Finite Group Databasebeta

← C2×#73507 #10964290

Subgroups — C2×#73507 #10964290

Order 768.

3678 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#930C211yesmaximal, characteristic
2C22×#885C212yesmaximal, complemented
3C22×#869C211yesmaximal, characteristic, complemented
4C2×C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
5C2×#5212C211yesmaximal, characteristic, complemented
6C2×#4961C212yesmaximal, complemented
7C2×#4939C211yesmaximal, characteristic, complemented
8C2×#4939C211yesmaximal, characteristic, complemented
9C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#73507C2116yesmaximal, direct product factor
11C2×#4673C212yesmaximal, complemented
12C2×#4670C212yesmaximal, complemented
13C2×#2491—31nomaximal, Sylow
14C24×C3×C4C2211yescharacteristic
15C23×#67C2212yes—
16C23×D12C2211yescharacteristic
17C23×Dic6C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C22×#217C2212yes—
23C2×#930—28no—
24C2×#930C2218yes—
25C22×(C2×D3)⋊C4C2212yes—
26C22×(C2×D3)⋊C4C2212yes—
27C22×(C2×D3)⋊C4C2212yes—
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2212yes—
30C22×C4⋊Dic3C2211yescharacteristic
31C22×C4⋊Dic3C2211yescharacteristic
32C22×C4⋊Dic3C2212yes—
33C22×C4⋊Dic3C2211yescharacteristic
34C22×C4×Dic3C2212yes—
35C2×#885—216no—
36C2×#885C22116yescomplemented
37C2×#869—28no—
38C2×#869C2218yescomplemented
39C3×C4×C22⋊C4C2218yescomplemented
40C2×C3×#95C2211yescharacteristic
41C2×C3×#95C2211yescharacteristic
42#5212C2218yescomplemented
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45#4961C22116yescomplemented
46#4939C2218yescomplemented
47#4939C2218yescomplemented
48C4×C4⋊Dic3C2218yescomplemented
49#4673C22116yescomplemented
50#4670C22116yescomplemented
51C22×C4⋊Q8—31no—
52C22×#124—32no—
53C22×#121—31no—
54#2491—316no—
55C2×#394—32no—
56C2×#393—32no—
57C2×#390—31no—
58C2×#390—31no—
59C2×#388—32no—
60C2×#380—31no—
61C2×C4×C4⋊C4—31no—
62C2×C4×C22⋊C4D311yescharacteristic, complemented
63C25×C3C2311yescharacteristic
64C24×D3C2312yes—
65C23×C3×C4—24no—
66C23×C3×C4—24no—
67C23×C3×C4—24no—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2314yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C23×C3×C4C2311yescharacteristic
75C22×#67—28no—
76C22×#67—28no—
77C22×#67—44no—
78C22×#67C2318yes—
79C23×Dic3C2312yes—
80C23×Dic3C2312yes—
81C23×Dic3C2312yes—
82C22×D12—24no—
83C22×D12—24no—
84C22×D12—24no—
85C22×D12C2314yes—
86C22×Dic6—28no—
87C22×Dic6—28no—
88C22×Dic6—44no—
89C22×Dic6C2318yes—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4D418yescomplemented
92C2×C3×C22⋊C4C2314yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C42—24no—
95C2×C3×C42—24no—
96C2×C3×C42C2314yes—
97C2×C3×C42C2314yes—
98C2×#217—28no—
99C2×#217C2318yes—
100#930—232no—
101C2×(C2×D3)⋊C4—28no—
102C2×(C2×D3)⋊C4—28no—
103C2×(C2×D3)⋊C4—28no—
104C2×(C2×D3)⋊C4—28no—
105C2×(C2×D3)⋊C4—44no—
106C2×(C2×D3)⋊C4C2318yes—
107C2×(C2×D3)⋊C4C2318yes—
108C2×(C2×D3)⋊C4C2318yes—
109C2×(C2×D3)⋊C4C2318yes—
110C2×(C2×D3)⋊C4C2318yes—
111C2×C4⋊Dic3—24no—
112C2×C4⋊Dic3—24no—
113C2×C4⋊Dic3—28no—
114C2×C4⋊Dic3D418yes—
115C2×C4⋊Dic3C2318yes—
116C2×C4⋊Dic3C2314yes—
117C2×C4⋊Dic3C2314yes—
118C2×C4⋊Dic3C2314yes—
119C2×C4×Dic3—28no—
120C2×C4×Dic3C2318yes—
121#885—264no—
122#869—232no—
123C3×#95C2314yes—
124C3×#95C2314yes—
125C3⋊#95C2318yes—
126C3⋊#95C2318yes—
127C23×Q8—32no—
128C23×D4—32no—
129C23×D4—31no—
130C24×C4C2×D311yescharacteristic
131C2×C4⋊Q8—38no—
132C2×C4⋊Q8—68no—
133C2×#124—316no—
134C2×#124—616no—
135C2×#121—38no—
136C2×#121—68no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4—32no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—31no—
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4—32no—
144C22×C22⋊C4—32no—
145C22×C22⋊C4—32no—
146C22×C22⋊C4—32no—
147C22×C22⋊C4C2×D311yescharacteristic
148C22×C22⋊C4C2×D311yescharacteristic
149C22×C42—32no—
150C22×C42C2×D311yescharacteristic
151C22×C42C2×D311yescharacteristic
152#394—316no—
153#393—316no—
154#390—38no—
155#390—38no—
156#388—316no—
157#380—38no—
158C4×C4⋊C4—38no—
159C4×C22⋊C4C2×D318yescomplemented
160C2×#95—32no—
161C2×#95—32no—
162C2×#95C2×D311yescharacteristic
163C2×#95C2×D311yescharacteristic
164C24×C3—22no—
165C24×C3—22no—
166C24×C3—22no—
167C24×C3—22no—
168C24×C3—22no—
169C24×C3—22no—
170C24×C3C2×D414yes—
171C24×C3C2412yes—
172C24×C3C2411yescharacteristic
173C23×D3—24no—
174C23×D3—24no—
175C23×D3—24no—
176C23×D3—24no—
177C23×D3—24no—
178C23×D3—24no—
179C23×D3—24no—
180C23×D3C2414yes—
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—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—28no—
194C22×C3×C4—28no—
195C22×C3×C4—28no—
196C22×C3×C4—416no—
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2×D414yes—
200C22×C3×C4#4114yes—
201C22×C3×C4#4114yes—
202C22×C3×C4#4114yes—
203C22×C3×C4C2412yes—
204C22×C3×C4C2412yes—
205C22×C3×C4C2412yes—
206C22×C3×C4C2412yes—
207C22×C3×C4C2412yes—
208C22×C3×C4C2412yes—
209C2×#67—216no—
210C2×#67—216no—
211C2×#67—48no—
212C2×#67—432no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—24no—
216C22×Dic3—24no—
217C22×Dic3—24no—
218C22×Dic3—24no—
219C22×Dic3—24no—
220C22×Dic3—24no—
221C22×Dic3C2×D418yes—
222C22×Dic3C2414yes—
223C22×Dic3C2414yes—
224C22×Dic3C2414yes—
225C2×D12—28no—
226C2×D12—28no—
227C2×D12—28no—
228C2×D12—416no—
229C2×Dic6—216no—
230C2×Dic6—216no—
231C2×Dic6—48no—
232C2×Dic6—432no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4C2×D4116yescomplemented
235C3×C42—28no—
236C3×C42—28no—
237#217—216no—
238(C2×D3)⋊C4—216no—
239(C2×D3)⋊C4—216no—
240(C2×D3)⋊C4—216no—
241(C2×D3)⋊C4—216no—
242(C2×D3)⋊C4—48no—
243C4⋊Dic3—28no—
244C4⋊Dic3—28no—
245C4⋊Dic3—216no—
246C4⋊Dic3C2×D4116yes—
247C4×Dic3—216no—
248C25—32no—
249C25C22×D311yescharacteristic
250C22×Q8—38no—
251C22×Q8—68no—
252C22×Q8—68no—
253C22×Q8—124no—
254C22×D4—38no—
255C22×D4—34no—
256C22×D4—68no—
257C22×D4—68no—
258C22×D4—64no—
259C22×D4—64no—
260C22×D4—64no—
261C22×D4—124no—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—24no—
265C23×C4—32no—
266C23×C4—32no—
267C23×C4—32no—
268C23×C4C22×D314yes—
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D311yescharacteristic
271C23×C4C22×D311yescharacteristic
272C23×C4C22×D311yescharacteristic
273C23×C4C22×D311yescharacteristic
274C23×C4C22×D311yescharacteristic
275C4⋊Q8—632no—
276#124—664no—
277#121—632no—
278C2×C4⋊C4—38no—
279C2×C4⋊C4—34no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—38no—
282C2×C4⋊C4—34no—
283C2×C4⋊C4—68no—
284C2×C4⋊C4—64no—
285C2×C4⋊C4—64no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—38no—
288C2×C22⋊C4—38no—
289C2×C22⋊C4—38no—
290C2×C22⋊C4—38no—
291C2×C22⋊C4—38no—
292C2×C22⋊C4—38no—
293C2×C22⋊C4—68no—
294C2×C22⋊C4—68no—
295C2×C22⋊C4—68no—
296C2×C22⋊C4—68no—
297C2×C22⋊C4—68no—
298C2×C22⋊C4—124no—
299C2×C22⋊C4D1218yescomplemented
300C2×C22⋊C4C22×D314yes—
301C2×C22⋊C4C22×D314yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—38no—
305C2×C42—68no—
306C2×C42C22×D314yes—
307C2×C42C22×D314yes—
308#95—38no—
309#95—38no—
310#95C22×D314yes—
311#95C22×D314yes—
312C23×C3—28no—
313C23×C3—28no—
314C23×C3—28no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—44no—
322C23×C3—44no—
323C23×C3—44no—
324C23×C3—44no—
325C23×C3C22×D414yes—
326C23×C3C22×D412yes—
327C23×C3C22×D412yes—
328C23×C3C2×#4112yes—
329C23×C3C2×#4112yes—
330C23×C3C2×#4112yes—
331C23×C3H(5,𝔽2)12yes—
332C23×C32−1+412yes—
333C23×C3C2511yescharacteristic
334C22×D3—24no—
335C22×D3—24no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3—24no—
339C22×D3—24no—
340C22×D3—24no—
341C22×D3—48no—
342C22×D3—48no—
343C22×D3—48no—
344C22×D3—48no—
345C22×D3—48no—
346C22×D3—48no—
347C22×D3—48no—
348C2×C3×C4—28no—
349C2×C3×C4—28no—
350C2×C3×C4—28no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—44no—
364C2×C3×C4—44no—
365C2×C3×C4—416no—
366C2×C3×C4#124116yescomplemented
367C2×C3×C4C22×D414yes—
368C2×C3×C4C22×D414yes—
369C2×C3×C4C22×D414yes—
370C2×C3×C4C2×#4114yes—
371C2×C3×C4C2×#4114yes—
372C2×C3×C4C2×#4114yes—
373#67—432no—
374C2×Dic3—216no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—48no—
384C2×Dic3—48no—
385C2×Dic3C22×D418yes—
386D12—416no—
387Dic6—432no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—34no—
395C24—64no—
396C24—64no—
397C24—64no—
398C24—64no—
399C24—64no—
400C24—64no—
401C24—64no—
402C24C2×D1214yes—
403C24C23×D312yes—
404C24C23×D311yescharacteristic
405C2×Q8—616no—
406C2×Q8—616no—
407C2×Q8—128no—
408C2×Q8—1232no—
409C2×D4—616no—
410C2×D4—616no—
411C2×D4—68no—
412C2×D4—68no—
413C2×D4—68no—
414C2×D4—128no—
415C2×D4—1232no—
416C2×D4—1216no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—38no—
433C22×C4—34no—
434C22×C4—34no—
435C22×C4—34no—
436C22×C4—416no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4C2×D1214yes—
446C22×C4C2×D1214yes—
447C22×C4#23514yes—
448C22×C4#23514yes—
449C22×C4D3×D414yes—
450C22×C4#23714yes—
451C22×C4C23×D312yes—
452C22×C4C23×D312yes—
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C22×C4C23×D312yes—
456C22×C4C23×D312yes—
457C4⋊C4—316no—
458C4⋊C4—616no—
459C4⋊C4—68no—
460C4⋊C4—68no—
461C22⋊C4—28no—
462C22⋊C4—616no—
463C22⋊C4—616no—
464C22⋊C4—616no—
465C22⋊C4—616no—
466C22⋊C4—616no—
467C22⋊C4—128no—
468C22⋊C4C2×D12116yescomplemented
469C42—28no—
470C42—28no—
471C42—616no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—416no—
480C22×C3C2×#12414yes—
481C22×C3#53214yes—
482C22×C3#54214yes—
483C22×C3D4214yes—
484C22×C3#54714yes—
485C22×C3#54814yes—
486C22×C3#55514yes—
487C22×C3C23×D411yescharacteristic
488C22×C3C23×D411yescharacteristic
489C22×C3C22×#4111yescharacteristic
490C22×C3C22×#4111yescharacteristic
491C22×C3C22×#4111yescharacteristic
492C22×C3C2×H(5,𝔽2)11yescharacteristic
493C22×C3C2×2−1+411yescharacteristic
494C2×D3—44no—
495C2×D3—44no—
496C2×D3—44no—
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—816no—
502C3×C4—24no—
503C3×C4—24no—
504C3×C4—24no—
505C3×C4—42no—
506C3×C4—42no—
507C3×C4C2×#12418yescomplemented
508Dic3—28no—
509Dic3—44no—
510Dic3—44no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23—64no—
528C23—64no—
529C23—64no—
530C23—64no—
531C23—128no—
532C23—128no—
533C23—128no—
534C23—128no—
535C23—128no—
536C23—128no—
537C23—128no—
538C23C22×D1214yes—
539C23C22×D1212yes—
540C23C2×#23512yes—
541C23C2×#23512yes—
542C23C2×D3×D412yes—
543C23C2×#23712yes—
544C23#100912yes—
545C23#101012yes—
546C23C24×D311yescharacteristic
547Q8—1232no—
548D4—1232no—
549D4—1216no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—38no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—416no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—616no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—64no—
576C2×C4—64no—
577C2×C4—64no—
578C2×C4—128no—
579C2×C4—128no—
580C2×C4#885116yescomplemented
581C2×C4C22×D1214yes—
582C2×C4C22×D1214yes—
583C2×C4C2×#23514yes—
584C2×C4C2×#23514yes—
585C2×C4C2×D3×D414yes—
586C2×C4C2×#23714yes—
587C2×C3—44no—
588C2×C3#249118yescomplemented
589C2×C3C22×#12411yescharacteristic
590C2×C3C2×#53211yescharacteristic
591C2×C3C2×#54211yescharacteristic
592C2×C3C2×D4211yescharacteristic
593C2×C3C2×#54711yescharacteristic
594C2×C3C2×#54811yescharacteristic
595C2×C3C2×#55511yescharacteristic
596D3—84no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—416no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—124no—
606C22—124no—
607C22—124no—
608C22—124no—
609C22—124no—
610C22—124no—
611C22—124no—
612C22—2416no—
613C22#547714yes—
614C22C2×#88514yes—
615C22#553414yes—
616C22D4×D1214yes—
617C22#555214yes—
618C22#555514yes—
619C22#556314yes—
620C22C23×D1211yescharacteristic
621C22C22×#23511yescharacteristic
622C22C22×#23511yescharacteristic
623C22C22×D3×D411yescharacteristic
624C22C22×#23711yescharacteristic
625C22C2×#100911yescharacteristic
626C22C2×#101011yescharacteristic
627C4—24no—
628C4—24no—
629C4—24no—
630C4—42no—
631C4—42no—
632C4—68no—
633C4—124no—
634C4—124no—
635C4C2×#88518yescomplemented
636C3C2×#249111yescharacteristic, complemented, Sylow
637C2—44no—
638C2—244no—
639C2#7350718yesdirect product factor
640C2C2×#547711yescharacteristic
641C2C22×#88511yescharacteristic
642C2C2×#553411yescharacteristic
643C2C2×#555511yescharacteristic
644C2C2×#555211yescharacteristic
645C2C2×#556311yescharacteristic
646C2C2×D4×D1211yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×C4×C22⋊C4C2
socleC24×C3C24
prime radicalC22×#930C2
hypercenterC2×C4×C22⋊C4D3
hypocenterC3C2×#2491
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×#885C2semidirect
3. C22×#869C2semidirect
4. C2×C3×C4×C22⋊C4C2semidirect
5. C2×#5212C2semidirect
6. C2×#4961C2semidirect
7. C2×#4939C2semidirect
8. C2×#4939C2semidirect
9. C2×C4×C4⋊Dic3C2semidirect
10. #73507C2direct (N × H)
11. C2×#4673C2semidirect
12. C2×#4670C2semidirect
36. C2×#885C22semidirect
38. C2×#869C22semidirect
39. C3×C4×C22⋊C4C22semidirect
42. #5212C22semidirect
45. #4961C22semidirect
46. #4939C22semidirect
47. #4939C22semidirect
48. C4×C4⋊Dic3C22semidirect
49. #4673C22semidirect
50. #4670C22semidirect
62. C2×C4×C22⋊C4D3semidirect
91. C2×C3×C22⋊C4D4semidirect
159. C4×C22⋊C4C2×D3semidirect
234. C3×C22⋊C4C2×D4semidirect
299. C2×C22⋊C4D12semidirect
366. C2×C3×C4#124semidirect
468. C22⋊C4C2×D12semidirect
507. C3×C4C2×#124semidirect
580. C2×C4#885semidirect
588. C2×C3#2491semidirect
635. C4C2×#885semidirect
636. C3C2×#2491semidirect
639. C2#73507direct (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×#8852
C22×#8692
C2×C3×C4×C22⋊C42
C2×#52122
C2×#49612
C2×#49392
C2×C4×C4⋊Dic32
#735072
C2×#46732
C2×#46702
C2×#24913