Finite Group Databasebeta

← C2×#74771 #10965699

Subgroups — C2×#74771 #10965699

Order 768.

3438 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
1C22×#939C211yesmaximal, characteristic
2C23×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#386C211yesmaximal, characteristic, complemented
4C2×#5220C211yesmaximal, characteristic, complemented
5C2×#5213C211yesmaximal, characteristic, complemented
6C2×#4961C212yesmaximal, complemented
7C2×#4951C212yesmaximal, complemented
8C2×#4958C211yesmaximal, characteristic, complemented
9C2×Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
10#74771C2116yesmaximal, direct product factor
11C2×#4660C212yesmaximal, complemented
12C2×#4670C212yesmaximal, complemented
13C2×#2591—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C23×#67C2211yescharacteristic
17C23×#67C2211yescharacteristic
18C24×Dic3C2211yescharacteristic
19C24×Dic3C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×#217C2211yescharacteristic
23C22×#217C2211yescharacteristic
24C22×#217C2212yes—
25C2×#939—28no—
26C2×#939C2218yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×(C2×D3)⋊C4C2211yescharacteristic
29C22×(C2×D3)⋊C4C2212yes—
30C22×C4⋊Dic3—28no—
31C22×C4⋊Dic3C2212yes—
32C22×C4⋊Dic3C2212yes—
33C22×C4⋊Dic3C2218yescomplemented
34C22×Dic3⋊C4C2211yescharacteristic
35C22×C4×Dic3C2212yes—
36C3×#386C2218yescomplemented
37C2×C3×#95C2212yes—
38#5220C2218yescomplemented
39#5213C2218yescomplemented
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44#4961C22116yescomplemented
45#4958C2218yescomplemented
46#4951C22116yescomplemented
47Dic3×C22⋊C4C2218yescomplemented
48#4670C22116yescomplemented
49#4660C22116yescomplemented
50C22×#121—31no—
51C23×C4⋊C4—31no—
52#2591—316no—
53C2×#400—32no—
54C2×#394—32no—
55C2×#388—32no—
56C2×#388—32no—
57C2×#386—31no—
58C2×#386—31no—
59C2×#386—31no—
60C2×#386D311yescharacteristic, complemented
61C2×C4×C22⋊C4—31no—
62C25×C3C2311yescharacteristic
63C25×C3C2311yescharacteristic
64C24×D3C2311yescharacteristic
65C22×C3×D4—24no—
66C22×C3×D4—24no—
67C22×C3×D4—24no—
68C22×C3×D4C2314yes—
69C23×C3×C4—24no—
70C23×C3×C4—28no—
71C23×C3×C4C2314yes—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2312yes—
74C23×C3×C4C2312yes—
75C22×#67—24no—
76C22×#67—24no—
77C22×#67—24no—
78C22×#67—24no—
79C22×#67—44no—
80C22×#67C2314yes—
81C22×#67C2314yes—
82C23×Dic3—24no—
83C23×Dic3—44no—
84C23×Dic3—44no—
85C23×Dic3D418yes—
86C23×Dic3C2312yes—
87C23×Dic3C2311yescharacteristic
88C23×Dic3C2312yes—
89C23×Dic3C2311yescharacteristic
90C23×Dic3C2311yescharacteristic
91C23×Dic3C2314yes—
92C23×Dic3C2314yes—
93C2×C3×C22⋊C4—24no—
94C2×C3×C22⋊C4—28no—
95C2×C3×C22⋊C4C2314yes—
96C2×C3×C22⋊C4C2318yes—
97C2×#217—24no—
98C2×#217—24no—
99C2×#217—44no—
100C2×#217C2314yes—
101C2×#217C2314yes—
102C2×#217C2318yes—
103#939—232no—
104C2×(C2×D3)⋊C4—28no—
105C2×(C2×D3)⋊C4—24no—
106C2×(C2×D3)⋊C4—44no—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2318yes—
109C2×(C2×D3)⋊C4C2318yes—
110C2×C4⋊Dic3—28no—
111C2×C4⋊Dic3—28no—
112C2×C4⋊Dic3—232no—
113C2×C4⋊Dic3C2318yes—
114C2×C4⋊Dic3C2318yes—
115C2×Dic3⋊C4—24no—
116C2×Dic3⋊C4C2314yes—
117C2×C4×Dic3—44no—
118C2×C4×Dic3C2318yes—
119C3×#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C23×D4—31no—
125C23×D4—31no—
126C23×D4C2×D311yescharacteristic
127C24×C4—31no—
128C24×C4—31no—
129C24×C4C2×D311yescharacteristic
130C2×#121—38no—
131C2×#121—68no—
132C22×C4⋊C4—38no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4—31no—
135C22×C4⋊C4—32no—
136C22×C4⋊C4—68no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—31no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—31no—
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4C2×D312yes—
144C22×C22⋊C4C2×D311yescharacteristic
145C22×C42—32no—
146#400—316no—
147#394—316no—
148#388—316no—
149#388—316no—
150#386—38no—
151#386—38no—
152#386—38no—
153#386C2×D318yescomplemented
154C4×C22⋊C4—38no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95—32no—
158C2×#95—32no—
159C2×#95C2×D312yes—
160C24×C3—24no—
161C24×C3—24no—
162C24×C3—22no—
163C24×C3—24no—
164C24×C3—22no—
165C24×C3—22no—
166C24×C3—24no—
167C24×C3C2×D414yes—
168C24×C3#4114yes—
169C24×C3#4114yes—
170C24×C3C2412yes—
171C24×C3C2412yes—
172C24×C3C2411yescharacteristic
173C23×D3—22no—
174C23×D3—24no—
175C23×D3—22no—
176C23×D3—22no—
177C23×D3—24no—
178C23×D3C2412yes—
179C2×C3×D4—28no—
180C2×C3×D4—28no—
181C2×C3×D4—28no—
182C2×C3×D4—416no—
183C22×C3×C4—232no—
184C22×C3×C4—28no—
185C22×C3×C4—24no—
186C22×C3×C4—24no—
187C22×C3×C4—216no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—24no—
194C22×C3×C4#4118yes—
195C22×C3×C4C2412yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2414yes—
198C2×#67—28no—
199C2×#67—28no—
200C2×#67—28no—
201C2×#67—28no—
202C2×#67—48no—
203C2×#67—416no—
204C2×#67—416no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—28no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—24no—
215C22×Dic3—24no—
216C22×Dic3—24no—
217C22×Dic3—416no—
218C22×Dic3—48no—
219C22×Dic3—48no—
220C22×Dic3—416no—
221C22×Dic3C2×D414yes—
222C22×Dic3C2×D414yes—
223C22×Dic3C2×D4116yes—
224C22×Dic3C2412yes—
225C22×Dic3C2414yes—
226C22×Dic3C2414yes—
227C22×Dic3C2412yes—
228C22×Dic3C2412yes—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—216no—
231#217—28no—
232#217—28no—
233#217—48no—
234(C2×D3)⋊C4—216no—
235(C2×D3)⋊C4—28no—
236(C2×D3)⋊C4—48no—
237C4⋊Dic3—216no—
238C4⋊Dic3—216no—
239Dic3⋊C4—28no—
240C4×Dic3—48no—
241C25—31no—
242C25C22×D311yescharacteristic
243C25C22×D311yescharacteristic
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—24no—
247C22×D4—34no—
248C22×D4—34no—
249C22×D4—64no—
250C22×D4—64no—
251C22×D4—64no—
252C22×D4—64no—
253C22×D4—124no—
254C22×D4C22×D314yes—
255C23×C4—24no—
256C23×C4—28no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—38no—
260C23×C4—34no—
261C23×C4—32no—
262C23×C4—32no—
263C23×C4—31no—
264C23×C4—34no—
265C23×C4—64no—
266C23×C4—124no—
267C23×C4—124no—
268C23×C4C22×D314yes—
269C23×C4C22×D312yes—
270C23×C4C22×D311yescharacteristic
271C23×C4C22×D312yes—
272#121—632no—
273C2×C4⋊C4—38no—
274C2×C4⋊C4—38no—
275C2×C4⋊C4—34no—
276C2×C4⋊C4—68no—
277C2×C4⋊C4—68no—
278C2×C4⋊C4—632no—
279C2×C4⋊C4—64no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—34no—
283C2×C22⋊C4—34no—
284C2×C22⋊C4—34no—
285C2×C22⋊C4—38no—
286C2×C22⋊C4—38no—
287C2×C22⋊C4—38no—
288C2×C22⋊C4—64no—
289C2×C22⋊C4—64no—
290C2×C22⋊C4—64no—
291C2×C22⋊C4—68no—
292C2×C22⋊C4—124no—
293C2×C22⋊C4—124no—
294C2×C22⋊C4C22×D314yes—
295C2×C22⋊C4C22×D318yes—
296C2×C42—38no—
297C2×C42—124no—
298#95—38no—
299#95—38no—
300#95—38no—
301#95—38no—
302#95C22×D318yes—
303C23×C3—28no—
304C23×C3—28no—
305C23×C3—216no—
306C23×C3—216no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—22no—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3—48no—
318C23×C3—48no—
319C23×C3#124116yes—
320C23×C3C22×D414yes—
321C23×C3C22×D412yes—
322C23×C3C2×#4114yes—
323C23×C3C2×#4112yes—
324C23×C3C2×#4114yes—
325C23×C3C2×#4114yes—
326C23×C3C2×#4112yes—
327C23×C3H(5,𝔽2)14yes—
328C23×C3C2511yescharacteristic
329C22×D3—22no—
330C22×D3—24no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—24no—
334C22×D3—44no—
335C22×D3—44no—
336C22×D3—44no—
337C22×D3—48no—
338C22×D3—48no—
339C3×D4—416no—
340C2×C3×C4—232no—
341C2×C3×C4—216no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—44no—
351C2×C3×C4—48no—
352C2×C3×C4C2×#4118yes—
353#67—416no—
354#67—416no—
355C2×Dic3—28no—
356C2×Dic3—28no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—22no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—416no—
369C2×Dic3—416no—
370C2×Dic3—44no—
371C2×Dic3—48no—
372C2×Dic3—48no—
373C2×Dic3C22×D414yes—
374C2×Dic3C22×D414yes—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—24no—
382C24—32no—
383C24—62no—
384C24—62no—
385C24—64no—
386C24—62no—
387C24—64no—
388C24#23514yes—
389C24D3×D414yes—
390C24#23714yes—
391C24C23×D312yes—
392C24C23×D312yes—
393C24C23×D311yescharacteristic
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—416no—
398C2×D4—68no—
399C2×D4—68no—
400C2×D4—68no—
401C2×D4—68no—
402C2×D4—128no—
403C2×D4—1216no—
404C2×D4—1216no—
405C22×C4—232no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—216no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—34no—
417C22×C4—34no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—316no—
421C22×C4—32no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—68no—
430C22×C4—64no—
431C22×C4—64no—
432C22×C4—62no—
433C22×C4—64no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—1216no—
437C22×C4—128no—
438C22×C4—128no—
439C22×C4—1216no—
440C22×C4#23718yes—
441C22×C4C23×D312yes—
442C22×C4C23×D314yes—
443C22×C4C23×D314yes—
444C4⋊C4—616no—
445C4⋊C4—616no—
446C4⋊C4—68no—
447C22⋊C4—28no—
448C22⋊C4—216no—
449C22⋊C4—68no—
450C22⋊C4—68no—
451C22⋊C4—68no—
452C22⋊C4—616no—
453C22⋊C4—128no—
454C22⋊C4—128no—
455C42—128no—
456C22×C3—232no—
457C22×C3—24no—
458C22×C3—24no—
459C22×C3—28no—
460C22×C3—28no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—44no—
465C22×C3—44no—
466C22×C3—88no—
467C22×C3#52514yes—
468C22×C3C2×#12418yes—
469C22×C3C2×#12414yes—
470C22×C3#53414yes—
471C22×C3#55318yes—
472C22×C3#55318yes—
473C22×C3C23×D411yescharacteristic
474C22×C3C22×#4111yescharacteristic
475C22×C3C22×#4112yes—
476C22×C3C22×#4111yescharacteristic
477C22×C3C2×H(5,𝔽2)12yes—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—42no—
481C2×D3—44no—
482C2×D3—44no—
483C2×D3—88no—
484C3×C4—28no—
485C3×C4—42no—
486C3×C4—44no—
487Dic3—24no—
488Dic3—24no—
489Dic3—42no—
490Dic3—44no—
491Dic3—44no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—216no—
496C23—28no—
497C23—216no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—44no—
504C23—44no—
505C23—48no—
506C23—44no—
507C23—48no—
508C23—62no—
509C23—62no—
510C23—64no—
511C23—62no—
512C23—64no—
513C23—124no—
514C23—124no—
515C23—128no—
516C23—124no—
517C23—128no—
518C23#885116yes—
519C23C2×#23514yes—
520C23C2×#23512yes—
521C23C2×D3×D414yes—
522C23C2×D3×D412yes—
523C23C2×#23714yes—
524C23C2×#23712yes—
525C23C2×#23714yes—
526C23#100914yes—
527C23C24×D311yescharacteristic
528D4—416no—
529D4—1216no—
530D4—1216no—
531C2×C4—232no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—216no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—34no—
542C2×C4—34no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—68no—
546C2×C4—68no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—1216no—
559C2×C4—1216no—
560C2×C4—128no—
561C2×C4—124no—
562C2×C4—128no—
563C2×C4C2×#23718yes—
564C2×C3—28no—
565C2×C3—82no—
566C2×C3#259118yescomplemented
567C2×C3C2×#52511yescharacteristic
568C2×C3C22×#12411yescharacteristic
569C2×C3C2×#53411yescharacteristic
570C2×C3C2×#55312yes—
571C2×C3C2×#55312yes—
572D3—82no—
573C22—232no—
574C22—24no—
575C22—28no—
576C22—24no—
577C22—28no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—88no—
584C22—122no—
585C22—124no—
586C22—122no—
587C22—122no—
588C22—124no—
589C22—248no—
590C22C2×#88518yes—
591C22C2×#88514yes—
592C22#556518yes—
593C22#558614yes—
594C22#561818yes—
595C22#565314yes—
596C22C22×#23511yescharacteristic
597C22C22×D3×D411yescharacteristic
598C22C22×#23712yes—
599C22C22×#23711yescharacteristic
600C22C2×#100912yes—
601C4—28no—
602C4—44no—
603C4—42no—
604C4—64no—
605C4—64no—
606C4—122no—
607C4—124no—
608C4—124no—
609C3C2×#259111yescharacteristic, complemented, Sylow
610C2—28no—
611C2—82no—
612C2—242no—
613C2#7477118yesdirect product factor
614C2C22×#88511yescharacteristic
615C2C2×#556512yes—
616C2C2×#558611yescharacteristic
617C2C2×#561812yes—
618C2C2×#565311yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#386C2
socleC24×C3C24
prime radicalC22×#939C2
hypercenterC2×#386D3
hypocenterC3C2×#2591
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C23×C4⋊Dic3C2semidirect
3. C2×C3×#386C2semidirect
4. C2×#5220C2semidirect
5. C2×#5213C2semidirect
6. C2×#4961C2semidirect
7. C2×#4951C2semidirect
8. C2×#4958C2semidirect
9. C2×Dic3×C22⋊C4C2semidirect
10. #74771C2direct (N × H)
11. C2×#4660C2semidirect
12. C2×#4670C2semidirect
33. C22×C4⋊Dic3C22semidirect
36. C3×#386C22semidirect
38. #5220C22semidirect
39. #5213C22semidirect
44. #4961C22semidirect
45. #4958C22semidirect
46. #4951C22semidirect
47. Dic3×C22⋊C4C22semidirect
48. #4670C22semidirect
49. #4660C22semidirect
60. C2×#386D3semidirect
153. #386C2×D3semidirect
566. C2×C3#2591semidirect
609. C3C2×#2591semidirect
613. C2#74771direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9392
C23×C4⋊Dic32
C2×C3×#3862
C2×#52202
C2×#52132
C2×#49612
C2×#49512
C2×#49582
C2×Dic3×C22⋊C42
#747712
C2×#46602
C2×#46702
C2×#25913