Finite Group Databasebeta

← C2×#83183 #11430398

Subgroups — C2×#83183 #11430398

Order 768.

3542 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#127C211yesmaximal, characteristic
2C22×#935C211yesmaximal, characteristic, complemented
3C2×#5222C212yesmaximal, complemented
4C2×#5222C212yesmaximal, complemented
5C2×#5222C212yesmaximal, complemented
6C2×#5220C212yesmaximal, complemented
7C2×#5220C212yesmaximal, complemented
8C2×#5220C211yesmaximal, characteristic, complemented
9#83183C2116yesmaximal, direct product factor
10C2×#4933C212yesmaximal, complemented
11C2×#2709—31nomaximal, Sylow
12C23×C3×D4C2212yes—
13C23×C3×D4C2212yes—
14C23×C3×D4C2211yescharacteristic
15C23×C3×D4C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C24×Dic3C2211yescharacteristic
18C2×C3×#127—28no—
19C2×C3×#127C2218yes—
20C22×C3×C42C2211yescharacteristic
21C22×#217C2212yes—
22C22×#217C2211yescharacteristic
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C22×#217C2211yescharacteristic
28C22×#217C2212yes—
29C2×#935—28no—
30C2×#935C2218yescomplemented
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33#5222C22116yescomplemented
34#5222C22116yescomplemented
35#5222C22116yescomplemented
36#5220C22116yescomplemented
37#5220C22116yescomplemented
38#5220C2218yescomplemented
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4933C22116yescomplemented
44C22×#127D311yescharacteristic
45C22×#123—31no—
46#2709—316no—
47C2×#394—32no—
48C2×#394—32no—
49C2×#394—32no—
50C2×#387—32no—
51C2×#386—32no—
52C2×#386—32no—
53C2×#386—31no—
54C25×C3C2312yes—
55C25×C3C2311yescharacteristic
56C25×C3C2311yescharacteristic
57C22×C3×D4—28no—
58C22×C3×D4—28no—
59C22×C3×D4—24no—
60C22×C3×D4—28no—
61C22×C3×D4—28no—
62C22×C3×D4—42no—
63C22×C3×D4—42no—
64C22×C3×D4—44no—
65C22×C3×D4—44no—
66C22×C3×D4—42no—
67C22×C3×D4—42no—
68C22×C3×D4D418yes—
69C22×C3×D4C2318yes—
70C22×C3×D4C2318yes—
71C22×C3×D4C2314yes—
72C22×C3×D4C2314yes—
73C23×C3×C4C2311yescharacteristic
74C23×C3×C4C2312yes—
75C23×Dic3—24no—
76C23×Dic3—42no—
77C23×Dic3—42no—
78C23×Dic3—42no—
79C23×Dic3—44no—
80C23×Dic3C2312yes—
81C23×Dic3C2312yes—
82C23×Dic3C2312yes—
83C23×Dic3C2312yes—
84C23×Dic3C2314yes—
85C23×Dic3C2314yes—
86C3×#127—232no—
87C2×C3×C42—24no—
88C2×C3×C42C2314yes—
89C2×#217—24no—
90C2×#217—28no—
91C2×#217—42no—
92C2×#217—44no—
93C2×#217—44no—
94C2×#217—44no—
95C2×#217—44no—
96C2×#217—44no—
97C2×#217C2318yes—
98C2×#217C2314yes—
99C2×#217C2318yes—
100C2×#217C2318yes—
101C2×#217C2318yes—
102C2×#217C2314yes—
103C2×#217C2318yes—
104C2×#217C2318yes—
105#935—232no—
106C2×Dic3⋊C4—28no—
107C2×Dic3⋊C4—28no—
108C2×Dic3⋊C4C2318yes—
109C2×Dic3⋊C4C2318yes—
110C3⋊#95C2318yes—
111C3⋊#95C2318yes—
112C3⋊#95C2318yes—
113C3⋊#95C2318yes—
114C23×D4C2×D312yes—
115C23×D4C2×D312yes—
116C23×D4C2×D311yescharacteristic
117C23×D4C2×D311yescharacteristic
118C24×C4—31no—
119C24×C4—31no—
120C2×#127—28no—
121C2×#127C2×D318yes—
122C2×#123—38no—
123C2×#123—68no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—32no—
126C22×C22⋊C4—31no—
127C22×C22⋊C4—32no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C22⋊C4—31no—
131C22×C22⋊C4—32no—
132C22×C22⋊C4—32no—
133C22×C22⋊C4—32no—
134C22×C42C2×D311yescharacteristic
135#394—316no—
136#394—316no—
137#394—316no—
138#387—316no—
139#386—316no—
140#386—316no—
141#386—38no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C24×C3—24no—
147C24×C3—24no—
148C24×C3—22no—
149C24×C3—24no—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—24no—
154C24×C3—24no—
155C24×C3—24no—
156C24×C3—24no—
157C24×C3—24no—
158C24×C3—24no—
159C24×C3—24no—
160C24×C3—24no—
161C24×C3C2×D414yes—
162C24×C3C2×D414yes—
163C24×C3C2414yes—
164C24×C3C2412yes—
165C24×C3C2412yes—
166C24×C3C2411yescharacteristic
167C2×C3×D4—216no—
168C2×C3×D4—216no—
169C2×C3×D4—28no—
170C2×C3×D4—216no—
171C2×C3×D4—216no—
172C2×C3×D4—416no—
173C2×C3×D4—44no—
174C2×C3×D4—44no—
175C2×C3×D4—48no—
176C2×C3×D4—48no—
177C2×C3×D4—44no—
178C2×C3×D4—44no—
179C2×C3×D4—432no—
180C2×C3×D4—432no—
181C2×C3×D4—416no—
182C2×C3×D4C2×D4116yes—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—24no—
187C22×C3×C4C2×D414yes—
188C22×C3×C4#4118yes—
189C22×C3×C4C2412yes—
190C22×C3×C4C2414yes—
191C22×Dic3—24no—
192C22×Dic3—24no—
193C22×Dic3—28no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—44no—
205C22×Dic3—44no—
206C22×Dic3—44no—
207C22×Dic3—48no—
208C22×Dic3—416no—
209C22×Dic3—416no—
210C22×Dic3C2414yes—
211C22×Dic3C2414yes—
212C22×Dic3C2414yes—
213C22×Dic3C2414yes—
214C3×C42—28no—
215#217—28no—
216#217—216no—
217#217—44no—
218#217—48no—
219#217—48no—
220#217—48no—
221#217—48no—
222#217—48no—
223Dic3⋊C4—216no—
224Dic3⋊C4—216no—
225C25C22×D312yes—
226C25C22×D311yescharacteristic
227C25C22×D311yescharacteristic
228C22×D4—28no—
229C22×D4—28no—
230C22×D4—28no—
231C22×D4—28no—
232C22×D4—24no—
233C22×D4—42no—
234C22×D4—42no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4#6718yes—
240C22×D4C22×D318yes—
241C22×D4C22×D318yes—
242C22×D4C22×D314yes—
243C22×D4C22×D314yes—
244C23×C4—32no—
245C23×C4—32no—
246C23×C4—32no—
247C23×C4—32no—
248C23×C4—34no—
249C23×C4—34no—
250C23×C4—64no—
251C23×C4—122no—
252C23×C4—122no—
253C23×C4—122no—
254C23×C4—124no—
255C23×C4C22×D312yes—
256C23×C4C22×D311yescharacteristic
257#127—232no—
258#123—632no—
259C2×C4⋊C4—38no—
260C2×C4⋊C4—38no—
261C2×C4⋊C4—68no—
262C2×C4⋊C4—68no—
263C2×C22⋊C4—38no—
264C2×C22⋊C4—34no—
265C2×C22⋊C4—38no—
266C2×C22⋊C4—38no—
267C2×C22⋊C4—38no—
268C2×C22⋊C4—34no—
269C2×C22⋊C4—38no—
270C2×C22⋊C4—38no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—68no—
273C2×C22⋊C4—122no—
274C2×C22⋊C4—124no—
275C2×C22⋊C4—124no—
276C2×C22⋊C4—124no—
277C2×C22⋊C4—124no—
278C2×C22⋊C4—124no—
279C2×C42—24no—
280C2×C42C22×D314yes—
281#95—38no—
282#95—38no—
283#95—38no—
284#95—38no—
285C23×C3—28no—
286C23×C3—28no—
287C23×C3—216no—
288C23×C3—216no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—22no—
292C23×C3—24no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—48no—
305C23×C3—48no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—44no—
309C23×C3—44no—
310C23×C3—48no—
311C23×C3—48no—
312C23×C3—48no—
313C23×C3—48no—
314C23×C3—48no—
315C23×C3—48no—
316C23×C3—48no—
317C23×C3C22×D414yes—
318C23×C3C22×D414yes—
319C23×C3C22×D412yes—
320C23×C3C2×#4114yes—
321C23×C3H(5,𝔽2)12yes—
322C23×C3H(5,𝔽2)12yes—
323C23×C3H(5,𝔽2)14yes—
324C23×C3C2511yescharacteristic
325C3×D4—416no—
326C3×D4—432no—
327C3×D4—432no—
328C3×D4—416no—
329C2×C3×C4—28no—
330C2×C3×C4—216no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4C22×D414yes—
336C2×C3×C4C2×#4118yes—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—416no—
350C2×Dic3—416no—
351C2×Dic3—48no—
352C2×Dic3—48no—
353C2×Dic3—48no—
354C2×Dic3—48no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24C2×#6714yes—
371C24C2×#6714yes—
372C24C23×D314yes—
373C24C23×D312yes—
374C24C23×D312yes—
375C24C23×D311yescharacteristic
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—28no—
381C2×D4—416no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—432no—
389C2×D4—432no—
390C2×D4—416no—
391C2×D4C2×#67116yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—34no—
397C22×C4—34no—
398C22×C4—34no—
399C22×C4—34no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—64no—
403C22×C4—64no—
404C22×C4—64no—
405C22×C4—64no—
406C22×C4—64no—
407C22×C4—64no—
408C22×C4—64no—
409C22×C4—64no—
410C22×C4—64no—
411C22×C4—64no—
412C22×C4—68no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4—124no—
416C22×C4—128no—
417C22×C4—1216no—
418C22×C4—1216no—
419C22×C4#23718yes—
420C22×C4C2×#6714yes—
421C22×C4C23×D314yes—
422C22×C4C23×D312yes—
423C4⋊C4—616no—
424C4⋊C4—616no—
425C22⋊C4—68no—
426C22⋊C4—616no—
427C22⋊C4—124no—
428C22⋊C4—128no—
429C22⋊C4—128no—
430C22⋊C4—128no—
431C22⋊C4—128no—
432C22⋊C4—128no—
433C42—28no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—28no—
437C22×C3—28no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—416no—
441C22×C3—416no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—44no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3—44no—
453C22×C3—816no—
454C22×C3#53414yes—
455C22×C3#54018yes—
456C22×C3#54618yes—
457C22×C3#55914yes—
458C22×C3#56014yes—
459C22×C3C23×D411yescharacteristic
460C22×C3C22×#4112yes—
461C22×C3C2×H(5,𝔽2)11yescharacteristic
462C22×C3C2×H(5,𝔽2)11yescharacteristic
463C22×C3C2×H(5,𝔽2)12yes—
464C3×C4—24no—
465C3×C4—28no—
466Dic3—44no—
467Dic3—44no—
468Dic3—44no—
469Dic3—44no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—28no—
479C23—28no—
480C23—216no—
481C23—216no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23C2×#23714yes—
503C23#100412yes—
504C23#100412yes—
505C23#100414yes—
506C23C22×#6714yes—
507C23C22×#6714yes—
508C23C22×#6712yes—
509C23C24×D311yescharacteristic
510D4—416no—
511D4—432no—
512D4—432no—
513D4—416no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—216no—
520C2×C4—64no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—64no—
524C2×C4—64no—
525C2×C4—64no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—64no—
529C2×C4—64no—
530C2×C4—64no—
531C2×C4—64no—
532C2×C4—1216no—
533C2×C4—1216no—
534C2×C4—128no—
535C2×C4—128no—
536C2×C4—128no—
537C2×C4—128no—
538C2×C4C2×#23718yes—
539C2×C4C22×#6714yes—
540C2×C3—44no—
541C2×C3—44no—
542C2×C3—84no—
543C2×C3#270918yescomplemented
544C2×C3C2×#53411yescharacteristic
545C2×C3C2×#54012yes—
546C2×C3C2×#54612yes—
547C2×C3C2×#55911yescharacteristic
548C2×C3C2×#56011yescharacteristic
549C22—28no—
550C22—28no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—416no—
560C22—416no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—816no—
569C22#571314yes—
570C22#571918yes—
571C22#572014yes—
572C22#580414yes—
573C22#580618yes—
574C22C22×#23712yes—
575C22C2×#100411yescharacteristic
576C22C2×#100411yescharacteristic
577C22C2×#100412yes—
578C22C23×#6711yescharacteristic
579C4—28no—
580C4—24no—
581C4—124no—
582C4—124no—
583C4—124no—
584C4—124no—
585C3C2×#270911yescharacteristic, complemented, Sylow
586C2—44no—
587C2—44no—
588C2—84no—
589C2#8318318yesdirect product factor
590C2C2×#571311yescharacteristic
591C2C2×#571912yes—
592C2C2×#572011yescharacteristic
593C2C2×#580612yes—
594C2C2×#580411yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
Fitting, prime radicalC22×C3×#127C2
socleC24×C3C24
hypercenterC22×#127D3
hypocenterC3C2×#2709
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#127

Complements

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

NH ≅ G/Ntype
2. C22×#935C2semidirect
3. C2×#5222C2semidirect
4. C2×#5222C2semidirect
5. C2×#5222C2semidirect
6. C2×#5220C2semidirect
7. C2×#5220C2semidirect
8. C2×#5220C2semidirect
9. #83183C2direct (N × H)
10. C2×#4933C2semidirect
30. C2×#935C22semidirect
33. #5222C22semidirect
34. #5222C22semidirect
35. #5222C22semidirect
36. #5220C22semidirect
37. #5220C22semidirect
38. #5220C22semidirect
43. #4933C22semidirect
543. C2×C3#2709semidirect
585. C3C2×#2709semidirect
589. C2#83183direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1272
C22×#9352
C2×#52222
C2×#52202
#831832
C2×#49332
C2×#27093