Finite Group Databasebeta

← C2×#70090 #10822372

Subgroups — C2×#70090 #10822372

Order 768.

2158 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#932C211yesmaximal, characteristic, complemented
2C23×#213C211yesmaximal, characteristic, complemented
3C22×#908C211yesmaximal, characteristic, complemented
4C22×C3×#100C211yesmaximal, characteristic, complemented
5C22×#823C211yesmaximal, characteristic, complemented
6C2×#5129C211yesmaximal, characteristic, complemented
7C2×#4558C211yesmaximal, characteristic, complemented
8#70090C218yesmaximal, direct product factor
9C2×#2018—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×#235C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×#214—21no—
16C2×#932—24no—
17C2×#932—22no—
18C2×#932—24no—
19C2×#932C2214yescomplemented
20C22×#213—24no—
21C22×#213—24no—
22C22×#213—21no—
23C22×#213—22no—
24C22×#213—22no—
25C22×#213C2214yescomplemented
26C22×#208C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28C2×#908—22no—
29C2×#908—24no—
30C2×#908—24no—
31C2×#908C2214yescomplemented
32C22×D24—21no—
33C22×#204—21no—
34C2×C3×#100—24no—
35C2×C3×#100C2214yescomplemented
36C2×#823—24no—
37C2×#823C2214yescomplemented
38#5129—22no—
39#5129C2214yescomplemented
40C2×(C2×D3)⋊C8—22no—
41#4558C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C22×AGL(1,ℤ/8)—31no—
44C23×D8—31no—
45C22×#100—31no—
46C22×#100D311yescharacteristic, complemented
47C2×#408—31no—
48#2018—38no—
49C2×#341—31no—
50C25×C3—21no—
51C24×D3—21no—
52C22×C3×D4—22no—
53C22×C3×D4—21no—
54C22×C3×D4—21no—
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4—21no—
60C22×C3×D4D414yes—
61C22×C3×D4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C22×#67—21no—
64C2×#235—21no—
65C2×#235—21no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235D412yes—
71C2×#235C2312yes—
72C22×D12—21no—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12—22no—
76C22×D12—21no—
77C22×D12—21no—
78C22×D12—22no—
79C22×D12—21no—
80C22×D12D414yes—
81C22×D12D411yescharacteristic
82C22×D12C2312yes—
83C22×C4×D3—21no—
84C22×Dic6D411yescharacteristic
85C2×C3×M16—22no—
86C2×C3×M16—21no—
87C2×C3×M16—21no—
88C2×C3×M16—22no—
89C2×C3×M16—22no—
90C2×C3×M16D414yes—
91C2×C3×M16C2312yes—
92C2×C3×M16C2312yes—
93C22×C3×C8—21no—
94C22×C3×C8—21no—
95C2×#214—28no—
96C2×#214—24no—
97#932—216no—
98#932—28no—
99#932—28no—
100C2×#213—216no—
101C2×#213—216no—
102C2×#213—28no—
103C2×#213—28no—
104C2×#213—28no—
105C2×#213—24no—
106C2×#213—28no—
107C2×#208—22no—
108C2×#208—21no—
109C2×#208D414yes—
110C2×#208C2312yes—
111C22×C3⋊C8—22no—
112C22×C3⋊C8—22no—
113C22×C3⋊C8—21no—
114C22×C3⋊C8D412yes—
115C22×C3⋊C8D412yes—
116C22×C3⋊C8C2312yes—
117#908—216no—
118#908—28no—
119#908—28no—
120C2×D24—28no—
121C2×D24—24no—
122C2×#204—28no—
123C2×#204—24no—
124C3×#100—28no—
125#823—28no—
126(C2×D3)⋊C8—28no—
127C22×#41—31no—
128C23×D4—31no—
129C23×D4C2×D311yescharacteristic
130C2×AGL(1,ℤ/8)—34no—
131C2×AGL(1,ℤ/8)—34no—
132C2×AGL(1,ℤ/8)—64no—
133C2×AGL(1,ℤ/8)—62no—
134C2×AGL(1,ℤ/8)—64no—
135C2×AGL(1,ℤ/8)—62no—
136C2×AGL(1,ℤ/8)—64no—
137C2×AGL(1,ℤ/8)—64no—
138C22×SD16—61no—
139C22×SD16—61no—
140C22×D8—34no—
141C22×D8—61no—
142C22×D8—62no—
143C22×D8—62no—
144C22×D8—64no—
145C22×D8—61no—
146C22×D8—64no—
147C22×M16—31no—
148C22×M16C2×D311yescharacteristic
149C22×M16C2×D311yescharacteristic
150C23×C8—31no—
151C2×#100—24no—
152C2×#100—34no—
153C2×#100—64no—
154C2×#100C2×D314yescomplemented
155#408—34no—
156#408—62no—
157C2×C22⋊C8—62no—
158#341—34no—
159C24×C3—24no—
160C24×C3—22no—
161C24×C3—41no—
162C24×C3—42no—
163C24×C3—41no—
164C24×C3—84no—
165C24×C3#4111yescharacteristic
166C23×D3—21no—
167C23×D3—24no—
168C23×D3—22no—
169C23×D3—41no—
170C23×D3—41no—
171C23×D3—42no—
172C23×D3—84no—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—42no—
184C2×C3×D4—42no—
185C2×C3×D4—41no—
186C2×C3×D4—41no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4C2×D414yes—
191C22×C3×C4C2×D412yes—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4#4111yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C2×#67—22no—
202C2×#67—22no—
203C2×#67—42no—
204C2×#67—42no—
205C22×Dic3—21no—
206#235—24no—
207#235—24no—
208#235—22no—
209#235—22no—
210#235—22no—
211#235—22no—
212#235—42no—
213#235—42no—
214#235—42no—
215#235—42no—
216C2×D12—24no—
217C2×D12—22no—
218C2×D12—24no—
219C2×D12—24no—
220C2×D12—24no—
221C2×D12—22no—
222C2×D12—22no—
223C2×D12—22no—
224C2×D12—22no—
225C2×D12—22no—
226C2×D12—24no—
227C2×D12—21no—
228C2×D12—22no—
229C2×D12—42no—
230C2×D12—42no—
231C2×D12—41no—
232C2×D12—41no—
233C2×D12—42no—
234C2×D12—42no—
235C2×D12—42no—
236C2×D12C2×D412yes—
237C2×D12C2×D414yes—
238C2×C4×D3—22no—
239C2×C4×D3—22no—
240C2×C4×D3—42no—
241C2×C4×D3—42no—
242C2×Dic6—22no—
243C2×Dic6—22no—
244C2×Dic6—21no—
245C2×Dic6C2×D412yes—
246C3×M16—24no—
247C3×M16—22no—
248C3×M16—22no—
249C3×M16—22no—
250C3×M16—42no—
251C3×M16C2×D414yes—
252C2×C3×C8—24no—
253C2×C3×C8—22no—
254C2×C3×C8—22no—
255C2×C3×C8—42no—
256#214—216no—
257#213—232no—
258#213—216no—
259#208—24no—
260#208—22no—
261#208C2×D414yes—
262C2×C3⋊C8—24no—
263C2×C3⋊C8—24no—
264C2×C3⋊C8—24no—
265C2×C3⋊C8—22no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8—22no—
268C2×C3⋊C8C2×D414yes—
269D24—216no—
270#204—216no—
271C25—21no—
272C25—61no—
273C2×#41—32no—
274C2×#41—32no—
275C2×#41—61no—
276C2×#41—61no—
277C2×#41—62no—
278C2×#41—62no—
279C2×#41—62no—
280C2×#41—62no—
281C22×Q8—31no—
282C22×D4—21no—
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4—21no—
287C22×D4—21no—
288C22×D4—22no—
289C22×D4—22no—
290C22×D4—31no—
291C22×D4—34no—
292C22×D4—32no—
293C22×D4—61no—
294C22×D4—62no—
295C22×D4—61no—
296C22×D4—61no—
297C22×D4—61no—
298C22×D4—61no—
299C22×D4—61no—
300C22×D4—62no—
301C22×D4—62no—
302C22×D4D1214yes—
303C22×D4C22×D312yes—
304C23×C4—61no—
305C23×C4C22×D311yescharacteristic
306AGL(1,ℤ/8)—68no—
307AGL(1,ℤ/8)—616no—
308AGL(1,ℤ/8)—68no—
309AGL(1,ℤ/8)—616no—
310AGL(1,ℤ/8)—68no—
311AGL(1,ℤ/8)—68no—
312C2×SD16—64no—
313C2×SD16—68no—
314C2×SD16—64no—
315C2×SD16—68no—
316C2×D8—64no—
317C2×D8—68no—
318C2×D8—68no—
319C2×D8—68no—
320C2×D8—64no—
321C2×D8—616no—
322C2×D8—616no—
323C2×D8—68no—
324C2×D8—68no—
325C2×M16—22no—
326C2×M16—21no—
327C2×M16—21no—
328C2×M16—22no—
329C2×M16—22no—
330C2×M16—34no—
331C2×M16—32no—
332C2×M16—62no—
333C2×M16—61no—
334C2×M16#6714yes—
335C2×M16C22×D312yes—
336C2×M16C22×D312yes—
337C22×C8—21no—
338C22×C8—21no—
339C22×C8—32no—
340C22×C8—32no—
341C22×C8—32no—
342C22×C8—62no—
343C22×C8—62no—
344C22×C8—61no—
345#100—28no—
346#100—68no—
347C22⋊C8—68no—
348C23×C3—24no—
349C23×C3—24no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—44no—
353C23×C3—44no—
354C23×C3—88no—
355C23×C3—84no—
356C23×C3—88no—
357C23×C3—84no—
358C23×C3—84no—
359C23×C3#12111yescharacteristic
360C23×C3#12112yes—
361C23×C3#12111yescharacteristic
362C23×C3#12112yes—
363C23×C3C2×#4111yescharacteristic
364C22×D3—22no—
365C22×D3—24no—
366C22×D3—42no—
367C22×D3—42no—
368C22×D3—42no—
369C22×D3—44no—
370C22×D3—44no—
371C22×D3—44no—
372C22×D3—88no—
373C22×D3—84no—
374C22×D3—84no—
375C22×D3—88no—
376C22×D3—84no—
377C3×D4—24no—
378C3×D4—24no—
379C3×D4—42no—
380C3×D4—42no—
381C3×D4—44no—
382C3×D4—44no—
383C2×C3×C4#12012yes—
384C2×C3×C4#12012yes—
385C2×C3×C4#12012yes—
386C2×C3×C4#12012yes—
387C2×C3×C4#12012yes—
388C2×C3×C4#12012yes—
389C2×C3×C4#12111yescharacteristic
390C2×C3×C4#12111yescharacteristic
391C2×C3×C4#12112yes—
392C2×C3×C4#12112yes—
393C2×C3×C4#12712yes—
394C2×C3×C4#12712yes—
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401#67—42no—
402#67—42no—
403#67—44no—
404C2×Dic3—22no—
405C2×Dic3—42no—
406D12—24no—
407D12—22no—
408D12—24no—
409D12—22no—
410D12—42no—
411D12—42no—
412D12—44no—
413D12—44no—
414D12—42no—
415C4×D3—42no—
416C4×D3—42no—
417C4×D3—44no—
418Dic6—22no—
419Dic6—22no—
420Dic6—42no—
421C3×C8—24no—
422C3×C8—42no—
423C3⋊C8—24no—
424C3⋊C8—24no—
425C24—24no—
426C24—22no—
427C24—41no—
428C24—42no—
429C24—41no—
430C24—61no—
431C24—64no—
432C24—62no—
433C24—84no—
434C24—121no—
435C24—121no—
436C24—122no—
437C24—244no—
438C24#23511yescharacteristic
439#41—64no—
440#41—64no—
441#41—62no—
442#41—62no—
443#41—62no—
444#41—62no—
445#41—122no—
446#41—122no—
447#41—122no—
448#41—122no—
449C2×Q8—32no—
450C2×Q8—61no—
451C2×Q8—62no—
452C2×Q8—62no—
453C2×D4—24no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—32no—
464C2×D4—34no—
465C2×D4—41no—
466C2×D4—42no—
467C2×D4—41no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—62no—
473C2×D4—62no—
474C2×D4—64no—
475C2×D4—64no—
476C2×D4—62no—
477C2×D4—64no—
478C2×D4—61no—
479C2×D4—62no—
480C2×D4—64no—
481C2×D4—64no—
482C2×D4—62no—
483C2×D4—62no—
484C2×D4—62no—
485C2×D4—62no—
486C2×D4—62no—
487C2×D4—122no—
488C2×D4—122no—
489C2×D4—122no—
490C2×D4—122no—
491C2×D4—121no—
492C2×D4—121no—
493C2×D4—122no—
494C2×D4—122no—
495C2×D4—122no—
496C2×D4C2×D1214yes—
497C22×C4—61no—
498C22×C4—62no—
499C22×C4—62no—
500C22×C4—122no—
501C22×C4—122no—
502C22×C4C2×D1212yes—
503C22×C4#23511yescharacteristic
504C22×C4D3×D412yes—
505C22×C4D3×D411yescharacteristic
506C22×C4D3×D411yescharacteristic
507C22×C4D3×D411yescharacteristic
508C22×C4D3×D412yes—
509C22×C4D3×D411yescharacteristic
510C22×C4C2×#6712yes—
511C22×C4C23×D311yescharacteristic
512SD16—616no—
513SD16—616no—
514D8—616no—
515D8—632no—
516D8—616no—
517M16—24no—
518M16—22no—
519M16—22no—
520M16—22no—
521M16—34no—
522M16—42no—
523M16—64no—
524M16—62no—
525M16C2×#6714yes—
526C2×C8—24no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—34no—
530C2×C8—42no—
531C2×C8—64no—
532C2×C8—64no—
533C2×C8—64no—
534C2×C8—62no—
535C2×C8—62no—
536C2×C8—62no—
537C22×C3—24no—
538C22×C3—24no—
539C22×C3—22no—
540C22×C3—22no—
541C22×C3—44no—
542C22×C3—84no—
543C22×C3—84no—
544C22×C3—88no—
545C22×C3—88no—
546C22×C3—84no—
547C22×C3#39212yes—
548C22×C3#39212yes—
549C22×C3#47214yes—
550C22×C3C2×#12111yescharacteristic
551C22×C3C2×#12111yescharacteristic
552C22×C3C2×#12111yescharacteristic
553C2×D3—42no—
554C2×D3—44no—
555C2×D3—44no—
556C2×D3—84no—
557C2×D3—88no—
558C2×D3—84no—
559C2×D3—88no—
560C2×D3—84no—
561C2×D3—84no—
562C3×C4#39212yes—
563C3×C4#39212yes—
564C3×C4C2×#12011yescharacteristic
565C3×C4C2×#12011yescharacteristic
566C3×C4C2×#12011yescharacteristic
567C3×C4C2×#12711yescharacteristic
568Dic3—42no—
569C23—24no—
570C23—24no—
571C23—44no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—62no—
576C23—64no—
577C23—84no—
578C23—88no—
579C23—84no—
580C23—88no—
581C23—84no—
582C23—122no—
583C23—124no—
584C23—124no—
585C23—122no—
586C23—124no—
587C23—122no—
588C23—244no—
589C23—248no—
590C23—244no—
591C23—248no—
592C23—244no—
593C23#88411yescharacteristic
594C23#88412yes—
595C23#88411yescharacteristic
596C23#93012yes—
597C23C2×#23511yescharacteristic
598Q8—62no—
599Q8—62no—
600Q8—122no—
601D4—24no—
602D4—24no—
603D4—44no—
604D4—44no—
605D4—42no—
606D4—42no—
607D4—64no—
608D4—62no—
609D4—62no—
610D4—64no—
611D4—124no—
612D4—122no—
613D4—122no—
614D4—124no—
615D4—124no—
616D4—122no—
617D4—122no—
618D4—122no—
619C2×C4—62no—
620C2×C4—122no—
621C2×C4—124no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4#88212yes—
625C2×C4#88212yes—
626C2×C4#88212yes—
627C2×C4#88212yes—
628C2×C4#88411yescharacteristic
629C2×C4#88412yes—
630C2×C4#88411yescharacteristic
631C2×C4#93012yes—
632C2×C4#93712yes—
633C2×C4#93712yes—
634C2×C4#94012yes—
635C2×C4#94012yes—
636C2×C4C22×D1211yescharacteristic
637C2×C4C2×D3×D411yescharacteristic
638C2×C4C2×D3×D411yescharacteristic
639C2×C4C2×D3×D411yescharacteristic
640C2×C4C2×D3×D411yescharacteristic
641C2×C4C22×#6711yescharacteristic
642C8—24no—
643C8—42no—
644C8—64no—
645C8—64no—
646C2×C3—21no—
647C2×C3—21no—
648C2×C3—22no—
649C2×C3—84no—
650C2×C3#201814yescomplemented
651C2×C3C2×#39211yescharacteristic
652C2×C3C2×#47212yes—
653D3—82no—
654D3—84no—
655C22—22no—
656C22—22no—
657C22—24no—
658C22—24no—
659C22—44no—
660C22—84no—
661C22—88no—
662C22—84no—
663C22—84no—
664C22—88no—
665C22—124no—
666C22—122no—
667C22—124no—
668C22—244no—
669C22—248no—
670C22—244no—
671C22—244no—
672C22—244no—
673C22—248no—
674C22#474914yes—
675C22#496012yes—
676C22#496012yes—
677C22C2×#88411yescharacteristic
678C22C2×#88411yescharacteristic
679C22C2×#93011yescharacteristic
680C4—122no—
681C4#496012yes—
682C4#496012yes—
683C4C2×#88211yescharacteristic
684C4C2×#88211yescharacteristic
685C4C2×#93711yescharacteristic
686C4C2×#94011yescharacteristic
687C3C2×#201811yescharacteristic, complemented, Sylow
688C2—22no—
689C2—21no—
690C2—21no—
691C2—84no—
692C2—242no—
693C2—244no—
694C2#7009014yesdirect product factor
695C2C2×#474912yes—
696C2C2×#496011yescharacteristic
subgroupG/H
centerC23#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×#100C2
socleC23×C3#121
hypercenterC22×#100D3
hypocenterC3C2×#2018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#100

Complements

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

NH ≅ G/Ntype
1. C22×#932C2semidirect
2. C23×#213C2semidirect
3. C22×#908C2semidirect
4. C22×C3×#100C2semidirect
5. C22×#823C2semidirect
6. C2×#5129C2semidirect
7. C2×#4558C2semidirect
8. #70090C2direct (N × H)
19. C2×#932C22semidirect
25. C22×#213C22semidirect
31. C2×#908C22semidirect
35. C2×C3×#100C22semidirect
37. C2×#823C22semidirect
39. #5129C22semidirect
41. #4558C22semidirect
46. C22×#100D3semidirect
154. C2×#100C2×D3semidirect
650. C2×C3#2018semidirect
687. C3C2×#2018semidirect
694. C2#70090direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9322
C23×#2132
C22×#9082
C22×C3×#1002
C22×#8232
C2×#51292
C2×#45582
#700902
C2×#20183