Finite Group Databasebeta

← C2×#73911 #10964235

Subgroups — C2×#73911 #10964235

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 644 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
3C22×#883C212yesmaximal
4C2×C3×#380C211yesmaximal, characteristic, complemented
5C2×#5212C211yesmaximal, characteristic, complemented
6C2×#4987C211yesmaximal, characteristic, complemented
7C2×#4984C211yesmaximal, characteristic, complemented
8C2×#4972C211yesmaximal, characteristic, complemented
9C2×#4952C212yesmaximal, complemented
10#73911C2116yesmaximal, direct product factor
11C2×#4664C212yesmaximal, complemented
12C2×#4657C212yesmaximal, complemented
13C2×#2486—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C23×C4×D3C2212yes—
17C23×C4×D3C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×#217C2212yes—
22C22×#217C2211yescharacteristic
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
36C2×#883—216no—
37C2×#883C22116yes—
38C3×#380C2218yescomplemented
39C2×C3×#95C2212yes—
40#5212C2218yescomplemented
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45#4987C2218yescomplemented
46#4984C2218yescomplemented
47#4972C2218yescomplemented
48#4952C22116yescomplemented
49#4664C22116yescomplemented
50#4657C22116yescomplemented
51C22×#123—32no—
52C22×C4×D4—31no—
53C23×C4⋊C4—31no—
54#2486—316no—
55C2×#398—32no—
56C2×#397—32no—
57C2×#389—31no—
58C2×#389—31no—
59C2×#381—32no—
60C2×#380—31no—
61C2×#380—31no—
62C2×#380D311yescharacteristic, complemented
63C25×C3C2311yescharacteristic
64C24×D3C2311yescharacteristic
65C23×C3×C4—24no—
66C23×C3×C4—24no—
67C23×C3×C4—24no—
68C23×C3×C4C2312yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2314yes—
71C23×C3×C4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C22×#67—24no—
74C22×#67—24no—
75C22×#67D418yes—
76C22×#67C2314yes—
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2312yes—
80C23×Dic3C2311yescharacteristic
81C23×Dic3C2311yescharacteristic
82C22×C4×D3—24no—
83C22×C4×D3—24no—
84C22×C4×D3—28no—
85C22×C4×D3—28no—
86C22×C4×D3—28no—
87C22×C4×D3D418yes—
88C22×C4×D3C2318yes—
89C22×C4×D3C2314yes—
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×#217—24no—
98C2×#217C2314yes—
99C2×#217C2318yes—
100C4×#67—232no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4—24no—
103C2×(C2×D3)⋊C4C2314yes—
104C2×(C2×D3)⋊C4C2318yes—
105C2×C4⋊Dic3—24no—
106C2×C4⋊Dic3—24no—
107C2×C4⋊Dic3—28no—
108C2×C4⋊Dic3C2318yes—
109C2×C4⋊Dic3C2314yes—
110C2×C4⋊Dic3C2314yes—
111C2×Dic3⋊C4—28no—
112C2×Dic3⋊C4—28no—
113C2×Dic3⋊C4—24no—
114C2×Dic3⋊C4C2314yes—
115C2×Dic3⋊C4C2318yes—
116C2×Dic3⋊C4C2318yes—
117C2×C4×Dic3—24no—
118C2×C4×Dic3C2314yes—
119D3×C4⋊C4—232no—
120#883—264no—
121C3×#95C2318yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C3⋊#95C2318yes—
125C3⋊#95C2318yes—
126C23×D4—31no—
127C24×C4—31no—
128C24×C4—32no—
129C24×C4C2×D311yescharacteristic
130C2×#123—316no—
131C2×#123—616no—
132C2×C4×D4—38no—
133C2×C4×D4—68no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4—32no—
138C22×C4⋊C4—31no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—38no—
141C22×C4⋊C4—68no—
142C22×C4⋊C4C2×D311yescharacteristic
143C22×C4⋊C4C2×D311yescharacteristic
144C22×C22⋊C4—31no—
145C22×C22⋊C4—32no—
146C22×C22⋊C4—31no—
147C22×C22⋊C4—32no—
148C22×C22⋊C4C2×D312yes—
149C22×C42—31no—
150#398—316no—
151#397—316no—
152#389—38no—
153#389—38no—
154#381—316no—
155#380—38no—
156#380—38no—
157#380C2×D318yescomplemented
158C2×#95—32no—
159C2×#95—32no—
160C2×#95—32no—
161C2×#95—32no—
162C2×#95C2×D312yes—
163C24×C3—22no—
164C24×C3—22no—
165C24×C3—22no—
166C24×C3—22no—
167C24×C3—22no—
168C24×C3—22no—
169C24×C3C2×D414yes—
170C24×C3C2412yes—
171C24×C3C2411yescharacteristic
172C23×D3—22no—
173C23×D3—22no—
174C23×D3—22no—
175C23×D3—22no—
176C23×D3C2×D414yes—
177C23×D3C2×D414yes—
178C23×D3#4114yes—
179C23×D3C2412yes—
180C22×C3×C4—22no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—24no—
189C22×C3×C4—28no—
190C22×C3×C4—28no—
191C22×C3×C4—28no—
192C22×C3×C4—416no—
193C22×C3×C4C2×D414yes—
194C22×C3×C4#4114yes—
195C22×C3×C4#4114yes—
196C22×C3×C4C2412yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2414yes—
199C22×C3×C4C2412yes—
200C2×#67—232no—
201C2×#67—28no—
202C2×#67—28no—
203C2×#67C2×D4116yes—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3—22no—
215C22×Dic3—24no—
216C22×Dic3—22no—
217C22×Dic3C2×D414yes—
218C22×Dic3C2×D414yes—
219C22×Dic3C2412yes—
220C22×Dic3C2414yes—
221C22×Dic3C2414yes—
222C22×Dic3C2412yes—
223C22×Dic3C2412yes—
224C2×C4×D3—232no—
225C2×C4×D3—28no—
226C2×C4×D3—28no—
227C2×C4×D3—216no—
228C2×C4×D3—216no—
229C2×C4×D3—216no—
230C2×C4×D3—432no—
231C2×C4×D3C2×D4116yes—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—28no—
234C3×C22⋊C4—216no—
235#217—216no—
236#217—28no—
237(C2×D3)⋊C4—216no—
238(C2×D3)⋊C4—28no—
239C4⋊Dic3—28no—
240C4⋊Dic3—28no—
241C4⋊Dic3—216no—
242Dic3⋊C4—216no—
243Dic3⋊C4—216no—
244Dic3⋊C4—28no—
245C4×Dic3—28no—
246C25—31no—
247C25C22×D311yescharacteristic
248C22×D4—38no—
249C22×D4—34no—
250C22×D4—64no—
251C22×D4—64no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—31no—
256C23×C4—31no—
257C23×C4—38no—
258C23×C4—34no—
259C23×C4—32no—
260C23×C4—31no—
261C23×C4—32no—
262C23×C4—38no—
263C23×C4—68no—
264C23×C4—68no—
265C23×C4—68no—
266C23×C4—64no—
267C23×C4—64no—
268C23×C4C22×D314yes—
269C23×C4C22×D312yes—
270C23×C4C22×D311yescharacteristic
271C23×C4C22×D312yes—
272C23×C4C22×D311yescharacteristic
273#123—664no—
274C4×D4—632no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—34no—
278C2×C4⋊C4—38no—
279C2×C4⋊C4—34no—
280C2×C4⋊C4—38no—
281C2×C4⋊C4—38no—
282C2×C4⋊C4—34no—
283C2×C4⋊C4—632no—
284C2×C4⋊C4—68no—
285C2×C4⋊C4—64no—
286C2×C4⋊C4—68no—
287C2×C4⋊C4—68no—
288C2×C4⋊C4—64no—
289C2×C4⋊C4—64no—
290C2×C4⋊C4C22×D314yes—
291C2×C4⋊C4C22×D314yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—34no—
294C2×C22⋊C4—38no—
295C2×C22⋊C4—34no—
296C2×C22⋊C4—38no—
297C2×C22⋊C4—64no—
298C2×C22⋊C4—68no—
299C2×C22⋊C4—64no—
300C2×C22⋊C4—68no—
301C2×C22⋊C4C22×D318yes—
302C2×C42—34no—
303C2×C42—64no—
304#95—38no—
305#95—38no—
306#95—38no—
307#95—38no—
308#95C22×D318yes—
309C23×C3—28no—
310C23×C3—28no—
311C23×C3—28no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3C22×D414yes—
323C23×C3C22×D412yes—
324C23×C3C22×D412yes—
325C23×C3C2×#4112yes—
326C23×C3C2×#4112yes—
327C23×C3C2×#4112yes—
328C23×C32−1+412yes—
329C23×C32−1+412yes—
330C23×C3C2511yescharacteristic
331C22×D3—28no—
332C22×D3—28no—
333C22×D3—28no—
334C22×D3—28no—
335C22×D3—28no—
336C22×D3—28no—
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—22no—
340C22×D3—22no—
341C22×D3#121116yes—
342C22×D3C22×D414yes—
343C22×D3C22×D414yes—
344C22×D3C2×#4114yes—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—48no—
355C2×C3×C4—44no—
356C2×C3×C4—48no—
357C2×C3×C4—416no—
358C2×C3×C4#123116yescomplemented
359C2×C3×C4C22×D414yes—
360C2×C3×C4C2×#4114yes—
361C2×C3×C4C2×#4114yes—
362#67—232no—
363C2×Dic3—28no—
364C2×Dic3—28no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377C2×Dic3—22no—
378C2×Dic3—48no—
379C2×Dic3—44no—
380C2×Dic3—48no—
381C2×Dic3C22×D414yes—
382C2×Dic3C22×D414yes—
383C4×D3—232no—
384C4×D3—432no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—34no—
392C24—34no—
393C24—34no—
394C24—32no—
395C24—62no—
396C24—62no—
397C24—62no—
398C24—62no—
399C24D3×D414yes—
400C24C23×D312yes—
401C24C23×D311yescharacteristic
402C2×D4—316no—
403C2×D4—68no—
404C2×D4—68no—
405C2×D4—632no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—34no—
419C22×C4—34no—
420C22×C4—32no—
421C22×C4—32no—
422C22×C4—316no—
423C22×C4—32no—
424C22×C4—34no—
425C22×C4—34no—
426C22×C4—416no—
427C22×C4—616no—
428C22×C4—616no—
429C22×C4—616no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—632no—
435C22×C4—68no—
436C22×C4—68no—
437C22×C4—64no—
438C22×C4—62no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—62no—
444C22×C4—64no—
445C22×C4—62no—
446C22×C4—1232no—
447C22×C4#23514yes—
448C22×C4D3×D414yes—
449C22×C4#23714yes—
450C22×C4C23×D312yes—
451C22×C4C23×D314yes—
452C22×C4C23×D314yes—
453C22×C4C23×D312yes—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—616no—
457C4⋊C4—68no—
458C4⋊C4—616no—
459C4⋊C4—616no—
460C4⋊C4—68no—
461C4⋊C4—68no—
462C22⋊C4—216no—
463C22⋊C4—68no—
464C22⋊C4—616no—
465C22⋊C4—68no—
466C22⋊C4—616no—
467C42—68no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—416no—
476C22×C3C2×#12114yes—
477C22×C3C2×#12314yes—
478C22×C3#53614yes—
479C22×C3#54714yes—
480C22×C3#54714yes—
481C22×C3#55214yes—
482C22×C3#55214yes—
483C22×C3C23×D411yescharacteristic
484C22×C3C23×D411yescharacteristic
485C22×C3C22×#4111yescharacteristic
486C22×C3C22×#4111yescharacteristic
487C22×C3C22×#4111yescharacteristic
488C22×C3C2×2−1+411yescharacteristic
489C22×C3C2×2−1+411yescharacteristic
490C2×D3—24no—
491C2×D3—24no—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—24no—
495C2×D3—24no—
496C2×D3—232no—
497C2×D3C2×#12118yes—
498C3×C4—44no—
499C3×C4—42no—
500C3×C4—44no—
501C3×C4C2×#12318yescomplemented
502Dic3—24no—
503Dic3—24no—
504Dic3—44no—
505Dic3—42no—
506Dic3—44no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—34no—
517C23—34no—
518C23—34no—
519C23—316no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—68no—
525C23—62no—
526C23—62no—
527C23—62no—
528C23—62no—
529C23—68no—
530C23—68no—
531C23—68no—
532C23—68no—
533C23—68no—
534C23C2×#23512yes—
535C23C2×D3×D414yes—
536C23C2×D3×D412yes—
537C23C2×D3×D412yes—
538C23C2×#23712yes—
539C23#100712yes—
540C23D3×#4112yes—
541C23#101012yes—
542C23C24×D311yescharacteristic
543D4—632no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—34no—
554C2×C4—34no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—416no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—68no—
564C2×C4—68no—
565C2×C4—632no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—1232no—
576C2×C4—128no—
577C2×C4—128no—
578C2×C4—124no—
579C2×C4#883116yescomplemented
580C2×C4C2×#23514yes—
581C2×C4C2×D3×D414yes—
582C2×C4C2×#23714yes—
583C2×C3—44no—
584C2×C3#248618yescomplemented
585C2×C3C22×#12111yescharacteristic
586C2×C3C22×#12311yescharacteristic
587C2×C3C2×#53611yescharacteristic
588C2×C3C2×#54711yescharacteristic
589C2×C3C2×#54711yescharacteristic
590C2×C3C2×#55211yescharacteristic
591C2×C3C2×#55211yescharacteristic
592D3—28no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—38no—
597C22—416no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—64no—
603C22—632no—
604C22—64no—
605C22—64no—
606C22—64no—
607C22—64no—
608C22—64no—
609C22C2×#88314yes—
610C22#550714yes—
611C22#552914yes—
612C22D3×#12114yes—
613C22#561114yes—
614C22#562814yes—
615C22#563914yes—
616C22C22×#23511yescharacteristic
617C22C22×D3×D411yescharacteristic
618C22C22×D3×D411yescharacteristic
619C22C22×#23711yescharacteristic
620C22C2×#100711yescharacteristic
621C22C2×D3×#4111yescharacteristic
622C22C2×#101011yescharacteristic
623C4—44no—
624C4—44no—
625C4—42no—
626C4—64no—
627C4—64no—
628C4—124no—
629C4—124no—
630C4—122no—
631C4C2×#88318yescomplemented
632C3C2×#248611yescharacteristic, complemented, Sylow
633C2—44no—
634C2—68no—
635C2#7391118yesdirect product factor
636C2C22×#88311yescharacteristic
637C2C2×#550711yescharacteristic
638C2C2×#552911yescharacteristic
639C2C2×#561111yescharacteristic
640C2C2×D3×#12111yescharacteristic
641C2C2×#562811yescharacteristic
642C2C2×#563911yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#380C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#380D3
hypocenterC3C2×#2486
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
4. C2×C3×#380C2semidirect
5. C2×#5212C2semidirect
6. C2×#4987C2semidirect
7. C2×#4984C2semidirect
8. C2×#4972C2semidirect
9. C2×#4952C2semidirect
10. #73911C2direct (N × H)
11. C2×#4664C2semidirect
12. C2×#4657C2semidirect
35. C2×D3×C4⋊C4C22semidirect
38. C3×#380C22semidirect
40. #5212C22semidirect
45. #4987C22semidirect
46. #4984C22semidirect
47. #4972C22semidirect
48. #4952C22semidirect
49. #4664C22semidirect
50. #4657C22semidirect
62. C2×#380D3semidirect
157. #380C2×D3semidirect
358. C2×C3×C4#123semidirect
501. C3×C4C2×#123semidirect
579. C2×C4#883semidirect
584. C2×C3#2486semidirect
631. C4C2×#883semidirect
632. C3C2×#2486semidirect
635. C2#73911direct (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
C22×#8832
C2×C3×#3802
C2×#52122
C2×#49872
C2×#49842
C2×#49722
C2×#49522
#739112
C2×#46642
C2×#46572
C2×#24863