Finite Group Databasebeta

← C2×#21456969 #30552061

Subgroups — C2×#21456969 #30552061

Order 15616.

2006 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×C4⋊Q8×D61C211yesmaximal, characteristic
3#21456969C2116yesmaximal, direct product factor
4C2×#16840166C212yesmaximal, complemented
5C2×#16840111C212yesmaximal, complemented
6C2×#16840099C212yesmaximal
7C2×#16840096C212yesmaximal
8C2×#16840058C211yesmaximal, characteristic, complemented
9C2×#16840057C211yesmaximal, characteristic, complemented
10C2×#16840054C211yesmaximal, characteristic, complemented
11C2×#16839814C211yesmaximal, characteristic
12C2×C61×#477C211yesmaximal, characteristic, complemented
13C22×Q8×D61—21no—
14#16840316—24no—
15#16840316C2218yes—
16C2×#14519777C2211yescharacteristic
17C2×#14519776—21no—
18C2×#14519776C2212yes—
19C2×#14519776C2211yescharacteristic
20C2×#14519775—21no—
21C2×#14519775C2211yescharacteristic
22C4⋊Q8×D61—24no—
23C4⋊Q8×D61C2218yes—
24C2×#14519767—21no—
25C2×#14519757C2211yescharacteristic
26C2×#14519755—21no—
27C2×#14519753C2211yescharacteristic
28C2×C4×Dic122—21no—
29C2×C4×Dic122C2211yescharacteristic
30C2×#14519734—21no—
31C2×C4⋊C4×D61—21no—
32C2×C4⋊C4×D61C2212yes—
33C2×C42×D61C2211yescharacteristic
34#16840166C22116yescomplemented
35#16840111C22116yescomplemented
36#16840099C22116yes—
37#16840096C22116yes—
38#16840058C2218yescomplemented
39#16840057C2218yescomplemented
40#16840054C2218yescomplemented
41#16839814C2218yes—
42C2×#14519711C2212yes—
43C2×#14519710C2212yes—
44C2×#14519710C2212yes—
45C2×C4⋊#92418C2211yescharacteristic
46C2×#14519702C2212yes—
47C2×#14519701C2212yes—
48C2×#14519700C2212yes—
49C2×#14519689C2212yes—
50C2×C8×Dic61C2212yes—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#477C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2212yes—
57C22×Dic122—21no—
58C22×Dic122—21no—
59C22×Dic122—22no—
60C22×Dic122C2311yescharacteristic
61C2×Q8×D61—28no—
62C2×Q8×D61—48no—
63C22×C4×D61—22no—
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C22×#92418C2312yes—
68#14519777—22no—
69#14519777C2314yes—
70#14519776—24no—
71#14519776—28no—
72#14519776—22no—
73#14519776C2318yes—
74#14519776C2314yes—
75#14519775—28no—
76#14519775—22no—
77#14519775C2314yes—
78#14519767—28no—
79#14519757—22no—
80#14519757C2314yes—
81#14519755—28no—
82#14519753—22no—
83#14519753C2314yes—
84C4×Dic122—22no—
85C4×Dic122—28no—
86C4×Dic122C2314yes—
87#14519734—28no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2312yes—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—22no—
94C2×#11758196—22no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C4⋊C4×D61—28no—
100C4⋊C4×D61—24no—
101C4⋊C4×D61C2318yes—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C42×D61D414yes—
109C42×D61C2314yes—
110#14519711C2318yes—
111#14519710C2318yes—
112#14519710C2318yes—
113C4⋊#92418C2314yes—
114#14519702C2318yes—
115#14519701C2318yes—
116#14519700C2318yes—
117#14519689C2318yes—
118C8×Dic61C2318yes—
119C22×Q8×C61—21no—
120C22×Q8×C61C2311yescharacteristic
121C22×C8×C61C2312yes—
122C4⋊Q8×C61—22no—
123C4⋊Q8×C61C2314yes—
124C4×Q8×C61—22no—
125C4×Q8×C61C2314yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2312yes—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#107×C61C2318yes—
133C4⋊C8×C61C2314yes—
134#103×C61C2318yes—
135C23×D61C2×D411yescharacteristic
136C2×Q8×C61—24no—
137C2×Q8×C61—21no—
138C2×Q8×C61—22no—
139C2×Q8×C61—22no—
140C2×Q8×C61—44no—
141C2×Q8×C61C2412yes—
142C22×C4×C61—22no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154Q8×D61—432no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—24no—
158C2×C4×D61—28no—
159C2×C4×D61—48no—
160C2×C4×D61Q1618yes—
161C2×C4×D61C2×D414yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D414yes—
164C2×C4×D61C2412yes—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—24no—
168C2×Dic122—21no—
169C2×Dic122—28no—
170C2×Dic122—24no—
171C2×Dic122—44no—
172C2×Dic122—48no—
173C2×Dic122—44no—
174C2×Dic122C2412yes—
175C2×C8×C61—24no—
176C2×C8×C61C2414yes—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C2414yes—
182C4⋊C4×C61C2412yes—
183C42×C61—24no—
184C42×C61C2×D412yes—
185C42×C61C2412yes—
186#11758198—24no—
187#11758198—24no—
188C4⋊Dic61—24no—
189C4⋊Dic61—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—24no—
192C4⋊Dic61C2414yes—
193C4⋊Dic61C2412yes—
194#11758196—24no—
195#11758196—28no—
196#11758196—28no—
197#11758196—24no—
198#11758196—24no—
199#11758196—24no—
200#11758196—24no—
201C4×Dic61—24no—
202C4×Dic61C2×D412yes—
203C4×Dic61C2×D414yes—
204C4×Dic61C2×D414yes—
205C4×Dic61C2412yes—
206C2×#92418—24no—
207C2×#92418C2414yes—
208C23×C61C22×D411yescharacteristic
209C22×D61—22no—
210C22×D61—22no—
211C22×D61C2×Q1614yes—
212C22×D61C22×D412yes—
213Q8×C61—22no—
214Q8×C61—22no—
215Q8×C61—48no—
216Q8×C61—42no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—42no—
222C2×C4×C61—44no—
223C2×C4×C61—42no—
224C2×C4×C61C2×Q1614yes—
225C2×C4×C61C22×D412yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D412yes—
228C2×C4×C612−1+412yes—
229C2×C4×C612−1+412yes—
230C2×C4×C61C2511yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—42no—
236C2×Dic61—44no—
237C2×Dic61—42no—
238C2×Dic61C2×Q1614yes—
239C2×Dic61AGL(1,ℤ/8)14yes—
240C2×Dic61C22×D412yes—
241C2×Dic61C22×D412yes—
242C2×Dic61C22×D412yes—
243C4×D61—24no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—28no—
247C4×D61—416no—
248C4×D61C2×Q16116yes—
249Dic122—22no—
250Dic122—22no—
251Dic122—48no—
252Dic122—42no—
253Dic122—48no—
254Dic122—416no—
255C8×C61—24no—
256#92418—24no—
257C2×#3230—611nomaximal, Sylow
258C22×C61#53612yes—
259C22×C61C22×Q1612yes—
260C22×C61C2×AGL(1,ℤ/8)12yes—
261C22×C61C23×D411yescharacteristic
262C2×D61—22no—
263C2×D61—28no—
264C2×D61—22no—
265C2×D61C22×Q1614yes—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—44no—
269C4×C61—42no—
270C4×C61#53612yes—
271C4×C61C22×Q1614yes—
272C4×C61C2×2−1+411yescharacteristic
273C4×C61C2×2−1+411yescharacteristic
274Dic61—22no—
275Dic61—42no—
276Dic61—44no—
277Dic61—42no—
278Dic61C22×Q1614yes—
279Dic61C2×AGL(1,ℤ/8)14yes—
280C2×#533—611no—
281C22×C4⋊Q8—611no—
282#3230—6116no—
283C2×#500—612no—
284C2×#487—612no—
285C2×#484—612no—
286C2×#484—612no—
287C2×#477—611no—
288C2×#477—611no—
289C2×#477—611no—
290C2×#477D6111yescharacteristic, complemented
291C2×#431—611no—
292C2×C61#323014yescomplemented
293C2×C61C2×#53611yescharacteristic
294C2×C61C23×Q1611yescharacteristic
295C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
296D61—24no—
297C23×Q8—1221no—
298#533—618no—
299#533—1224no—
300C2×C4⋊Q8—618no—
301C2×C4⋊Q8—612no—
302C2×C4⋊Q8—611no—
303C2×C4⋊Q8—611no—
304C2×C4⋊Q8—611no—
305C2×C4⋊Q8—1221no—
306C2×C4⋊Q8—1224no—
307C2×C4⋊Q8—1221no—
308C2×C4⋊Q8C2×D6111yescharacteristic
309C2×#125—1221no—
310C2×C22⋊Q8—611no—
311C2×C22⋊Q8—611no—
312C2×C22⋊Q8—1221no—
313C2×C4×Q8—611no—
314C2×C4×Q8—1221no—
315C2×C4×Q8C2×D6111yescharacteristic
316C2×#117—1221no—
317C22×C4⋊C4—612no—
318C22×C4⋊C4—1221no—
319C22×C42—611no—
320#500—6116no—
321#487—6116no—
322#484—6116no—
323#484—6116no—
324#477—618no—
325#477—618no—
326#477—618no—
327#477C2×D6118yescomplemented
328#431—618no—
329C2×#107—612no—
330C2×#107—612no—
331C2×#107—612no—
332C2×#107C2×D6112yes—
333C2×C4⋊C8—611no—
334C2×C4⋊C8C2×D6111yescharacteristic
335C2×#103—612no—
336C2×#103—612no—
337C2×#103—612no—
338C2×#103C2×D6112yes—
339C2×C22⋊C8—612no—
340C2×C4×C8—612no—
341C61C2×#323011yescharacteristic, complemented, Sylow
342C22×Q8—21no—
343C22×Q8—611no—
344C22×Q8—1221no—
345C22×Q8—1222no—
346C22×Q8—1228no—
347C22×Q8—1221no—
348C22×Q8—2448no—
349C22×Q8C22×D6111yescharacteristic
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—1222no—
354C22×C8—612no—
355C22×C8C22×D6112yes—
356C4⋊Q8—22no—
357C4⋊Q8—618no—
358C4⋊Q8—614no—
359C4⋊Q8—614no—
360C4⋊Q8—614no—
361C4⋊Q8—1228no—
362C4⋊Q8—1224no—
363C4⋊Q8—1222no—
364C4⋊Q8—1228no—
365C4⋊Q8—1222no—
366C4⋊Q8—1222no—
367C4⋊Q8C22×D6114yes—
368#125—1228no—
369C22⋊Q8—614no—
370C22⋊Q8—614no—
371C22⋊Q8—1222no—
372C22⋊Q8—1228no—
373C22⋊Q8—1222no—
374C4×Q8—22no—
375C4×Q8—614no—
376C4×Q8—1228no—
377C4×Q8—1222no—
378C4×Q8C22×D6114yes—
379#117—1228no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—612no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—618no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4C22×D6112yes—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—21no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—614no—
405C2×C42—614no—
406C2×C42—1221no—
407C2×C42C22×D6111yescharacteristic
408#107—618no—
409#107—618no—
410#107—618no—
411#107C22×D6118yes—
412C4⋊C8—614no—
413C4⋊C8C22×D6114yes—
414#103—618no—
415#103—618no—
416#103—618no—
417#103C22×D6118yes—
418C22⋊C8—618no—
419C4×C8—618no—
420C24—611no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—612no—
427C2×Q8—1224no—
428C2×Q8—1228no—
429C2×Q8—1221no—
430C2×Q8—1224no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—2444no—
434C2×Q8—2444no—
435C2×Q8—2448no—
436C2×Q8—24432no—
437C2×Q8C23×D6112yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—614no—
444C22×C4—618no—
445C22×C4—611no—
446C22×C4—612no—
447C22×C4—614no—
448C22×C4—612no—
449C22×C4—1224no—
450C22×C4—1228no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1222no—
456C22×C4—2448no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—24no—
461C2×C8—614no—
462C2×C8—1224no—
463C2×C8C23×D6114yes—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—614no—
469C4⋊C4—612no—
470C4⋊C4—1228no—
471C4⋊C4—1228no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C4⋊C4—1224no—
475C4⋊C4—1224no—
476C4⋊C4—1224no—
477C4⋊C4—1222no—
478C4⋊C4—1221no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4C23×D6114yes—
482C4⋊C4C23×D6112yes—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C42—24no—
486C42—614no—
487C42—614no—
488C42—612no—
489C42—612no—
490C42—1224no—
491C42D4×D6112yes—
492C42C23×D6112yes—
493C23—614no—
494C23—612no—
495C23—1222no—
496C23—1222no—
497C23C2×D4×D6111yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—48no—
501Q8—42no—
502Q8—1222no—
503Q8—1222no—
504Q8—2448no—
505Q8—2448no—
506Q8—24416no—
507Q8—2442no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—614no—
516C2×C4—614no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—6116no—
520C2×C4—612no—
521C2×C4—1228no—
522C2×C4—1222no—
523C2×C4—1224no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1224no—
529C2×C4—24416no—
530C2×C4—2444no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4Q16×D6114yes—
534C2×C4C2×D4×D6112yes—
535C2×C4C2×D4×D6112yes—
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4#1451982412yes—
538C2×C4#1451982512yes—
539C2×C4C24×D6111yescharacteristic
540C8—24no—
541C8—1224no—
542C22—614no—
543C22—1222no—
544C22—1228no—
545C22—1222no—
546C22#1684033512yes—
547C22C2×Q16×D6112yes—
548C22AGL(1,ℤ/8)×D6112yes—
549C22C22×D4×D6111yescharacteristic
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—614no—
555C4—614no—
556C4—1222no—
557C4—2444no—
558C4—2442no—
559C4—2442no—
560C4#1684033512yes—
561C4C2×Q16×D6114yes—
562C4C2×#1451982411yescharacteristic
563C4C2×#1451982511yescharacteristic
564C2—1224no—
565C2#2145696914yesdirect product factor
566C2C2×#1684033511yescharacteristic
567C2C22×Q16×D6111yescharacteristic
568C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#477D61
hypocenterC61C2×#3230
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #21456969C2direct (N × H)
4. C2×#16840166C2semidirect
5. C2×#16840111C2semidirect
8. C2×#16840058C2semidirect
9. C2×#16840057C2semidirect
10. C2×#16840054C2semidirect
12. C2×C61×#477C2semidirect
34. #16840166C22semidirect
35. #16840111C22semidirect
38. #16840058C22semidirect
39. #16840057C22semidirect
40. #16840054C22semidirect
53. C61×#477C22semidirect
290. C2×#477D61semidirect
292. C2×C61#3230semidirect
327. #477C2×D61semidirect
341. C61C2×#3230semidirect
565. C2#21456969direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×C4⋊Q8×D612
#214569692
C2×#168401662
C2×#168401112
C2×#168400992
C2×#168400962
C2×#168400582
C2×#168400572
C2×#168400542
C2×#168398142
C2×C61×#4772
C2×#323061