Finite Group Databasebeta

← C2×#70060 #10947965

Subgroups — C2×#70060 #10947965

Order 768.

1338 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2C2×C3×#410C211yesmaximal, characteristic, complemented
3C2×#5240C211yesmaximal, characteristic, complemented
4#70060C218yesmaximal, direct product factor
5C2×#4474C212yesmaximal, complemented
6C2×#4473C212yesmaximal, complemented
7C2×#2126—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C22×#235C411yescharacteristic, complemented
10C2×C3×#117—21no—
11#5832—22no—
12#5832—22no—
13#5832C414yescomplemented
14#5832C2214yes—
15C2×#947C411yescharacteristic, complemented
16C2×#944C411yescharacteristic, complemented
17C2×#940C2211yescharacteristic
18C2×#939—21no—
19C2×#939—21no—
20C2×#936C2211yescharacteristic
21C2×C4×#67—21no—
22C2×C4×#67—21no—
23C2×#926—21no—
24C22×C4×Dic3C411yescharacteristic, complemented
25C3×#410—22no—
26C3×#410C2214yescomplemented
27C2×C3×#99—21no—
28C2×C3×#99C2212yes—
29#5240—22no—
30#5240C2214yescomplemented
31C2×#834—21no—
32C2×#834C2212yes—
33#4474C2218yescomplemented
34#4473C2218yescomplemented
35C2×#532—31no—
36C2×#410—31no—
37C2×#410D311yescharacteristic, complemented
38#2126—38no—
39C2×#354—32no—
40C2×#353—32no—
41C2×C3×#41—41no—
42C2×C3×#41—41no—
43C2×C3×#41—42no—
44C2×C3×#41C2×C412yes—
45C2×C3×#41D412yes—
46C2×C3×#41D412yes—
47C2×C3×#41C2312yes—
48C22×C3×Q8C2×C411yescharacteristic
49C22×C3×D4D411yescharacteristic
50C22×C3×D4D411yescharacteristic
51C22×C3×D4C2311yescharacteristic
52C23×C3×C4C2×C411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C22×#67—21no—
56C22×#67—21no—
57C22×#67—21no—
58C23×Dic3—21no—
59C2×#235—22no—
60C2×#235—42no—
61C2×#235—41no—
62C2×#235—41no—
63C2×#235C2×C414yescomplemented
64C22×D12C2×C411yescharacteristic
65C22×C4×D3—21no—
66C22×Dic6C2×C411yescharacteristic
67C3×#117—24no—
68C3×#117—42no—
69C2×C3×C4⋊C4—41no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C42—41no—
72C2×#217—21no—
73C2×#217—21no—
74C2×#217—22no—
75#947—22no—
76#947C2×C414yescomplemented
77#944—22no—
78#944C2×C414yescomplemented
79#940—21no—
80#940—21no—
81#940C2×C412yes—
82#940C2312yes—
83#939—24no—
84#939—24no—
85#939—24no—
86#939—24no—
87#936—21no—
88#936—21no—
89#936C2×C412yes—
90#936C2312yes—
91C4×#67—24no—
92C4×#67—24no—
93C4×#67—24no—
94C4×#67—24no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97#926—24no—
98#926—42no—
99C2×C4⋊Dic3—41no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3—41no—
105C2×C4×Dic3C2×C411yescharacteristic
106C2×C4×Dic3C2×C411yescharacteristic
107C2×C4×Dic3C2×C414yescomplemented
108C3×#99—24no—
109C3×#99—22no—
110C3×#99C2314yes—
111#834—24no—
112#834—22no—
113#834C2314yes—
114C22×#41—31no—
115C22×#41C2×D311yescharacteristic
116#532—34no—
117#532—34no—
118#532—62no—
119#532—62no—
120C2×C4⋊Q8—31no—
121C2×#127—31no—
122C2×#124—31no—
123C2×#124—31no—
124C2×#121—61no—
125C2×#121—61no—
126C2×C4×D4—61no—
127C2×C4×D4—61no—
128C2×#117—21no—
129C2×#117—61no—
130C22×C42—31no—
131#410—22no—
132#410—34no—
133#410—62no—
134#410C2×D314yescomplemented
135C2×#99—21no—
136C2×#99—32no—
137C2×#99—61no—
138C2×#99C2×D312yes—
139#354—38no—
140#353—38no—
141C24×C3C22⋊C411yescharacteristic
142C24×C3C2×D411yescharacteristic
143C24×C3C2×D411yescharacteristic
144C23×D3—21no—
145C3×#41—42no—
146C3×#41—42no—
147C3×#41—42no—
148C3×#41—42no—
149C3×#41—42no—
150C3×#41—42no—
151C3×#41—42no—
152C3×#41—42no—
153C2×C3×Q8—41no—
154C2×C3×Q8—41no—
155C2×C3×Q8C22⋊C412yes—
156C2×C3×Q8C22×C412yes—
157C2×C3×D4—41no—
158C2×C3×D4—41no—
159C2×C3×D4—41no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C2×C3×D4—41no—
163C2×C3×D4C22⋊C412yes—
164C2×C3×D4C22⋊C412yes—
165C2×C3×D4C22×C411yescharacteristic
166C2×C3×D4C2×D411yescharacteristic
167C2×C3×D4C2×D412yes—
168C2×C3×D4C2×D411yescharacteristic
169C2×C3×D4C2×D412yes—
170C2×C3×D4C2411yescharacteristic
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—42no—
181C22×C3×C4C22⋊C412yes—
182C22×C3×C4C22⋊C411yescharacteristic
183C22×C3×C4C22⋊C411yescharacteristic
184C22×C3×C4C22⋊C412yes—
185C22×C3×C4C22⋊C412yes—
186C22×C3×C4C22⋊C411yescharacteristic
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C22×C412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C2×#67—22no—
192C2×#67—22no—
193C2×#67—24no—
194C2×#67—22no—
195C2×#67—22no—
196C2×#67—44no—
197C2×#67—44no—
198C2×#67—42no—
199C2×#67—42no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—24no—
203C22×Dic3—21no—
204C22×Dic3—44no—
205C22×Dic3—42no—
206#235—44no—
207#235—44no—
208#235—44no—
209#235—44no—
210C2×D12—21no—
211C2×D12—41no—
212C2×D12—41no—
213C2×D12C22×C412yes—
214C2×C4×D3—24no—
215C2×C4×D3—42no—
216C2×C4×D3—41no—
217C2×C4×D3—41no—
218C2×Dic6—21no—
219C2×Dic6—41no—
220C2×Dic6—41no—
221C2×Dic6C22×C412yes—
222C3×C4⋊C4—44no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—42no—
225C3×C42—44no—
226#217—22no—
227#217—22no—
228#217—22no—
229#217—22no—
230#217—24no—
231#217—42no—
232(C2×D3)⋊C4—24no—
233(C2×D3)⋊C4—24no—
234C4⋊Dic3—44no—
235Dic3⋊C4—24no—
236Dic3⋊C4—24no—
237C4×Dic3—21no—
238C4×Dic3—21no—
239C4×Dic3—24no—
240C4×Dic3—44no—
241C4×Dic3C22×C412yes—
242C4×Dic3C22×C412yes—
243C2×#41—34no—
244C2×#41—41no—
245C2×#41—42no—
246C2×#41—41no—
247C2×#41—62no—
248C2×#41—122no—
249C2×#41—121no—
250C2×#41—121no—
251C2×#41C4×D312yes—
252C2×#41D1212yes—
253C2×#41#6712yes—
254C2×#41C22×D312yes—
255C22×Q8—31no—
256C22×Q8C4×D311yescharacteristic
257C22×D4—31no—
258C22×D4—61no—
259C22×D4—61no—
260C22×D4—61no—
261C22×D4D1211yescharacteristic
262C22×D4#6711yescharacteristic
263C22×D4C22×D311yescharacteristic
264C23×C4—61no—
265C23×C4—61no—
266C23×C4C4×D311yescharacteristic
267C23×C4D1211yescharacteristic
268C23×C4#6711yescharacteristic
269C4⋊Q8—34no—
270C4⋊Q8—62no—
271#127—32no—
272#127—32no—
273#127—61no—
274#127—61no—
275#124—34no—
276#124—32no—
277#124—32no—
278#124—62no—
279#124—61no—
280#124—61no—
281#121—64no—
282#121—64no—
283#121—64no—
284#121—64no—
285C4×D4—64no—
286C4×D4—64no—
287C4×D4—64no—
288C4×D4—64no—
289#117—24no—
290#117—42no—
291#117—64no—
292#117—122no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—121no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—62no—
303C2×C42—31no—
304C2×C42—31no—
305C2×C42—34no—
306C2×C42—41no—
307C2×C42—62no—
308C2×C42—61no—
309C2×C42—121no—
310#99—24no—
311#99—22no—
312#99—34no—
313#99—62no—
314#99—64no—
315#99C22×D314yes—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3—42no—
325C23×C3C2×C22⋊C411yescharacteristic
326C23×C3C2×C22⋊C411yescharacteristic
327C23×C3C2×C22⋊C411yescharacteristic
328C23×C3C2×C22⋊C412yes—
329C23×C3C22×D411yescharacteristic
330C23×C3C22×D411yescharacteristic
331C22×D3—22no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—41no—
335C22×D3—41no—
336C3×Q8—42no—
337C3×Q8—42no—
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—41no—
341C3×D4—41no—
342C3×D4—42no—
343C3×D4—41no—
344C3×D4—42no—
345C3×D4—41no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
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—22no—
364C2×C3×C4—44no—
365C2×C3×C4—84no—
366C2×C3×C4#9912yes—
367C2×C3×C4#9911yescharacteristic
368C2×C3×C4#9911yescharacteristic
369C2×C3×C4C2×C22⋊C411yescharacteristic
370C2×C3×C4C2×C22⋊C412yes—
371C2×C3×C4C2×C22⋊C412yes—
372C2×C3×C4C2×C22⋊C412yes—
373C2×C3×C4C23×C411yescharacteristic
374#67—44no—
375#67—44no—
376#67—44no—
377#67—44no—
378#67—48no—
379C2×Dic3—24no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—48no—
384C2×Dic3—42no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—44no—
388C2×Dic3—84no—
389D12—42no—
390D12—42no—
391C4×D3—44no—
392C4×D3—44no—
393Dic6—42no—
394Dic6—42no—
395C24—61no—
396C24(C2×D3)⋊C411yescharacteristic
397C24C2×D1211yescharacteristic
398C24C2×#6711yescharacteristic
399#41—42no—
400#41—42no—
401#41—42no—
402#41—42no—
403#41—42no—
404#41—42no—
405#41—42no—
406#41—42no—
407#41—124no—
408#41—124no—
409#41—124no—
410#41—124no—
411C2×Q8—32no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8—61no—
415C2×Q8—121no—
416C2×Q8—121no—
417C2×Q8(C2×D3)⋊C412yes—
418C2×Q8C2×C4×D312yes—
419C2×D4—32no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—64no—
429C2×D4—62no—
430C2×D4—62no—
431C2×D4—61no—
432C2×D4—121no—
433C2×D4—121no—
434C2×D4—124no—
435C2×D4—124no—
436C2×D4—122no—
437C2×D4—122no—
438C2×D4(C2×D3)⋊C412yes—
439C2×D4(C2×D3)⋊C412yes—
440C2×D4C2×C4×D311yescharacteristic
441C2×D4C2×D1212yes—
442C2×D4C2×D1211yescharacteristic
443C2×D4C2×#6712yes—
444C2×D4C2×#6711yescharacteristic
445C2×D4C23×D311yescharacteristic
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—64no—
459C22×C4—61no—
460C22×C4—64no—
461C22×C4—124no—
462C22×C4—122no—
463C22×C4—121no—
464C22×C4—121no—
465C22×C4—122no—
466C22×C4(C2×D3)⋊C412yes—
467C22×C4(C2×D3)⋊C411yescharacteristic
468C22×C4(C2×D3)⋊C411yescharacteristic
469C22×C4(C2×D3)⋊C412yes—
470C22×C4(C2×D3)⋊C412yes—
471C22×C4(C2×D3)⋊C411yescharacteristic
472C22×C4C2×C4×D311yescharacteristic
473C22×C4C2×C4×D312yes—
474C22×C4C2×D1212yes—
475C22×C4C2×#6712yes—
476C4⋊C4—44no—
477C4⋊C4—64no—
478C4⋊C4—64no—
479C4⋊C4—124no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—64no—
487C22⋊C4—64no—
488C22⋊C4—64no—
489C22⋊C4—122no—
490C42—32no—
491C42—32no—
492C42—44no—
493C42—64no—
494C42—61no—
495C42—61no—
496C42—124no—
497C22×C3—22no—
498C22×C3—21no—
499C22×C3—22no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—42no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3C2×#9911yescharacteristic
515C22×C3C2×#9911yescharacteristic
516C22×C3C22×C22⋊C411yescharacteristic
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—82no—
521C2×D3—82no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—22no—
525C3×C4—22no—
526C3×C4—84no—
527C3×C4C2×#9912yes—
528C3×C4C2×#9912yes—
529Dic3—24no—
530Dic3—42no—
531Dic3—42no—
532Dic3—84no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—62no—
543C23—122no—
544C23—122no—
545C23—121no—
546C23—121no—
547C23C2×(C2×D3)⋊C411yescharacteristic
548C23C2×(C2×D3)⋊C411yescharacteristic
549C23C2×(C2×D3)⋊C411yescharacteristic
550C23C2×(C2×D3)⋊C412yes—
551C23C22×D1211yescharacteristic
552C23C22×#6711yescharacteristic
553Q8—42no—
554Q8—42no—
555Q8—122no—
556Q8—122no—
557D4—42no—
558D4—42no—
559D4—41no—
560D4—41no—
561D4—42no—
562D4—41no—
563D4—42no—
564D4—41no—
565D4—122no—
566D4—122no—
567D4—124no—
568D4—124no—
569D4—124no—
570D4—124no—
571D4—128no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—44no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—84no—
596C2×C4—128no—
597C2×C4—122no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—124no—
601C2×C4—124no—
602C2×C4—124no—
603C2×C4—244no—
604C2×C4#80611yescharacteristic
605C2×C4#80612yes—
606C2×C4#80611yescharacteristic
607C2×C4C2×(C2×D3)⋊C411yescharacteristic
608C2×C4C2×(C2×D3)⋊C412yes—
609C2×C4C2×(C2×D3)⋊C412yes—
610C2×C4C2×(C2×D3)⋊C412yes—
611C2×C4C22×C4×D311yescharacteristic
612C2×C3—21no—
613C2×C3—21no—
614C2×C3—41no—
615C2×C3—41no—
616C2×C3—42no—
617C2×C3—41no—
618C2×C3—41no—
619C2×C3#212612yescomplemented
620C2×C3C22×#9911yescharacteristic
621D3—82no—
622C22—22no—
623C22—21no—
624C22—21no—
625C22—22no—
626C22—21no—
627C22—21no—
628C22—42no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—242no—
643C22—242no—
644C22C2×#80611yescharacteristic
645C22C2×#80611yescharacteristic
646C22C22×(C2×D3)⋊C411yescharacteristic
647C4—21no—
648C4—22no—
649C4—22no—
650C4—21no—
651C4—64no—
652C4—84no—
653C4—122no—
654C4—122no—
655C4—244no—
656C4C2×#80612yes—
657C4C2×#80612yes—
658C3C2×#212611yescharacteristic, complemented, Sylow
659C2—21no—
660C2—21no—
661C2—41no—
662C2—41no—
663C2—42no—
664C2—41no—
665C2—41no—
666C2—242no—
667C2#7006012yesdirect product factor
668C2C22×#80611yescharacteristic
subgroupG/H
centerC2×C4#806
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC2×C3×#410C2
socleC22×C3C2×#99
prime radicalC2×#5832C2
cosocleC2×C3×D4C24
hypercenterC2×#410D3
hypocenterC3C2×#2126
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×C3×#410C2semidirect
3. C2×#5240C2semidirect
4. #70060C2direct (N × H)
5. C2×#4474C2semidirect
6. C2×#4473C2semidirect
9. C22×#235C4semidirect
13. #5832C4semidirect
15. C2×#947C4semidirect
16. C2×#944C4semidirect
24. C22×C4×Dic3C4semidirect
26. C3×#410C22semidirect
30. #5240C22semidirect
33. #4474C22semidirect
34. #4473C22semidirect
37. C2×#410D3semidirect
63. C2×#235C2×C4semidirect
76. #947C2×C4semidirect
78. #944C2×C4semidirect
107. C2×C4×Dic3C2×C4semidirect
134. #410C2×D3semidirect
619. C2×C3#2126semidirect
658. C3C2×#2126semidirect
667. C2#70060direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#58322
C2×C3×#4102
C2×#52402
#700602
C2×#44742
C2×#44732
C2×#21263