Finite Group Databasebeta

← C2×#80393 #11670358

Subgroups — C2×#80393 #11670358

Order 768.

2794 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#915C212yesmaximal, complemented
2C2×#5718C211yesmaximal, characteristic
3C2×Q8×D12C211yesmaximal, characteristic, complemented
4C2×C3×#460C211yesmaximal, characteristic, complemented
5C2×#5170C212yesmaximal, complemented
6C2×#5076C211yesmaximal, characteristic, complemented
7C2×C4×#216C211yesmaximal, characteristic, complemented
8#80393C2116yesmaximal, direct product factor
9C2×#4849C211yesmaximal, characteristic, complemented
10C2×#4838C211yesmaximal, characteristic, complemented
11C2×#4780C212yesmaximal, complemented
12C2×#4783C212yesmaximal, complemented
13C2×#3278—31nomaximal, Sylow
14C22×D3×Q8C2212yes—
15C22×#237C2212yes—
16C22×C3×SD16C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C2×#940—21no—
21C2×#939—22no—
22C2×#938C2212yes—
23C2×D4×Dic3—21no—
24C22×#214C2212yes—
25C22×#214C2212yes—
26C2×#915—216no—
27C2×#915C22116yescomplemented
28C22×Dic12C2212yes—
29C22×ΓL(1,𝔽25)C2212yes—
30#5718—24no—
31#5718C2218yes—
32Q8×D12—24no—
33Q8×D12C2218yescomplemented
34C2×#897—22no—
35C2×#897C2212yes—
36C2×C4×D12—21no—
37C2×C4×D12C2211yescharacteristic
38C2×#869—21no—
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#460C2218yescomplemented
42C2×C3×C4⋊C8C2211yescharacteristic
43C2×C3×#102C2212yes—
44C2×#835C2211yescharacteristic
45#5170C22116yescomplemented
46C2×(C2×D3)⋊C8C2212yes—
47C2×#816C2212yes—
48#5076C2218yescomplemented
49C4×#216C2218yescomplemented
50C2×#810C2211yescharacteristic
51C2×#807C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4849C2218yescomplemented
54#4838C2218yescomplemented
55#4783C22116yescomplemented
56#4780C22116yescomplemented
57C22×#137—32no—
58C2×#550—31no—
59C2×D4×Q8—31no—
60#3278—316no—
61C2×#492—31no—
62C2×#468—32no—
63C2×#462—31no—
64C2×#460D311yescharacteristic, complemented
65C2×#452—32no—
66C2×#448—32no—
67C2×C4×Q16—31no—
68C2×#436—31no—
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4—21no—
71C22×C3×D4—21no—
72C22×C3×D4C2312yes—
73C22×#67—22no—
74C22×#67—22no—
75C23×Dic3—22no—
76C2×D3×Q8—24no—
77C2×D3×Q8—28no—
78C2×D3×Q8—44no—
79C2×D3×Q8C2318yes—
80C2×#237—24no—
81C2×#237—28no—
82C2×#237—44no—
83C2×#237C2318yes—
84C22×D12D411yescharacteristic
85C22×C4×D3—22no—
86C22×C4×D3C2312yes—
87C22×Dic6—22no—
88C22×Dic6C2312yes—
89C22×Dic6C2311yescharacteristic
90C2×C3×SD16—28no—
91C2×C3×SD16C2318yes—
92C22×C3×C8C2312yes—
93C3×#127—22no—
94C3×#127C2314yes—
95C3×C4×Q8—22no—
96C3×C4×Q8C2314yes—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C42—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—22no—
102C2×#216D418yescomplemented
103C2×#216C2314yes—
104#940—28no—
105#939—216no—
106#938—24no—
107#938C2318yes—
108D4×Dic3—28no—
109C2×#214—28no—
110C2×#214—44no—
111C2×#214C2318yes—
112C2×#214C2318yes—
113C2×(C2×D3)⋊C4—22no—
114C2×(C2×D3)⋊C4D412yes—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—22no—
118C2×C4⋊Dic3D411yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×Dic3⋊C4—21no—
121C2×C4×Dic3—21no—
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#915—264no—
125C2×Dic12—28no—
126C2×Dic12C2318yes—
127C2×ΓL(1,𝔽25)—28no—
128C2×ΓL(1,𝔽25)C2318yes—
129#897—24no—
130#897—216no—
131#897C2318yes—
132C4×D12—28no—
133C4×D12D414yes—
134C4×D12C2314yes—
135#869—22no—
136#869—28no—
137#869C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140C3×C4⋊C8C2314yes—
141C3×#102C2318yes—
142#835C2314yes—
143(C2×D3)⋊C8C2318yes—
144#816C2318yes—
145#810C2314yes—
146#807C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41—32no—
149C23×Q8—32no—
150C2×#137—316no—
151C2×#137—616no—
152C22×Q16—31no—
153C22×Q16—32no—
154C22×SD16—32no—
155C22×SD16—32no—
156C22×SD16C2×D312yes—
157C22×M16—32no—
158#550—38no—
159#550—64no—
160D4×Q8—38no—
161D4×Q8—64no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—61no—
164C2×#127—61no—
165C2×#127C2×D311yescharacteristic
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—62no—
168C2×#121—32no—
169C2×#121—62no—
170C2×C4×Q8—31no—
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×D4—31no—
173C2×C4×D4—61no—
174C2×C4×D4—61no—
175#492—38no—
176#468—316no—
177#462—38no—
178#460C2×D318yescomplemented
179#452—316no—
180#448—316no—
181C4×Q16—38no—
182#436—38no—
183C2×#107—31no—
184C2×C4⋊C8C2×D311yescharacteristic
185C2×#103—31no—
186C2×#103—32no—
187C2×#103—31no—
188C2×#102C2×D312yes—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3—22no—
192C23×D3C2×D412yes—
193C2×C3×Q8—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8C2×D414yes—
196C2×C3×Q8C2412yes—
197C2×C3×D4—24no—
198C2×C3×D4—24no—
199C2×C3×D4—22no—
200C2×C3×D4—24no—
201C2×C3×D4—24no—
202C2×C3×D4—44no—
203C2×C3×D4—82no—
204C2×C3×D4C2414yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C2×#67—28no—
211C2×#67—28no—
212C2×#67—48no—
213C2×#67—84no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—22no—
217C22×Dic3—28no—
218C22×Dic3—48no—
219C22×Dic3C2×D412yes—
220D3×Q8—216no—
221D3×Q8—48no—
222D3×Q8—416no—
223#237—216no—
224#237—48no—
225#237—416no—
226C2×D12—24no—
227C2×D12C2×D414yes—
228C2×C4×D3—24no—
229C2×C4×D3—24no—
230C2×C4×D3—28no—
231C2×C4×D3—48no—
232C2×C4×D3C2×D414yes—
233C2×C4×D3C2414yes—
234C2×Dic6—21no—
235C2×Dic6—22no—
236C2×Dic6—24no—
237C2×Dic6—24no—
238C2×Dic6—22no—
239C2×Dic6—28no—
240C2×Dic6—48no—
241C2×Dic6C2×D414yes—
242C2×Dic6C2414yes—
243C2×Dic6C2412yes—
244C3×SD16—216no—
245C2×C3×C8—24no—
246C2×C3×C8C2414yes—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2412yes—
250C3×C42—24no—
251C3×C42C2×D412yes—
252C3×C42C2412yes—
253#217—28no—
254#216C2×D4116yescomplemented
255#214—216no—
256#214—48no—
257(C2×D3)⋊C4—28no—
258(C2×D3)⋊C4C2×D418yes—
259C4⋊Dic3—21no—
260C4⋊Dic3—24no—
261C4⋊Dic3—24no—
262C4⋊Dic3—28no—
263C4⋊Dic3C2×D414yes—
264C4⋊Dic3C2412yes—
265Dic3⋊C4—24no—
266C4×Dic3—24no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8C2×D414yes—
269C2×C3⋊C8C2412yes—
270C2×C3⋊C8C2412yes—
271Dic12—216no—
272ΓL(1,𝔽25)—216no—
273C2×#41—38no—
274C2×#41—64no—
275C2×#41—68no—
276C2×#41—124no—
277C22×Q8—38no—
278C22×Q8—31no—
279C22×Q8—32no—
280C22×Q8—62no—
281C22×Q8—64no—
282C22×Q8—68no—
283C22×Q8—124no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—21no—
286C22×D4—21no—
287C22×D4—31no—
288C22×D4—62no—
289C22×D4—62no—
290C22×D4C22×D312yes—
291C23×C4—32no—
292C23×C4—62no—
293C23×C4—62no—
294#137—664no—
295C2×Q16—38no—
296C2×Q16—34no—
297C2×Q16—38no—
298C2×Q16—68no—
299C2×SD16—28no—
300C2×SD16—38no—
301C2×SD16—38no—
302C2×SD16—68no—
303C2×SD16—124no—
304C2×SD16C22×D318yes—
305C2×M16—38no—
306C2×M16—68no—
307C22×C8—31no—
308C22×C8—31no—
309C22×C8C22×D312yes—
310C4⋊Q8—34no—
311C4⋊Q8—68no—
312C4⋊Q8—62no—
313#127—22no—
314#127—68no—
315#127C22×D314yes—
316C22⋊Q8—38no—
317C22⋊Q8—616no—
318C22⋊Q8—64no—
319#121—38no—
320#121—616no—
321#121—64no—
322C4×Q8—22no—
323C4×Q8—34no—
324C4×Q8—62no—
325C4×Q8C22×D314yes—
326C4×D4—34no—
327C4×D4—34no—
328C4×D4—68no—
329C4×D4—68no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—62no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C22⋊C4—32no—
339C2×C22⋊C4—62no—
340C2×C22⋊C4—62no—
341C2×C42—21no—
342C2×C42—61no—
343C2×C42C22×D311yescharacteristic
344#107—34no—
345C4⋊C8C22×D314yes—
346#103—34no—
347#103—38no—
348#103—34no—
349#102C22×D318yes—
350C22⋊C8—38no—
351C4×C8—34no—
352C23×C3—24no—
353C23×C3—44no—
354C23×C3—44no—
355C23×C3—44no—
356C23×C3C22×D411yescharacteristic
357C22×D3—24no—
358C22×D3—24no—
359C22×D3—24no—
360C22×D3C22×D414yes—
361C3×Q8—24no—
362C3×Q8—22no—
363C3×Q8C22×D414yes—
364C3×D4—24no—
365C3×D4—24no—
366C3×D4—44no—
367C3×D4—48no—
368C3×D4—84no—
369C2×C3×C4—24no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—42no—
374C2×C3×C4#13714yescomplemented
375C2×C3×C4C22×D412yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4H(5,𝔽2)12yes—
380C2×C3×C4C2511yescharacteristic
381#67—416no—
382#67—88no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—416no—
389C2×Dic3—44no—
390C2×Dic3—42no—
391C2×Dic3C22×D414yes—
392D12—28no—
393C4×D3—28no—
394C4×D3—24no—
395C4×D3—24no—
396C4×D3—416no—
397Dic6—24no—
398Dic6—22no—
399Dic6—24no—
400Dic6—24no—
401Dic6—416no—
402Dic6—44no—
403Dic6C22×D414yes—
404C3×C8—24no—
405C3⋊C8—22no—
406C3⋊C8C22×D414yes—
407C24—22no—
408C24—32no—
409#41—616no—
410#41—128no—
411#41—1216no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—34no—
415C2×Q8—32no—
416C2×Q8—34no—
417C2×Q8—68no—
418C2×Q8—61no—
419C2×Q8—62no—
420C2×Q8—62no—
421C2×Q8—616no—
422C2×Q8—64no—
423C2×Q8—64no—
424C2×Q8—128no—
425C2×Q8—128no—
426C2×Q8—1216no—
427C2×Q8D3×D414yes—
428C2×Q8C23×D312yes—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—34no—
435C2×D4—44no—
436C2×D4—68no—
437C2×D4—68no—
438C2×D4—64no—
439C2×D4—82no—
440C2×D4—128no—
441C2×D4—244no—
442C2×D4C23×D314yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—32no—
446C22×C4—34no—
447C22×C4—34no—
448C22×C4—68no—
449C22×C4—68no—
450C22×C4—61no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—61no—
454C22×C4—62no—
455C22×C4—128no—
456C22×C4—128no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460Q16—316no—
461Q16—616no—
462SD16—216no—
463SD16—616no—
464SD16—128no—
465M16—616no—
466C2×C8—24no—
467C2×C8—34no—
468C2×C8—32no—
469C2×C8—32no—
470C2×C8—62no—
471C2×C8C23×D314yes—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—34no—
475C4⋊C4—32no—
476C4⋊C4—68no—
477C4⋊C4—61no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—64no—
481C4⋊C4C23×D312yes—
482C22⋊C4—38no—
483C22⋊C4—68no—
484C22⋊C4—68no—
485C42—24no—
486C42—64no—
487C42D3×D412yes—
488C42C23×D312yes—
489C22×C3—44no—
490C22×C3—44no—
491C22×C3—44no—
492C22×C3—88no—
493C22×C3D4212yes—
494C22×C3C2×#13712yes—
495C22×C3#57712yes—
496C22×C3C23×D411yescharacteristic
497C2×D3—24no—
498C2×D3—24no—
499C2×D3—24no—
500C2×D3—48no—
501C3×C4—24no—
502C3×C4—22no—
503C3×C4—42no—
504C3×C4D4212yes—
505C3×C4C2×#13714yescomplemented
506C3×C4C23×D411yescharacteristic
507C3×C4C2×H(5,𝔽2)11yescharacteristic
508Dic3—24no—
509Dic3—24no—
510Dic3—44no—
511Dic3—42no—
512C23—24no—
513C23—34no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—64no—
518C23—64no—
519C23—64no—
520C23C2×D3×D411yescharacteristic
521Q8—22no—
522Q8—24no—
523Q8—34no—
524Q8—62no—
525Q8—64no—
526Q8—64no—
527Q8—64no—
528Q8—1216no—
529Q8—124no—
530Q8C2×D3×D414yes—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—44no—
535D4—68no—
536D4—84no—
537D4—1216no—
538D4—248no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—34no—
544C2×C4—42no—
545C2×C4—62no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—68no—
551C2×C4—62no—
552C2×C4—64no—
553C2×C4—1216no—
554C2×C4—1216no—
555C2×C4—122no—
556C2×C4—124no—
557C2×C4#91514yescomplemented
558C2×C4C2×D3×D412yes—
559C2×C4C2×D3×D412yes—
560C2×C4C2×D3×D412yes—
561C2×C4C2×D3×D411yescharacteristic
562C2×C4#100912yes—
563C2×C4C24×D311yescharacteristic
564C8—24no—
565C8—34no—
566C8—62no—
567C2×C3—84no—
568C2×C3#327814yescomplemented
569C2×C3C2×D4211yescharacteristic
570C2×C3C22×#13711yescharacteristic
571C2×C3C2×#57711yescharacteristic
572D3—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—64no—
577C22—64no—
578C22—64no—
579C22—88no—
580C22—128no—
581C22#560912yes—
582C22C2×#91512yes—
583C22#577512yes—
584C22C22×D3×D411yescharacteristic
585C4—22no—
586C4—24no—
587C4—42no—
588C4—64no—
589C4—64no—
590C4—122no—
591C4—124no—
592C4#560912yes—
593C4C2×#91514yescomplemented
594C4C22×D3×D411yescharacteristic
595C4C2×#100911yescharacteristic
596C3C2×#327811yescharacteristic, complemented, Sylow
597C2—84no—
598C2—124no—
599C2#8039314yesdirect product factor
600C2C2×#560911yescharacteristic
601C2C22×#91511yescharacteristic
602C2C2×#577511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
prime radicalC2×#5718C2
hypercenterC2×#460D3
hypocenterC3C2×#3278
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C22×#915C2semidirect
3. C2×Q8×D12C2semidirect
4. C2×C3×#460C2semidirect
5. C2×#5170C2semidirect
6. C2×#5076C2semidirect
7. C2×C4×#216C2semidirect
8. #80393C2direct (N × H)
9. C2×#4849C2semidirect
10. C2×#4838C2semidirect
11. C2×#4780C2semidirect
12. C2×#4783C2semidirect
27. C2×#915C22semidirect
33. Q8×D12C22semidirect
41. C3×#460C22semidirect
45. #5170C22semidirect
48. #5076C22semidirect
49. C4×#216C22semidirect
53. #4849C22semidirect
54. #4838C22semidirect
55. #4783C22semidirect
56. #4780C22semidirect
64. C2×#460D3semidirect
102. C2×#216D4semidirect
178. #460C2×D3semidirect
254. #216C2×D4semidirect
374. C2×C3×C4#137semidirect
505. C3×C4C2×#137semidirect
557. C2×C4#915semidirect
568. C2×C3#3278semidirect
593. C4C2×#915semidirect
596. C3C2×#3278semidirect
599. C2#80393direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9152
C2×#57182
C2×Q8×D122
C2×C3×#4602
C2×#51702
C2×#50762
C2×C4×#2162
#803932
C2×#48492
C2×#48382
C2×#47802
C2×#47832
C2×#32783