Finite Group Databasebeta

← C2×#74025 #10964705

Subgroups — C2×#74025 #10964705

Order 768.

3070 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×#67C211yesmaximal, characteristic
2C22×#892C211yesmaximal, characteristic, complemented
3C2×C3×#380C211yesmaximal, characteristic, complemented
4C2×#5212C211yesmaximal, characteristic, complemented
5C2×#4987C211yesmaximal, characteristic, complemented
6C2×#4984C211yesmaximal, characteristic, complemented
7C2×#4972C211yesmaximal, characteristic, complemented
8C2×#4961C212yesmaximal
9C2×#4945C212yesmaximal, complemented
10#74025C2116yesmaximal, direct product factor
11C2×#4675C212yesmaximal, complemented
12C2×#4654C212yesmaximal, complemented
13C2×#2485—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×#217C2212yes—
21C22×#217C2211yescharacteristic
22C2×C4×#67—28no—
23C2×C4×#67C2218yes—
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×C4⋊Dic3C2211yescharacteristic
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2212yes—
30C22×Dic3⋊C4C2211yescharacteristic
31C22×Dic3⋊C4C2212yes—
32C22×C4×Dic3C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C2×#892—28no—
35C2×#892C2218yescomplemented
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#4961C22116yes—
47#4945C22116yescomplemented
48#4675C22116yescomplemented
49#4654C22116yescomplemented
50C22×C4×D4—31no—
51C22×#117—31no—
52#2485—316no—
53C2×#397—32no—
54C2×#394—32no—
55C2×#389—31no—
56C2×#389—31no—
57C2×#388—32no—
58C2×#387—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×C4C2311yescharacteristic
68C23×C3×C4C2314yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C22×#67—24no—
73C22×#67—24no—
74C22×#67—24no—
75C22×#67C2314yes—
76C23×Dic3C2312yes—
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2312yes—
79C23×Dic3C2311yescharacteristic
80C23×Dic3C2311yescharacteristic
81C22×C4×D3—24no—
82C22×C4×D3—24no—
83C22×C4×D3D418yes—
84C22×C4×D3C2314yes—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C22⋊C4—44no—
90C2×C3×C22⋊C4C2318yes—
91C2×#217—28no—
92C2×#217—24no—
93C2×#217C2314yes—
94C2×#217C2318yes—
95C4×#67—232no—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—44no—
99C2×(C2×D3)⋊C4C2314yes—
100C2×(C2×D3)⋊C4C2318yes—
101C2×(C2×D3)⋊C4C2318yes—
102C2×C4⋊Dic3—24no—
103C2×C4⋊Dic3—24no—
104C2×C4⋊Dic3—28no—
105C2×C4⋊Dic3C2318yes—
106C2×C4⋊Dic3C2314yes—
107C2×C4⋊Dic3C2314yes—
108C2×Dic3⋊C4—24no—
109C2×Dic3⋊C4—44no—
110C2×Dic3⋊C4C2314yes—
111C2×Dic3⋊C4C2318yes—
112C2×C4×Dic3—24no—
113C2×C4×Dic3—28no—
114C2×C4×Dic3C2318yes—
115C2×C4×Dic3C2314yes—
116#892—232no—
117C3×#95C2318yes—
118C3⋊#95C2318yes—
119C3⋊#95C2318yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C23×D4—31no—
123C24×C4—31no—
124C24×C4C2×D311yescharacteristic
125C2×C4×D4—38no—
126C2×C4×D4—68no—
127C2×#117—38no—
128C2×#117—68no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—31no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4C2×D311yescharacteristic
135C22×C4⋊C4C2×D311yescharacteristic
136C22×C22⋊C4—32no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—31no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4C2×D312yes—
142C22×C42—32no—
143C22×C42—31no—
144#397—316no—
145#394—316no—
146#389—38no—
147#389—38no—
148#388—316no—
149#387—316no—
150#380—38no—
151#380—38no—
152#380C2×D318yescomplemented
153C2×#95—32no—
154C2×#95—32no—
155C2×#95—32no—
156C2×#95—32no—
157C2×#95C2×D312yes—
158C24×C3—22no—
159C24×C3—22no—
160C24×C3—22no—
161C24×C3—22no—
162C24×C3—22no—
163C24×C3—22no—
164C24×C3#4114yes—
165C24×C3C2412yes—
166C24×C3C2411yescharacteristic
167C23×D3—22no—
168C23×D3—22no—
169C23×D3—22no—
170C23×D3—22no—
171C23×D3—22no—
172C23×D3—22no—
173C23×D3C2×D414yes—
174C23×D3C2412yes—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—24no—
178C22×C3×C4—24no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4—28no—
185C22×C3×C4—28no—
186C22×C3×C4—28no—
187C22×C3×C4—416no—
188C22×C3×C4C2×D414yes—
189C22×C3×C4#4114yes—
190C22×C3×C4#4114yes—
191C22×C3×C4C2412yes—
192C22×C3×C4C2414yes—
193C22×C3×C4C2414yes—
194C22×C3×C4C2412yes—
195C2×#67—232no—
196C2×#67—28no—
197C2×#67—28no—
198C2×#67—28no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—24no—
204C22×Dic3—22no—
205C22×Dic3—24no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—22no—
212C22×Dic3C2×D414yes—
213C22×Dic3#4114yes—
214C22×Dic3C2414yes—
215C22×Dic3C2412yes—
216C22×Dic3C2414yes—
217C22×Dic3C2412yes—
218C22×Dic3C2412yes—
219C2×C4×D3—232no—
220C2×C4×D3—28no—
221C2×C4×D3—28no—
222C2×C4×D3C2×D4116yes—
223C3×C4⋊C4—28no—
224C3×C4⋊C4—28no—
225C3×C22⋊C4—48no—
226#217—216no—
227#217—28no—
228(C2×D3)⋊C4—216no—
229(C2×D3)⋊C4—28no—
230(C2×D3)⋊C4—48no—
231C4⋊Dic3—28no—
232C4⋊Dic3—28no—
233C4⋊Dic3—216no—
234Dic3⋊C4—28no—
235Dic3⋊C4—48no—
236C4×Dic3—28no—
237C4×Dic3—216no—
238C25—31no—
239C25C22×D311yescharacteristic
240C22×D4—34no—
241C22×D4—64no—
242C22×D4—64no—
243C22×D4—64no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—38no—
250C23×C4—34no—
251C23×C4—32no—
252C23×C4—31no—
253C23×C4—32no—
254C23×C4—64no—
255C23×C4—64no—
256C23×C4C22×D314yes—
257C23×C4C22×D312yes—
258C23×C4C22×D312yes—
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C4×D4—632no—
262#117—632no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—34no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—38no—
269C2×C4⋊C4—38no—
270C2×C4⋊C4—64no—
271C2×C4⋊C4—68no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—124no—
275C2×C4⋊C4C22×D314yes—
276C2×C4⋊C4C22×D314yes—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—34no—
281C2×C22⋊C4—38no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—68no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—68no—
287C2×C22⋊C4—124no—
288C2×C22⋊C4C22×D318yes—
289C2×C42—38no—
290C2×C42—34no—
291C2×C42—68no—
292C2×C42—64no—
293#95—38no—
294#95—38no—
295#95—38no—
296#95—38no—
297#95C22×D318yes—
298C23×C3—28no—
299C23×C3—28no—
300C23×C3—28no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—44no—
308C23×C3—44no—
309C23×C3—44no—
310C23×C3—44no—
311C23×C3C22×D412yes—
312C23×C3C2×#4112yes—
313C23×C3C2×#4112yes—
314C23×C3C2×#4114yes—
315C23×C3C2×#4112yes—
316C23×C3C2×#4112yes—
317C23×C3H(5,𝔽2)12yes—
318C23×C32−1+412yes—
319C23×C3C2511yescharacteristic
320C22×D3—28no—
321C22×D3—28no—
322C22×D3—28no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—44no—
330C22×D3—44no—
331C22×D3—44no—
332C22×D3—44no—
333C22×D3C22×D414yes—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—44no—
344C2×C3×C4—48no—
345C2×C3×C4—48no—
346C2×C3×C4—416no—
347C2×C3×C4#123116yes—
348C2×C3×C4C22×D414yes—
349C2×C3×C4C2×#4114yes—
350C2×C3×C4C2×#4114yes—
351#67—232no—
352C2×Dic3—28no—
353C2×Dic3—28no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—22no—
358C2×Dic3—24no—
359C2×Dic3—22no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—24no—
363C2×Dic3—24no—
364C2×Dic3—22no—
365C2×Dic3—24no—
366C2×Dic3—22no—
367C2×Dic3—48no—
368C2×Dic3—44no—
369C2×Dic3—48no—
370C2×Dic3C22×D414yes—
371C2×Dic3C2×#4114yes—
372C4×D3—232no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—34no—
380C24—32no—
381C24—62no—
382C24—62no—
383C24—62no—
384C24—62no—
385C24—62no—
386C24—62no—
387C24#23914yes—
388C24C23×D312yes—
389C24C23×D311yescharacteristic
390C2×D4—68no—
391C2×D4—68no—
392C2×D4—632no—
393C2×D4—68no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—34no—
407C22×C4—34no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—316no—
411C22×C4—34no—
412C22×C4—32no—
413C22×C4—34no—
414C22×C4—416no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—632no—
420C22×C4—68no—
421C22×C4—68no—
422C22×C4—64no—
423C22×C4—62no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—62no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—62no—
431C22×C4#23514yes—
432C22×C4D3×D414yes—
433C22×C4#23714yes—
434C22×C4C23×D312yes—
435C22×C4C23×D314yes—
436C22×C4C23×D314yes—
437C22×C4C23×D312yes—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—68no—
441C4⋊C4—616no—
442C4⋊C4—68no—
443C4⋊C4—68no—
444C4⋊C4—128no—
445C22⋊C4—48no—
446C22⋊C4—616no—
447C22⋊C4—68no—
448C22⋊C4—68no—
449C22⋊C4—616no—
450C22⋊C4—128no—
451C42—616no—
452C42—68no—
453C22×C3—24no—
454C22×C3—24no—
455C22×C3—24no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—416no—
461C22×C3C2×#12314yes—
462C22×C3#52514yes—
463C22×C3#53714yes—
464C22×C3#55214yes—
465C22×C3#55314yes—
466C22×C3#55514yes—
467C22×C3#55614yes—
468C22×C3C23×D411yescharacteristic
469C22×C3C22×#4111yescharacteristic
470C22×C3C22×#4111yescharacteristic
471C22×C3C22×#4111yescharacteristic
472C22×C3C22×#4111yescharacteristic
473C22×C3C2×H(5,𝔽2)11yescharacteristic
474C22×C3C2×2−1+411yescharacteristic
475C2×D3—24no—
476C2×D3—24no—
477C2×D3—24no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—42no—
481C2×D3—42no—
482C2×D3—416no—
483C3×C4—42no—
484C3×C4—44no—
485C3×C4—44no—
486C3×C4C2×#12318yes—
487Dic3—24no—
488Dic3—24no—
489Dic3—44no—
490Dic3—42no—
491Dic3—44no—
492C23—28no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—28no—
500C23—28no—
501C23—34no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—62no—
507C23—62no—
508C23—62no—
509C23—62no—
510C23—62no—
511C23—62no—
512C23—68no—
513C23—68no—
514C23—68no—
515C23—124no—
516C23—124no—
517C23—124no—
518C23—124no—
519C23C2×#23512yes—
520C23C2×D3×D412yes—
521C23C2×#23712yes—
522C23#100412yes—
523C23C2×#23914yes—
524C23C2×#23912yes—
525C23D3×#4112yes—
526C23#101012yes—
527C23C24×D311yescharacteristic
528D4—632no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—34no—
539C2×C4—34no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—416no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—68no—
549C2×C4—68no—
550C2×C4—632no—
551C2×C4—64no—
552C2×C4—62no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—62no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—62no—
560C2×C4—128no—
561C2×C4—124no—
562C2×C4—128no—
563C2×C4#883116yes—
564C2×C4C2×#23514yes—
565C2×C4C2×D3×D414yes—
566C2×C4C2×#23714yes—
567C2×C3—44no—
568C2×C3#248518yescomplemented
569C2×C3C22×#12311yescharacteristic
570C2×C3C2×#52511yescharacteristic
571C2×C3C2×#53711yescharacteristic
572C2×C3C2×#55211yescharacteristic
573C2×C3C2×#55311yescharacteristic
574C2×C3C2×#55511yescharacteristic
575C2×C3C2×#55611yescharacteristic
576D3—44no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—416no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—64no—
586C22—64no—
587C22—64no—
588C22—1216no—
589C22—122no—
590C22—122no—
591C22—122no—
592C22—122no—
593C22C2×#88314yes—
594C22#551414yes—
595C22#553314yes—
596C22#560814yes—
597C22#561614yes—
598C22#562714yes—
599C22#564214yes—
600C22C22×#23511yescharacteristic
601C22C22×D3×D411yescharacteristic
602C22C22×#23711yescharacteristic
603C22C2×#100411yescharacteristic
604C22C22×#23911yescharacteristic
605C22C2×D3×#4111yescharacteristic
606C22C2×#101011yescharacteristic
607C4—44no—
608C4—44no—
609C4—42no—
610C4—64no—
611C4—64no—
612C4—122no—
613C4—124no—
614C4—124no—
615C4C2×#88318yes—
616C3C2×#248511yescharacteristic, complemented, Sylow
617C2—44no—
618C2—124no—
619C2#7402518yesdirect product factor
620C2C22×#88311yescharacteristic
621C2C2×#551411yescharacteristic
622C2C2×#553311yescharacteristic
623C2C2×#561611yescharacteristic
624C2C2×#560811yescharacteristic
625C2C2×#562711yescharacteristic
626C2C2×#564211yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#380C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC2×#380D3
hypocenterC3C2×#2485
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#892C2semidirect
3. C2×C3×#380C2semidirect
4. C2×#5212C2semidirect
5. C2×#4987C2semidirect
6. C2×#4984C2semidirect
7. C2×#4972C2semidirect
9. C2×#4945C2semidirect
10. #74025C2direct (N × H)
11. C2×#4675C2semidirect
12. C2×#4654C2semidirect
35. C2×#892C22semidirect
36. C3×#380C22semidirect
38. #5212C22semidirect
43. #4987C22semidirect
44. #4984C22semidirect
45. #4972C22semidirect
47. #4945C22semidirect
48. #4675C22semidirect
49. #4654C22semidirect
61. C2×#380D3semidirect
152. #380C2×D3semidirect
568. C2×C3#2485semidirect
616. C3C2×#2485semidirect
619. C2#74025direct (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×#8922
C2×C3×#3802
C2×#52122
C2×#49872
C2×#49842
C2×#49722
C2×#49612
C2×#49452
#740252
C2×#46752
C2×#46542
C2×#24853