Finite Group Databasebeta

← C2×#81064 #11667227

Subgroups — C2×#81064 #11667227

Order 768.

2286 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#5548C211yesmaximal, characteristic
3C2×C3×#476C211yesmaximal, characteristic, complemented
4C2×#5084C211yesmaximal, characteristic, complemented
5C2×#5008C211yesmaximal, characteristic, complemented
6#81064C2116yesmaximal, direct product factor
7C2×#4863C212yesmaximal, complemented
8C2×#4855C212yesmaximal, complemented
9C2×#4842C211yesmaximal, characteristic, complemented
10C2×#4833C211yesmaximal, characteristic, complemented
11C2×#4777C212yesmaximal, complemented
12C2×#4760C212yesmaximal, complemented
13C2×#3227—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C22×#235—21no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4×D4C2211yescharacteristic
18C2×#944—21no—
19C2×#940C2211yescharacteristic
20C2×#938C2211yescharacteristic
21C2×C4×#67—21no—
22D3×C4⋊Q8—24no—
23D3×C4⋊Q8C2218yescomplemented
24#5548—24no—
25#5548C2218yes—
26C2×#895C2211yescharacteristic
27C2×D3×C4⋊C4—21no—
28C2×D3×C4⋊C4C2212yes—
29C2×#888—21no—
30C2×#888C2211yescharacteristic
31C2×#888C2212yes—
32C2×#885—21no—
33C2×#884—21no—
34C2×C4×D12C2211yescharacteristic
35C2×C42×D3C2211yescharacteristic
36C2×#869C2211yescharacteristic
37C3×#476C2218yescomplemented
38C2×C3×#106C2212yes—
39C2×C3×C4⋊C8C2211yescharacteristic
40C2×C3×#102C2212yes—
41C2×#832C2212yes—
42C2×#821C2212yes—
43C2×(C2×D3)⋊C8C2212yes—
44C2×#817C2212yes—
45C2×C8×Dic3C2212yes—
46#5084C2218yescomplemented
47#5008C2218yescomplemented
48C2×#809C2212yes—
49C2×#808C2212yes—
50C2×#808C2212yes—
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4863C22116yescomplemented
53#4855C22116yescomplemented
54#4842C2218yescomplemented
55#4833C2218yescomplemented
56#4777C22116yescomplemented
57#4760C22116yescomplemented
58C2×#532—31no—
59C22×C4⋊Q8—31no—
60#3227—316no—
61C2×#498—32no—
62C2×#486—32no—
63C2×#481—32no—
64C2×#481—32no—
65C2×#476—31no—
66C2×#476—31no—
67C2×#476—31no—
68C2×#476D311yescharacteristic, complemented
69C2×#431—31no—
70C22×C3×Q8—21no—
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C22×#67—21no—
74C22×#67—21no—
75C22×#67—21no—
76C2×D3×Q8—28no—
77C2×D3×Q8—48no—
78C2×#235—28no—
79C2×#235—48no—
80C22×D12—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—22no—
83C22×C4×D3—21no—
84C22×C4×D3D411yescharacteristic
85C22×C4×D3D411yescharacteristic
86C22×C4×D3C2311yescharacteristic
87C22×Dic6—21no—
88C22×Dic6—21no—
89C22×Dic6—22no—
90C22×C3×C8C2312yes—
91C3×C4⋊Q8—22no—
92C3×C4⋊Q8C2314yes—
93C3×C4×D4—22no—
94C3×C4×D4C2314yes—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101#944—28no—
102#940—22no—
103#940C2314yes—
104#938—22no—
105#938C2314yes—
106C4×#67—28no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2312yes—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—22no—
114C2×Dic3⋊C4—22no—
115C2×C4×Dic3D411yescharacteristic
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2312yes—
119#895—22no—
120#895C2314yes—
121D3×C4⋊C4—24no—
122D3×C4⋊C4—28no—
123D3×C4⋊C4C2318yes—
124#888—24no—
125#888—28no—
126#888—22no—
127#888C2318yes—
128#888C2314yes—
129#885—28no—
130#884—28no—
131C4×D12—22no—
132C4×D12C2314yes—
133C42×D3D414yes—
134C42×D3C2314yes—
135#869—22no—
136#869C2314yes—
137C3×#106C2318yes—
138C3×C4⋊C8C2314yes—
139C3×#102C2318yes—
140#832C2318yes—
141#821C2318yes—
142(C2×D3)⋊C8C2318yes—
143#817C2318yes—
144C8×Dic3C2318yes—
145#809C2318yes—
146#808C2318yes—
147#808C2318yes—
148C4⋊(C3⋊C8)C2314yes—
149C22×#41—61no—
150C23×Q8—61no—
151#532—38no—
152#532—64no—
153C2×C4⋊Q8—38no—
154C2×C4⋊Q8—32no—
155C2×C4⋊Q8—31no—
156C2×C4⋊Q8—31no—
157C2×C4⋊Q8—61no—
158C2×C4⋊Q8—64no—
159C2×C4⋊Q8—61no—
160C2×C4⋊Q8C2×D311yescharacteristic
161C2×#127—31no—
162C2×#124—61no—
163C2×#121—31no—
164C2×#121—31no—
165C2×#121—61no—
166C2×C4×D4—31no—
167C2×C4×D4—61no—
168C2×C4×D4C2×D311yescharacteristic
169C22×C4⋊C4—32no—
170C22×C4⋊C4—61no—
171C22×C42—31no—
172#498—316no—
173#486—316no—
174#481—316no—
175#481—316no—
176#476—38no—
177#476—38no—
178#476—38no—
179#476C2×D318yescomplemented
180#431—38no—
181C2×#106—32no—
182C2×#106—32no—
183C2×#106—32no—
184C2×#106C2×D312yes—
185C2×C4⋊C8—31no—
186C2×C4⋊C8C2×D311yescharacteristic
187C2×#102—32no—
188C2×#102—32no—
189C2×#102—32no—
190C2×#102C2×D312yes—
191C2×C22⋊C8—32no—
192C2×C4×C8—32no—
193C24×C3—21no—
194C23×D3—21no—
195C23×D3C2×D411yescharacteristic
196C2×C3×Q8—24no—
197C2×C3×Q8—44no—
198C2×C3×D4—21no—
199C2×C3×D4—22no—
200C2×C3×D4—22no—
201C2×C3×D4C2412yes—
202C22×C3×C4—22no—
203C22×C3×C4—21no—
204C22×C3×C4—24no—
205C22×C3×C4—44no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—24no—
210C2×#67—24no—
211C2×#67—24no—
212C2×#67—44no—
213C2×#67—44no—
214C2×#67—44no—
215C22×Dic3—22no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220D3×Q8—432no—
221#235—432no—
222C2×D12—22no—
223C2×D12—22no—
224C2×D12—24no—
225C2×D12—21no—
226C2×D12—44no—
227C2×D12C2412yes—
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—24no—
231C2×C4×D3—28no—
232C2×C4×D3—24no—
233C2×C4×D3—48no—
234C2×C4×D3—44no—
235C2×C4×D3SD1618yes—
236C2×C4×D3C2×D412yes—
237C2×C4×D3C2×D414yes—
238C2×C4×D3C2×D414yes—
239C2×C4×D3C2412yes—
240C2×Dic6—24no—
241C2×Dic6—24no—
242C2×Dic6—28no—
243C2×Dic6—44no—
244C2×Dic6—44no—
245C2×Dic6—48no—
246C2×C3×C8—24no—
247C2×C3×C8C2414yes—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2412yes—
252C3×C4⋊C4C2414yes—
253C3×C22⋊C4—24no—
254C3×C42C2×D412yes—
255C3×C42C2412yes—
256#217—24no—
257(C2×D3)⋊C4—24no—
258(C2×D3)⋊C4—24no—
259C4⋊Dic3—22no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3C2412yes—
263C4⋊Dic3C2414yes—
264Dic3⋊C4—24no—
265Dic3⋊C4—28no—
266Dic3⋊C4—28no—
267C4×Dic3C2×D412yes—
268C4×Dic3C2×D414yes—
269C4×Dic3C2×D414yes—
270C4×Dic3C2412yes—
271C2×C3⋊C8—24no—
272C2×C3⋊C8C2414yes—
273C2×#41—68no—
274C2×#41—128no—
275C22×Q8—21no—
276C22×Q8—61no—
277C22×Q8—62no—
278C22×Q8—68no—
279C22×Q8—61no—
280C22×Q8—128no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4—61no—
284C22×D4—61no—
285C22×D4—61no—
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C23×C4—31no—
289C23×C4—31no—
290C23×C4—31no—
291C23×C4—62no—
292C23×C4—61no—
293C22×C8—32no—
294C22×C8C22×D312yes—
295C4⋊Q8—22no—
296C4⋊Q8—38no—
297C4⋊Q8—34no—
298C4⋊Q8—34no—
299C4⋊Q8—68no—
300C4⋊Q8—64no—
301C4⋊Q8—62no—
302C4⋊Q8—68no—
303C4⋊Q8—62no—
304C4⋊Q8C22×D314yes—
305#127—34no—
306#127—62no—
307#124—68no—
308#121—34no—
309#121—34no—
310#121—68no—
311#121—62no—
312#121—62no—
313C4×D4—22no—
314C4×D4—34no—
315C4×D4—68no—
316C4×D4—62no—
317C4×D4C22×D314yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—38no—
322C2×C4⋊C4—64no—
323C2×C4⋊C4—68no—
324C2×C4⋊C4—62no—
325C2×C4⋊C4—62no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4C22×D312yes—
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—34no—
338C2×C42—34no—
339C2×C42C22×D311yescharacteristic
340#106—38no—
341#106—38no—
342#106—38no—
343#106C22×D318yes—
344C4⋊C8—34no—
345C4⋊C8C22×D314yes—
346#102—38no—
347#102—38no—
348#102—38no—
349#102C22×D318yes—
350C22⋊C8—38no—
351C4×C8—38no—
352C23×C3—22no—
353C23×C3—42no—
354C23×C3—42no—
355C23×C3—42no—
356C23×C3C22×D411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—42no—
361C22×D3—42no—
362C22×D3—42no—
363C22×D3C2×SD1614yes—
364C22×D3C22×D412yes—
365C3×Q8—48no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—42no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—42no—
373C2×C3×C4—44no—
374C2×C3×C4—48no—
375C2×C3×C4C2×SD1614yes—
376C2×C3×C4C22×D412yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D412yes—
379C2×C3×C42−1+412yes—
380C2×C3×C42−1+412yes—
381C2×C3×C4C2511yescharacteristic
382#67—48no—
383#67—48no—
384#67—48no—
385C2×Dic3—22no—
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—42no—
389C2×Dic3—44no—
390C2×Dic3C2×SD1614yes—
391C2×Dic3AGL(1,ℤ/8)14yes—
392C2×Dic3C22×D412yes—
393C2×Dic3C22×D412yes—
394C2×Dic3C22×D412yes—
395D12—22no—
396D12—22no—
397D12—48no—
398D12—42no—
399C4×D3—24no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—28no—
403C4×D3—416no—
404C4×D3—48no—
405C4×D3C2×SD16116yes—
406Dic6—48no—
407Dic6—48no—
408Dic6—416no—
409C3×C8—24no—
410C3⋊C8—24no—
411C24—21no—
412C24—31no—
413C24—61no—
414#41—1232no—
415C2×Q8—24no—
416C2×Q8—44no—
417C2×Q8—64no—
418C2×Q8—68no—
419C2×Q8—64no—
420C2×Q8—124no—
421C2×Q8—128no—
422C2×Q8—124no—
423C2×Q8—1232no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—32no—
428C2×D4—64no—
429C2×D4—64no—
430C2×D4—64no—
431C2×D4—64no—
432C2×D4—61no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—124no—
436C2×D4—124no—
437C2×D4—124no—
438C2×D4—124no—
439C2×D4C23×D312yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—38no—
446C22×C4—34no—
447C22×C4—31no—
448C22×C4—32no—
449C22×C4—34no—
450C22×C4—32no—
451C22×C4—44no—
452C22×C4—64no—
453C22×C4—68no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—61no—
457C22×C4—62no—
458C22×C4—64no—
459C22×C4—128no—
460C22×C4—124no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464C2×C8—24no—
465C2×C8—34no—
466C2×C8—64no—
467C2×C8C23×D314yes—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—34no—
472C4⋊C4—32no—
473C4⋊C4—68no—
474C4⋊C4—68no—
475C4⋊C4—64no—
476C4⋊C4—62no—
477C4⋊C4—61no—
478C4⋊C4—64no—
479C4⋊C4C23×D314yes—
480C4⋊C4C23×D312yes—
481C22⋊C4—24no—
482C22⋊C4—64no—
483C22⋊C4—64no—
484C22⋊C4—64no—
485C42—34no—
486C42—34no—
487C42—32no—
488C42—32no—
489C42D3×D412yes—
490C42C23×D312yes—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—84no—
495C22×C3#53612yes—
496C22×C3C22×SD1612yes—
497C22×C3C2×AGL(1,ℤ/8)12yes—
498C22×C3C23×D411yescharacteristic
499C2×D3—22no—
500C2×D3—28no—
501C2×D3—22no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—84no—
506C2×D3C22×SD1614yes—
507C3×C4—22no—
508C3×C4—42no—
509C3×C4—44no—
510C3×C4#53612yes—
511C3×C4C22×SD1614yes—
512C3×C4C2×2−1+411yescharacteristic
513C3×C4C2×2−1+411yescharacteristic
514Dic3—22no—
515Dic3—42no—
516Dic3—44no—
517Dic3C22×SD1614yes—
518Dic3C2×AGL(1,ℤ/8)14yes—
519C23—22no—
520C23—32no—
521C23—34no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23—122no—
529C23—122no—
530C23—122no—
531C23C2×D3×D411yescharacteristic
532Q8—48no—
533Q8—128no—
534Q8—1216no—
535Q8—128no—
536D4—22no—
537D4—22no—
538D4—42no—
539D4—62no—
540D4—62no—
541D4—128no—
542D4—128no—
543D4—128no—
544D4—128no—
545D4—122no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—32no—
550C2×C4—32no—
551C2×C4—34no—
552C2×C4—34no—
553C2×C4—316no—
554C2×C4—32no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—68no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—1216no—
566C2×C4—122no—
567C2×C4—124no—
568C2×C4—128no—
569C2×C4D3×SD1614yes—
570C2×C4C2×D3×D412yes—
571C2×C4C2×D3×D412yes—
572C2×C4C2×D3×D411yescharacteristic
573C2×C4#100712yes—
574C2×C4#101012yes—
575C2×C4C24×D311yescharacteristic
576C8—24no—
577C8—64no—
578C2×C3—82no—
579C2×C3#322714yescomplemented
580C2×C3C2×#53611yescharacteristic
581C2×C3C23×SD1611yescharacteristic
582C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
583D3—24no—
584D3—82no—
585C22—34no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—62no—
590C22—68no—
591C22—62no—
592C22—84no—
593C22—122no—
594C22—122no—
595C22—122no—
596C22—244no—
597C22#562812yes—
598C22C2×D3×SD1612yes—
599C22D3×AGL(1,ℤ/8)12yes—
600C22C22×D3×D411yescharacteristic
601C4—22no—
602C4—34no—
603C4—34no—
604C4—42no—
605C4—44no—
606C4—62no—
607C4—122no—
608C4—124no—
609C4#562812yes—
610C4C2×D3×SD1614yes—
611C4C2×#100711yescharacteristic
612C4C2×#101011yescharacteristic
613C3C2×#322711yescharacteristic, complemented, Sylow
614C2—64no—
615C2—82no—
616C2—242no—
617C2#8106414yesdirect product factor
618C2C2×#562811yescharacteristic
619C2C22×D3×SD1611yescharacteristic
620C2C2×D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#476C2
socleC23×C3C22×D4
prime radicalC2×#5548C2
hypercenterC2×#476D3
hypocenterC3C2×#3227
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊Q8C2semidirect
3. C2×C3×#476C2semidirect
4. C2×#5084C2semidirect
5. C2×#5008C2semidirect
6. #81064C2direct (N × H)
7. C2×#4863C2semidirect
8. C2×#4855C2semidirect
9. C2×#4842C2semidirect
10. C2×#4833C2semidirect
11. C2×#4777C2semidirect
12. C2×#4760C2semidirect
23. D3×C4⋊Q8C22semidirect
37. C3×#476C22semidirect
46. #5084C22semidirect
47. #5008C22semidirect
52. #4863C22semidirect
53. #4855C22semidirect
54. #4842C22semidirect
55. #4833C22semidirect
56. #4777C22semidirect
57. #4760C22semidirect
68. C2×#476D3semidirect
179. #476C2×D3semidirect
579. C2×C3#3227semidirect
613. C3C2×#3227semidirect
617. C2#81064direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#55482
C2×C3×#4762
C2×#50842
C2×#50082
#810642
C2×#48632
C2×#48552
C2×#48422
C2×#48332
C2×#47772
C2×#47602
C2×#32273