Finite Group Databasebeta

← C2×#68959 #10794581

Subgroups — C2×#68959 #10794581

Order 768.

2066 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67⋊C4C211yesmaximal, characteristic
2C22×#806C212yesmaximal
3C2×#4942C212yesmaximal, complemented
4C2×C3×#342C211yesmaximal, characteristic, complemented
5C2×#4526C211yesmaximal, characteristic, complemented
6#68959C218yesmaximal, direct product factor
7C2×#1799—31nomaximal, Sylow
8C23×#67C2211yescharacteristic
9C24×Dic3C2211yescharacteristic
10C23×C4×D3C411yescharacteristic, complemented
11C22×C3×C22⋊C4C2212yes—
12C22×C3×C22⋊C4C2211yescharacteristic
13C22×#217C2212yes—
14C22×(C2×D3)⋊C4C411yescharacteristic, complemented
15C22×Dic3⋊C4—22no—
16C22×Dic3⋊C4C411yescharacteristic, complemented
17C22×C4×Dic3C411yescharacteristic, complemented
18C2×#67⋊C4—22no—
19C2×#67⋊C4—24no—
20C2×#67⋊C4—22no—
21C2×#67⋊C4C414yescomplemented
22C2×#67⋊C4C2214yes—
23C2×C3⋊#95—22no—
24C2×#806—28no—
25C2×#806C2218yes—
26#4942—24no—
27#4942C2218yescomplemented
28C3×#342C2214yescomplemented
29#4526C2214yescomplemented
30C22×C4×D4—31no—
31C22×#99—32no—
32C2×#380—32no—
33#1799—38no—
34C2×#342—31no—
35C2×#342D311yescharacteristic, complemented
36C25×C3C2311yescharacteristic
37C24×D3C2×C411yescharacteristic
38C23×C3×C4—22no—
39C23×C3×C4C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C22×#67—22no—
42C22×#67—22no—
43C22×#67—24no—
44C22×#67C2×C412yes—
45C22×#67D414yes—
46C22×#67Q814yes—
47C22×#67C2312yes—
48C23×Dic3—22no—
49C23×Dic3—44no—
50C23×Dic3C2×C411yescharacteristic
51C23×Dic3C2×C411yescharacteristic
52C23×Dic3C2×C412yes—
53C23×Dic3C2×C411yescharacteristic
54C23×Dic3D412yes—
55C23×Dic3D414yes—
56C23×Dic3D412yes—
57C23×Dic3C2312yes—
58C22×C4×D3—22no—
59C22×C4×D3—21no—
60C22×C4×D3—21no—
61C22×C4×D3—44no—
62C22×C4×D3C2×C414yescomplemented
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4C2×C412yes—
69C2×C3×C22⋊C4C2314yes—
70C2×C3×C22⋊C4C2312yes—
71C2×#217—24no—
72C2×#217—24no—
73C2×#217—22no—
74C2×#217C2314yes—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4C2×C414yescomplemented
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—28no—
81C2×Dic3⋊C4—48no—
82C2×Dic3⋊C4C2×C414yescomplemented
83C2×C4×Dic3—22no—
84C2×C4×Dic3—21no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3C2×C414yescomplemented
87#67⋊C4—28no—
88#67⋊C4—28no—
89#67⋊C4—28no—
90#67⋊C4—28no—
91C3⋊#95—28no—
92#806—216no—
93C23×D4—31no—
94C24×C4—31no—
95C24×C4—31no—
96C2×C4×D4—34no—
97C2×C4×D4—34no—
98C2×C4×D4—62no—
99C2×C4×D4—62no—
100C2×C4×D4—64no—
101C22×C4⋊C4—31no—
102C22×C4⋊C4—62no—
103C22×C22⋊C4—31no—
104C22×C22⋊C4—32no—
105C22×C22⋊C4C2×D312yes—
106C22×C22⋊C4C2×D311yescharacteristic
107C22×C42—31no—
108C2×#99—38no—
109C2×#99—68no—
110#380—38no—
111#380—64no—
112C2×#95—62no—
113#342—34no—
114#342C2×D314yescomplemented
115C24×C3—42no—
116C24×C3—42no—
117C24×C3C22×C411yescharacteristic
118C24×C3C22×C411yescharacteristic
119C24×C3C2×D412yes—
120C24×C3C2×D412yes—
121C24×C3C2×D412yes—
122C24×C3C2×Q812yes—
123C24×C3#4114yes—
124C24×C3C2411yescharacteristic
125C23×D3—21no—
126C23×D3—21no—
127C23×D3—22no—
128C23×D3—42no—
129C23×D3—42no—
130C23×D3C4⋊C414yescomplemented
131C23×D3C22×C412yes—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—24no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—44no—
140C22×C3×C4—44no—
141C22×C3×C4—44no—
142C22×C3×C4C22×C412yes—
143C22×C3×C4C22×C412yes—
144C2×#67—24no—
145C2×#67—24no—
146C2×#67—28no—
147C2×#67—28no—
148C2×#67—28no—
149C2×#67—28no—
150C2×#67—24no—
151C2×#67—24no—
152C2×#67C4⋊C418yescomplemented
153C2×#67C2×D414yes—
154C2×#67C2×Q814yes—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—28no—
159C22×Dic3—24no—
160C22×Dic3—24no—
161C22×Dic3—44no—
162C22×Dic3—48no—
163C22×Dic3—44no—
164C22×Dic3—42no—
165C22×Dic3—44no—
166C22×Dic3—44no—
167C22×Dic3—44no—
168C22×Dic3—44no—
169C22×Dic3C22⋊C414yes—
170C22×Dic3C22⋊C412yes—
171C22×Dic3C22⋊C411yescharacteristic
172C22×Dic3C22⋊C412yes—
173C22×Dic3C22⋊C414yes—
174C22×Dic3C22⋊C412yes—
175C22×Dic3C22⋊C411yescharacteristic
176C22×Dic3C4⋊C414yescomplemented
177C22×Dic3C22×C412yes—
178C22×Dic3C22×C412yes—
179C22×Dic3C22×C412yes—
180C22×Dic3C2×D414yes—
181C22×Dic3C2×D414yes—
182C2×C4×D3—24no—
183C2×C4×D3—24no—
184C2×C4×D3—44no—
185C2×C4×D3—48no—
186C2×C4×D3—44no—
187C2×C4×D3—48no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—44no—
193#217—24no—
194#217—24no—
195#217—44no—
196(C2×D3)⋊C4—28no—
197Dic3⋊C4—28no—
198Dic3⋊C4—416no—
199C4×Dic3—24no—
200C4×Dic3—24no—
201C25—31no—
202C25C22×D311yescharacteristic
203C22×D4—32no—
204C22×D4—34no—
205C22×D4—34no—
206C22×D4—32no—
207C22×D4—64no—
208C22×D4—62no—
209C22×D4—62no—
210C23×C4—22no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—32no—
214C23×C4—31no—
215C23×C4—32no—
216C23×C4—32no—
217C23×C4—34no—
218C23×C4—32no—
219C23×C4—34no—
220C23×C4—62no—
221C23×C4—61no—
222C23×C4—61no—
223C23×C4—62no—
224C23×C4—124no—
225C23×C4—124no—
226C23×C4C4×D311yescharacteristic
227C23×C4C4×D311yescharacteristic
228C4×D4—68no—
229C4×D4—68no—
230C4×D4—68no—
231C4×D4—68no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—68no—
236C2×C4⋊C4—128no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—34no—
243C2×C22⋊C4—34no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—64no—
247C2×C22⋊C4—64no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4C4×D312yes—
250C2×C22⋊C4C22×D314yes—
251C2×C22⋊C4C22×D312yes—
252C2×C42—34no—
253C2×C42—62no—
254C2×C42—61no—
255C2×C42—61no—
256#99—616no—
257#95—68no—
258C23×C3—24no—
259C23×C3—22no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—44no—
267C23×C3—42no—
268C23×C3—44no—
269C23×C3—48no—
270C23×C3—41no—
271C23×C3C2×C22⋊C412yes—
272C23×C3C2×C22⋊C411yescharacteristic
273C23×C3C2×C22⋊C411yescharacteristic
274C23×C3C2×C4⋊C412yes—
275C23×C3C2×C4⋊C412yes—
276C23×C3#11712yes—
277C23×C3#11712yes—
278C23×C3#12114yes—
279C23×C3#12114yes—
280C23×C3C22⋊Q814yes—
281C23×C3C22⋊Q814yes—
282C23×C3C23×C411yescharacteristic
283C23×C3C22×D412yes—
284C23×C3C22×D411yescharacteristic
285C23×C3C22×Q811yescharacteristic
286C23×C3C2×#4112yes—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—24no—
290C22×D3—24no—
291C22×D3—42no—
292C22×D3—48no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—42no—
297C22×D3—48no—
298C22×D3—42no—
299C22×D3C2×C4⋊C414yescomplemented
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—44no—
309C2×C3×C4—44no—
310C2×C3×C4—44no—
311C2×C3×C4—88no—
312#67—28no—
313#67—28no—
314#67—216no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—24no—
318C2×Dic3—24no—
319C2×Dic3—22no—
320C2×Dic3—24no—
321C2×Dic3—28no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—28no—
325C2×Dic3—44no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—44no—
329C2×Dic3—44no—
330C2×Dic3—44no—
331C2×Dic3—44no—
332C2×Dic3—88no—
333C2×Dic3#9914yescomplemented
334C2×Dic3C2×C22⋊C412yes—
335C2×Dic3C2×C22⋊C414yes—
336C2×Dic3C2×C22⋊C412yes—
337C2×Dic3C2×C4⋊C414yescomplemented
338C4×D3—48no—
339C4×D3—48no—
340C24—34no—
341C24—32no—
342C24—42no—
343C24—42no—
344C24—62no—
345C24—61no—
346C24—61no—
347C24—122no—
348C24—122no—
349C24C2×C4×D311yescharacteristic
350C24C2×C4×D311yescharacteristic
351C24#23514yes—
352C24D3×D412yes—
353C24D3×D412yes—
354C24D3×D412yes—
355C24D3×Q812yes—
356C24C23×D311yescharacteristic
357C2×D4—38no—
358C2×D4—34no—
359C2×D4—34no—
360C2×D4—64no—
361C2×D4—64no—
362C2×D4—68no—
363C2×D4—68no—
364C2×D4—68no—
365C2×D4—68no—
366C2×D4—64no—
367C2×D4—64no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—31no—
374C22×C4—32no—
375C22×C4—34no—
376C22×C4—31no—
377C22×C4—32no—
378C22×C4—34no—
379C22×C4—32no—
380C22×C4—32no—
381C22×C4—34no—
382C22×C4—32no—
383C22×C4—34no—
384C22×C4—32no—
385C22×C4—34no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—64no—
392C22×C4—64no—
393C22×C4—68no—
394C22×C4—64no—
395C22×C4—62no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—64no—
399C22×C4—124no—
400C22×C4—128no—
401C22×C4—124no—
402C22×C4—128no—
403C22×C4—124no—
404C22×C4—128no—
405C22×C4—124no—
406C22×C4—124no—
407C22×C4—122no—
408C22×C4—124no—
409C22×C4—124no—
410C22×C4—124no—
411C22×C4C2×C4×D312yes—
412C22×C4C2×C4×D312yes—
413C4⋊C4—68no—
414C4⋊C4—1216no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—68no—
421C22⋊C4—64no—
422C22⋊C4—64no—
423C22⋊C4—124no—
424C42—64no—
425C42—64no—
426C22×C3—28no—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—28no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—22no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—42no—
445C22×C3#38012yes—
446C22×C3#38014yes—
447C22×C3#38012yes—
448C22×C3C2×#9912yes—
449C22×C3#41012yes—
450C22×C3C22×C22⋊C411yescharacteristic
451C22×C3C22×C4⋊C411yescharacteristic
452C22×C3C2×#11711yescharacteristic
453C22×C3C2×#12112yes—
454C22×C3C2×C22⋊Q812yes—
455C2×D3—24no—
456C2×D3—44no—
457C2×D3—48no—
458C2×D3—48no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—44no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—84no—
465Dic3—22no—
466Dic3—24no—
467Dic3—24no—
468Dic3—84no—
469Dic3C2×#9914yescomplemented
470C23—24no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—34no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—48no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—41no—
484C23—64no—
485C23—64no—
486C23—62no—
487C23—62no—
488C23—122no—
489C23—122no—
490C23—128no—
491C23—122no—
492C23—122no—
493C23—128no—
494C23—122no—
495C23—122no—
496C23#87212yes—
497C23#87212yes—
498C23D3×C22⋊C412yes—
499C23D3×C22⋊C411yescharacteristic
500C23D3×C22⋊C411yescharacteristic
501C23#88414yes—
502C23#88414yes—
503C23D3×C4⋊C412yes—
504C23D3×C4⋊C412yes—
505C23#89614yes—
506C23#89614yes—
507C23C22×C4×D311yescharacteristic
508C23C2×#23512yes—
509C23C2×D3×D412yes—
510C23C2×D3×D411yescharacteristic
511C23C2×D3×Q811yescharacteristic
512D4—616no—
513D4—68no—
514D4—68no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—32no—
518C2×C4—34no—
519C2×C4—32no—
520C2×C4—34no—
521C2×C4—34no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—68no—
532C2×C4—64no—
533C2×C4—64no—
534C2×C4—68no—
535C2×C4—64no—
536C2×C4—64no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—64no—
541C2×C4—88no—
542C2×C4—128no—
543C2×C4—128no—
544C2×C4—124no—
545C2×C4—124no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—124no—
549C2×C4—124no—
550C2×C4—124no—
551C2×C4—248no—
552C2×C3—22no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—24no—
558C2×C3#179914yescomplemented
559C2×C3C2×#38011yescharacteristic
560C2×C3C22×#9911yescharacteristic
561C2×C3C2×#41011yescharacteristic
562D3—44no—
563C22—22no—
564C22—21no—
565C22—21no—
566C22—22no—
567C22—28no—
568C22—24no—
569C22—24no—
570C22—28no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—41no—
581C22—41no—
582C22—64no—
583C22—124no—
584C22—128no—
585C22—124no—
586C22—122no—
587C22—128no—
588C22—122no—
589C22#466712yes—
590C22#466714yes—
591C22#466712yes—
592C22#474012yes—
593C22D3×#9912yes—
594C22C2×#87211yescharacteristic
595C22C2×D3×C22⋊C411yescharacteristic
596C22C2×#88412yes—
597C22C2×D3×C4⋊C411yescharacteristic
598C22C2×#89612yes—
599C4—34no—
600C4—42no—
601C4—42no—
602C4—62no—
603C4—64no—
604C4—64no—
605C4—84no—
606C4—244no—
607C3C2×#179911yescharacteristic, complemented, Sylow
608C2—22no—
609C2—22no—
610C2—21no—
611C2—21no—
612C2—24no—
613C2—22no—
614C2—124no—
615C2#6895914yesdirect product factor
616C2C2×#466711yescharacteristic
617C2C2×#474011yescharacteristic
618C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC2×C3×#342C2
socleC23×C3C2×C22⋊C4
prime radicalC23×#67C22
cosocleC24×C3C24
hypercenterC2×#342D3
hypocenterC3C2×#1799
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. C2×#4942C2semidirect
4. C2×C3×#342C2semidirect
5. C2×#4526C2semidirect
6. #68959C2direct (N × H)
10. C23×C4×D3C4semidirect
14. C22×(C2×D3)⋊C4C4semidirect
16. C22×Dic3⋊C4C4semidirect
17. C22×C4×Dic3C4semidirect
21. C2×#67⋊C4C4semidirect
27. #4942C22semidirect
28. C3×#342C22semidirect
29. #4526C22semidirect
35. C2×#342D3semidirect
62. C22×C4×D3C2×C4semidirect
77. C2×(C2×D3)⋊C4C2×C4semidirect
82. C2×Dic3⋊C4C2×C4semidirect
86. C2×C4×Dic3C2×C4semidirect
114. #342C2×D3semidirect
130. C23×D3C4⋊C4semidirect
152. C2×#67C4⋊C4semidirect
176. C22×Dic3C4⋊C4semidirect
299. C22×D3C2×C4⋊C4semidirect
333. C2×Dic3#99semidirect
337. C2×Dic3C2×C4⋊C4semidirect
469. Dic3C2×#99semidirect
558. C2×C3#1799semidirect
607. C3C2×#1799semidirect
615. C2#68959direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#67⋊C42
C22×#8062
C2×#49422
C2×C3×#3422
C2×#45262
#689592
C2×#17993