Finite Group Databasebeta

← C2×#74457 #10964359

Subgroups — C2×#74457 #10964359

Order 768.

3374 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic3⋊C4C211yesmaximal, characteristic, complemented
2C22×#67⋊C4C211yesmaximal, characteristic
3C2×C3×#385C211yesmaximal, characteristic, complemented
4C2×#5213C211yesmaximal, characteristic
5C2×#4993C212yesmaximal, complemented
6C2×#4976C212yesmaximal, complemented
7C2×#4958C211yesmaximal, characteristic
8C2×#4946C211yesmaximal, characteristic, complemented
9C2×#4942C211yesmaximal, characteristic, complemented
10#74457C2116yesmaximal, direct product factor
11C2×#4667C212yesmaximal, complemented
12C2×#4658C212yesmaximal, complemented
13C2×#2487—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C24×Dic3C2211yescharacteristic
18C23×C4×D3C2211yescharacteristic
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×#217C2211yescharacteristic
23C22×(C2×D3)⋊C4C2211yescharacteristic
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×Dic3⋊C4—28no—
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2211yescharacteristic
29C22×Dic3⋊C4C2212yes—
30C22×Dic3⋊C4C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2218yescomplemented
33C22×C4×Dic3C2211yescharacteristic
34C2×#67⋊C4—28no—
35C2×#67⋊C4C2218yes—
36C3×#385C2218yescomplemented
37C2×C3×#95C2212yes—
38#5213C2218yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4993C22116yescomplemented
44#4976C22116yescomplemented
45#4958C2218yes—
46#4946C2218yescomplemented
47#4942C2218yescomplemented
48#4667C22116yescomplemented
49#4658C22116yescomplemented
50C22×C4×D4—31no—
51C23×C4⋊C4—31no—
52#2487—316no—
53C2×#400—32no—
54C2×#399—32no—
55C2×#389—32no—
56C2×#386—31no—
57C2×#386—31no—
58C2×#385—31no—
59C2×#385D311yescharacteristic, complemented
60C2×#380—32no—
61C2×#380—31no—
62C25×C3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C23×C3×C4—24no—
65C23×C3×C4—44no—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2314yes—
71C22×#67—24no—
72C22×#67—24no—
73C22×#67Q818yes—
74C22×#67C2314yes—
75C23×Dic3—28no—
76C23×Dic3—24no—
77C23×Dic3—44no—
78C23×Dic3D418yes—
79C23×Dic3C2312yes—
80C23×Dic3C2312yes—
81C23×Dic3C2314yes—
82C23×Dic3C2311yescharacteristic
83C23×Dic3C2311yescharacteristic
84C23×Dic3C2314yes—
85C23×Dic3C2311yescharacteristic
86C22×C4×D3—24no—
87C22×C4×D3—28no—
88C22×C4×D3C2314yes—
89C2×C3×C4⋊C4—44no—
90C2×C3×C4⋊C4C2318yes—
91C2×C3×C22⋊C4—24no—
92C2×C3×C22⋊C4—24no—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C22⋊C4C2314yes—
95C2×#217—24no—
96C2×#217C2314yes—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—44no—
99C2×(C2×D3)⋊C4—44no—
100C2×(C2×D3)⋊C4C2314yes—
101C2×(C2×D3)⋊C4C2318yes—
102C2×(C2×D3)⋊C4C2318yes—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4—28no—
106C2×Dic3⋊C4—232no—
107C2×Dic3⋊C4—28no—
108C2×Dic3⋊C4—44no—
109C2×Dic3⋊C4C2314yes—
110C2×Dic3⋊C4C2318yes—
111C2×Dic3⋊C4C2318yes—
112C2×Dic3⋊C4C2318yes—
113C2×Dic3⋊C4C2318yes—
114C2×C4×Dic3—24no—
115C2×C4×Dic3C2314yes—
116#67⋊C4—232no—
117C3×#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C23×D4—31no—
123C24×C4—31no—
124C24×C4—31no—
125C24×C4—31no—
126C24×C4C2×D311yescharacteristic
127C2×C4×D4—38no—
128C2×C4×D4—68no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—38no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—68no—
136C22×C4⋊C4C2×D312yes—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—31no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4C2×D311yescharacteristic
142C22×C22⋊C4C2×D311yescharacteristic
143C22×C42—31no—
144#400—316no—
145#399—316no—
146#389—316no—
147#386—38no—
148#386—38no—
149#385—38no—
150#385C2×D318yescomplemented
151#380—38no—
152#380—316no—
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95C2×D312yes—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3C2×D414yes—
161C24×C3C2×Q814yes—
162C24×C3#4114yes—
163C24×C3C2412yes—
164C24×C3C2411yescharacteristic
165C23×D3—22no—
166C23×D3—24no—
167C23×D3—22no—
168C23×D3—24no—
169C23×D3C2×Q814yes—
170C23×D3C2412yes—
171C22×C3×C4—22no—
172C22×C3×C4—28no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—48no—
183C22×C3×C4—416no—
184C22×C3×C4C2412yes—
185C22×C3×C4C2414yes—
186C22×C3×C4C2414yes—
187C22×C3×C4C2412yes—
188C2×#67—232no—
189C2×#67—28no—
190C2×#67—28no—
191C2×#67C2×Q8116yes—
192C22×Dic3—22no—
193C22×Dic3—24no—
194C22×Dic3—22no—
195C22×Dic3—232no—
196C22×Dic3—216no—
197C22×Dic3—28no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—48no—
205C22×Dic3—416no—
206C22×Dic3C2×D4116yes—
207C22×Dic3C2×D414yes—
208C22×Dic3C2×D414yes—
209C22×Dic3C2×Q814yes—
210C22×Dic3#4118yes—
211C22×Dic3C2414yes—
212C22×Dic3C2414yes—
213C22×Dic3C2412yes—
214C22×Dic3C2412yes—
215C22×Dic3C2412yes—
216C2×C4×D3—28no—
217C2×C4×D3—216no—
218C2×C4×D3—416no—
219C3×C4⋊C4—48no—
220C3×C22⋊C4—28no—
221C3×C22⋊C4—28no—
222#217—28no—
223(C2×D3)⋊C4—28no—
224(C2×D3)⋊C4—48no—
225(C2×D3)⋊C4—48no—
226Dic3⋊C4—28no—
227Dic3⋊C4—216no—
228Dic3⋊C4—216no—
229Dic3⋊C4—216no—
230Dic3⋊C4—48no—
231C4×Dic3—28no—
232C25—31no—
233C25C22×D311yescharacteristic
234C22×D4—38no—
235C22×D4—34no—
236C22×D4—64no—
237C22×D4—64no—
238C23×C4—24no—
239C23×C4—31no—
240C23×C4—38no—
241C23×C4—34no—
242C23×C4—31no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4—32no—
246C23×C4—34no—
247C23×C4—34no—
248C23×C4—44no—
249C23×C4—64no—
250C23×C4—68no—
251C23×C4—68no—
252C23×C4—64no—
253C23×C4—124no—
254C23×C4C22×D311yescharacteristic
255C23×C4C22×D312yes—
256C23×C4C22×D312yes—
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D314yes—
259C4×D4—632no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—38no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—38no—
264C2×C4⋊C4—38no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—68no—
267C2×C4⋊C4—68no—
268C2×C4⋊C4—632no—
269C2×C4⋊C4—64no—
270C2×C4⋊C4—68no—
271C2×C4⋊C4—124no—
272C2×C4⋊C4C22×D318yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—34no—
276C2×C22⋊C4—34no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—124no—
282C2×C22⋊C4—124no—
283C2×C22⋊C4C22×D314yes—
284C2×C22⋊C4C22×D314yes—
285C2×C42—34no—
286C2×C42—64no—
287#95—38no—
288#95—38no—
289#95—38no—
290#95—38no—
291#95C22×D318yes—
292C23×C3—28no—
293C23×C3—28no—
294C23×C3—216no—
295C23×C3—216no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3C22⋊Q8116yes—
299C23×C3C22×D414yes—
300C23×C3C22×D412yes—
301C23×C3C22×Q814yes—
302C23×C3C22×Q812yes—
303C23×C3C2×#4114yes—
304C23×C3C2×#4112yes—
305C23×C3C2×#4114yes—
306C23×C3H(5,𝔽2)14yes—
307C23×C3C2511yescharacteristic
308C22×D3—28no—
309C22×D3—216no—
310C22×D3—22no—
311C22×D3—24no—
312C22×D3—22no—
313C22×D3—24no—
314C22×D3—44no—
315C22×D3—44no—
316C22×D3—48no—
317C22×D3C22×Q814yes—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—416no—
329C2×C3×C4—44no—
330C2×C3×C4—44no—
331C2×C3×C4—48no—
332C2×C3×C4—48no—
333#67—232no—
334C2×Dic3—28no—
335C2×Dic3—28no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—22no—
339C2×Dic3—232no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—416no—
347C2×Dic3—48no—
348C2×Dic3—48no—
349C2×Dic3#123116yes—
350C2×Dic3C22×D414yes—
351C2×Dic3C22×D414yes—
352C2×Dic3C22×Q814yes—
353C2×Dic3C2×#4118yes—
354C4×D3—416no—
355C24—24no—
356C24—24no—
357C24—34no—
358C24—32no—
359C24—62no—
360C24—62no—
361C24—64no—
362C24—64no—
363C24#23514yes—
364C24D3×D414yes—
365C24D3×Q814yes—
366C24C23×D312yes—
367C24C23×D311yescharacteristic
368C2×D4—316no—
369C2×D4—632no—
370C2×D4—68no—
371C2×D4—68no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—34no—
384C22×C4—38no—
385C22×C4—32no—
386C22×C4—34no—
387C22×C4—316no—
388C22×C4—34no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—48no—
394C22×C4—416no—
395C22×C4—68no—
396C22×C4—616no—
397C22×C4—632no—
398C22×C4—616no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—68no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—1216no—
410C22×C4—128no—
411C22×C4—1216no—
412C22×C4C23×D312yes—
413C22×C4C23×D314yes—
414C22×C4C23×D314yes—
415C22×C4C23×D312yes—
416C4⋊C4—48no—
417C4⋊C4—616no—
418C4⋊C4—616no—
419C4⋊C4—68no—
420C4⋊C4—616no—
421C4⋊C4—128no—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—68no—
425C22⋊C4—68no—
426C22⋊C4—128no—
427C22⋊C4—128no—
428C42—68no—
429C22×C3—24no—
430C22×C3—24no—
431C22×C3—28no—
432C22×C3—28no—
433C22×C3—232no—
434C22×C3C2×C22⋊Q818yes—
435C22×C3C2×C22⋊Q814yes—
436C22×C3C2×#12314yes—
437C22×C3#53514yes—
438C22×C3#55318yes—
439C22×C3#55418yes—
440C22×C3C23×D411yescharacteristic
441C22×C3C23×Q811yescharacteristic
442C22×C3C22×#4111yescharacteristic
443C22×C3C22×#4112yes—
444C22×C3C2×H(5,𝔽2)12yes—
445C2×D3—24no—
446C2×D3—28no—
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—44no—
450C2×D3—416no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—44no—
454C3×C4—44no—
455Dic3—24no—
456Dic3—24no—
457Dic3—44no—
458Dic3—44no—
459Dic3C2×#12318yes—
460C23—216no—
461C23—28no—
462C23—24no—
463C23—24no—
464C23—216no—
465C23—28no—
466C23—34no—
467C23—616no—
468C23—62no—
469C23—62no—
470C23—64no—
471C23—64no—
472C23—68no—
473C23—124no—
474C23—128no—
475C23—124no—
476C23#896116yes—
477C23C2×#23514yes—
478C23C2×#23512yes—
479C23C2×D3×D414yes—
480C23C2×D3×D412yes—
481C23#100414yes—
482C23C2×D3×Q814yes—
483C23C2×D3×Q812yes—
484C23D3×#4114yes—
485C23C24×D311yescharacteristic
486D4—632no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—316no—
498C2×C4—34no—
499C2×C4—38no—
500C2×C4—34no—
501C2×C4—34no—
502C2×C4—416no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—44no—
507C2×C4—632no—
508C2×C4—68no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—64no—
512C2×C4—68no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—1216no—
520C2×C4—1216no—
521C2×C4—128no—
522C2×C4—128no—
523C2×C3—28no—
524C2×C3#248718yescomplemented
525C2×C3C22×C22⋊Q811yescharacteristic
526C2×C3C22×#12311yescharacteristic
527C2×C3C2×#53511yescharacteristic
528C2×C3C2×#55312yes—
529C2×C3C2×#55412yes—
530D3—44no—
531C22—24no—
532C22—28no—
533C22—232no—
534C22—24no—
535C22—28no—
536C22—64no—
537C22—68no—
538C22—122no—
539C22—1216no—
540C22—122no—
541C22—124no—
542C22C2×#89618yes—
543C22C2×#89614yes—
544C22#554018yes—
545C22#559214yes—
546C22#563618yes—
547C22D3×#12314yes—
548C22C22×#23511yescharacteristic
549C22C22×D3×D411yescharacteristic
550C22C2×#100412yes—
551C22C22×D3×Q811yescharacteristic
552C22C2×D3×#4112yes—
553C4—38no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4—64no—
559C4—64no—
560C4—124no—
561C4—124no—
562C3C2×#248711yescharacteristic, complemented, Sylow
563C2—28no—
564C2—124no—
565C2#7445718yesdirect product factor
566C2C22×#89611yescharacteristic
567C2C2×#554012yes—
568C2C2×#559211yescharacteristic
569C2C2×#563612yes—
570C2C2×D3×#12311yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#385C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#385D3
hypocenterC3C2×#2487
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. C23×Dic3⋊C4C2semidirect
3. C2×C3×#385C2semidirect
5. C2×#4993C2semidirect
6. C2×#4976C2semidirect
8. C2×#4946C2semidirect
9. C2×#4942C2semidirect
10. #74457C2direct (N × H)
11. C2×#4667C2semidirect
12. C2×#4658C2semidirect
32. C22×Dic3⋊C4C22semidirect
36. C3×#385C22semidirect
43. #4993C22semidirect
44. #4976C22semidirect
46. #4946C22semidirect
47. #4942C22semidirect
48. #4667C22semidirect
49. #4658C22semidirect
59. C2×#385D3semidirect
150. #385C2×D3semidirect
524. C2×C3#2487semidirect
562. C3C2×#2487semidirect
565. C2#74457direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Dic3⋊C42
C22×#67⋊C42
C2×C3×#3852
C2×#52132
C2×#49932
C2×#49762
C2×#49582
C2×#49462
C2×#49422
#744572
C2×#46672
C2×#46582
C2×#24873