Finite Group Databasebeta

← C2×#70027 #10794312

Subgroups — C2×#70027 #10794312

Order 768.

4062 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#237C211yesmaximal, characteristic
2C22×C3×#99C211yesmaximal, characteristic, complemented
3C22×#834C211yesmaximal, characteristic, complemented
4C22×#806C212yesmaximal, complemented
5C2×#4943C212yesmaximal, complemented
6#70027C218yesmaximal, direct product factor
7C2×#1782—31nomaximal, Sylow
8C23×C3×D4C2211yescharacteristic
9C23×#67C411yescharacteristic, complemented
10C23×#67C2211yescharacteristic
11C24×Dic3C411yescharacteristic, complemented
12C24×Dic3C2211yescharacteristic
13C22×#237—24no—
14C22×#237—24no—
15C22×#237—28no—
16C22×#237—24no—
17C22×#237—24no—
18C22×#237C414yescomplemented
19C22×#237C2214yes—
20C23×C4×D3C411yescharacteristic, complemented
21C23×Dic6C411yescharacteristic, complemented
22C22×C3×C22⋊C4C2212yes—
23C22×#217C2212yes—
24C2×C3×#99—24no—
25C2×C3×#99C2214yescomplemented
26C2×#834—24no—
27C2×#834C2214yescomplemented
28C2×C3⋊#95—24no—
29C2×#806—28no—
30C2×#806C2218yescomplemented
31#4943—24no—
32#4943C2218yescomplemented
33C23×#41—31no—
34C22×#99—31no—
35C22×#99—32no—
36C22×#99D311yescharacteristic, complemented
37C2×#381—32no—
38#1782—38no—
39C25×C3C2×C411yescharacteristic
40C25×C3C2311yescharacteristic
41C24×D3C2×C411yescharacteristic
42C22×C3×D4—22no—
43C22×C3×D4—24no—
44C22×C3×D4—22no—
45C22×C3×D4C2×C412yes—
46C22×C3×D4D414yes—
47C22×C3×D4D414yes—
48C22×C3×D4C2312yes—
49C23×C3×C4—22no—
50C23×C3×C4C2×C411yescharacteristic
51C22×#67—24no—
52C22×#67—22no—
53C22×#67—22no—
54C22×#67—22no—
55C22×#67—22no—
56C22×#67—24no—
57C22×#67—22no—
58C22×#67—22no—
59C22×#67C2×C412yes—
60C22×#67C2×C414yescomplemented
61C22×#67D418yes—
62C22×#67C2312yes—
63C23×Dic3—24no—
64C23×Dic3—22no—
65C23×Dic3—24no—
66C23×Dic3—22no—
67C23×Dic3—22no—
68C23×Dic3—22no—
69C23×Dic3—24no—
70C23×Dic3—22no—
71C23×Dic3C2×C411yescharacteristic
72C23×Dic3C2×C411yescharacteristic
73C23×Dic3C2×C411yescharacteristic
74C23×Dic3C2×C412yes—
75C23×Dic3C2×C414yescomplemented
76C23×Dic3D414yes—
77C23×Dic3D414yes—
78C23×Dic3C2312yes—
79C2×#237—232no—
80C2×#237—232no—
81C2×#237—232no—
82C2×#237—232no—
83C2×#237—216no—
84C2×#237—28no—
85C2×#237—28no—
86C2×#237—216no—
87C2×#237—216no—
88C2×#237—216no—
89C2×#237—216no—
90C22×C4×D3—24no—
91C22×C4×D3—24no—
92C22×C4×D3—22no—
93C22×C4×D3—22no—
94C22×C4×D3C2×C414yescomplemented
95C22×Dic6—24no—
96C22×Dic6—24no—
97C22×Dic6—22no—
98C22×Dic6—22no—
99C22×Dic6C2×C414yescomplemented
100C2×C3×C22⋊C4—24no—
101C2×C3×C22⋊C4—24no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4C2314yes—
104C2×#217—24no—
105C2×#217—24no—
106C2×#217—22no—
107C2×#217C2314yes—
108C3×#99—28no—
109#834—28no—
110C3⋊#95—216no—
111#806—216no—
112C22×#41—34no—
113C22×#41—34no—
114C22×#41—64no—
115C22×#41—64no—
116C22×#41—68no—
117C22×#41—64no—
118C22×#41—64no—
119C23×Q8—31no—
120C23×D4—31no—
121C23×D4—31no—
122C23×D4C2×D311yescharacteristic
123C24×C4—31no—
124C24×C4—31no—
125C24×C4—31no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4C2×D312yes—
128C2×#99—24no—
129C2×#99—34no—
130C2×#99—38no—
131C2×#99—68no—
132C2×#99—64no—
133C2×#99C2×D314yescomplemented
134#381—38no—
135#381—64no—
136C2×#95—64no—
137C24×C3—22no—
138C24×C3—21no—
139C24×C3—21no—
140C24×C3—44no—
141C24×C3—44no—
142C24×C3C22⋊C414yes—
143C24×C3C22×C411yescharacteristic
144C24×C3C22×C412yes—
145C24×C3C22×C411yescharacteristic
146C24×C3C2×D412yes—
147C24×C3C2×D412yes—
148C24×C3C2×D414yes—
149C24×C3C2×D412yes—
150C24×C3C2×D412yes—
151C24×C3C2411yescharacteristic
152C23×D3—21no—
153C23×D3—21no—
154C23×D3—22no—
155C23×D3—44no—
156C23×D3C22⋊C414yes—
157C23×D3C22×C412yes—
158C2×C3×D4—24no—
159C2×C3×D4—24no—
160C2×C3×D4—28no—
161C2×C3×D4—28no—
162C2×C3×D4—28no—
163C2×C3×D4—28no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4C22⋊C418yes—
167C2×C3×D4C2×D414yes—
168C2×C3×D4C2×D414yes—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—24no—
173C22×C3×C4—48no—
174C22×C3×C4—44no—
175C22×C3×C4C22⋊C414yes—
176C22×C3×C4C22×C412yes—
177C2×#67—216no—
178C2×#67—216no—
179C2×#67—24no—
180C2×#67—24no—
181C2×#67—28no—
182C2×#67—28no—
183C2×#67—28no—
184C2×#67—28no—
185C2×#67—28no—
186C2×#67—22no—
187C2×#67—24no—
188C2×#67—22no—
189C2×#67—28no—
190C2×#67—28no—
191C2×#67C22⋊C418yes—
192C2×#67C2×D418yes—
193C22×Dic3—216no—
194C22×Dic3—216no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—24no—
203C22×Dic3—24no—
204C22×Dic3—28no—
205C22×Dic3—28no—
206C22×Dic3—28no—
207C22×Dic3—28no—
208C22×Dic3—28no—
209C22×Dic3—22no—
210C22×Dic3—24no—
211C22×Dic3—22no—
212C22×Dic3—28no—
213C22×Dic3—28no—
214C22×Dic3—24no—
215C22×Dic3—48no—
216C22×Dic3—44no—
217C22×Dic3C22⋊C414yes—
218C22×Dic3C22⋊C412yes—
219C22×Dic3C22⋊C412yes—
220C22×Dic3C22⋊C412yes—
221C22×Dic3C22⋊C412yes—
222C22×Dic3C22⋊C418yes—
223C22×Dic3C22×C412yes—
224C22×Dic3C22×C412yes—
225C22×Dic3C22×C412yes—
226C22×Dic3C2×D418yes—
227#237—264no—
228#237—264no—
229#237—2128no—
230C2×C4×D3—216no—
231C2×C4×D3—216no—
232C2×C4×D3—28no—
233C2×C4×D3—28no—
234C2×C4×D3—28no—
235C2×Dic6—216no—
236C2×Dic6—216no—
237C2×Dic6—28no—
238C2×Dic6—28no—
239C2×Dic6—28no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—44no—
243#217—24no—
244#217—24no—
245#217—44no—
246C25—31no—
247C25C4×D311yescharacteristic
248C25C22×D311yescharacteristic
249C2×#41—616no—
250C2×#41—616no—
251C2×#41—616no—
252C2×#41—616no—
253C2×#41—632no—
254C2×#41—632no—
255C2×#41—632no—
256C2×#41—632no—
257C2×#41—616no—
258C2×#41—68no—
259C2×#41—68no—
260C22×Q8—34no—
261C22×Q8—62no—
262C22×Q8—64no—
263C22×Q8—62no—
264C22×Q8—64no—
265C22×D4—22no—
266C22×D4—24no—
267C22×D4—22no—
268C22×D4—32no—
269C22×D4—38no—
270C22×D4—32no—
271C22×D4—34no—
272C22×D4—62no—
273C22×D4—62no—
274C22×D4—64no—
275C22×D4—64no—
276C22×D4—62no—
277C22×D4—62no—
278C22×D4—62no—
279C22×D4—62no—
280C22×D4C4×D312yes—
281C22×D4D1214yes—
282C22×D4#6714yes—
283C22×D4C22×D312yes—
284C23×C4—22no—
285C23×C4—31no—
286C23×C4—31no—
287C23×C4—31no—
288C23×C4—32no—
289C23×C4—34no—
290C23×C4—34no—
291C23×C4—32no—
292C23×C4—34no—
293C23×C4—34no—
294C23×C4—62no—
295C23×C4—62no—
296C23×C4—64no—
297C23×C4—62no—
298C23×C4—64no—
299C23×C4—62no—
300C23×C4—64no—
301C23×C4—62no—
302C23×C4—64no—
303C23×C4—62no—
304C23×C4—64no—
305C23×C4—62no—
306C23×C4C4×D311yescharacteristic
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—34no—
311C2×C22⋊C4—64no—
312C2×C22⋊C4—64no—
313C2×C22⋊C4—62no—
314C2×C22⋊C4C22×D314yes—
315#99—28no—
316#99—616no—
317#99—68no—
318#95—616no—
319C23×C3—28no—
320C23×C3—24no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—42no—
326C23×C3—44no—
327C23×C3—44no—
328C23×C3—48no—
329C23×C3—44no—
330C23×C3—42no—
331C23×C3—48no—
332C23×C3—44no—
333C23×C3—44no—
334C23×C3—48no—
335C23×C3—44no—
336C23×C3C2×C22⋊C412yes—
337C23×C3C2×C22⋊C411yescharacteristic
338C23×C3C2×C22⋊C412yes—
339C23×C3C2×C22⋊C412yes—
340C23×C3C2×C22⋊C411yescharacteristic
341C23×C3C2×C22⋊C412yes—
342C23×C3C2×C22⋊C414yes—
343C23×C3C2×C22⋊C412yes—
344C23×C3#12014yes—
345C23×C3#12014yes—
346C23×C3#12014yes—
347C23×C3#12014yes—
348C23×C3C23×C411yescharacteristic
349C23×C3C22×D411yescharacteristic
350C23×C3C22×D412yes—
351C23×C3C22×D412yes—
352C23×C3C22×D411yescharacteristic
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—24no—
356C22×D3—24no—
357C22×D3—44no—
358C22×D3—48no—
359C22×D3—44no—
360C22×D3—48no—
361C22×D3—44no—
362C22×D3C2×C22⋊C414yes—
363C3×D4—28no—
364C3×D4—28no—
365C3×D4—216no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—48no—
371C2×C3×C4—44no—
372C2×C3×C4—88no—
373C2×C3×C4C2×C22⋊C414yes—
374#67—232no—
375#67—28no—
376#67—28no—
377#67—216no—
378C2×Dic3—232no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—28no—
386C2×Dic3—28no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—216no—
394C2×Dic3—48no—
395C2×Dic3—44no—
396C2×Dic3—88no—
397C2×Dic3C2×C22⋊C414yes—
398C2×Dic3C2×C22⋊C414yes—
399C2×Dic3C2×C22⋊C414yes—
400C4×D3—232no—
401Dic6—232no—
402C24—21no—
403C24—22no—
404C24—21no—
405C24—34no—
406C24—32no—
407C24—44no—
408C24—44no—
409C24—61no—
410C24—61no—
411C24—62no—
412C24—124no—
413C24(C2×D3)⋊C414yes—
414C24C2×C4×D311yescharacteristic
415C24C2×C4×D312yes—
416C24C2×C4×D311yescharacteristic
417C24C2×D1212yes—
418C24D3×D412yes—
419C24D3×D412yes—
420C24D3×D414yes—
421C24C2×#6712yes—
422C24C23×D311yescharacteristic
423#41—6128no—
424#41—664no—
425#41—664no—
426C2×Q8—68no—
427C2×Q8—68no—
428C2×Q8—616no—
429C2×Q8—616no—
430C2×Q8—68no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—28no—
436C2×D4—28no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—38no—
440C2×D4—38no—
441C2×D4—68no—
442C2×D4—68no—
443C2×D4—616no—
444C2×D4—616no—
445C2×D4—64no—
446C2×D4—68no—
447C2×D4—68no—
448C2×D4—68no—
449C2×D4—64no—
450C2×D4—68no—
451C2×D4—68no—
452C2×D4—62no—
453C2×D4—64no—
454C2×D4—62no—
455C2×D4(C2×D3)⋊C418yes—
456C2×D4C2×D1214yes—
457C2×D4C2×#6714yes—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—32no—
463C22×C4—32no—
464C22×C4—38no—
465C22×C4—34no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—32no—
469C22×C4—32no—
470C22×C4—32no—
471C22×C4—38no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—68no—
475C22×C4—68no—
476C22×C4—616no—
477C22×C4—616no—
478C22×C4—68no—
479C22×C4—68no—
480C22×C4—616no—
481C22×C4—616no—
482C22×C4—62no—
483C22×C4—61no—
484C22×C4—62no—
485C22×C4—64no—
486C22×C4—61no—
487C22×C4—68no—
488C22×C4—68no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—62no—
492C22×C4—68no—
493C22×C4—64no—
494C22×C4—68no—
495C22×C4—68no—
496C22×C4—62no—
497C22×C4—64no—
498C22×C4—62no—
499C22×C4—68no—
500C22×C4—64no—
501C22×C4—128no—
502C22×C4—124no—
503C22×C4(C2×D3)⋊C414yes—
504C22×C4C2×C4×D312yes—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—44no—
508C22⋊C4—64no—
509C22⋊C4—64no—
510C22⋊C4—124no—
511C22×C3—24no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—24no—
515C22×C3—28no—
516C22×C3—28no—
517C22×C3—24no—
518C22×C3—24no—
519C22×C3—24no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—44no—
523C22×C3—44no—
524C22×C3—41no—
525C22×C3—42no—
526C22×C3—41no—
527C22×C3—44no—
528C22×C3—44no—
529C22×C3—48no—
530C22×C3—44no—
531C22×C3—48no—
532C22×C3#37912yes—
533C22×C3#37914yes—
534C22×C3#37912yes—
535C22×C3#41014yes—
536C22×C3C22×C22⋊C411yescharacteristic
537C22×C3C22×C22⋊C411yescharacteristic
538C22×C3C22×C22⋊C411yescharacteristic
539C22×C3C2×#12012yes—
540C22×C3C2×#12012yes—
541C2×D3—24no—
542C2×D3—44no—
543C2×D3—48no—
544C2×D3—44no—
545C2×D3—48no—
546C2×D3—44no—
547C3×C4—24no—
548C3×C4—84no—
549Dic3—24no—
550Dic3—24no—
551Dic3—24no—
552Dic3—84no—
553C23—28no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—34no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—48no—
564C23—44no—
565C23—42no—
566C23—48no—
567C23—44no—
568C23—44no—
569C23—48no—
570C23—44no—
571C23—64no—
572C23—64no—
573C23—62no—
574C23—62no—
575C23—124no—
576C23—128no—
577C23—124no—
578C23—128no—
579C23—124no—
580C23D3×C22⋊C411yescharacteristic
581C23D3×C22⋊C412yes—
582C23D3×C22⋊C412yes—
583C23D3×C22⋊C412yes—
584C23D3×C22⋊C411yescharacteristic
585C23#88214yes—
586C23#88214yes—
587C23C2×(C2×D3)⋊C412yes—
588C23C2×(C2×D3)⋊C414yes—
589C23C2×(C2×D3)⋊C412yes—
590C23#93714yes—
591C23#93714yes—
592C23C22×C4×D311yescharacteristic
593C23C22×D1211yescharacteristic
594C23C2×D3×D412yes—
595C23C2×D3×D412yes—
596C23C22×#6711yescharacteristic
597Q8—632no—
598D4—216no—
599D4—28no—
600D4—28no—
601D4—632no—
602D4—616no—
603D4—68no—
604D4—68no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—34no—
610C2×C4—34no—
611C2×C4—34no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—632no—
615C2×C4—632no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—64no—
619C2×C4—64no—
620C2×C4—64no—
621C2×C4—64no—
622C2×C4—616no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—68no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—68no—
631C2×C4—88no—
632C2×C4—128no—
633C2×C4—124no—
634C2×C4—248no—
635C2×C4C2×(C2×D3)⋊C414yes—
636C2×C3—22no—
637C2×C3—22no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—22no—
641C2×C3—24no—
642C2×C3—44no—
643C2×C3#178214yescomplemented
644C2×C3C2×#37911yescharacteristic
645C2×C3C2×#41012yes—
646D3—44no—
647C22—24no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—28no—
652C22—28no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—42no—
657C22—44no—
658C22—42no—
659C22—44no—
660C22—41no—
661C22—42no—
662C22—41no—
663C22—44no—
664C22—48no—
665C22—44no—
666C22—44no—
667C22—48no—
668C22—64no—
669C22—124no—
670C22—128no—
671C22—124no—
672C22—124no—
673C22—128no—
674C22#474014yes—
675C22#495512yes—
676C22#495514yes—
677C22#495512yes—
678C22C2×D3×C22⋊C411yescharacteristic
679C22C2×D3×C22⋊C411yescharacteristic
680C22C2×#88212yes—
681C22C22×(C2×D3)⋊C411yescharacteristic
682C22C2×#93712yes—
683C4—24no—
684C4—64no—
685C4—64no—
686C4—64no—
687C4—84no—
688C4—244no—
689C3C2×#178211yescharacteristic, complemented, Sylow
690C2—22no—
691C2—22no—
692C2—22no—
693C2—21no—
694C2—21no—
695C2—24no—
696C2—44no—
697C2—124no—
698C2#7002714yesdirect product factor
699C2C2×#474012yes—
700C2C2×#495511yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC22×C3×#99C2
socleC23×C3C2×C22⋊C4
prime radicalC23×#237C2
cosocleC24×C3C24
hypercenterC22×#99D3
hypocenterC3C2×#1782
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
2. C22×C3×#99C2semidirect
3. C22×#834C2semidirect
4. C22×#806C2semidirect
5. C2×#4943C2semidirect
6. #70027C2direct (N × H)
9. C23×#67C4semidirect
11. C24×Dic3C4semidirect
18. C22×#237C4semidirect
20. C23×C4×D3C4semidirect
21. C23×Dic6C4semidirect
25. C2×C3×#99C22semidirect
27. C2×#834C22semidirect
30. C2×#806C22semidirect
32. #4943C22semidirect
36. C22×#99D3semidirect
60. C22×#67C2×C4semidirect
75. C23×Dic3C2×C4semidirect
94. C22×C4×D3C2×C4semidirect
99. C22×Dic6C2×C4semidirect
133. C2×#99C2×D3semidirect
643. C2×C3#1782semidirect
689. C3C2×#1782semidirect
698. C2#70027direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#2372
C22×C3×#992
C22×#8342
C22×#8062
C2×#49432
#700272
C2×#17823