—
61C2×#492—31no—
62C2×#465—32no—
63C2×#462—31no—
64C2×#459—31no—
65C2×#452—32no—
66C2×#449—32no—
67C2×C4×SD16—31no—
68C2×C8×D4—31no—
69C2×C3×#41—24no—
70C2×C3×#41—28no—
71C2×C3×#41—44no—
72C2×C3×#41C2318yes—
73C22×C3×Q8C2312yes—
74C22×C3×D4—22no—
75C22×C3×D4D411yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—22no—
78C23×C3×C4—21no—
79C23×C3×C4C2312yes—
80C22×#67—22no—
81C22×#67—22no—
82C2×#235—24no—
83C2×#235—28no—
84C2×#235—44no—
85C2×#235C2318yes—
86C22×D12—21no—
87C22×D12—21no—
88C22×D12C2312yes—
89C22×C4×D3—22no—
90C22×Dic6C2311yescharacteristic
91C3×C4⋊Q8—22no—
92C3×C4⋊Q8C2314yes—
93C3×#123—216no—
94C3×C22⋊Q8—24no—
95C3×C22⋊Q8C2318yes—
96C3×#121—28no—
97C3×C4×D4—22no—
98C3×C4×D4D414yes—
99C3×C4×D4C2314yes—
100C3×C4×D4C2314yes—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—28no—
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4D412yes—
110C2×C3×C42C2311yescharacteristic
111#950—264no—
112C2×#216—28no—
113C2×#216C2318yes—
114C2×#215—28no—
115C2×#215—44no—
116C2×#215C2318yes—
117C2×#215C2318yes—
118#940—28no—
119C2×#214D418yescomplemented
120C2×#214C2314yes—
121C2×#213—28no—
122C2×#213C2318yes—
123#930—24no—
124#930C2318yes—
125C2×(C2×D3)⋊C4—22no—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×Dic3⋊C4—21no—
128C2×C4×Dic3—21no—
129C22×C3⋊C8—28no—
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2312yes—
132C22×C3⋊C8C2318yes—
133C22×C3⋊C8C2311yescharacteristic
134#893—28no—
135#884—216no—
136#874—22no—
137#874C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140#835C2318yes—
141#832C2314yes—
142#830C2318yes—
143#810C2314yes—
144#809C2318yes—
145#808C2314yes—
146C4⋊(C3⋊C8)C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41—32no—
149C22×#41C2×D312yes—
150C2×#135—316no—
151C2×#135—616no—
152C22×Q16—32no—
153C22×SD16—31no—
154C22×SD16—32no—
155C22×SD16—32no—
156C22×D8—32no—
157C23×C8—32no—
158#550—38no—
159#550—64no—
160#547—24no—
161#547C2×D318yescomplemented
162C2×C4⋊Q8C2×D311yescharacteristic
163C2×#127—31no—
164C2×#127—61no—
165C2×#123—22no—
166C2×C22⋊Q8C2×D312yes—
167C2×#121—21no—
168C2×#121—32no—
169C2×#121—62no—
170C2×C4×Q8—31no—
171C2×C4×D4—61no—
172C2×C4×D4C2×D311yescharacteristic
173C2×C4×D4C2×D311yescharacteristic
174C22×C4⋊C4—21no—
175#492—38no—
176#465—316no—
177#462—38no—
178#459—38no—
179#452—316no—
180#449—316no—
181C4×SD16—38no—
182C8×D4—38no—
183C2×#106—31no—
184C2×C4⋊C8—31no—
185C2×#103—32no—
186C2×#103—31no—
187C2×#102—31no—
188C2×#102—32no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3—21no—
192C24×C3C2×D412yes—
193C23×D3—22no—
194C3×#41—216no—
195C3×#41—48no—
196C3×#41—416no—
197C2×C3×Q8—22no—
198C2×C3×Q8—24no—
199C2×C3×Q8—24no—
200C2×C3×Q8C2414yes—
201C2×C3×D4—28no—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—22no—
205C2×C3×D4—48no—
206C2×C3×D4C2×D414yes—
207C2×C3×D4C2×D414yes—
208C2×C3×D4C2412yes—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—24no—
214C22×C3×C4—28no—
215C22×C3×C4—48no—
216C22×C3×C4—44no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D414yes—
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2414yes—
222C22×C3×C4C2411yescharacteristic
223C2×#67—28no—
224C2×#67—28no—
225C2×#67—48no—
226C2×#67—84no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229#235—216no—
230#235—48no—
231#235—416no—
232C2×D12—24no—
233C2×D12—24no—
234C2×D12—22no—
235C2×D12—24no—
236C2×D12—24no—
237C2×D12—44no—
238C2×D12—82no—
239C2×D12C2414yes—
240C2×C4×D3—28no—
241C2×C4×D3—48no—
242C2×Dic6—21no—
243C2×Dic6—22no—
244C2×Dic6C2×D414yes—
245C2×Dic6C2412yes—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—28no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C2×D414yes—
251C3×C4⋊C4C2412yes—
252C3×C22⋊C4—28no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4C2×D418yes—
255C3×C42C2×D412yes—
256C3×C42C2412yes—
257#216—216no—
258#215—216no—
259#215—48no—
260#214C2×D4116yescomplemented
261#213—216no—
262(C2×D3)⋊C4—28no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2412yes—
265Dic3⋊C4—24no—
266C4×Dic3—24no—
267C2×C3⋊C8—24no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—216no—
270C2×C3⋊C8C2×D414yes—
271C2×C3⋊C8C2412yes—
272C2×C3⋊C8C2414yes—
273C2×C3⋊C8C2412yes—
274C2×#41—24no—
275C2×#41—28no—
276C2×#41—38no—
277C2×#41—44no—
278C2×#41—68no—
279C2×#41—64no—
280C2×#41—124no—
281C2×#41C22×D318yes—
282C22×Q8—31no—
283C22×Q8C22×D312yes—
284C22×D4—22no—
285C22×D4—32no—
286C22×D4—62no—
287C22×D4—62no—
288C22×D4—61no—
289C22×D4—61no—
290C22×D4#6711yescharacteristic
291C22×D4C22×D311yescharacteristic
292C23×C4—21no—
293C23×C4—22no—
294C23×C4—62no—
295C23×C4C22×D312yes—
296#135—664no—
297C2×Q16—38no—
298C2×Q16—68no—
299C2×SD16—38no—
300C2×SD16—34no—
301C2×SD16—38no—
302C2×SD16—38no—
303C2×SD16—68no—
304C2×SD16—124no—
305C2×D8—38no—
306C2×D8—68no—
307C22×C8—31no—
308C22×C8—32no—
309C22×C8—38no—
310C22×C8—31no—
311C22×C8—68no—
312C4⋊Q8—22no—
313C4⋊Q8C22×D314yes—
314#127—34no—
315#127—68no—
316#127—62no—
317#123—216no—
318C22⋊Q8—24no—
319C22⋊Q8C22×D318yes—
320#121—28no—
321#121—38no—
322#121—64no—
323#121—616no—
324C4×Q8—34no—
325C4×Q8—62no—
326C4×D4—22no—
327C4×D4—68no—
328C4×D4#6714yes—
329C4×D4C22×D314yes—
330C4×D4C22×D314yes—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4#6711yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—62no—
342C2×C22⋊C4#6712yes—
343C2×C42—61no—
344C2×C42C22×D311yescharacteristic
345#106—34no—
346C4⋊C8—34no—
347#103—38no—
348#103—34no—
349#102—34no—
350#102—38no—
351C22⋊C8—38no—
352C4×C8—34no—
353C23×C3—24no—
354C23×C3—24no—
355C23×C3—24no—
356C23×C3—24no—
357C23×C3—22no—
358C23×C3—42no—
359C23×C3—42no—
360C23×C3C22×D414yes—
361C23×C3C22×D411yescharacteristic
362C22×D3—24no—
363C22×D3—44no—
364C22×D3—44no—
365C22×D3—44no—
366C3×Q8—24no—
367C3×Q8—24no—
368C3×Q8—44no—
369C3×D4—24no—
370C3×D4—28no—
371C3×D4—22no—
372C3×D4—416no—
373C3×D4C22×D414yes—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—28no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—22no—
380C2×C3×C4—24no—
381C2×C3×C4—416no—
382C2×C3×C4—44no—
383C2×C3×C4—42no—
384C2×C3×C4—48no—
385C2×C3×C4#13514yescomplemented
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D414yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D412yes—
391C2×C3×C4H(5,𝔽2)12yes—
392C2×C3×C4C2511yescharacteristic
393#67—416no—
394#67—88no—
395C2×Dic3—24no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—42no—
399D12—24no—
400D12—24no—
401D12—44no—
402D12—48no—
403D12—84no—
404C4×D3—416no—
405Dic6—24no—
406Dic6—22no—
407Dic6C22×D414yes—
408C3⋊C8—24no—
409C3⋊C8—22no—
410C3⋊C8C22×D414yes—
411C24—21no—
412C24—62no—
413C24C2×#6712yes—
414#41—216no—
415#41—48no—
416#41—416no—
417#41—616no—
418#41—128no—
419#41—1216no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—34no—
424C2×Q8—32no—
425C2×Q8—62no—
426C2×Q8—61no—
427C2×Q8C23×D314yes—
428C2×D4—28no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—34no—
433C2×D4—48no—
434C2×D4—68no—
435C2×D4—68no—
436C2×D4—64no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—62no—
440C2×D4—64no—
441C2×D4—128no—
442C2×D4—124no—
443C2×D4—242no—
444C2×D4—244no—
445C2×D4D3×D414yes—
446C2×D4C2×#6714yes—
447C2×D4C23×D312yes—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—68no—
459C22×C4—128no—
460C22×C4C2×#6714yes—
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6712yes—
463C22×C4C2×#6711yescharacteristic
464C22×C4C23×D314yes—
465C22×C4C23×D311yescharacteristic
466Q16—616no—
467SD16—316no—
468SD16—616no—
469SD16—128no—
470D8—616no—
471C2×C8—34no—
472C2×C8—32no—
473C2×C8—32no—
474C2×C8—34no—
475C2×C8—616no—
476C2×C8—64no—
477C2×C8—62no—
478C4⋊C4—24no—
479C4⋊C4—28no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—61no—
484C4⋊C4—64no—
485C4⋊C4C2×#6714yes—
486C4⋊C4C23×D312yes—
487C22⋊C4—28no—
488C22⋊C4—24no—
489C22⋊C4—68no—
490C22⋊C4C2×#6718yes—
491C42—64no—
492C42C2×#6712yes—
493C42C23×D312yes—
494C22×C3—24no—
495C22×C3—24no—
496C22×C3—24no—
497C22×C3—24no—
498C22×C3—48no—
499C22×C3—48no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3D4212yes—
503C22×C3C2×#13512yes—
504C22×C3#57712yes—
505C22×C3C23×D411yescharacteristic
506C2×D3—44no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—88no—
510C3×C4—24no—
511C3×C4—22no—
512C3×C4—44no—
513C3×C4—42no—
514C3×C4D4212yes—
515C3×C4C2×#13514yescomplemented
516C3×C4C23×D411yescharacteristic
517C3×C4C2×H(5,𝔽2)11yescharacteristic
518Dic3—24no—
519Dic3—42no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—64no—
528C23—124no—
529C23—124no—
530C23—124no—
531C23C22×#6714yes—
532C23C22×#6711yescharacteristic
533Q8—24no—
534Q8—24no—
535Q8—34no—
536Q8—44no—
537Q8—62no—
538Q8—64no—
539D4—22no—
540D4—24no—
541D4—28no—
542D4—416no—
543D4—64no—
544D4—64no—
545D4—1216no—
546D4—124no—
547D4—128no—
548D4—244no—
549D4—248no—
550D4C2×D3×D414yes—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—416no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—48no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—62no—
565C2×C4—1216no—
566C2×C4—122no—
567C2×C4#95014yescomplemented
568C2×C4C2×D3×D412yes—
569C2×C4#100412yes—
570C2×C4C22×#6712yes—
571C2×C4C22×#6714yes—
572C2×C4C22×#6711yescharacteristic
573C2×C4C22×#6712yes—
574C2×C4C24×D311yescharacteristic
575C8—34no—
576C8—64no—
577C8—62no—
578C2×C3—44no—
579C2×C3—44no—
580C2×C3#328414yescomplemented
581C2×C3C2×D4211yescharacteristic
582C2×C3C22×#13511yescharacteristic
583C2×C3C2×#57711yescharacteristic
584D3—84no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—48no—
590C22—48no—
591C22—42no—
592C22—42no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—248no—
597C22D4×#6712yes—
598C22C2×#95012yes—
599C22#583712yes—
600C22C23×#6711yescharacteristic
601C4—24no—
602C4—22no—
603C4—44no—
604C4—42no—
605C4—64no—
606C4—122no—
607C4D4×#6712yes—
608C4C2×#95014yescomplemented
609C4C22×D3×D411yescharacteristic
610C4C2×#100411yescharacteristic
611C3C2×#328411yescharacteristic, complemented, Sylow
612C2—44no—
613C2—44no—
614C2—244no—
615C2#8353014yesdirect product factor
616C2C2×D4×#6711yescharacteristic
617C2C22×#95011yescharacteristic
618C2C2×#583711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#547C2
socleC23×C3C22×D4
hypercenterC2×#547D3
hypocenterC3C2×#3284
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×C3×#547C2semidirect
2. C22×#950C2semidirect
3. C2×#5553C2semidirect
4. C2×#5238C2semidirect
5. #83530C2direct (N × H)
6. C2×#5090C2semidirect
7. C2×#5085C2semidirect
8. C2×#5047C2semidirect
9. C2×#5048C2semidirect
10. C2×C4×#214C2semidirect
11. C2×#5015C2semidirect
12. C2×D4×C3⋊C8C2semidirect
17. C3×#547C22semidirect
26. C2×#950C22semidirect
36. #5553C22semidirect
41. #5238C22semidirect
45. #5090C22semidirect
46. #5085C22semidirect
47. #5048C22semidirect
48. #5047C22semidirect
49. C4×#214C22semidirect
50. #5015C22semidirect
51. D4×C3⋊C8C22semidirect
59. C2×#547D3semidirect
119. C2×#214D4semidirect
161. #547C2×D3semidirect
260. #214C2×D4semidirect
385. C2×C3×C4#135semidirect
515. C3×C4C2×#135semidirect
567. C2×C4#950semidirect
580. C2×C3#3284semidirect
608. C4C2×#950semidirect
611. C3C2×#3284semidirect
615. C2#83530direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C22×#9502
C2×#55532
C2×#52382
#835302
C2×#50902
C2×#50852
C2×#50472
C2×#50482
C2×C4×#2142
C2×#50152
C2×D4×C3⋊C82
C2×#32843