Finite Group Databasebeta

← C2×#70028 #10794580

Subgroups — C2×#70028 #10794580

Order 768.

2066 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic3C211yesmaximal, characteristic
2C22×C3×#99C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic
4C2×#4942C212yesmaximal, complemented
5C2×#4527C212yesmaximal, complemented
6#70028C218yesmaximal, direct product factor
7C2×#1799—31nomaximal, Sylow
8C23×C3×D4C2211yescharacteristic
9C24×Dic3C411yescharacteristic, complemented
10C24×Dic3C2211yescharacteristic
11C22×C3×C22⋊C4C2212yes—
12C22×#217C411yescharacteristic, complemented
13C22×#217C2212yes—
14C22×#217C2211yescharacteristic
15C2×D4×Dic3—22no—
16C2×D4×Dic3—24no—
17C2×D4×Dic3—22no—
18C2×D4×Dic3C414yescomplemented
19C2×D4×Dic3C2214yes—
20C22×C4⋊Dic3C411yescharacteristic, complemented
21C22×Dic3⋊C4—22no—
22C22×C4×Dic3C411yescharacteristic, complemented
23C2×C3×#99—24no—
24C2×C3×#99C2214yes—
25C2×#834—24no—
26C2×#834C2214yes—
27C2×C3⋊#95—22no—
28#4942—24no—
29#4942C2218yescomplemented
30#4527C2218yescomplemented
31C22×C4×D4—31no—
32C22×#99—31no—
33C22×#99D311yescharacteristic
34C2×#380—32no—
35#1799—38no—
36C2×#342—32no—
37C25×C3C2×C411yescharacteristic
38C25×C3C2311yescharacteristic
39C22×C3×D4—22no—
40C22×C3×D4—22no—
41C22×C3×D4—24no—
42C22×C3×D4C2×C412yes—
43C22×C3×D4D414yes—
44C22×C3×D4Q814yes—
45C22×C3×D4C2312yes—
46C23×C3×C4—22no—
47C23×C3×C4C2×C411yescharacteristic
48C23×Dic3—22no—
49C23×Dic3—21no—
50C23×Dic3—21no—
51C23×Dic3—22no—
52C23×Dic3—44no—
53C23×Dic3—44no—
54C23×Dic3C2×C411yescharacteristic
55C23×Dic3C2×C411yescharacteristic
56C23×Dic3C2×C411yescharacteristic
57C23×Dic3C2×C411yescharacteristic
58C23×Dic3C2×C412yes—
59C23×Dic3C2×C414yescomplemented
60C23×Dic3D412yes—
61C23×Dic3D414yes—
62C23×Dic3D412yes—
63C23×Dic3C2312yes—
64C2×C3×C22⋊C4—24no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4C2314yes—
68C2×#217—22no—
69C2×#217—22no—
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—24no—
73C2×#217—24no—
74C2×#217—22no—
75C2×#217C2×C412yes—
76C2×#217C2×C414yescomplemented
77C2×#217C2314yes—
78C2×#217C2312yes—
79D4×Dic3—28no—
80D4×Dic3—28no—
81D4×Dic3—28no—
82D4×Dic3—28no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3C2×C414yescomplemented
86C2×Dic3⋊C4—28no—
87C2×Dic3⋊C4—48no—
88C2×C4×Dic3—22no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3C2×C414yescomplemented
92C3×#99—28no—
93#834—28no—
94C3⋊#95—28no—
95C23×D4C2×D311yescharacteristic
96C24×C4—31no—
97C24×C4—31no—
98C2×C4×D4—34no—
99C2×C4×D4—34no—
100C2×C4×D4—62no—
101C2×C4×D4—62no—
102C2×C4×D4—64no—
103C22×C4⋊C4—31no—
104C22×C4⋊C4—62no—
105C22×C22⋊C4—31no—
106C22×C22⋊C4—31no—
107C22×C22⋊C4—32no—
108C22×C22⋊C4C2×D312yes—
109C22×C42—31no—
110C2×#99—24no—
111C2×#99—34no—
112C2×#99—64no—
113C2×#99C2×D314yes—
114#380—38no—
115#380—64no—
116C2×#95—62no—
117#342—38no—
118C24×C3—21no—
119C24×C3—21no—
120C24×C3—22no—
121C24×C3—42no—
122C24×C3—42no—
123C24×C3—42no—
124C24×C3—42no—
125C24×C3C4⋊C414yescomplemented
126C24×C3C22×C411yescharacteristic
127C24×C3C22×C412yes—
128C24×C3C22×C411yescharacteristic
129C24×C3C2×D412yes—
130C24×C3C2×D412yes—
131C24×C3C2×D412yes—
132C24×C3C2×Q812yes—
133C24×C3#4114yes—
134C24×C3C2411yescharacteristic
135C2×C3×D4—24no—
136C2×C3×D4—24no—
137C2×C3×D4—28no—
138C2×C3×D4—28no—
139C2×C3×D4—28no—
140C2×C3×D4—28no—
141C2×C3×D4—24no—
142C2×C3×D4—24no—
143C2×C3×D4C4⋊C418yescomplemented
144C2×C3×D4C2×D414yes—
145C2×C3×D4C2×Q814yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—24no—
150C22×C3×C4—44no—
151C22×C3×C4—44no—
152C22×C3×C4—44no—
153C22×C3×C4C4⋊C414yescomplemented
154C22×C3×C4C22×C412yes—
155C22×Dic3—28no—
156C22×Dic3—24no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—24no—
162C22×Dic3—24no—
163C22×Dic3—24no—
164C22×Dic3—44no—
165C22×Dic3—48no—
166C22×Dic3—44no—
167C22×Dic3—42no—
168C22×Dic3—44no—
169C22×Dic3—44no—
170C22×Dic3—48no—
171C22×Dic3—48no—
172C22×Dic3—44no—
173C22×Dic3—42no—
174C22×Dic3—42no—
175C22×Dic3—44no—
176C22×Dic3—44no—
177C22×Dic3—44no—
178C22×Dic3C22⋊C412yes—
179C22×Dic3C22⋊C411yescharacteristic
180C22×Dic3C22⋊C412yes—
181C22×Dic3C22⋊C414yes—
182C22×Dic3C22⋊C412yes—
183C22×Dic3C22⋊C414yes—
184C22×Dic3C22⋊C411yescharacteristic
185C22×Dic3C22×C412yes—
186C22×Dic3C22×C412yes—
187C22×Dic3C22×C412yes—
188C22×Dic3C22×C412yes—
189C22×Dic3C2×D414yes—
190C22×Dic3C2×D414yes—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—44no—
194#217—24no—
195#217—24no—
196#217—28no—
197#217—24no—
198#217—24no—
199#217—44no—
200C4⋊Dic3—28no—
201Dic3⋊C4—416no—
202C4×Dic3—24no—
203C4×Dic3—24no—
204C25C4×D311yescharacteristic
205C25C22×D311yescharacteristic
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4Dic614yes—
210C22×D4C4×D312yes—
211C22×D4#6714yes—
212C22×D4C22×D312yes—
213C23×C4—22no—
214C23×C4—31no—
215C23×C4—32no—
216C23×C4—31no—
217C23×C4—32no—
218C23×C4—32no—
219C23×C4—34no—
220C23×C4—32no—
221C23×C4—31no—
222C23×C4—31no—
223C23×C4—34no—
224C23×C4—61no—
225C23×C4—62no—
226C23×C4—61no—
227C23×C4—62no—
228C23×C4—124no—
229C23×C4—124no—
230C23×C4C4×D311yescharacteristic
231C4×D4—68no—
232C4×D4—68no—
233C4×D4—68no—
234C4×D4—68no—
235C2×C4⋊C4—34no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—68no—
239C2×C4⋊C4—128no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—32no—
244C2×C22⋊C4—34no—
245C2×C22⋊C4—32no—
246C2×C22⋊C4—34no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—62no—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—64no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4C22×D314yes—
255C2×C42—34no—
256C2×C42—62no—
257C2×C42—61no—
258C2×C42—61no—
259#99—28no—
260#99—68no—
261#95—68no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—24no—
269C23×C3—24no—
270C23×C3—41no—
271C23×C3—42no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—42no—
275C23×C3—44no—
276C23×C3—44no—
277C23×C3—48no—
278C23×C3—41no—
279C23×C3—42no—
280C23×C3—48no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—48no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3C2×C22⋊C412yes—
288C23×C3C2×C22⋊C411yescharacteristic
289C23×C3C2×C22⋊C411yescharacteristic
290C23×C3C2×C4⋊C412yes—
291C23×C3C2×C4⋊C412yes—
292C23×C3C2×C4⋊C414yescomplemented
293C23×C3#11712yes—
294C23×C3#11712yes—
295C23×C3#12114yes—
296C23×C3#12114yes—
297C23×C3C22⋊Q814yes—
298C23×C3C22⋊Q814yes—
299C23×C3C23×C411yescharacteristic
300C23×C3C22×D412yes—
301C23×C3C22×D411yescharacteristic
302C23×C3C22×Q811yescharacteristic
303C23×C3C2×#4112yes—
304C3×D4—28no—
305C3×D4—28no—
306C3×D4—216no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—44no—
312C2×C3×C4—44no—
313C2×C3×C4—44no—
314C2×C3×C4—88no—
315C2×C3×C4C2×C4⋊C414yescomplemented
316C2×Dic3—22no—
317C2×Dic3—24no—
318C2×Dic3—28no—
319C2×Dic3—24no—
320C2×Dic3—24no—
321C2×Dic3—28no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—44no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—44no—
328C2×Dic3—48no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—48no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—44no—
337C2×Dic3—44no—
338C2×Dic3—44no—
339C2×Dic3—88no—
340C2×Dic3#9914yescomplemented
341C2×Dic3C2×C22⋊C412yes—
342C2×Dic3C2×C22⋊C414yes—
343C2×Dic3C2×C22⋊C412yes—
344C24—22no—
345C24—21no—
346C24—21no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24Dic3⋊C414yescomplemented
352C24C2×Dic612yes—
353C24C2×C4×D311yescharacteristic
354C24C2×C4×D312yes—
355C24C2×C4×D311yescharacteristic
356C24D3×D412yes—
357C24D3×D412yes—
358C24#23714yes—
359C24C2×#6712yes—
360C24C23×D311yescharacteristic
361C2×D4—24no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—24no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4Dic3⋊C418yescomplemented
370C2×D4C2×Dic614yes—
371C2×D4C2×#6714yes—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—31no—
377C22×C4—32no—
378C22×C4—34no—
379C22×C4—31no—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—34no—
383C22×C4—32no—
384C22×C4—34no—
385C22×C4—32no—
386C22×C4—34no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—68no—
393C22×C4—64no—
394C22×C4—64no—
395C22×C4—64no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—64no—
401C22×C4—124no—
402C22×C4—124no—
403C22×C4—128no—
404C22×C4—124no—
405C22×C4—124no—
406C22×C4—122no—
407C22×C4—128no—
408C22×C4—124no—
409C22×C4—128no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4—124no—
413C22×C4—124no—
414C22×C4—124no—
415C22×C4Dic3⋊C414yescomplemented
416C22×C4C2×C4×D312yes—
417C4⋊C4—68no—
418C4⋊C4—1216no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—44no—
422C22⋊C4—68no—
423C22⋊C4—64no—
424C22⋊C4—64no—
425C22⋊C4—64no—
426C22⋊C4—64no—
427C22⋊C4—124no—
428C42—64no—
429C42—64no—
430C22×C3—28no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—28no—
434C22×C3—22no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—22no—
442C22×C3—24no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—44no—
446C22×C3—44no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—42no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3—48no—
453C22×C3—48no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3#38012yes—
457C22×C3#38014yes—
458C22×C3#38012yes—
459C22×C3C2×#9912yes—
460C22×C3#41012yes—
461C22×C3C22×C22⋊C411yescharacteristic
462C22×C3C22×C4⋊C411yescharacteristic
463C22×C3C2×#11711yescharacteristic
464C22×C3C2×#12112yes—
465C22×C3C2×C22⋊Q812yes—
466C3×C4—24no—
467C3×C4—84no—
468Dic3—22no—
469Dic3—24no—
470Dic3—42no—
471Dic3—42no—
472Dic3—84no—
473Dic3C2×#9914yescomplemented
474C23—24no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23—44no—
486C23—41no—
487C23—42no—
488C23—48no—
489C23—44no—
490C23—41no—
491C23—42no—
492C23—48no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—48no—
497C23—42no—
498C23—42no—
499C23#87712yes—
500C23#87712yes—
501C23#87814yes—
502C23#87814yes—
503C23D3×C22⋊C412yes—
504C23D3×C22⋊C411yescharacteristic
505C23D3×C22⋊C411yescharacteristic
506C23C2×Dic3⋊C412yes—
507C23C2×Dic3⋊C412yes—
508C23C2×Dic3⋊C414yescomplemented
509C23#93914yes—
510C23#93914yes—
511C23C22×Dic611yescharacteristic
512C23C22×C4×D311yescharacteristic
513C23C2×D3×D412yes—
514C23C2×#23712yes—
515C23C22×#6711yescharacteristic
516D4—216no—
517D4—28no—
518D4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—32no—
524C2×C4—34no—
525C2×C4—32no—
526C2×C4—34no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—68no—
531C2×C4—64no—
532C2×C4—64no—
533C2×C4—68no—
534C2×C4—62no—
535C2×C4—64no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—88no—
539C2×C4—128no—
540C2×C4—124no—
541C2×C4—124no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—128no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4—122no—
551C2×C4—124no—
552C2×C4—124no—
553C2×C4—124no—
554C2×C4—248no—
555C2×C4C2×Dic3⋊C414yescomplemented
556C2×C3—22no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—22no—
560C2×C3—22no—
561C2×C3—24no—
562C2×C3—44no—
563C2×C3#179914yescomplemented
564C2×C3C2×#38011yescharacteristic
565C2×C3C22×#9911yescharacteristic
566C2×C3C2×#41011yescharacteristic
567C22—22no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—28no—
576C22—24no—
577C22—24no—
578C22—28no—
579C22—24no—
580C22—42no—
581C22—44no—
582C22—42no—
583C22—44no—
584C22—41no—
585C22—42no—
586C22—41no—
587C22—44no—
588C22—48no—
589C22—44no—
590C22—42no—
591C22—42no—
592C22—48no—
593C22#474012yes—
594C22D3×#9912yes—
595C22#494212yes—
596C22#494214yes—
597C22#494212yes—
598C22C2×#87711yescharacteristic
599C22C2×#87812yes—
600C22C2×D3×C22⋊C411yescharacteristic
601C22C22×Dic3⋊C411yescharacteristic
602C22C2×#93912yes—
603C4—24no—
604C4—34no—
605C4—62no—
606C4—64no—
607C4—84no—
608C4—122no—
609C4—122no—
610C4—244no—
611C3C2×#179911yescharacteristic, complemented, Sylow
612C2—22no—
613C2—22no—
614C2—21no—
615C2—21no—
616C2—24no—
617C2—22no—
618C2—44no—
619C2#7002814yesdirect product factor
620C2C2×#474011yescharacteristic
621C2C2×D3×#9911yescharacteristic
622C2C2×#494211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC22×C3×#99C2
socleC23×C3C2×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC24×C3C24
hypercenterC22×#99D3
hypocenterC3C2×#1799
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
4. C2×#4942C2semidirect
5. C2×#4527C2semidirect
6. #70028C2direct (N × H)
9. C24×Dic3C4semidirect
12. C22×#217C4semidirect
18. C2×D4×Dic3C4semidirect
20. C22×C4⋊Dic3C4semidirect
22. C22×C4×Dic3C4semidirect
29. #4942C22semidirect
30. #4527C22semidirect
59. C23×Dic3C2×C4semidirect
76. C2×#217C2×C4semidirect
85. C2×C4⋊Dic3C2×C4semidirect
91. C2×C4×Dic3C2×C4semidirect
125. C24×C3C4⋊C4semidirect
143. C2×C3×D4C4⋊C4semidirect
153. C22×C3×C4C4⋊C4semidirect
292. C23×C3C2×C4⋊C4semidirect
315. C2×C3×C4C2×C4⋊C4semidirect
340. C2×Dic3#99semidirect
351. C24Dic3⋊C4semidirect
369. C2×D4Dic3⋊C4semidirect
415. C22×C4Dic3⋊C4semidirect
473. Dic3C2×#99semidirect
508. C23C2×Dic3⋊C4semidirect
555. C2×C4C2×Dic3⋊C4semidirect
563. C2×C3#1799semidirect
611. C3C2×#1799semidirect
619. C2#70028direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Dic32
C22×C3×#992
C22×#8342
C2×#49422
C2×#45272
#700282
C2×#17993