Finite Group Databasebeta

← C2×#74001 #11001121

Subgroups — C2×#74001 #11001121

Order 768.

3790 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Dic3C211yesmaximal, characteristic
2C22×#888C211yesmaximal, characteristic, complemented
3C22×#878C212yesmaximal
4C2×C3×#380C211yesmaximal, characteristic
5C2×#5212C211yesmaximal, characteristic
6C2×#4984C211yesmaximal, characteristic, complemented
7C2×#4972C211yesmaximal, characteristic, complemented
8C2×#4975C211yesmaximal, characteristic, complemented
9C2×#4942C212yesmaximal
10#74001C2116yesmaximal, direct product factor
11C2×#4646C212yesmaximal, complemented
12C2×#4647C212yesmaximal, complemented
13C2×#2857—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C24×Dic3C2211yescharacteristic
17C23×Dic6C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×#217C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2212yes—
27C22×Dic3⋊C4C2212yes—
28C22×Dic3⋊C4C2212yes—
29C22×C4×Dic3—28no—
30C22×C4×Dic3C2218yes—
31C22×C4×Dic3C2211yescharacteristic
32C22×C4×Dic3C2211yescharacteristic
33C22×C4×Dic3C2211yescharacteristic
34C22×C4×Dic3C2211yescharacteristic
35C2×#888—28no—
36C2×#888C2218yescomplemented
37C2×#878—216no—
38C2×#878C22116yes—
39C3×#380C2218yes—
40C2×C3×#95C2212yes—
41#5212C2218yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45#4984C2218yescomplemented
46#4975C2218yescomplemented
47#4972C2218yescomplemented
48#4942C22116yes—
49#4647C22116yescomplemented
50#4646C22116yescomplemented
51C22×C4⋊Q8—31no—
52C22×C22⋊Q8—32no—
53C23×C42—31no—
54#2857—316no—
55C2×#391—32no—
56C2×#389—32no—
57C2×#389—31no—
58C2×#389—31no—
59C2×#384—31no—
60C2×#380—31no—
61C2×#380—32no—
62C2×#380D311yescharacteristic
63C25×C3C2311yescharacteristic
64C23×C3×C4—24no—
65C23×C3×C4—24no—
66C23×C3×C4Q818yescomplemented
67C23×C3×C4C2312yes—
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2314yes—
71C23×C3×C4C2311yescharacteristic
72C23×Dic3—24no—
73C23×Dic3—24no—
74C23×Dic3—24no—
75C23×Dic3—24no—
76C23×Dic3D418yes—
77C23×Dic3Q818yescomplemented
78C23×Dic3C2312yes—
79C23×Dic3C2312yes—
80C23×Dic3C2314yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C2311yescharacteristic
83C23×Dic3C2314yes—
84C23×Dic3C2311yescharacteristic
85C23×Dic3C2311yescharacteristic
86C22×Dic6—28no—
87C22×Dic6—28no—
88C22×Dic6—28no—
89C22×Dic6C2318yes—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4—24no—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C4⋊C4C2314yes—
94C2×C3×C22⋊C4—28no—
95C2×C3×C22⋊C4C2318yes—
96C2×#217—28no—
97C2×#217C2318yes—
98C2×C4⋊Dic3—24no—
99C2×C4⋊Dic3—24no—
100C2×C4⋊Dic3—28no—
101C2×C4⋊Dic3C2318yes—
102C2×C4⋊Dic3C2314yes—
103C2×C4⋊Dic3C2314yes—
104C2×Dic3⋊C4—28no—
105C2×Dic3⋊C4—28no—
106C2×Dic3⋊C4—28no—
107C2×Dic3⋊C4—28no—
108C2×Dic3⋊C4C2318yes—
109C2×Dic3⋊C4C2318yes—
110C2×Dic3⋊C4C2318yes—
111C2×Dic3⋊C4C2318yes—
112C2×C4×Dic3—24no—
113C2×C4×Dic3—24no—
114C2×C4×Dic3—24no—
115C2×C4×Dic3—232no—
116C2×C4×Dic3D418yes—
117C2×C4×Dic3C2314yes—
118C2×C4×Dic3C2314yes—
119C2×C4×Dic3C2314yes—
120C2×C4×Dic3C2314yes—
121#888—232no—
122#878—264no—
123C3×#95C2318yes—
124C3⋊#95C2318yes—
125C3⋊#95C2318yes—
126C3⋊#95C2318yes—
127C23×Q8—32no—
128C24×C4—31no—
129C24×C4—31no—
130C24×C4C2×D311yescharacteristic
131C2×C4⋊Q8—38no—
132C2×C4⋊Q8—68no—
133C2×C22⋊Q8—316no—
134C2×C22⋊Q8—616no—
135C22×C4⋊C4—32no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4—32no—
138C22×C4⋊C4—32no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—32no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4C2×D311yescharacteristic
143C22×C4⋊C4C2×D311yescharacteristic
144C22×C22⋊C4—32no—
145C22×C22⋊C4C2×D312yes—
146C22×C42—31no—
147C22×C42—38no—
148C22×C42—31no—
149C22×C42—31no—
150C22×C42—31no—
151C22×C42—68no—
152#391—316no—
153#389—38no—
154#389—38no—
155#389—316no—
156#384—38no—
157#380—38no—
158#380—316no—
159#380C2×D318yes—
160C2×#95—32no—
161C2×#95—32no—
162C2×#95—32no—
163C2×#95C2×D312yes—
164C24×C3—22no—
165C24×C3—22no—
166C24×C3—22no—
167C24×C3—22no—
168C24×C3C2×D414yes—
169C24×C3C2×Q814yes—
170C24×C3C2×Q814yes—
171C24×C3C2412yes—
172C24×C3C2411yescharacteristic
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—232no—
176C22×C3×C4—28no—
177C22×C3×C4—28no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2×Q814yes—
186C22×C3×C4C2×Q814yes—
187C22×C3×C4C2×Q8116yescomplemented
188C22×C3×C4#4114yes—
189C22×C3×C4C2414yes—
190C22×C3×C4C2414yes—
191C22×C3×C4C2412yes—
192C22×C3×C4C2412yes—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—232no—
198C22×Dic3—232no—
199C22×Dic3—28no—
200C22×Dic3—28no—
201C22×Dic3—28no—
202C22×Dic3—28no—
203C22×Dic3—24no—
204C22×Dic3—24no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3C2×D4116yes—
210C22×Dic3C2×D414yes—
211C22×Dic3C2×D414yes—
212C22×Dic3C2×D414yes—
213C22×Dic3C2×D414yes—
214C22×Dic3C2×Q814yes—
215C22×Dic3C2×Q814yes—
216C22×Dic3C2×Q8116yescomplemented
217C22×Dic3#4114yes—
218C22×Dic3#4114yes—
219C22×Dic3C2414yes—
220C22×Dic3C2414yes—
221C22×Dic3C2412yes—
222C22×Dic3C2412yes—
223C22×Dic3C2412yes—
224C22×Dic3C2412yes—
225C2×Dic6—216no—
226C2×Dic6—216no—
227C2×Dic6—216no—
228C2×Dic6—432no—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—28no—
231C3×C22⋊C4—216no—
232#217—216no—
233C4⋊Dic3—28no—
234C4⋊Dic3—28no—
235C4⋊Dic3—216no—
236Dic3⋊C4—216no—
237Dic3⋊C4—216no—
238Dic3⋊C4—216no—
239Dic3⋊C4—216no—
240C4×Dic3—28no—
241C4×Dic3—28no—
242C4×Dic3—28no—
243C4×Dic3C2×D4116yes—
244C25C22×D311yescharacteristic
245C22×Q8—38no—
246C22×Q8—68no—
247C22×Q8—68no—
248C22×Q8—68no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—31no—
252C23×C4—31no—
253C23×C4—38no—
254C23×C4—38no—
255C23×C4—34no—
256C23×C4—34no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—32no—
260C23×C4—32no—
261C23×C4—64no—
262C23×C4—64no—
263C23×C4—64no—
264C23×C4—64no—
265C23×C4Dic618yescomplemented
266C23×C4C22×D312yes—
267C23×C4C22×D312yes—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D314yes—
270C23×C4C22×D311yescharacteristic
271C4⋊Q8—632no—
272C22⋊Q8—664no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—38no—
276C2×C4⋊C4—38no—
277C2×C4⋊C4—38no—
278C2×C4⋊C4—38no—
279C2×C4⋊C4—34no—
280C2×C4⋊C4—38no—
281C2×C4⋊C4—34no—
282C2×C4⋊C4—68no—
283C2×C4⋊C4—68no—
284C2×C4⋊C4—68no—
285C2×C4⋊C4—68no—
286C2×C4⋊C4—68no—
287C2×C4⋊C4—64no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4C22×D314yes—
290C2×C4⋊C4C22×D314yes—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—38no—
293C2×C22⋊C4—68no—
294C2×C22⋊C4C22×D318yes—
295C2×C42—38no—
296C2×C42—34no—
297C2×C42—34no—
298C2×C42—34no—
299C2×C42—34no—
300C2×C42—632no—
301C2×C42—64no—
302C2×C42—64no—
303C2×C42—64no—
304#95—38no—
305#95—38no—
306#95—38no—
307#95C22×D318yes—
308C23×C3—28no—
309C23×C3—28no—
310C23×C3—28no—
311C23×C3—28no—
312C23×C3—28no—
313C23×C3—28no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3C4⋊Q8116yescomplemented
319C23×C3C22×D414yes—
320C23×C3C22×D412yes—
321C23×C3C22×D412yes—
322C23×C3C22×Q814yes—
323C23×C3C22×Q812yes—
324C23×C3C22×Q814yes—
325C23×C3C22×Q812yes—
326C23×C3C2×#4112yes—
327C23×C3C2×#4112yes—
328C23×C3C2×#4112yes—
329C23×C3C2511yescharacteristic
330C2×C3×C4—28no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—232no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—48no—
341C2×C3×C4—48no—
342C2×C3×C4C22⋊Q8116yescomplemented
343C2×C3×C4C22×D414yes—
344C2×C3×C4C22×Q814yes—
345C2×C3×C4C22×Q814yes—
346C2×C3×C4C2×#4114yes—
347C2×Dic3—28no—
348C2×Dic3—28no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—232no—
354C2×Dic3—232no—
355C2×Dic3—24no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—48no—
362C2×Dic3—48no—
363C2×Dic3#121116yes—
364C2×Dic3C22⋊Q8116yescomplemented
365C2×Dic3C22×D414yes—
366C2×Dic3C22×D414yes—
367C2×Dic3C22×D414yes—
368C2×Dic3C22×D414yes—
369C2×Dic3C22×Q814yes—
370C2×Dic3C22×Q814yes—
371C2×Dic3C2×#4114yes—
372C2×Dic3C2×#4114yes—
373Dic6—432no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24C2×Dic614yes—
379C24D3×D414yes—
380C24D3×Q814yes—
381C24C23×D312yes—
382C24C23×D311yescharacteristic
383C2×Q8—616no—
384C2×Q8—616no—
385C2×Q8—616no—
386C2×Q8—1232no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—232no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—34no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—34no—
402C22×C4—34no—
403C22×C4—34no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—34no—
407C22×C4—316no—
408C22×C4—34no—
409C22×C4—316no—
410C22×C4—32no—
411C22×C4—32no—
412C22×C4—34no—
413C22×C4—34no—
414C22×C4—68no—
415C22×C4—68no—
416C22×C4—68no—
417C22×C4—68no—
418C22×C4—632no—
419C22×C4—632no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4C2×Dic614yes—
431C22×C4C2×Dic614yes—
432C22×C4C2×Dic6116yescomplemented
433C22×C4D3×D414yes—
434C22×C4#23714yes—
435C22×C4C23×D314yes—
436C22×C4C23×D314yes—
437C22×C4C23×D312yes—
438C22×C4C23×D312yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—616no—
442C4⋊C4—616no—
443C4⋊C4—616no—
444C4⋊C4—616no—
445C4⋊C4—616no—
446C4⋊C4—68no—
447C4⋊C4—68no—
448C22⋊C4—216no—
449C22⋊C4—616no—
450C42—316no—
451C42—68no—
452C42—68no—
453C42—68no—
454C22×C3—232no—
455C22×C3—24no—
456C22×C3—24no—
457C22×C3—24no—
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—24no—
461C22×C3C2×#12114yes—
462C22×C3C2×C22⋊Q814yes—
463C22×C3C2×C22⋊Q814yes—
464C22×C3C2×C4⋊Q814yes—
465C22×C3C2×C4⋊Q818yescomplemented
466C22×C3#53214yes—
467C22×C3#53314yes—
468C22×C3#53314yes—
469C22×C3C23×D411yescharacteristic
470C22×C3C23×D411yescharacteristic
471C22×C3C23×Q811yescharacteristic
472C22×C3C23×Q811yescharacteristic
473C22×C3C22×#4111yescharacteristic
474C22×C3C22×#4111yescharacteristic
475C22×C3C22×#4111yescharacteristic
476C3×C4—24no—
477C3×C4—44no—
478C3×C4—44no—
479C3×C4C2×C22⋊Q818yescomplemented
480Dic3—24no—
481Dic3—24no—
482Dic3—44no—
483Dic3—44no—
484Dic3C2×#12118yes—
485Dic3C2×C22⋊Q818yescomplemented
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23#888116yescomplemented
497C23C22×Dic614yes—
498C23C22×Dic612yes—
499C23C2×D3×D414yes—
500C23C2×D3×D412yes—
501C23C2×D3×D412yes—
502C23C2×#23712yes—
503C23C2×D3×Q814yes—
504C23C2×D3×Q812yes—
505C23D3×#4112yes—
506C23D3×#4112yes—
507C23C24×D311yescharacteristic
508Q8—1232no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—232no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—34no—
520C2×C4—34no—
521C2×C4—34no—
522C2×C4—34no—
523C2×C4—34no—
524C2×C4—34no—
525C2×C4—316no—
526C2×C4—316no—
527C2×C4—34no—
528C2×C4—34no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—632no—
532C2×C4—632no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—68no—
538C2×C4—68no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—128no—
546C2×C4—128no—
547C2×C4#878116yescomplemented
548C2×C4C22×Dic614yes—
549C2×C4C22×Dic614yes—
550C2×C4C2×D3×D414yes—
551C2×C4C2×#23714yes—
552C2×C3—28no—
553C2×C3#285718yescomplemented
554C2×C3C22×#12111yescharacteristic
555C2×C3C22×C22⋊Q811yescharacteristic
556C2×C3C22×C22⋊Q811yescharacteristic
557C2×C3C22×C4⋊Q811yescharacteristic
558C2×C3C2×#53211yescharacteristic
559C2×C3C2×#53311yescharacteristic
560C2×C3C2×#53311yescharacteristic
561C22—232no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22C2×#87814yes—
569C22C2×#88814yes—
570C22C2×#88818yescomplemented
571C22#551714yes—
572C22#559614yes—
573C22D3×#12114yes—
574C22#562214yes—
575C22D3×C22⋊Q814yes—
576C22C23×Dic611yescharacteristic
577C22C22×D3×D411yescharacteristic
578C22C22×D3×D411yescharacteristic
579C22C22×#23711yescharacteristic
580C22C22×D3×Q811yescharacteristic
581C22C2×D3×#4111yescharacteristic
582C22C2×D3×#4111yescharacteristic
583C4—24no—
584C4—38no—
585C4—38no—
586C4—44no—
587C4—44no—
588C4—64no—
589C4—64no—
590C4—124no—
591C4—124no—
592C4C2×#87818yescomplemented
593C3C2×#285711yescharacteristic, complemented, Sylow
594C2—28no—
595C2#7400118yesdirect product factor
596C2C22×#87811yescharacteristic
597C2C22×#88811yescharacteristic
598C2C2×#551711yescharacteristic
599C2C2×#559611yescharacteristic
600C2C2×D3×#12111yescharacteristic
601C2C2×#562211yescharacteristic
602C2C2×D3×C22⋊Q811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#380C2
socleC24×C3C24
prime radicalC25×C3C23
hypercenterC2×#380D3
hypocenterC3C2×#2857
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#888C2semidirect
6. C2×#4984C2semidirect
7. C2×#4972C2semidirect
8. C2×#4975C2semidirect
10. #74001C2direct (N × H)
11. C2×#4646C2semidirect
12. C2×#4647C2semidirect
36. C2×#888C22semidirect
45. #4984C22semidirect
46. #4975C22semidirect
47. #4972C22semidirect
49. #4647C22semidirect
50. #4646C22semidirect
66. C23×C3×C4Q8semidirect
77. C23×Dic3Q8semidirect
187. C22×C3×C4C2×Q8semidirect
216. C22×Dic3C2×Q8semidirect
265. C23×C4Dic6semidirect
318. C23×C3C4⋊Q8semidirect
342. C2×C3×C4C22⋊Q8semidirect
364. C2×Dic3C22⋊Q8semidirect
432. C22×C4C2×Dic6semidirect
465. C22×C3C2×C4⋊Q8semidirect
479. C3×C4C2×C22⋊Q8semidirect
485. Dic3C2×C22⋊Q8semidirect
496. C23#888semidirect
547. C2×C4#878semidirect
553. C2×C3#2857semidirect
570. C22C2×#888semidirect
592. C4C2×#878semidirect
593. C3C2×#2857semidirect
595. C2#74001direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×Dic32
C22×#8882
C22×#8782
C2×C3×#3802
C2×#52122
C2×#49842
C2×#49722
C2×#49752
C2×#49422
#740012
C2×#46462
C2×#46472
C2×#28573