Finite Group Databasebeta

← C2×#68325 #10767366

Subgroups — C2×#68325 #10767366

Order 768.

1372 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5623C211yesmaximal, characteristic, complemented
2C2×#5046C211yesmaximal, characteristic
3C2×#4731C211yesmaximal, characteristic
4C2×#4672C211yesmaximal, characteristic
5C2×C3×#328C211yesmaximal, characteristic, complemented
6C2×#4469C211yesmaximal, characteristic, complemented
7C2×#4452C211yesmaximal, characteristic, complemented
8#68325C218yesmaximal, direct product factor
9C2×#1599—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#944—21no—
13C22×#215—21no—
14C22×(C2×D3)⋊C4—21no—
15C22×(C2×D3)⋊C4—21no—
16C2×#924C2211yescharacteristic
17C22×Dic3⋊C4C2211yescharacteristic
18C22×#204—21no—
19#5623—22no—
20#5623—22no—
21#5623—22no—
22#5623C2214yescomplemented
23C2×#889—21no—
24C2×#889—21no—
25C2×#888—21no—
26C2×#878—21no—
27C2×#877—21no—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×#830C2211yescharacteristic
31C2×#821—21no—
32#5046—22no—
33#5046C2214yes—
34C2×#809—21no—
35#4731—22no—
36#4731C2214yes—
37#4672—22no—
38#4672C2214yes—
39C3×#328C2214yescomplemented
40#4469C2214yescomplemented
41#4452C2214yescomplemented
42C2×#544—31no—
43C2×#450—31no—
44C2×#450—31no—
45C2×#394—31no—
46#1599—38no—
47C2×#328—31no—
48C2×#328D311yescharacteristic, complemented
49C2×#324—31no—
50C24×D3—21no—
51C22×C3×Q8D411yescharacteristic
52C23×C3×C4—21no—
53C23×C3×C4C2311yescharacteristic
54C23×Dic3D411yescharacteristic
55C23×Dic3D411yescharacteristic
56C22×D12—21no—
57C22×D12—21no—
58C22×D12—21no—
59C22×D12—21no—
60C22×D12—21no—
61C22×D12—44no—
62C22×D12C2312yes—
63C22×Dic6—21no—
64C22×Dic6D411yescharacteristic
65C22×C3×C8—21no—
66C3×C22⋊Q8—21no—
67C3×C22⋊Q8—21no—
68C3×C22⋊Q8D412yes—
69C3×C22⋊Q8C2312yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×#217—21no—
74#944—24no—
75#944—24no—
76C2×#215—24no—
77C2×#215—44no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—44no—
81C2×(C2×D3)⋊C4—44no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3D411yescharacteristic
84#924—21no—
85#924—21no—
86#924D412yes—
87#924C2312yes—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—21no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4D412yes—
98C2×Dic3⋊C4C2312yes—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C22×C3⋊C8—21no—
102C2×#204—24no—
103C2×#204—44no—
104#889—24no—
105#889—24no—
106#889—24no—
107#889—24no—
108#888—24no—
109#888—24no—
110#878—24no—
111#878—24no—
112#877—24no—
113#877—24no—
114C3×C22⋊C8—21no—
115C3×C22⋊C8C2312yes—
116C3×#95—21no—
117C3×#95C2312yes—
118#830—21no—
119#830C2312yes—
120#821—24no—
121#809—24no—
122C23×D4—31no—
123C22×SD16—61no—
124C22×SD16—61no—
125#544—34no—
126#544—62no—
127#544—62no—
128#544—62no—
129C2×C4⋊Q8—61no—
130C2×C4⋊Q8—61no—
131C2×#125—61no—
132C2×#125—61no—
133C2×C22⋊Q8—31no—
134C2×C22⋊Q8—61no—
135C2×C22⋊Q8C2×D311yescharacteristic
136C2×#117—61no—
137C22×C4⋊C4—31no—
138C22×C22⋊C4—61no—
139C22×C22⋊C4—61no—
140#450—34no—
141#450—34no—
142#450—62no—
143#450—62no—
144C2×#102—61no—
145C2×#102—61no—
146C2×C22⋊C8—31no—
147C2×C22⋊C8C2×D311yescharacteristic
148#394—34no—
149#394—62no—
150C2×#95C2×D311yescharacteristic
151#328—34no—
152#328C2×D314yescomplemented
153#324—34no—
154C24×C3C2×D411yescharacteristic
155C23×D3—24no—
156C23×D3—22no—
157C23×D3—42no—
158C23×D3—42no—
159C23×D3—42no—
160C23×D3—41no—
161C23×D3—41no—
162C23×D3—42no—
163C23×D3—42no—
164C2×C3×Q8—21no—
165C2×C3×Q8—42no—
166C2×C3×Q8C2×D412yes—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—42no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—42no—
188C22×Dic3SD1614yescomplemented
189C22×Dic3C2×D412yes—
190C22×Dic3C2×D412yes—
191C2×D12—24no—
192C2×D12—24no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—44no—
196C2×D12—44no—
197C2×D12—44no—
198C2×D12—42no—
199C2×D12—42no—
200C2×D12—44no—
201C2×D12—41no—
202C2×D12—41no—
203C2×Dic6—24no—
204C2×Dic6—21no—
205C2×Dic6—42no—
206C2×Dic6—82no—
207C2×Dic6C2×D412yes—
208C2×C3×C8—22no—
209C2×C3×C8—42no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4C2×D412yes—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—22no—
216#217—22no—
217#217—22no—
218#215—48no—
219(C2×D3)⋊C4—48no—
220(C2×D3)⋊C4—48no—
221C4⋊Dic3—24no—
222C4⋊Dic3—21no—
223C4⋊Dic3C2×D412yes—
224Dic3⋊C4—24no—
225Dic3⋊C4—24no—
226Dic3⋊C4—24no—
227Dic3⋊C4—24no—
228Dic3⋊C4—22no—
229Dic3⋊C4—22no—
230Dic3⋊C4—22no—
231Dic3⋊C4—22no—
232Dic3⋊C4—22no—
233Dic3⋊C4—22no—
234C4×Dic3—24no—
235C4×Dic3—24no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—42no—
238#204—48no—
239C25—61no—
240C22×Q8—31no—
241C22×Q8—61no—
242C22×Q8D1211yescharacteristic
243C22×D4—32no—
244C22×D4—61no—
245C22×D4—61no—
246C22×D4—61no—
247C22×D4—61no—
248C22×D4—61no—
249C22×D4—124no—
250C23×C4—21no—
251C23×C4—31no—
252C23×C4—31no—
253C23×C4C22×D311yescharacteristic
254C2×SD16—64no—
255C2×SD16—64no—
256C2×SD16—124no—
257C2×SD16—124no—
258C22×C8—21no—
259C22×C8—61no—
260C4⋊Q8—64no—
261C4⋊Q8—64no—
262C4⋊Q8—64no—
263C4⋊Q8—64no—
264#125—64no—
265#125—64no—
266#125—64no—
267#125—64no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—32no—
271C22⋊Q8—32no—
272C22⋊Q8—64no—
273C22⋊Q8—64no—
274C22⋊Q8—61no—
275C22⋊Q8—61no—
276C22⋊Q8D1212yes—
277C22⋊Q8C22×D312yes—
278#117—64no—
279#117—64no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4D1211yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4—64no—
299C2×C22⋊C4—124no—
300C2×C22⋊C4—124no—
301C2×C42—61no—
302C2×C42—61no—
303#102—64no—
304#102—64no—
305C22⋊C8—21no—
306C22⋊C8—32no—
307C22⋊C8—61no—
308C22⋊C8C22×D312yes—
309#95—21no—
310#95C22×D312yes—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3#12011yescharacteristic
314C23×C3#12011yescharacteristic
315C23×C3C2×SD1612yes—
316C23×C3#13712yes—
317C23×C3C22×D411yescharacteristic
318C22×D3—24no—
319C22×D3—42no—
320C22×D3—44no—
321C22×D3—48no—
322C22×D3—48no—
323C22×D3—42no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—42no—
327C22×D3—42no—
328C22×D3—44no—
329C22×D3—84no—
330C22×D3—84no—
331C22×D3—82no—
332C22×D3—82no—
333C22×D3—82no—
334C22×D3—82no—
335C3×Q8—42no—
336C3×Q8—81no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—84no—
351C2×C3×C4#12011yescharacteristic
352C2×C3×C4#12011yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—22no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—81no—
367C2×Dic3C2×SD1614yescomplemented
368D12—24no—
369D12—24no—
370D12—42no—
371D12—42no—
372D12—44no—
373D12—44no—
374Dic6—42no—
375Dic6—81no—
376Dic6—84no—
377C3×C8—42no—
378C3⋊C8—42no—
379C24—64no—
380C24—62no—
381C24—122no—
382C24—122no—
383C24—122no—
384C24—121no—
385C24—121no—
386C24—122no—
387C24—122no—
388C24D3×D411yescharacteristic
389C2×Q8—21no—
390C2×Q8—32no—
391C2×Q8—42no—
392C2×Q8—64no—
393C2×Q8—61no—
394C2×Q8—122no—
395C2×Q8—242no—
396C2×Q8C2×D1212yes—
397C2×D4—64no—
398C2×D4—64no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—122no—
402C2×D4—122no—
403C2×D4—124no—
404C2×D4—124no—
405C2×D4—124no—
406C2×D4—124no—
407C2×D4—121no—
408C2×D4—121no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—34no—
413C22×C4—32no—
414C22×C4—32no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—122no—
427C22×C4C2×D1211yescharacteristic
428C22×C4C2×D1211yescharacteristic
429C22×C4D3×D411yescharacteristic
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4C23×D311yescharacteristic
433SD16—128no—
434SD16—128no—
435C2×C8—22no—
436C2×C8—42no—
437C2×C8—62no—
438C2×C8—122no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—32no—
443C4⋊C4—64no—
444C4⋊C4—64no—
445C4⋊C4—64no—
446C4⋊C4—64no—
447C4⋊C4—64no—
448C4⋊C4—62no—
449C4⋊C4—62no—
450C4⋊C4—61no—
451C4⋊C4—62no—
452C4⋊C4—62no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4C2×D1212yes—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—62no—
459C22⋊C4—62no—
460C22⋊C4—128no—
461C22⋊C4—128no—
462C42—64no—
463C42—64no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3#44812yes—
471C22×C3#44812yes—
472C22×C3#45412yes—
473C22×C3B(4,𝔽2)12yes—
474C22×C3C2×#12011yescharacteristic
475C22×C3C22×SD1611yescharacteristic
476C22×C3C2×#13711yescharacteristic
477C2×D3—44no—
478C2×D3—44no—
479C2×D3—44no—
480C2×D3—82no—
481C2×D3—82no—
482C2×D3—88no—
483C2×D3—88no—
484C2×D3—82no—
485C2×D3—82no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—82no—
493Dic3—24no—
494Dic3—42no—
495Dic3—42no—
496Dic3—42no—
497C23—22no—
498C23—22no—
499C23—64no—
500C23—122no—
501C23—124no—
502C23—122no—
503C23—122no—
504C23—128no—
505C23—128no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—124no—
510C23—242no—
511C23—244no—
512C23—244no—
513C23—242no—
514C23—242no—
515C23—242no—
516C23#88211yescharacteristic
517C23#88211yescharacteristic
518C23D3×SD1612yes—
519C23#91812yes—
520C23C2×D3×D411yescharacteristic
521Q8—42no—
522Q8—81no—
523Q8—122no—
524Q8—241no—
525Q8—244no—
526D4—64no—
527D4—64no—
528D4—122no—
529D4—122no—
530D4—124no—
531D4—124no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—34no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—84no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—241no—
559C2×C4#88211yescharacteristic
560C2×C4#88211yescharacteristic
561C2×C4C22×D1211yescharacteristic
562C2×C4C2×D3×D411yescharacteristic
563C8—42no—
564C8—122no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—22no—
568C2×C3#159914yescomplemented
569C2×C3C2×#44811yescharacteristic
570C2×C3C2×#45411yescharacteristic
571C2×C3C2×B(4,𝔽2)11yescharacteristic
572D3—84no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—24no—
578C22—24no—
579C22—124no—
580C22—124no—
581C22—124no—
582C22—242no—
583C22—242no—
584C22—248no—
585C22—242no—
586C22—242no—
587C22—248no—
588C22#474112yes—
589C22#480612yes—
590C22#480612yes—
591C22#482712yes—
592C22C2×#88211yescharacteristic
593C22C2×D3×SD1611yescharacteristic
594C22C2×#91811yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—42no—
600C4—42no—
601C4—64no—
602C4—82no—
603C4—122no—
604C4—122no—
605C4—122no—
606C3C2×#159911yescharacteristic, complemented, Sylow
607C2—21no—
608C2—21no—
609C2—22no—
610C2—244no—
611C2#6832514yesdirect product factor
612C2C2×#474111yescharacteristic
613C2C2×#480611yescharacteristic
614C2C2×#482711yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC23×D12C22
hypercenterC2×#328D3
hypocenterC3C2×#1599
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
1. C2×#5623C2semidirect
5. C2×C3×#328C2semidirect
6. C2×#4469C2semidirect
7. C2×#4452C2semidirect
8. #68325C2direct (N × H)
22. #5623C22semidirect
39. C3×#328C22semidirect
40. #4469C22semidirect
41. #4452C22semidirect
48. C2×#328D3semidirect
152. #328C2×D3semidirect
188. C22×Dic3SD16semidirect
367. C2×Dic3C2×SD16semidirect
568. C2×C3#1599semidirect
606. C3C2×#1599semidirect
611. C2#68325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56232
C2×#50462
C2×#47312
C2×#46722
C2×C3×#3282
C2×#44692
C2×#44522
#683252
C2×#15993