Finite Group Databasebeta

← C2×#83524 #11670427

Subgroups — C2×#83524 #11670427

Order 768.

2806 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C22×#950C212yesmaximal, complemented
3C2×#5551C211yesmaximal, characteristic, complemented
4C2×#5239C212yesmaximal, complemented
5#83524C2116yesmaximal, direct product factor
6C2×#5083C211yesmaximal, characteristic, complemented
7C2×#5092C211yesmaximal, characteristic, complemented
8C2×#5047C212yesmaximal, complemented
9C2×#5051C212yesmaximal, complemented
10C2×#5019C211yesmaximal, characteristic, complemented
11C2×D4×C3⋊C8C211yesmaximal, characteristic, complemented
12C2×C4×#213C211yesmaximal, characteristic, complemented
13C2×#3283—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C22×#235C2212yes—
16C3×#547—24no—
17C3×#547C2218yescomplemented
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C2×#950—216no—
26C2×#950C22116yescomplemented
27C22×#216C2212yes—
28C22×#216C2212yes—
29C22×#215C2212yes—
30C2×#938—21no—
31C22×#214C2212yes—
32C22×#213C2211yescharacteristic
33C2×#924C2212yes—
34C23×C3⋊C8C2212yes—
35#5551—24no—
36#5551C2218yescomplemented
37C2×D3×C4⋊C4—21no—
38C2×#883—22no—
39C2×C4×D12C2211yescharacteristic
40C2×#869C2211yescharacteristic
41#5239C22116yescomplemented
42C2×#835C2212yes—
43C2×#832C2211yescharacteristic
44C2×#830C2212yes—
45#5092C2218yescomplemented
46#5083C2218yescomplemented
47#5051C22116yescomplemented
48#5047C22116yescomplemented
49#5019C2218yescomplemented
50C4×#213C2218yescomplemented
51D4×C3⋊C8C2218yescomplemented
52C2×#810C2212yes—
53C2×#809C2211yescharacteristic
54C2×#807C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57C22×#135—32no—
58C2×#547—31no—
59C2×#547D311yescharacteristic, complemented
60#3283—316no—
61C2×#494—31no—
62C2×#467—32no—
63C2×#461—31no—
64C2×#461—31no—
65C2×#452—32no—
66C2×#452—32no—
67C2×C4×D8—31no—
68C2×C8×D4—31no—
69C2×C3×#41—24no—
70C2×C3×#41—28no—
71C2×C3×#41—44no—
72C2×C3×#41C2318yes—
73C22×C3×Q8C2312yes—
74C22×C3×D4—22no—
75C22×C3×D4D411yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—22no—
78C23×C3×C4—21no—
79C23×C3×C4C2312yes—
80C22×#67—22no—
81C2×#235—24no—
82C2×#235—28no—
83C2×#235—44no—
84C2×#235C2318yes—
85C22×D12C2311yescharacteristic
86C22×C4×D3—22no—
87C22×C4×D3—21no—
88C22×Dic6C2312yes—
89C3×C4⋊Q8—22no—
90C3×C4⋊Q8C2314yes—
91C3×#123—216no—
92C3×C22⋊Q8—24no—
93C3×C22⋊Q8C2318yes—
94C3×#121—28no—
95C3×C4×D4—22no—
96C3×C4×D4D414yes—
97C3×C4×D4C2314yes—
98C3×C4×D4C2314yes—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—28no—
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4D412yes—
108C2×C3×C42C2311yescharacteristic
109C2×#217—22no—
110#950—264no—
111C2×#216—28no—
112C2×#216—44no—
113C2×#216C2318yes—
114C2×#216C2318yes—
115C2×#215—28no—
116C2×#215C2318yes—
117#938—28no—
118C2×#214—28no—
119C2×#214C2318yes—
120C2×#213D418yescomplemented
121C2×#213C2314yes—
122C2×(C2×D3)⋊C4—21no—
123C2×C4⋊Dic3—21no—
124C2×C4⋊Dic3—21no—
125C2×C4⋊Dic3C2311yescharacteristic
126#924—24no—
127#924C2318yes—
128C2×Dic3⋊C4—22no—
129C22×C3⋊C8—28no—
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2312yes—
132C22×C3⋊C8C2318yes—
133C22×C3⋊C8C2311yescharacteristic
134D3×C4⋊C4—28no—
135#883—216no—
136C4×D12—22no—
137C4×D12C2314yes—
138#869—22no—
139#869C2314yes—
140#835C2318yes—
141#832C2314yes—
142#830C2318yes—
143#810C2318yes—
144#809C2314yes—
145#807C2314yes—
146C4⋊(C3⋊C8)C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41—32no—
149C22×#41C2×D312yes—
150C2×#135—316no—
151C2×#135—616no—
152C22×Q16—32no—
153C22×Q16—32no—
154C22×SD16—32no—
155C22×SD16—32no—
156C22×D8—31no—
157C23×C8—32no—
158#547—24no—
159#547—38no—
160#547—64no—
161#547C2×D318yescomplemented
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8C2×D311yescharacteristic
164C2×#123—22no—
165C2×#123—62no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8C2×D312yes—
168C2×#121—21no—
169C2×#121—61no—
170C2×C4×D4—31no—
171C2×C4×D4C2×D311yescharacteristic
172C2×C4×D4C2×D311yescharacteristic
173C22×C4⋊C4—21no—
174C22×C4⋊C4—61no—
175#494—38no—
176#467—316no—
177#461—38no—
178#461—38no—
179#452—316no—
180#452—316no—
181C4×D8—38no—
182C8×D4—38no—
183C2×#107—31no—
184C2×C4⋊C8—31no—
185C2×#103—32no—
186C2×#103—32no—
187C2×#102—31no—
188C2×#102—31no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3—21no—
192C24×C3C2×D412yes—
193C23×D3—21no—
194C3×#41—216no—
195C3×#41—48no—
196C3×#41—416no—
197C2×C3×Q8—22no—
198C2×C3×Q8—24no—
199C2×C3×Q8—24no—
200C2×C3×Q8C2414yes—
201C2×C3×D4—28no—
202C2×C3×D4—24no—
203C2×C3×D4—21no—
204C2×C3×D4—22no—
205C2×C3×D4—48no—
206C2×C3×D4C2×D414yes—
207C2×C3×D4C2×D414yes—
208C2×C3×D4C2412yes—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—24no—
214C22×C3×C4—28no—
215C22×C3×C4—48no—
216C22×C3×C4—44no—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D414yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2414yes—
222C22×C3×C4C2411yescharacteristic
223C2×#67—28no—
224C2×#67—48no—
225C22×Dic3—21no—
226C22×Dic3—22no—
227#235—216no—
228#235—48no—
229#235—416no—
230C2×D12—21no—
231C2×D12—22no—
232C2×D12C2×D414yes—
233C2×D12C2412yes—
234C2×C4×D3—24no—
235C2×C4×D3—28no—
236C2×C4×D3—48no—
237C2×C4×D3—44no—
238C2×Dic6—24no—
239C2×Dic6—24no—
240C2×Dic6—22no—
241C2×Dic6C2414yes—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—28no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2×D414yes—
247C3×C4⋊C4C2412yes—
248C3×C22⋊C4—28no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4C2×D418yes—
251C3×C42C2×D412yes—
252C3×C42C2412yes—
253#217—28no—
254#216—216no—
255#216—48no—
256#215—216no—
257#214—216no—
258#213C2×D4116yescomplemented
259(C2×D3)⋊C4—24no—
260C4⋊Dic3—24no—
261C4⋊Dic3—24no—
262C4⋊Dic3—21no—
263C4⋊Dic3C2412yes—
264Dic3⋊C4—28no—
265C2×C3⋊C8—24no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8—216no—
268C2×C3⋊C8C2×D414yes—
269C2×C3⋊C8C2412yes—
270C2×C3⋊C8C2414yes—
271C2×C3⋊C8C2412yes—
272C2×#41—24no—
273C2×#41—28no—
274C2×#41—38no—
275C2×#41—44no—
276C2×#41—68no—
277C2×#41—64no—
278C2×#41—124no—
279C2×#41C22×D318yes—
280C22×Q8—32no—
281C22×Q8C22×D312yes—
282C22×D4—22no—
283C22×D4—31no—
284C22×D4—62no—
285C22×D4#6711yescharacteristic
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C23×C4—22no—
289C23×C4—61no—
290C23×C4—62no—
291C23×C4C22×D312yes—
292#135—664no—
293C2×Q16—38no—
294C2×Q16—38no—
295C2×Q16—68no—
296C2×Q16—124no—
297C2×SD16—38no—
298C2×SD16—38no—
299C2×SD16—68no—
300C2×SD16—68no—
301C2×D8—38no—
302C2×D8—34no—
303C22×C8—31no—
304C22×C8—32no—
305C22×C8—38no—
306C22×C8—31no—
307C22×C8—68no—
308C4⋊Q8—22no—
309C4⋊Q8—34no—
310C4⋊Q8—62no—
311C4⋊Q8C22×D314yes—
312#123—216no—
313#123—616no—
314C22⋊Q8—24no—
315C22⋊Q8—38no—
316C22⋊Q8—64no—
317C22⋊Q8C22×D318yes—
318#121—28no—
319#121—68no—
320C4×D4—22no—
321C4×D4—34no—
322C4×D4—62no—
323C4×D4#6714yes—
324C4×D4C22×D314yes—
325C4×D4C22×D314yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—62no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—68no—
335C2×C4⋊C4#6711yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—62no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4#6712yes—
342C2×C42C22×D311yescharacteristic
343#107—34no—
344C4⋊C8—34no—
345#103—38no—
346#103—38no—
347#102—34no—
348#102—34no—
349C22⋊C8—38no—
350C4×C8—34no—
351C23×C3—24no—
352C23×C3—24no—
353C23×C3—24no—
354C23×C3—24no—
355C23×C3—22no—
356C23×C3—42no—
357C23×C3—42no—
358C23×C3C22×D414yes—
359C23×C3C22×D411yescharacteristic
360C22×D3—24no—
361C22×D3—22no—
362C22×D3—42no—
363C22×D3—42no—
364C3×Q8—24no—
365C3×Q8—24no—
366C3×Q8—44no—
367C3×D4—24no—
368C3×D4—28no—
369C3×D4—22no—
370C3×D4—416no—
371C3×D4C22×D414yes—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—28no—
375C2×C3×C4—24no—
376C2×C3×C4—24no—
377C2×C3×C4—22no—
378C2×C3×C4—24no—
379C2×C3×C4—416no—
380C2×C3×C4—44no—
381C2×C3×C4—42no—
382C2×C3×C4—48no—
383C2×C3×C4#13514yescomplemented
384C2×C3×C4C22×D412yes—
385C2×C3×C4C22×D414yes—
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C22×D412yes—
389C2×C3×C4H(5,𝔽2)12yes—
390C2×C3×C4C2511yescharacteristic
391#67—416no—
392C2×Dic3—22no—
393C2×Dic3—24no—
394C2×Dic3—44no—
395C2×Dic3—42no—
396D12—24no—
397D12—22no—
398D12C22×D414yes—
399C4×D3—416no—
400C4×D3—48no—
401Dic6—24no—
402Dic6—24no—
403Dic6—44no—
404C3⋊C8—24no—
405C3⋊C8—22no—
406C3⋊C8C22×D414yes—
407C24—21no—
408C24—61no—
409C24C2×#6712yes—
410#41—216no—
411#41—48no—
412#41—416no—
413#41—616no—
414#41—128no—
415#41—1216no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—34no—
420C2×Q8—62no—
421C2×Q8—64no—
422C2×Q8—64no—
423C2×Q8C23×D314yes—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—34no—
429C2×D4—32no—
430C2×D4—48no—
431C2×D4—68no—
432C2×D4—62no—
433C2×D4—61no—
434C2×D4—128no—
435C2×D4D3×D414yes—
436C2×D4C2×#6714yes—
437C2×D4C23×D312yes—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—64no—
449C22×C4—68no—
450C22×C4—128no—
451C22×C4—124no—
452C22×C4C2×#6714yes—
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6712yes—
455C22×C4C2×#6711yescharacteristic
456C22×C4C23×D314yes—
457C22×C4C23×D311yescharacteristic
458Q16—616no—
459Q16—128no—
460SD16—616no—
461SD16—616no—
462D8—316no—
463C2×C8—34no—
464C2×C8—32no—
465C2×C8—32no—
466C2×C8—34no—
467C2×C8—616no—
468C2×C8—64no—
469C2×C8—62no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—32no—
475C4⋊C4—61no—
476C4⋊C4—64no—
477C4⋊C4—68no—
478C4⋊C4—64no—
479C4⋊C4C2×#6714yes—
480C4⋊C4C23×D312yes—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—68no—
484C22⋊C4—64no—
485C22⋊C4C2×#6718yes—
486C42C2×#6712yes—
487C42C23×D312yes—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—24no—
492C22×C3—48no—
493C22×C3—48no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3D4212yes—
497C22×C3C2×#13512yes—
498C22×C3#57812yes—
499C22×C3C23×D411yescharacteristic
500C2×D3—24no—
501C2×D3—48no—
502C2×D3—42no—
503C2×D3—42no—
504C3×C4—24no—
505C3×C4—22no—
506C3×C4—44no—
507C3×C4—42no—
508C3×C4D4212yes—
509C3×C4C2×#13514yescomplemented
510C3×C4C23×D411yescharacteristic
511C3×C4C2×H(5,𝔽2)11yescharacteristic
512Dic3—44no—
513Dic3—42no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—62no—
522C23—64no—
523C23—122no—
524C23—122no—
525C23C22×#6714yes—
526C23C22×#6711yescharacteristic
527Q8—24no—
528Q8—24no—
529Q8—44no—
530Q8—64no—
531Q8—64no—
532Q8—124no—
533D4—22no—
534D4—24no—
535D4—28no—
536D4—34no—
537D4—416no—
538D4—62no—
539D4—64no—
540D4—1216no—
541D4C2×D3×D414yes—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—416no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—48no—
553C2×C4—62no—
554C2×C4—64no—
555C2×C4—1216no—
556C2×C4—124no—
557C2×C4—122no—
558C2×C4—128no—
559C2×C4#95014yescomplemented
560C2×C4C2×D3×D412yes—
561C2×C4#100412yes—
562C2×C4C22×#6712yes—
563C2×C4C22×#6714yes—
564C2×C4C22×#6711yescharacteristic
565C2×C4C22×#6712yes—
566C2×C4C24×D311yescharacteristic
567C8—34no—
568C8—64no—
569C8—62no—
570C2×C3—44no—
571C2×C3—44no—
572C2×C3#328314yescomplemented
573C2×C3C2×D4211yescharacteristic
574C2×C3C22×#13511yescharacteristic
575C2×C3C2×#57811yescharacteristic
576D3—44no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—48no—
582C22—48no—
583C22—42no—
584C22—42no—
585C22—64no—
586C22—128no—
587C22—122no—
588C22—122no—
589C22D4×#6712yes—
590C22C2×#95012yes—
591C22#583812yes—
592C22C23×#6711yescharacteristic
593C4—24no—
594C4—22no—
595C4—44no—
596C4—42no—
597C4—124no—
598C4—122no—
599C4D4×#6712yes—
600C4C2×#95014yescomplemented
601C4C22×D3×D411yescharacteristic
602C4C2×#100411yescharacteristic
603C3C2×#328311yescharacteristic, complemented, Sylow
604C2—44no—
605C2—44no—
606C2—124no—
607C2#8352414yesdirect product factor
608C2C2×D4×#6711yescharacteristic
609C2C22×#95011yescharacteristic
610C2C2×#583811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#547C2
socleC23×C3C22×D4
hypercenterC2×#547D3
hypocenterC3C2×#3283
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×C3×#547C2semidirect
2. C22×#950C2semidirect
3. C2×#5551C2semidirect
4. C2×#5239C2semidirect
5. #83524C2direct (N × H)
6. C2×#5083C2semidirect
7. C2×#5092C2semidirect
8. C2×#5047C2semidirect
9. C2×#5051C2semidirect
10. C2×#5019C2semidirect
11. C2×D4×C3⋊C8C2semidirect
12. C2×C4×#213C2semidirect
17. C3×#547C22semidirect
26. C2×#950C22semidirect
36. #5551C22semidirect
41. #5239C22semidirect
45. #5092C22semidirect
46. #5083C22semidirect
47. #5051C22semidirect
48. #5047C22semidirect
49. #5019C22semidirect
50. C4×#213C22semidirect
51. D4×C3⋊C8C22semidirect
59. C2×#547D3semidirect
120. C2×#213D4semidirect
161. #547C2×D3semidirect
258. #213C2×D4semidirect
383. C2×C3×C4#135semidirect
509. C3×C4C2×#135semidirect
559. C2×C4#950semidirect
572. C2×C3#3283semidirect
600. C4C2×#950semidirect
603. C3C2×#3283semidirect
607. C2#83524direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C22×#9502
C2×#55512
C2×#52392
#835242
C2×#50832
C2×#50922
C2×#50472
C2×#50512
C2×#50192
C2×D4×C3⋊C82
C2×C4×#2132
C2×#32833