Finite Group Databasebeta

← C2×#21462082 #30623918

Subgroups — C2×#21462082 #30623918

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840749C211yesmaximal, characteristic
2C2×#16840742C211yesmaximal, characteristic
3#21462082C218yesmaximal, direct product factor
4C2×#16840665C212yesmaximal, complemented
5C2×#16840664C212yesmaximal, complemented
6C2×#16840334C211yesmaximal, characteristic
7C22×#11758217—21no—
8C2×#14519856—21no—
9#16840749—22no—
10#16840749C2214yes—
11C2×#14519838—21no—
12C2×#14519837C2212yes—
13#16840742—22no—
14#16840742C2214yes—
15C2×#14519835—21no—
16C2×#14519835C2212yes—
17#16840665C2218yescomplemented
18#16840664C2218yescomplemented
19C22×#11758224C2211yescharacteristic
20C23×Dic122C411yescharacteristic, complemented
21#16840334—22no—
22#16840334—22no—
23#16840334C414yescomplemented
24#16840334C2214yes—
25C2×#14519776—21no—
26C2×#14519763—21no—
27C2×#14519761C2211yescharacteristic
28C2×#14519760C2211yescharacteristic
29C2×#14519758C411yescharacteristic, complemented
30C2×#14519755—21no—
31C2×#14519754—21no—
32C2×#14519733C411yescharacteristic, complemented
33C2×#123×C61C411yescharacteristic, complemented
34C2×#11758217—22no—
35C2×#11758217—24no—
36C2×#11758217—41no—
37C2×#11758217—42no—
38C22×#92420—41no—
39#14519856—24no—
40#14519856—42no—
41C2×C4⋊#6923—41no—
42C2×C22⋊#6923—22no—
43C2×C4×#6923—41no—
44#14519838—24no—
45#14519837—22no—
46#14519837C2314yes—
47#14519835—24no—
48#14519835—22no—
49#14519835C2314yes—
50C2×#11758224—41no—
51C2×#11758224—41no—
52C2×#11758224—42no—
53C2×#11758224C2×C412yes—
54C2×#11758224D412yes—
55C2×#11758224D412yes—
56C2×#11758224C2312yes—
57C22×Dic122—21no—
58C22×Dic122—21no—
59C22×Dic122—22no—
60C22×Dic122—42no—
61C22×Dic122—41no—
62C22×Dic122—41no—
63C22×Dic122C2×C414yescomplemented
64C22×Dic122D411yescharacteristic
65C22×C22⋊D61D411yescharacteristic
66C22×C22⋊D61C2311yescharacteristic
67C23×Dic61C2×C411yescharacteristic
68C23×Dic61C2×C411yescharacteristic
69C23×Dic61D411yescharacteristic
70C22×C4×D61D411yescharacteristic
71#14519776—24no—
72#14519776—24no—
73#14519763—24no—
74#14519763—24no—
75#14519761—21no—
76#14519761—21no—
77#14519761C2×C412yes—
78#14519761C2312yes—
79#14519760—21no—
80#14519760—21no—
81#14519760C2×C412yes—
82#14519760C2312yes—
83#14519758—22no—
84#14519758C2×C414yescomplemented
85#14519755—24no—
86#14519755—24no—
87#14519754—24no—
88#14519754—24no—
89#14519733—22no—
90#14519733C2×C414yescomplemented
91C2×C4⋊Dic61—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×C22⋊Dic61—21no—
96C2×C22⋊Dic61C2×C411yescharacteristic
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C22×D4×C61C2×C411yescharacteristic
101C23×C4×C61C2×C411yescharacteristic
102#123×C61—22no—
103#123×C61C2×C414yescomplemented
104C2×C4⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61C2×C411yescharacteristic
107C24×C61C22⋊C411yescharacteristic
108C24×C61C22×C411yescharacteristic
109C23×D61C2×D411yescharacteristic
110C22×#6923—42no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61C22⋊C412yes—
114C2×D4×C61C22×C412yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61C22⋊C411yescharacteristic
121C22×C4×C61C22×C412yes—
122C2×C22⋊D61—41no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61C22⋊C412yes—
127C2×C22⋊D61C22×C411yescharacteristic
128C2×C22⋊D61C2×D412yes—
129C2×C22⋊D61C2×D411yescharacteristic
130C2×C22⋊D61C2×D411yescharacteristic
131C2×C22⋊D61C2411yescharacteristic
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—41no—
142C22×Dic61—41no—
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C22⋊C412yes—
145C22×Dic61C22⋊C412yes—
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22×C412yes—
148C22×Dic61C22×C412yes—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D412yes—
151#11758224—42no—
152#11758224—42no—
153#11758224—42no—
154#11758224—42no—
155#11758224—42no—
156#11758224—42no—
157#11758224—42no—
158#11758224—42no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—22no—
162C2×C4×D61C22⋊C412yes—
163C2×C4×D61C2×D412yes—
164C2×Dic122—24no—
165C2×Dic122—24no—
166C2×Dic122—41no—
167C2×Dic122—41no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×Dic122—44no—
171C2×Dic122—44no—
172C2×Dic122—42no—
173C2×Dic122—41no—
174C2×Dic122—41no—
175C2×Dic122—44no—
176C2×Dic122—44no—
177C2×Dic122C22⋊C412yes—
178C2×Dic122C2×D412yes—
179C22⋊#6923—24no—
180C22⋊#6923—42no—
181#11758217—24no—
182#11758217—42no—
183#11758217—44no—
184C2×#92420—42no—
185C2×#92420—44no—
186C4⋊#6923—44no—
187C4×#6923—44no—
188C4⋊C4×C61—24no—
189C22⋊C4×C61—24no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61C22×C412yes—
192C22⋊Dic61—24no—
193C22⋊Dic61—21no—
194C22⋊Dic61C22×C412yes—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61—22no—
201#11758196—24no—
202#11758196—24no—
203#11758196—22no—
204#11758196—22no—
205C4×Dic61—24no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61C2×C22⋊C411yescharacteristic
213C23×C61C2×C22⋊C412yes—
214C23×C61C2×C22⋊C411yescharacteristic
215C23×C61C23×C411yescharacteristic
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—42no—
219C22×D61C2×C22⋊C411yescharacteristic
220C22×D61C22×D411yescharacteristic
221C2×#6923—44no—
222C2×#6923—84no—
223D4×C61—42no—
224D4×C61—42no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—42no—
234C2×C4×C61C2×C22⋊C412yes—
235C22⋊D61—42no—
236C22⋊D61—41no—
237C22⋊D61—41no—
238C22⋊D61—42no—
239C22⋊D61—41no—
240C22⋊D61—41no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—42no—
260C2×Dic61#9912yes—
261C2×Dic61#9911yescharacteristic
262C2×Dic61#9911yescharacteristic
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C22×D411yescharacteristic
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271Dic122—42no—
272Dic122—42no—
273Dic122—48no—
274Dic122—44no—
275Dic122—44no—
276#92420—44no—
277C2×#2123—611nomaximal, Sylow
278C22×C61—22no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—22no—
282C22×C61—41no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—81no—
289C22×C61C2×#9911yescharacteristic
290C22×C61C2×#9911yescharacteristic
291C22×C61C22×C22⋊C411yescharacteristic
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—42no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—81no—
298#6923—84no—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—42no—
302Dic61—21no—
303Dic61—22no—
304Dic61—21no—
305Dic61—42no—
306Dic61—42no—
307Dic61C2×#9912yes—
308Dic61C2×#9912yes—
309C2×#536—611no—
310C2×#413—611no—
311C2×#410—611no—
312#2123—618no—
313C2×#352—612no—
314C2×#352—612no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—41no—
318C2×C61—41no—
319C2×C61—42no—
320C2×C61#212312yescomplemented
321C2×C61C22×#9911yescharacteristic
322D61—41no—
323D61—41no—
324C22×#41—611no—
325C23×Q8—611no—
326C22×M16—1221no—
327#536—614no—
328#536—614no—
329#536—1222no—
330#536—1222no—
331C2×C4⋊Q8—1221no—
332C2×#124—1221no—
333C2×#123—611no—
334C2×#123—611no—
335C2×#123—611no—
336C2×#123#692311yescharacteristic, complemented
337C2×C22⋊Q8—1221no—
338C2×C22⋊Q8—1221no—
339C2×#117—611no—
340C2×#117—1221no—
341#413—614no—
342#413—1222no—
343C2×#101—1221no—
344C2×#100—612no—
345#410—614no—
346#410—1222no—
347C2×#99—612no—
348C2×#99—1221no—
349#352—618no—
350#352—618no—
351C61C2×#212311yescharacteristic, complemented, Sylow
352C2×#41—612no—
353C2×#41—612no—
354C2×#41—612no—
355C2×#41—612no—
356C2×#41—2441no—
357C2×#41—2441no—
358C2×#41—2442no—
359C22×Q8—611no—
360C22×Q8—614no—
361C22×Q8—1221no—
362C22×Q8—1221no—
363C22×Q8—1222no—
364C22×Q8—2442no—
365C22×Q8—2441no—
366C22×Q8—2441no—
367C22×D4—611no—
368C22×D4—611no—
369C22×D4C2×#692311yescharacteristic
370C23×C4—611no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4C2×#692311yescharacteristic
375C2×M16—1222no—
376C2×M16—1224no—
377C2×M16—2442no—
378C2×M16—2441no—
379C22×C8—2441no—
380C4⋊Q8—1224no—
381C4⋊Q8—1224no—
382#124—1224no—
383#124—1224no—
384#123—22no—
385#123—614no—
386#123—612no—
387#123—612no—
388#123—612no—
389#123—612no—
390#123—1221no—
391#123—1221no—
392#123—1221no—
393#123—1221no—
394#123—1222no—
395#123C2×#692314yescomplemented
396C22⋊Q8—1224no—
397C22⋊Q8—1224no—
398C22⋊Q8—1224no—
399C22⋊Q8—1224no—
400#117—614no—
401#117—1222no—
402#117—1224no—
403#117—2442no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—2441no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—611no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1222no—
416C2×C22⋊C4C2×#692311yescharacteristic
417C2×C42—1221no—
418C2×C42—2441no—
419#101—1224no—
420#100—614no—
421#100—1222no—
422#99—614no—
423#99—1222no—
424#99—1224no—
425C24—611no—
426C24C22⋊#692311yescharacteristic
427C24C22×#692311yescharacteristic
428#41—2442no—
429#41—2442no—
430#41—2442no—
431#41—2442no—
432#41—2442no—
433#41—2442no—
434#41—2442no—
435#41—2442no—
436C2×Q8—612no—
437C2×Q8—612no—
438C2×Q8—1224no—
439C2×Q8—1224no—
440C2×Q8—2441no—
441C2×Q8—2441no—
442C2×Q8—2442no—
443C2×Q8—2441no—
444C2×Q8—2441no—
445C2×Q8—2444no—
446C2×Q8—2444no—
447C2×Q8—2444no—
448C2×Q8—2444no—
449C2×Q8—2442no—
450C2×Q8—2442no—
451C2×D4—41no—
452C2×D4—41no—
453C2×D4—611no—
454C2×D4—612no—
455C2×D4—611no—
456C2×D4—612no—
457C2×D4—611no—
458C2×D4—611no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4C22⋊#692312yes—
464C2×D4C22×#692312yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—611no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2441no—
494C22×C4—2441no—
495C22×C4C22⋊#692311yescharacteristic
496C22×C4C22×#692312yes—
497M16—1224no—
498M16—2444no—
499M16—2442no—
500C2×C8—2444no—
501C2×C8—2442no—
502C4⋊C4—24no—
503C4⋊C4—1224no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1224no—
509C4⋊C4—2444no—
510C22⋊C4—24no—
511C22⋊C4—21no—
512C22⋊C4—612no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1221no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—2442no—
521C22⋊C4C22×#692312yes—
522C42—1224no—
523C42—2444no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—42no—
529C23—42no—
530C23—611no—
531C23—611no—
532C23—1221no—
533C23—1221no—
534C23—2442no—
535C23C2×C22⋊#692312yes—
536C23C2×C22⋊#692311yescharacteristic
537C23C2×C22⋊#692311yescharacteristic
538C23C23×#692311yescharacteristic
539Q8—2442no—
540Q8—2442no—
541Q8—2444no—
542Q8—2444no—
543Q8—2448no—
544D4—42no—
545D4—42no—
546D4—2441no—
547D4—2441no—
548D4—2441no—
549D4—2441no—
550D4—2442no—
551D4—2442no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—612no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—612no—
565C2×C4—612no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—4884no—
587C2×C4—2444no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4C2×C22⋊#692312yes—
594C8—2444no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—42no—
601C22—41no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—81no—
606C22—1221no—
607C22—1221no—
608C22—4881no—
609C22—2441no—
610C22—2441no—
611C22—2442no—
612C22D61⋊#9911yescharacteristic
613C22D61⋊#9911yescharacteristic
614C22C22×C22⋊#692311yescharacteristic
615C4—22no—
616C4—42no—
617C4—42no—
618C4—612no—
619C4—612no—
620C4—1221no—
621C4—1221no—
622C4—1222no—
623C4—4884no—
624C4—2442no—
625C4—2442no—
626C2—21no—
627C2—21no—
628C2—41no—
629C2—41no—
630C2—42no—
631C2—2441no—
632C2—2441no—
633C2#2146208212yesdirect product factor
634C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#123×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758224C22
cosocleC2×C22⋊D61C24
hypercenterC2×#123#6923
hypocenterC61C2×#2123
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #21462082C2direct (N × H)
4. C2×#16840665C2semidirect
5. C2×#16840664C2semidirect
17. #16840665C22semidirect
18. #16840664C22semidirect
20. C23×Dic122C4semidirect
23. #16840334C4semidirect
29. C2×#14519758C4semidirect
32. C2×#14519733C4semidirect
33. C2×#123×C61C4semidirect
63. C22×Dic122C2×C4semidirect
84. #14519758C2×C4semidirect
90. #14519733C2×C4semidirect
103. #123×C61C2×C4semidirect
320. C2×C61#2123semidirect
336. C2×#123#6923semidirect
351. C61C2×#2123semidirect
395. #123C2×#6923semidirect
633. C2#21462082direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407492
C2×#168407422
#214620822
C2×#168406652
C2×#168406642
C2×#168403342
C2×#212361