Finite Group Databasebeta

← C2×#70128 #10948020

Subgroups — C2×#70128 #10948020

Order 768.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5816C211yesmaximal, characteristic
2C22×C3×#101C211yesmaximal, characteristic, complemented
3C22×#836C211yesmaximal, characteristic, complemented
4C2×#4478C212yesmaximal, complemented
5#70128C218yesmaximal, direct product factor
6C2×#4477C212yesmaximal, complemented
7C2×#2133—31nomaximal, Sylow
8C23×C3×Q8C2211yescharacteristic
9C22×#235C411yescharacteristic
10C22×C3×M16—21no—
11#5816—22no—
12#5816—22no—
13#5816C414yes—
14#5816C2214yes—
15C2×#947C411yescharacteristic
16C2×#947C2211yescharacteristic
17C2×#946—21no—
18C2×#946—21no—
19C2×#944C411yescharacteristic
20C2×#944C2211yescharacteristic
21C2×#929—21no—
22C2×#929—21no—
23C2×#926C411yescharacteristic
24C22×#208—21no—
25C2×C3×#101—22no—
26C2×C3×#101—21no—
27C2×C3×#101C2212yes—
28C2×C3×#101C2214yescomplemented
29C2×#836—22no—
30C2×#836—21no—
31C2×#836C2212yes—
32C2×#836C2214yescomplemented
33#4478C2218yescomplemented
34#4477C2218yescomplemented
35C2×#536—31no—
36C22×#101—31no—
37C22×#101D311yescharacteristic, complemented
38#2133—38no—
39C2×#356—32no—
40C2×#355—32no—
41C22×C3×Q8—41no—
42C22×C3×Q8—41no—
43C22×C3×Q8—42no—
44C22×C3×Q8C2×C411yescharacteristic
45C22×C3×Q8C2×C412yes—
46C22×C3×Q8D411yescharacteristic
47C22×C3×Q8D412yes—
48C22×C3×Q8D411yescharacteristic
49C22×C3×Q8D412yes—
50C22×C3×Q8C2311yescharacteristic
51C22×C3×Q8C2312yes—
52C23×C3×C4C2×C411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C22×#67—21no—
56C2×#235—22no—
57C2×#235—41no—
58C2×#235—41no—
59C2×#235—42no—
60C2×#235C2×C414yes—
61C22×D12C2×C411yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6C2×C411yescharacteristic
64C2×C3×M16—22no—
65C2×C3×M16—24no—
66C2×C3×M16—41no—
67C2×C3×M16—42no—
68C22×C3×C8—41no—
69C2×#217—21no—
70#947—22no—
71#947—21no—
72#947—21no—
73#947C2×C414yes—
74#947C2×C412yes—
75#947C2312yes—
76#946—24no—
77#946—24no—
78#946—24no—
79#946—24no—
80#944—22no—
81#944—21no—
82#944—21no—
83#944C2×C414yes—
84#944C2×C412yes—
85#944C2312yes—
86#929—24no—
87#929—24no—
88#929—24no—
89#929—24no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94#926—22no—
95#926C2×C414yes—
96C2×C4⋊Dic3—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103C2×#208—22no—
104C2×#208—24no—
105C2×#208—41no—
106C2×#208—42no—
107C22×C3⋊C8—41no—
108C3×#101—22no—
109C3×#101—24no—
110C3×#101C2314yes—
111#836—22no—
112#836—24no—
113#836C2314yes—
114C22×#41—31no—
115C23×Q8C2×D311yescharacteristic
116C22×M16—21no—
117C22×M16—61no—
118#536—34no—
119#536—34no—
120#536—62no—
121#536—62no—
122C2×C4⋊Q8—31no—
123C2×C4⋊Q8—31no—
124C2×#124—31no—
125C2×#124—31no—
126C2×#123—61no—
127C2×#123—61no—
128C2×C22⋊Q8—61no—
129C2×C22⋊Q8—61no—
130C2×#117—31no—
131C2×#101—22no—
132C2×#101—21no—
133C2×#101—32no—
134C2×#101—34no—
135C2×#101—61no—
136C2×#101—62no—
137C2×#101C2×D312yes—
138C2×#101C2×D314yescomplemented
139#356—38no—
140#355—38no—
141C24×C3C22⋊C411yescharacteristic
142C23×D3—21no—
143C2×C3×Q8—41no—
144C2×C3×Q8—41no—
145C2×C3×Q8—42no—
146C2×C3×Q8—42no—
147C2×C3×Q8—41no—
148C2×C3×Q8—42no—
149C2×C3×Q8—41no—
150C2×C3×Q8—42no—
151C2×C3×Q8—41no—
152C2×C3×Q8—42no—
153C2×C3×Q8—41no—
154C2×C3×Q8—42no—
155C2×C3×Q8—41no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C2×C3×Q8—41no—
159C2×C3×Q8C22⋊C412yes—
160C2×C3×Q8C22⋊C412yes—
161C2×C3×Q8C22⋊C412yes—
162C2×C3×Q8C22×C412yes—
163C2×C3×Q8C22×C411yescharacteristic
164C2×C3×Q8C2×D411yescharacteristic
165C2×C3×Q8C2×D412yes—
166C2×C3×Q8C2×D411yescharacteristic
167C2×C3×Q8C2×D412yes—
168C2×C3×Q8C2411yescharacteristic
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4C22⋊C411yescharacteristic
179C22×C3×C4C22⋊C411yescharacteristic
180C22×C3×C4C22⋊C411yescharacteristic
181C22×C3×C4C22⋊C412yes—
182C22×C3×C4C22⋊C412yes—
183C22×C3×C4C22⋊C412yes—
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C22×C412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C2×#67—24no—
191C2×#67—41no—
192C2×#67—41no—
193C2×#67—42no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197#235—44no—
198#235—44no—
199#235—44no—
200#235—44no—
201C2×D12—21no—
202C2×D12—41no—
203C2×D12—41no—
204C2×D12C22×C412yes—
205C2×C4×D3—24no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×Dic6—21no—
209C2×Dic6—41no—
210C2×Dic6—41no—
211C2×Dic6C22×C412yes—
212C3×M16—24no—
213C3×M16—42no—
214C3×M16—44no—
215C2×C3×C8—42no—
216C2×C3×C8—44no—
217#217—24no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—22no—
222(C2×D3)⋊C4—22no—
223(C2×D3)⋊C4—22no—
224C4⋊Dic3—24no—
225Dic3⋊C4—24no—
226Dic3⋊C4—24no—
227Dic3⋊C4—22no—
228Dic3⋊C4—22no—
229Dic3⋊C4—22no—
230Dic3⋊C4—22no—
231C4×Dic3—21no—
232C4×Dic3—21no—
233C4×Dic3C22×C412yes—
234C4×Dic3C22×C412yes—
235#208—24no—
236#208—42no—
237#208—44no—
238C2×C3⋊C8—42no—
239C2×C3⋊C8—44no—
240C2×#41—34no—
241C2×#41—62no—
242C2×#41—122no—
243C2×#41—121no—
244C2×#41—121no—
245C22×Q8—31no—
246C22×Q8—41no—
247C22×Q8—41no—
248C22×Q8—42no—
249C22×Q8C4×D311yescharacteristic
250C22×Q8C4×D312yes—
251C22×Q8D1211yescharacteristic
252C22×Q8D1212yes—
253C22×Q8#6711yescharacteristic
254C22×Q8#6712yes—
255C22×Q8C22×D311yescharacteristic
256C22×Q8C22×D312yes—
257C22×D4—31no—
258C22×D4—61no—
259C23×C4—61no—
260C23×C4C4×D311yescharacteristic
261C23×C4D1211yescharacteristic
262C23×C4#6711yescharacteristic
263C2×M16—22no—
264C2×M16—24no—
265C2×M16—42no—
266C2×M16—41no—
267C2×M16—62no—
268C2×M16—64no—
269C2×M16—122no—
270C2×M16—121no—
271C22×C8—41no—
272C22×C8—121no—
273C4⋊Q8—34no—
274C4⋊Q8—32no—
275C4⋊Q8—32no—
276C4⋊Q8—62no—
277C4⋊Q8—61no—
278C4⋊Q8—61no—
279#124—34no—
280#124—32no—
281#124—32no—
282#124—62no—
283#124—61no—
284#124—61no—
285#123—64no—
286#123—64no—
287#123—64no—
288#123—64no—
289C22⋊Q8—64no—
290C22⋊Q8—64no—
291C22⋊Q8—64no—
292C22⋊Q8—64no—
293#117—34no—
294#117—62no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C42—31no—
306C2×C42—31no—
307#101—22no—
308#101—24no—
309#101—34no—
310#101—64no—
311#101—62no—
312#101C22×D314yes—
313C23×C3—42no—
314C23×C3#9911yescharacteristic
315C23×C3#9911yescharacteristic
316C23×C3#9912yes—
317C23×C3C2×C22⋊C411yescharacteristic
318C23×C3C2×C22⋊C412yes—
319C22×D3—22no—
320C22×D3—44no—
321C22×D3—41no—
322C22×D3—41no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—41no—
328C3×Q8—41no—
329C3×Q8—42no—
330C3×Q8—41no—
331C3×Q8—42no—
332C3×Q8—41no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4C2×C22⋊C411yescharacteristic
358C2×C3×C4C2×C22⋊C412yes—
359C2×C3×C4C2×C22⋊C411yescharacteristic
360C2×C3×C4C2×C22⋊C412yes—
361C2×C3×C4C2×C22⋊C411yescharacteristic
362C2×C3×C4C2×C22⋊C412yes—
363C2×C3×C4C23×C411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366#67—44no—
367#67—44no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375D12—42no—
376D12—42no—
377C4×D3—48no—
378Dic6—42no—
379Dic6—42no—
380C3×C8—44no—
381C3⋊C8—44no—
382C24—61no—
383C24(C2×D3)⋊C411yescharacteristic
384#41—124no—
385#41—124no—
386#41—124no—
387#41—124no—
388C2×Q8—32no—
389C2×Q8—41no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—41no—
393C2×Q8—42no—
394C2×Q8—41no—
395C2×Q8—42no—
396C2×Q8—41no—
397C2×Q8—42no—
398C2×Q8—41no—
399C2×Q8—42no—
400C2×Q8—41no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8—61no—
406C2×Q8—121no—
407C2×Q8—121no—
408C2×Q8(C2×D3)⋊C412yes—
409C2×Q8(C2×D3)⋊C412yes—
410C2×Q8(C2×D3)⋊C412yes—
411C2×Q8C2×C4×D312yes—
412C2×Q8C2×C4×D311yescharacteristic
413C2×Q8C2×D1212yes—
414C2×Q8C2×D1211yescharacteristic
415C2×Q8C2×#6712yes—
416C2×Q8C2×#6711yescharacteristic
417C2×Q8C23×D311yescharacteristic
418C2×D4—32no—
419C2×D4—64no—
420C2×D4—61no—
421C2×D4—121no—
422C2×D4—121no—
423C2×D4—122no—
424C2×D4—121no—
425C2×D4—121no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—64no—
439C22×C4—122no—
440C22×C4—122no—
441C22×C4(C2×D3)⋊C411yescharacteristic
442C22×C4(C2×D3)⋊C411yescharacteristic
443C22×C4(C2×D3)⋊C411yescharacteristic
444C22×C4(C2×D3)⋊C412yes—
445C22×C4(C2×D3)⋊C412yes—
446C22×C4(C2×D3)⋊C412yes—
447C22×C4C2×C4×D311yescharacteristic
448C22×C4C2×C4×D312yes—
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1212yes—
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6712yes—
453M16—24no—
454M16—42no—
455M16—44no—
456M16—64no—
457M16—122no—
458M16—124no—
459C2×C8—44no—
460C2×C8—42no—
461C2×C8—124no—
462C2×C8—122no—
463C4⋊C4—64no—
464C4⋊C4—64no—
465C4⋊C4—62no—
466C4⋊C4—64no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C22⋊C4—64no—
471C22⋊C4—64no—
472C22⋊C4—62no—
473C22⋊C4—64no—
474C22⋊C4—62no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C42—32no—
478C42—32no—
479C42—61no—
480C42—61no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3C2×#9911yescharacteristic
487C22×C3C2×#9912yes—
488C22×C3C2×#9912yes—
489C22×C3C2×#9911yescharacteristic
490C22×C3C22×C22⋊C411yescharacteristic
491C2×D3—44no—
492C2×D3—42no—
493C2×D3—82no—
494C2×D3—82no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—22no—
500C3×C4—22no—
501C3×C4—21no—
502C3×C4—22no—
503C3×C4—21no—
504Dic3—42no—
505Dic3—42no—
506Dic3—42no—
507C23—42no—
508C23—62no—
509C23—124no—
510C23—121no—
511C23—121no—
512C23#80611yescharacteristic
513C23#80611yescharacteristic
514C23#80612yes—
515C23C2×(C2×D3)⋊C411yescharacteristic
516C23C2×(C2×D3)⋊C412yes—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—42no—
521Q8—41no—
522Q8—42no—
523Q8—41no—
524Q8—42no—
525Q8—41no—
526Q8—41no—
527Q8—122no—
528Q8—122no—
529D4—122no—
530D4—122no—
531D4—124no—
532D4—124no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—128no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4C2×(C2×D3)⋊C411yescharacteristic
566C2×C4C2×(C2×D3)⋊C412yes—
567C2×C4C2×(C2×D3)⋊C411yescharacteristic
568C2×C4C2×(C2×D3)⋊C412yes—
569C2×C4C2×(C2×D3)⋊C411yescharacteristic
570C2×C4C2×(C2×D3)⋊C412yes—
571C2×C4C22×C4×D311yescharacteristic
572C2×C4C22×D1211yescharacteristic
573C2×C4C22×#6711yescharacteristic
574C8—44no—
575C8—124no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—22no—
579C2×C3—22no—
580C2×C3#213312yescomplemented
581C2×C3C22×#9911yescharacteristic
582D3—82no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—124no—
589C22—122no—
590C22—242no—
591C22—242no—
592C22C2×#80611yescharacteristic
593C22C2×#80612yes—
594C22C2×#80612yes—
595C22C2×#80611yescharacteristic
596C22C22×(C2×D3)⋊C411yescharacteristic
597C4—21no—
598C4—21no—
599C4—22no—
600C4—21no—
601C4—22no—
602C4—22no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—122no—
607C4—122no—
608C4—122no—
609C3C2×#213311yescharacteristic, complemented, Sylow
610C2—21no—
611C2—21no—
612C2—22no—
613C2—22no—
614C2—242no—
615C2#7012812yesdirect product factor
616C2C22×#80611yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC2×C3×C4C23×C4
FrattiniC2×Q8C23×D3
FittingC22×C3×#101C2
socleC22×C3C2×#99
prime radicalC22×#235C4
cosocleC2×C3×Q8C24
hypercenterC22×#101D3
hypocenterC3C2×#2133
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
2. C22×C3×#101C2semidirect
3. C22×#836C2semidirect
4. C2×#4478C2semidirect
5. #70128C2direct (N × H)
6. C2×#4477C2semidirect
28. C2×C3×#101C22semidirect
32. C2×#836C22semidirect
33. #4478C22semidirect
34. #4477C22semidirect
37. C22×#101D3semidirect
138. C2×#101C2×D3semidirect
580. C2×C3#2133semidirect
609. C3C2×#2133semidirect
615. C2#70128direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#58162
C22×C3×#1012
C22×#8362
C2×#44782
#701282
C2×#44772
C2×#21333