Finite Group Databasebeta

← C2×#73912 #10964356

Subgroups — C2×#73912 #10964356

Order 768.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#67C211yesmaximal, characteristic
2C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#380C211yesmaximal, characteristic, complemented
4C2×#5212C211yesmaximal, characteristic, complemented
5C2×#4987C211yesmaximal, characteristic, complemented
6C2×#4984C211yesmaximal, characteristic, complemented
7C2×#4972C211yesmaximal, characteristic, complemented
8C2×#4956C212yesmaximal
9C2×#4949C212yesmaximal, complemented
10#73912C2116yesmaximal, direct product factor
11C2×#4668C212yesmaximal, complemented
12C2×#4658C212yesmaximal, complemented
13C2×#2487—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C23×C4×D3C2212yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×#217C2211yescharacteristic
22C22×#217C2212yes—
23C2×C4×#67—28no—
24C2×C4×#67C2218yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2212yes—
30C22×Dic3⋊C4C2211yescharacteristic
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C2×D3×C4⋊C4—28no—
35C2×D3×C4⋊C4C2218yescomplemented
36C3×#380C2218yescomplemented
37C2×C3×#95C2212yes—
38#5212C2218yescomplemented
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4987C2218yescomplemented
44#4984C2218yescomplemented
45#4972C2218yescomplemented
46#4956C22116yes—
47#4949C22116yescomplemented
48#4668C22116yescomplemented
49#4658C22116yescomplemented
50C22×C4×D4—31no—
51C23×C4⋊C4—31no—
52#2487—316no—
53C2×#400—32no—
54C2×#399—32no—
55C2×#389—31no—
56C2×#389—31no—
57C2×#386—32no—
58C2×#385—32no—
59C2×#380—31no—
60C2×#380—31no—
61C2×#380D311yescharacteristic, complemented
62C25×C3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C23×C3×C4—24no—
65C23×C3×C4—24no—
66C23×C3×C4—24no—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2314yes—
70C23×C3×C4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C22×#67—24no—
73C22×#67—24no—
74C22×#67Q818yes—
75C22×#67C2314yes—
76C23×Dic3C2312yes—
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2311yescharacteristic
81C22×C4×D3—24no—
82C22×C4×D3—24no—
83C22×C4×D3—28no—
84C22×C4×D3—44no—
85C22×C4×D3—44no—
86C22×C4×D3D418yes—
87C22×C4×D3C2314yes—
88C22×C4×D3C2318yes—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4C2314yes—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C22⋊C4—44no—
94C2×C3×C22⋊C4C2318yes—
95C2×#217—24no—
96C2×#217—44no—
97C2×#217C2314yes—
98C2×#217C2318yes—
99C4×#67—232no—
100C2×(C2×D3)⋊C4—24no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4C2314yes—
103C2×(C2×D3)⋊C4C2318yes—
104C2×C4⋊Dic3—24no—
105C2×C4⋊Dic3—24no—
106C2×C4⋊Dic3—44no—
107C2×C4⋊Dic3C2314yes—
108C2×C4⋊Dic3C2314yes—
109C2×C4⋊Dic3C2318yes—
110C2×Dic3⋊C4—28no—
111C2×Dic3⋊C4—24no—
112C2×Dic3⋊C4—44no—
113C2×Dic3⋊C4C2314yes—
114C2×Dic3⋊C4C2318yes—
115C2×Dic3⋊C4C2318yes—
116C2×C4×Dic3—24no—
117C2×C4×Dic3C2314yes—
118D3×C4⋊C4—232no—
119C3×#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C23×D4—31no—
125C24×C4—31no—
126C24×C4—32no—
127C24×C4C2×D311yescharacteristic
128C2×C4×D4—38no—
129C2×C4×D4—68no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—32no—
136C22×C4⋊C4—38no—
137C22×C4⋊C4—68no—
138C22×C4⋊C4C2×D311yescharacteristic
139C22×C4⋊C4C2×D311yescharacteristic
140C22×C22⋊C4—31no—
141C22×C22⋊C4—31no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—32no—
144C22×C22⋊C4C2×D312yes—
145C22×C42—31no—
146#400—316no—
147#399—316no—
148#389—38no—
149#389—38no—
150#386—316no—
151#385—316no—
152#380—38no—
153#380—38no—
154#380C2×D318yescomplemented
155C2×#95—32no—
156C2×#95—32no—
157C2×#95—32no—
158C2×#95—32no—
159C2×#95C2×D312yes—
160C24×C3—22no—
161C24×C3—22no—
162C24×C3—22no—
163C24×C3—22no—
164C24×C3—22no—
165C24×C3—22no—
166C24×C3C2×Q814yes—
167C24×C3C2412yes—
168C24×C3C2411yescharacteristic
169C23×D3—22no—
170C23×D3—22no—
171C23×D3—22no—
172C23×D3—22no—
173C23×D3C2×D414yes—
174C23×D3C2×Q814yes—
175C23×D3#4114yes—
176C23×D3C2412yes—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—28no—
187C22×C3×C4—28no—
188C22×C3×C4—28no—
189C22×C3×C4—416no—
190C22×C3×C4C2×D414yes—
191C22×C3×C4#4114yes—
192C22×C3×C4#4114yes—
193C22×C3×C4C2412yes—
194C22×C3×C4C2414yes—
195C22×C3×C4C2414yes—
196C22×C3×C4C2412yes—
197C2×#67—232no—
198C2×#67—28no—
199C2×#67—28no—
200C2×#67C2×Q8116yes—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—22no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3C2×D414yes—
215C22×Dic3C2×Q814yes—
216C22×Dic3C2414yes—
217C22×Dic3C2414yes—
218C22×Dic3C2412yes—
219C22×Dic3C2412yes—
220C22×Dic3C2412yes—
221C2×C4×D3—232no—
222C2×C4×D3—28no—
223C2×C4×D3—28no—
224C2×C4×D3—216no—
225C2×C4×D3—48no—
226C2×C4×D3—48no—
227C2×C4×D3—432no—
228C2×C4×D3C2×D4116yes—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—28no—
231C3×C22⋊C4—48no—
232#217—28no—
233#217—48no—
234(C2×D3)⋊C4—28no—
235(C2×D3)⋊C4—216no—
236C4⋊Dic3—28no—
237C4⋊Dic3—28no—
238C4⋊Dic3—48no—
239Dic3⋊C4—216no—
240Dic3⋊C4—28no—
241Dic3⋊C4—48no—
242C4×Dic3—28no—
243C25—31no—
244C25C22×D311yescharacteristic
245C22×D4—38no—
246C22×D4—34no—
247C22×D4—64no—
248C22×D4—64no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—31no—
253C23×C4—31no—
254C23×C4—38no—
255C23×C4—34no—
256C23×C4—31no—
257C23×C4—32no—
258C23×C4—32no—
259C23×C4—38no—
260C23×C4—68no—
261C23×C4—64no—
262C23×C4—64no—
263C23×C4—124no—
264C23×C4—124no—
265C23×C4C22×D314yes—
266C23×C4C22×D312yes—
267C23×C4C22×D311yescharacteristic
268C23×C4C22×D312yes—
269C23×C4C22×D311yescharacteristic
270C4×D4—632no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—34no—
274C2×C4⋊C4—38no—
275C2×C4⋊C4—34no—
276C2×C4⋊C4—34no—
277C2×C4⋊C4—38no—
278C2×C4⋊C4—38no—
279C2×C4⋊C4—632no—
280C2×C4⋊C4—68no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—124no—
285C2×C4⋊C4—124no—
286C2×C4⋊C4C22×D314yes—
287C2×C4⋊C4C22×D314yes—
288C2×C22⋊C4—34no—
289C2×C22⋊C4—34no—
290C2×C22⋊C4—38no—
291C2×C22⋊C4—38no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—68no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4—124no—
297C2×C22⋊C4C22×D318yes—
298C2×C42—34no—
299C2×C42—64no—
300#95—38no—
301#95—38no—
302#95—38no—
303#95—38no—
304#95C22×D318yes—
305C23×C3—28no—
306C23×C3—28no—
307C23×C3—28no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3C22×D412yes—
319C23×C3C22×Q814yes—
320C23×C3C22×Q812yes—
321C23×C3C2×#4112yes—
322C23×C3C2×#4112yes—
323C23×C3C2×#4112yes—
324C23×C3H(5,𝔽2)12yes—
325C23×C3H(5,𝔽2)12yes—
326C23×C3C2511yescharacteristic
327C22×D3—28no—
328C22×D3—28no—
329C22×D3—28no—
330C22×D3—28no—
331C22×D3—28no—
332C22×D3—28no—
333C22×D3—22no—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—22no—
337C22×D3C22⋊Q8116yes—
338C22×D3C22×D414yes—
339C22×D3C22×Q814yes—
340C22×D3C2×#4114yes—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—44no—
351C2×C3×C4—48no—
352C2×C3×C4—48no—
353C2×C3×C4—416no—
354C2×C3×C4#123116yes—
355C2×C3×C4C22×D414yes—
356C2×C3×C4C2×#4114yes—
357C2×C3×C4C2×#4114yes—
358#67—232no—
359C2×Dic3—28no—
360C2×Dic3—28no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—22no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—48no—
375C2×Dic3—44no—
376C2×Dic3—48no—
377C2×Dic3C22×D414yes—
378C2×Dic3C22×Q814yes—
379C4×D3—232no—
380C4×D3—432no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—34no—
388C24—34no—
389C24—34no—
390C24—32no—
391C24—62no—
392C24—62no—
393C24—62no—
394C24—62no—
395C24D3×Q814yes—
396C24C23×D312yes—
397C24C23×D311yescharacteristic
398C2×D4—316no—
399C2×D4—68no—
400C2×D4—68no—
401C2×D4—632no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—34no—
415C22×C4—34no—
416C22×C4—32no—
417C22×C4—32no—
418C22×C4—316no—
419C22×C4—34no—
420C22×C4—34no—
421C22×C4—32no—
422C22×C4—416no—
423C22×C4—616no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—62no—
428C22×C4—632no—
429C22×C4—68no—
430C22×C4—68no—
431C22×C4—64no—
432C22×C4—62no—
433C22×C4—64no—
434C22×C4—62no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—62no—
440C22×C4—128no—
441C22×C4—128no—
442C22×C4—1232no—
443C22×C4#23514yes—
444C22×C4D3×D414yes—
445C22×C4#23714yes—
446C22×C4C23×D312yes—
447C22×C4C23×D314yes—
448C22×C4C23×D314yes—
449C22×C4C23×D312yes—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—616no—
453C4⋊C4—68no—
454C4⋊C4—68no—
455C4⋊C4—68no—
456C4⋊C4—128no—
457C4⋊C4—128no—
458C22⋊C4—48no—
459C22⋊C4—68no—
460C22⋊C4—616no—
461C22⋊C4—68no—
462C22⋊C4—128no—
463C42—68no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—416no—
472C22×C3C2×C22⋊Q814yes—
473C22×C3C2×#12314yes—
474C22×C3#53514yes—
475C22×C3#55314yes—
476C22×C3#55314yes—
477C22×C3#55414yes—
478C22×C3#55414yes—
479C22×C3C23×D411yescharacteristic
480C22×C3C23×Q811yescharacteristic
481C22×C3C22×#4111yescharacteristic
482C22×C3C22×#4111yescharacteristic
483C22×C3C22×#4111yescharacteristic
484C22×C3C2×H(5,𝔽2)11yescharacteristic
485C22×C3C2×H(5,𝔽2)11yescharacteristic
486C2×D3—24no—
487C2×D3—24no—
488C2×D3—24no—
489C2×D3—24no—
490C2×D3—24no—
491C2×D3—24no—
492C2×D3—232no—
493C2×D3C2×C22⋊Q818yes—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4—44no—
497C3×C4C2×#12318yes—
498Dic3—24no—
499Dic3—24no—
500Dic3—44no—
501Dic3—42no—
502Dic3—44no—
503C23—28no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—28no—
511C23—28no—
512C23—34no—
513C23—34no—
514C23—34no—
515C23—316no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—68no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—68no—
526C23—68no—
527C23—68no—
528C23—68no—
529C23—68no—
530C23C2×#23512yes—
531C23C2×D3×D412yes—
532C23C2×#23712yes—
533C23#100412yes—
534C23C2×D3×Q814yes—
535C23C2×D3×Q812yes—
536C23D3×#4112yes—
537C23#100912yes—
538C23C24×D311yescharacteristic
539D4—632no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—34no—
550C2×C4—34no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—416no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—68no—
560C2×C4—68no—
561C2×C4—632no—
562C2×C4—64no—
563C2×C4—62no—
564C2×C4—64no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—62no—
571C2×C4—1232no—
572C2×C4—128no—
573C2×C4—128no—
574C2×C4—124no—
575C2×C4#883116yes—
576C2×C4C2×#23514yes—
577C2×C4C2×D3×D414yes—
578C2×C4C2×#23714yes—
579C2×C3—44no—
580C2×C3#248718yescomplemented
581C2×C3C22×C22⋊Q811yescharacteristic
582C2×C3C22×#12311yescharacteristic
583C2×C3C2×#53511yescharacteristic
584C2×C3C2×#55311yescharacteristic
585C2×C3C2×#55311yescharacteristic
586C2×C3C2×#55411yescharacteristic
587C2×C3C2×#55411yescharacteristic
588D3—28no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—38no—
593C22—42no—
594C22—416no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—64no—
599C22—632no—
600C22—64no—
601C22—64no—
602C22—64no—
603C22—64no—
604C22—64no—
605C22C2×#88314yes—
606C22#551114yes—
607C22#553014yes—
608C22#560714yes—
609C22#561414yes—
610C22D3×C22⋊Q814yes—
611C22#563514yes—
612C22C22×#23511yescharacteristic
613C22C22×D3×D411yescharacteristic
614C22C22×#23711yescharacteristic
615C22C2×#100411yescharacteristic
616C22C22×D3×Q811yescharacteristic
617C22C2×D3×#4111yescharacteristic
618C22C2×#100911yescharacteristic
619C4—44no—
620C4—44no—
621C4—42no—
622C4—64no—
623C4—64no—
624C4—124no—
625C4—122no—
626C4—124no—
627C4C2×#88318yes—
628C3C2×#248711yescharacteristic, complemented, Sylow
629C2—44no—
630C2—68no—
631C2#7391218yesdirect product factor
632C2C22×#88311yescharacteristic
633C2C2×#551111yescharacteristic
634C2C2×#553011yescharacteristic
635C2C2×#560711yescharacteristic
636C2C2×#561411yescharacteristic
637C2C2×#563511yescharacteristic
638C2C2×D3×C22⋊Q811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#380C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#380D3
hypocenterC3C2×#2487
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×D3×C4⋊C4C2semidirect
3. C2×C3×#380C2semidirect
4. C2×#5212C2semidirect
5. C2×#4987C2semidirect
6. C2×#4984C2semidirect
7. C2×#4972C2semidirect
9. C2×#4949C2semidirect
10. #73912C2direct (N × H)
11. C2×#4668C2semidirect
12. C2×#4658C2semidirect
35. C2×D3×C4⋊C4C22semidirect
36. C3×#380C22semidirect
38. #5212C22semidirect
43. #4987C22semidirect
44. #4984C22semidirect
45. #4972C22semidirect
47. #4949C22semidirect
48. #4668C22semidirect
49. #4658C22semidirect
61. C2×#380D3semidirect
154. #380C2×D3semidirect
580. C2×C3#2487semidirect
628. C3C2×#2487semidirect
631. C2#73912direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#672
C22×D3×C4⋊C42
C2×C3×#3802
C2×#52122
C2×#49872
C2×#49842
C2×#49722
C2×#49562
C2×#49492
#739122
C2×#46682
C2×#46582
C2×#24873