Finite Group Databasebeta

← C2×#73506 #10964633

Subgroups — C2×#73506 #10964633

Order 768.

3158 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#930C211yesmaximal, characteristic
2C22×#870C211yesmaximal, characteristic, complemented
3C2×C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×#5212C211yesmaximal, characteristic, complemented
5C2×#4959C212yesmaximal, complemented
6C2×#4957C212yesmaximal, complemented
7C2×#4939C211yesmaximal, characteristic, complemented
8C2×#4939C211yesmaximal, characteristic, complemented
9C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#73506C2116yesmaximal, direct product factor
11C2×#4675C212yesmaximal, complemented
12C2×#4669C212yesmaximal, complemented
13C2×#2490—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2212yes—
16C23×D12C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×#217C2212yes—
22C2×#930—28no—
23C2×#930C2218yes—
24C22×(C2×D3)⋊C4C2212yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×(C2×D3)⋊C4C2212yes—
29C22×C4⋊Dic3C2211yescharacteristic
30C22×C4⋊Dic3C2211yescharacteristic
31C22×C4⋊Dic3C2211yescharacteristic
32C22×Dic3⋊C4C2212yes—
33C22×Dic3⋊C4C2212yes—
34C22×C4×Dic3C2212yes—
35C2×#870—28no—
36C2×#870C2218yescomplemented
37C3×C4×C22⋊C4C2218yescomplemented
38C2×C3×#95C2211yescharacteristic
39C2×C3×#95C2211yescharacteristic
40#5212C2218yescomplemented
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2212yes—
43#4959C22116yescomplemented
44#4957C22116yescomplemented
45#4939C2218yescomplemented
46#4939C2218yescomplemented
47C4×C4⋊Dic3C2218yescomplemented
48#4675C22116yescomplemented
49#4669C22116yescomplemented
50C22×#125—31no—
51C22×#121—31no—
52#2490—316no—
53C2×#397—32no—
54C2×#394—32no—
55C2×#390—31no—
56C2×#390—31no—
57C2×#390—32no—
58C2×#388—32no—
59C2×#380—31no—
60C2×C4×C4⋊C4—31no—
61C2×C4×C22⋊C4D311yescharacteristic, complemented
62C25×C3C2311yescharacteristic
63C24×D3C2312yes—
64C23×C3×C4—24no—
65C23×C3×C4—24no—
66C23×C3×C4—24no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2314yes—
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C22×#67—28no—
75C22×#67—28no—
76C22×#67—44no—
77C22×#67C2318yes—
78C23×Dic3C2312yes—
79C23×Dic3C2312yes—
80C23×Dic3C2312yes—
81C22×D12—24no—
82C22×D12—24no—
83C22×D12—24no—
84C22×D12C2314yes—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4C2314yes—
88C2×C3×C22⋊C4C2314yes—
89C2×C3×C42—24no—
90C2×C3×C42—24no—
91C2×C3×C42C2314yes—
92C2×C3×C42C2314yes—
93C2×#217—44no—
94C2×#217C2318yes—
95#930—232no—
96C2×(C2×D3)⋊C4—28no—
97C2×(C2×D3)⋊C4—28no—
98C2×(C2×D3)⋊C4—44no—
99C2×(C2×D3)⋊C4—44no—
100C2×(C2×D3)⋊C4—44no—
101C2×(C2×D3)⋊C4C2318yes—
102C2×(C2×D3)⋊C4C2318yes—
103C2×(C2×D3)⋊C4C2318yes—
104C2×(C2×D3)⋊C4C2318yes—
105C2×(C2×D3)⋊C4C2318yes—
106C2×C4⋊Dic3—24no—
107C2×C4⋊Dic3—24no—
108C2×C4⋊Dic3D418yes—
109C2×C4⋊Dic3C2314yes—
110C2×C4⋊Dic3C2314yes—
111C2×C4⋊Dic3C2314yes—
112C2×Dic3⋊C4—28no—
113C2×Dic3⋊C4—28no—
114C2×Dic3⋊C4C2318yes—
115C2×Dic3⋊C4C2318yes—
116C2×C4×Dic3—44no—
117C2×C4×Dic3C2318yes—
118#870—232no—
119C3×#95C2314yes—
120C3×#95C2314yes—
121C3⋊#95C2318yes—
122C3⋊#95C2318yes—
123C23×D4—32no—
124C23×D4—31no—
125C24×C4C2×D311yescharacteristic
126C2×#125—38no—
127C2×#125—68no—
128C2×#121—38no—
129C2×#121—68no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4—31no—
134C22×C4⋊C4—31no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—32no—
139C22×C22⋊C4—32no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4C2×D311yescharacteristic
142C22×C22⋊C4C2×D311yescharacteristic
143C22×C42—32no—
144C22×C42C2×D311yescharacteristic
145C22×C42C2×D311yescharacteristic
146#397—316no—
147#394—316no—
148#390—316no—
149#390—38no—
150#390—38no—
151#388—316no—
152#380—38no—
153C4×C4⋊C4—38no—
154C4×C22⋊C4C2×D318yescomplemented
155C2×#95—32no—
156C2×#95—32no—
157C2×#95C2×D311yescharacteristic
158C2×#95C2×D311yescharacteristic
159C24×C3—22no—
160C24×C3—22no—
161C24×C3—22no—
162C24×C3—22no—
163C24×C3—22no—
164C24×C3—22no—
165C24×C3#4114yes—
166C24×C3C2412yes—
167C24×C3C2411yescharacteristic
168C23×D3—24no—
169C23×D3—24no—
170C23×D3—24no—
171C23×D3—24no—
172C23×D3—24no—
173C23×D3—24no—
174C23×D3—24no—
175C23×D3C2414yes—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—28no—
191C22×C3×C4—28no—
192C22×C3×C4—28no—
193C22×C3×C4—416no—
194C22×C3×C4C2×D414yes—
195C22×C3×C4#4114yes—
196C22×C3×C4#4114yes—
197C22×C3×C4#4114yes—
198C22×C3×C4C2412yes—
199C22×C3×C4C2412yes—
200C22×C3×C4C2412yes—
201C22×C3×C4C2412yes—
202C22×C3×C4C2412yes—
203C22×C3×C4C2412yes—
204C2×#67—216no—
205C2×#67—216no—
206C2×#67—48no—
207C2×#67—432no—
208C22×Dic3—24no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—24no—
216C22×Dic3C2×D418yes—
217C22×Dic3C2414yes—
218C22×Dic3C2414yes—
219C22×Dic3C2414yes—
220C2×D12—28no—
221C2×D12—28no—
222C2×D12—28no—
223C2×D12—416no—
224C3×C22⋊C4—28no—
225C3×C22⋊C4—28no—
226C3×C42—28no—
227C3×C42—28no—
228#217—48no—
229(C2×D3)⋊C4—216no—
230(C2×D3)⋊C4—216no—
231(C2×D3)⋊C4—48no—
232(C2×D3)⋊C4—48no—
233(C2×D3)⋊C4—48no—
234C4⋊Dic3—28no—
235C4⋊Dic3—28no—
236C4⋊Dic3C2×D4116yes—
237Dic3⋊C4—216no—
238Dic3⋊C4—216no—
239C4×Dic3—48no—
240C25—32no—
241C25C22×D311yescharacteristic
242C22×D4—38no—
243C22×D4—34no—
244C22×D4—68no—
245C22×D4—68no—
246C22×D4—64no—
247C22×D4—64no—
248C22×D4—64no—
249C22×D4—124no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—32no—
254C23×C4—32no—
255C23×C4—32no—
256C23×C4C22×D314yes—
257C23×C4C22×D311yescharacteristic
258C23×C4C22×D311yescharacteristic
259C23×C4C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261C23×C4C22×D311yescharacteristic
262C23×C4C22×D311yescharacteristic
263#125—632no—
264#121—632no—
265C2×C4⋊C4—38no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—38no—
269C2×C4⋊C4—38no—
270C2×C4⋊C4—34no—
271C2×C4⋊C4—68no—
272C2×C4⋊C4—68no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—38no—
280C2×C22⋊C4—38no—
281C2×C22⋊C4—38no—
282C2×C22⋊C4—38no—
283C2×C22⋊C4—68no—
284C2×C22⋊C4—68no—
285C2×C22⋊C4—124no—
286C2×C22⋊C4—124no—
287C2×C22⋊C4—124no—
288C2×C22⋊C4—124no—
289C2×C22⋊C4C22×D314yes—
290C2×C22⋊C4C22×D314yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—38no—
294C2×C42—124no—
295C2×C42C22×D314yes—
296C2×C42C22×D314yes—
297#95—38no—
298#95—38no—
299#95C22×D314yes—
300#95C22×D314yes—
301C23×C3—28no—
302C23×C3—28no—
303C23×C3—28no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—44no—
311C23×C3—44no—
312C23×C3—44no—
313C23×C3—44no—
314C23×C3C22×D412yes—
315C23×C3C2×#4114yes—
316C23×C3C2×#4112yes—
317C23×C3C2×#4112yes—
318C23×C3C2×#4112yes—
319C23×C3C2×#4112yes—
320C23×C3H(5,𝔽2)12yes—
321C23×C3H(5,𝔽2)12yes—
322C23×C3C2511yescharacteristic
323C22×D3—24no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—24no—
329C22×D3—24no—
330C22×D3—48no—
331C22×D3—48no—
332C22×D3—48no—
333C22×D3—48no—
334C22×D3—48no—
335C22×D3—48no—
336C22×D3—48no—
337C2×C3×C4—28no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—44no—
353C2×C3×C4—44no—
354C2×C3×C4—44no—
355C2×C3×C4—44no—
356C2×C3×C4—416no—
357C2×C3×C4#124116yes—
358C2×C3×C4C22×D414yes—
359C2×C3×C4C2×#4114yes—
360C2×C3×C4C2×#4114yes—
361C2×C3×C4C2×#4114yes—
362#67—432no—
363C2×Dic3—216no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—48no—
373C2×Dic3—48no—
374C2×Dic3C22×D418yes—
375D12—416no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—34no—
383C24—64no—
384C24—64no—
385C24—64no—
386C24—64no—
387C24—64no—
388C24—64no—
389C24—64no—
390C24#23514yes—
391C24C23×D312yes—
392C24C23×D311yescharacteristic
393C2×D4—616no—
394C2×D4—616no—
395C2×D4—68no—
396C2×D4—68no—
397C2×D4—68no—
398C2×D4—128no—
399C2×D4—1232no—
400C2×D4—1216no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—38no—
419C22×C4—34no—
420C22×C4—34no—
421C22×C4—34no—
422C22×C4—416no—
423C22×C4—64no—
424C22×C4—64no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—64no—
428C22×C4—64no—
429C22×C4—64no—
430C22×C4—64no—
431C22×C4#23514yes—
432C22×C4#23514yes—
433C22×C4D3×D414yes—
434C22×C4#23714yes—
435C22×C4C23×D312yes—
436C22×C4C23×D312yes—
437C22×C4C23×D312yes—
438C22×C4C23×D312yes—
439C22×C4C23×D312yes—
440C22×C4C23×D312yes—
441C4⋊C4—316no—
442C4⋊C4—616no—
443C4⋊C4—616no—
444C4⋊C4—68no—
445C4⋊C4—68no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—616no—
449C22⋊C4—616no—
450C22⋊C4—128no—
451C22⋊C4—128no—
452C22⋊C4—128no—
453C22⋊C4—128no—
454C42—28no—
455C42—28no—
456C42—128no—
457C22×C3—24no—
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—416no—
465C22×C3C2×#12414yes—
466C22×C3#52914yes—
467C22×C3#54014yes—
468C22×C3#54614yes—
469C22×C3#55014yes—
470C22×C3#55314yes—
471C22×C3#55314yes—
472C22×C3C23×D411yescharacteristic
473C22×C3C22×#4111yescharacteristic
474C22×C3C22×#4111yescharacteristic
475C22×C3C22×#4111yescharacteristic
476C22×C3C22×#4111yescharacteristic
477C22×C3C2×H(5,𝔽2)11yescharacteristic
478C22×C3C2×H(5,𝔽2)11yescharacteristic
479C2×D3—44no—
480C2×D3—44no—
481C2×D3—44no—
482C2×D3—44no—
483C2×D3—44no—
484C2×D3—44no—
485C2×D3—44no—
486C2×D3—816no—
487C3×C4—24no—
488C3×C4—42no—
489C3×C4—42no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4C2×#12418yes—
493Dic3—28no—
494Dic3—44no—
495Dic3—44no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—64no—
510C23—64no—
511C23—64no—
512C23—64no—
513C23—64no—
514C23—64no—
515C23—64no—
516C23—128no—
517C23—128no—
518C23—128no—
519C23—128no—
520C23—128no—
521C23—128no—
522C23—128no—
523C23C2×#23514yes—
524C23C2×#23512yes—
525C23C2×#23512yes—
526C23C2×#23512yes—
527C23C2×D3×D412yes—
528C23C2×#23712yes—
529C23#100912yes—
530C23#100912yes—
531C23C24×D311yescharacteristic
532D4—1232no—
533D4—1216no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—38no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—416no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—616no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—128no—
565C2×C4—128no—
566C2×C4#885116yes—
567C2×C4C2×#23514yes—
568C2×C4C2×#23514yes—
569C2×C4C2×D3×D414yes—
570C2×C4C2×#23714yes—
571C2×C3—44no—
572C2×C3#249018yescomplemented
573C2×C3C22×#12411yescharacteristic
574C2×C3C2×#52911yescharacteristic
575C2×C3C2×#54011yescharacteristic
576C2×C3C2×#54611yescharacteristic
577C2×C3C2×#55011yescharacteristic
578C2×C3C2×#55311yescharacteristic
579C2×C3C2×#55311yescharacteristic
580D3—84no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—416no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—124no—
590C22—124no—
591C22—124no—
592C22—124no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—2416no—
597C22#547914yes—
598C22C2×#88514yes—
599C22#553514yes—
600C22#555014yes—
601C22#555314yes—
602C22#556514yes—
603C22#556514yes—
604C22C22×#23511yescharacteristic
605C22C22×#23511yescharacteristic
606C22C22×#23511yescharacteristic
607C22C22×D3×D411yescharacteristic
608C22C22×#23711yescharacteristic
609C22C2×#100911yescharacteristic
610C22C2×#100911yescharacteristic
611C4—24no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—68no—
617C4—124no—
618C4—124no—
619C4C2×#88518yes—
620C3C2×#249011yescharacteristic, complemented, Sylow
621C2—44no—
622C2—244no—
623C2#7350618yesdirect product factor
624C2C2×#547911yescharacteristic
625C2C22×#88511yescharacteristic
626C2C2×#553511yescharacteristic
627C2C2×#555311yescharacteristic
628C2C2×#556511yescharacteristic
629C2C2×#556511yescharacteristic
630C2C2×#555011yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×C4×C22⋊C4C2
socleC24×C3C24
prime radicalC22×#930C2
hypercenterC2×C4×C22⋊C4D3
hypocenterC3C2×#2490
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#870C2semidirect
3. C2×C3×C4×C22⋊C4C2semidirect
4. C2×#5212C2semidirect
5. C2×#4959C2semidirect
6. C2×#4957C2semidirect
7. C2×#4939C2semidirect
8. C2×#4939C2semidirect
9. C2×C4×C4⋊Dic3C2semidirect
10. #73506C2direct (N × H)
11. C2×#4675C2semidirect
12. C2×#4669C2semidirect
36. C2×#870C22semidirect
37. C3×C4×C22⋊C4C22semidirect
40. #5212C22semidirect
43. #4959C22semidirect
44. #4957C22semidirect
45. #4939C22semidirect
46. #4939C22semidirect
47. C4×C4⋊Dic3C22semidirect
48. #4675C22semidirect
49. #4669C22semidirect
61. C2×C4×C22⋊C4D3semidirect
154. C4×C22⋊C4C2×D3semidirect
572. C2×C3#2490semidirect
620. C3C2×#2490semidirect
623. C2#73506direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9302
C22×#8702
C2×C3×C4×C22⋊C42
C2×#52122
C2×#49592
C2×#49572
C2×#49392
C2×C4×C4⋊Dic32
#735062
C2×#46752
C2×#46692
C2×#24903