Finite Group Databasebeta

← C2×#21457041 #30552136

Subgroups — C2×#21457041 #30552136

Order 15616.

2018 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×C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#16840297C211yesmaximal, characteristic
3#21457041C2116yesmaximal, direct product factor
4C2×#16840172C211yesmaximal, characteristic, complemented
5C2×#16840124C211yesmaximal, characteristic, complemented
6C2×#16840122C211yesmaximal, characteristic, complemented
7C2×#16840094C212yesmaximal, complemented
8C2×#16840077C212yesmaximal
9C2×#16840056C212yesmaximal, complemented
10C2×#16840051C212yesmaximal, complemented
11C2×#16839817C211yesmaximal, characteristic, complemented
12C2×C61×#501C211yesmaximal, characteristic, complemented
13C22×Q8×D61—21no—
14C2×#14519777C2211yescharacteristic
15C2×#14519776—21no—
16C2×#14519776C2212yes—
17C2×#14519775—21no—
18C4⋊Q8×D61—24no—
19C4⋊Q8×D61C2218yescomplemented
20#16840297—24no—
21#16840297C2218yes—
22C2×#14519772—21no—
23C2×#14519765C2211yescharacteristic
24C2×#14519762—21no—
25C2×#14519757C2211yescharacteristic
26C2×#14519752C2211yescharacteristic
27C2×#14519742C2211yescharacteristic
28C2×#14519739—21no—
29C2×#14519739C2211yescharacteristic
30C2×#14519735—21no—
31C2×C4⋊C4×D61—21no—
32C2×C4⋊C4×D61C2212yes—
33C2×C42×D61C2211yescharacteristic
34#16840172C2218yescomplemented
35#16840124C2218yescomplemented
36#16840122C2218yescomplemented
37#16840094C22116yescomplemented
38#16840077C22116yes—
39#16840056C22116yescomplemented
40#16840051C22116yescomplemented
41#16839817C2218yescomplemented
42C2×#14519710C2212yes—
43C2×#14519708C2212yes—
44C2×#14519706C2212yes—
45C2×#14519702C2212yes—
46C2×#14519700C2212yes—
47C2×#14519699C2212yes—
48C2×#14519697C2212yes—
49C2×#14519689C2212yes—
50C2×#14519688C2211yescharacteristic
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#501C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×#106×C61C2212yes—
56C2×#97×C61C2211yescharacteristic
57C22×Dic122—21no—
58C22×Dic122—22no—
59C22×Dic122C2311yescharacteristic
60C2×Q8×D61—28no—
61C2×Q8×D61—48no—
62C22×D244—21no—
63C22×D244C2311yescharacteristic
64C22×C4×D61—22no—
65C22×C4×D61—21no—
66C22×C4×D61D411yescharacteristic
67C22×C4×D61D411yescharacteristic
68C22×C4×D61C2311yescharacteristic
69C22×#92418C2312yes—
70#14519777—22no—
71#14519777C2314yes—
72#14519776—24no—
73#14519776—28no—
74#14519776C2318yes—
75#14519775—28no—
76#14519772—28no—
77#14519765—22no—
78#14519765C2314yes—
79#14519762—28no—
80#14519757—22no—
81#14519757C2314yes—
82#14519752—22no—
83#14519752C2314yes—
84#14519742—22no—
85#14519742C2314yes—
86#14519739—28no—
87#14519739—22no—
88#14519739C2314yes—
89#14519735—28no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2312yes—
93C2×#11758196—22no—
94C2×#11758196—22no—
95C2×#11758196—21no—
96C4⋊C4×D61—28no—
97C4⋊C4×D61—24no—
98C4⋊C4×D61C2318yes—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C42×D61D414yes—
109C42×D61C2314yes—
110#14519710C2318yes—
111#14519708C2318yes—
112#14519706C2318yes—
113#14519702C2318yes—
114#14519700C2318yes—
115#14519699C2318yes—
116#14519697C2318yes—
117#14519689C2318yes—
118#14519688C2314yes—
119C22×Q8×C61—21no—
120C22×C8×C61C2312yes—
121C4⋊Q8×C61—22no—
122C4⋊Q8×C61C2314yes—
123#125×C61—22no—
124#125×C61C2314yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2312yes—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132#107×C61C2318yes—
133#106×C61C2318yes—
134#97×C61C2314yes—
135C23×D61—21no—
136C23×D61C2×D411yescharacteristic
137C2×Q8×C61—24no—
138C2×Q8×C61—44no—
139C22×C4×C61—22no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150Q8×D61—432no—
151C2×D244—24no—
152C2×D244—24no—
153C2×D244—21no—
154C2×D244—44no—
155C2×D244C2412yes—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—28no—
160C2×C4×D61—24no—
161C2×C4×D61—48no—
162C2×C4×D61—44no—
163C2×C4×D61C2×D414yes—
164C2×C4×D61C2×D414yes—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2412yes—
167C2×Dic122—24no—
168C2×Dic122—24no—
169C2×Dic122—21no—
170C2×Dic122—28no—
171C2×Dic122—48no—
172C2×Dic122—44no—
173C2×Dic122C2412yes—
174C2×C8×C61—24no—
175C2×C8×C61C2414yes—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C2412yes—
183C4⋊C4×C61C2414yes—
184C4⋊C4×C61C2412yes—
185C42×C61C2×D412yes—
186C42×C61C2412yes—
187#11758198—24no—
188#11758198—24no—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192C4⋊Dic61—24no—
193C4⋊Dic61—22no—
194C4⋊Dic61—24no—
195C4⋊Dic61C2414yes—
196#11758196—28no—
197#11758196—28no—
198#11758196—24no—
199C4×Dic61—24no—
200C4×Dic61C2×D412yes—
201C4×Dic61C2×D414yes—
202C4×Dic61C2×D414yes—
203C4×Dic61C2412yes—
204C2×#92418—24no—
205C2×#92418C2414yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61AGL(1,ℤ/8)14yes—
214C22×D61C22×D412yes—
215Q8×C61—48no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—22no—
221C2×C4×C61—44no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61C22×D412yes—
225C2×C4×C61C22×D412yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C612−1+414yes—
228C2×C4×C61C2511yescharacteristic
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—42no—
234C2×Dic61—44no—
235C2×Dic61#13714yes—
236C2×Dic61C22×D412yes—
237C2×Dic61C22×D412yes—
238C2×Dic61C22×D412yes—
239D244—24no—
240D244—48no—
241D244—42no—
242C4×D61—28no—
243C4×D61—24no—
244C4×D61—24no—
245C4×D61—28no—
246C4×D61—416no—
247C4×D61—48no—
248Dic122—24no—
249Dic122—416no—
250Dic122—48no—
251Dic122—42no—
252C8×C61—24no—
253#92418—24no—
254C2×#3238—611nomaximal, Sylow
255C22×C61#53612yes—
256C22×C61C2×AGL(1,ℤ/8)12yes—
257C22×C61C2×#13712yes—
258C22×C61C23×D411yescharacteristic
259C2×D61—22no—
260C2×D61—28no—
261C2×D61—22no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—84no—
266C2×D61C2×AGL(1,ℤ/8)14yes—
267C4×C61—22no—
268C4×C61—22no—
269C4×C61—44no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61#53612yes—
273C4×C61C2×2−1+412yes—
274Dic61—22no—
275Dic61—22no—
276Dic61—42no—
277Dic61—44no—
278Dic61C2×#13714yes—
279C22×C4⋊Q8—611no—
280C2×#529—611no—
281#3238—6116no—
282C2×#501—611no—
283C2×#501D6111yescharacteristic, complemented
284C2×#489—611no—
285C2×#489—611no—
286C2×#483—612no—
287C2×#480—612no—
288C2×#477—612no—
289C2×#476—612no—
290C2×#432—611no—
291C2×C61#323814yescomplemented
292C2×C61C2×#53611yescharacteristic
293C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
294C2×C61C22×#13711yescharacteristic
295D61—24no—
296D61—82no—
297C23×Q8—1221no—
298C2×C4⋊Q8—618no—
299C2×C4⋊Q8—612no—
300C2×C4⋊Q8—611no—
301C2×C4⋊Q8—1221no—
302C2×C4⋊Q8—1224no—
303C2×C4⋊Q8—1221no—
304C2×C4⋊Q8C2×D6111yescharacteristic
305#529—618no—
306#529—1224no—
307C2×#126—1221no—
308C2×#125C2×D6111yescharacteristic
309C2×#124—611no—
310C2×#123—1221no—
311C2×C22⋊Q8—611no—
312C2×#121—611no—
313C2×C4×Q8—611no—
314C2×C4×D4—611no—
315C2×C4×D4—1221no—
316C2×#117—1221no—
317C22×C4⋊C4—612no—
318C22×C4⋊C4—1221no—
319C22×C42—611no—
320#501—618no—
321#501C2×D6118yescomplemented
322#489—618no—
323#489—618no—
324#483—6116no—
325#480—6116no—
326#477—6116no—
327#476—6116no—
328#432—618no—
329C2×#107—612no—
330C2×#107C2×D6112yes—
331C2×#106—612no—
332C2×#106C2×D6112yes—
333C2×C4⋊C8—612no—
334C2×#103—612no—
335C2×#103—612no—
336C2×#102—612no—
337C2×#102—612no—
338C2×C22⋊C8—612no—
339C2×#97—611no—
340C2×#97C2×D6111yescharacteristic
341C61C2×#323811yescharacteristic, complemented, Sylow
342C22×Q8—21no—
343C22×Q8—611no—
344C22×Q8—1222no—
345C22×Q8—1228no—
346C22×Q8—1221no—
347C22×Q8—2448no—
348C22×D4—611no—
349C22×D4—1221no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—1221no—
354C23×C4—1222no—
355C22×C8—612no—
356C22×C8C22×D6112yes—
357C4⋊Q8—22no—
358C4⋊Q8—618no—
359C4⋊Q8—614no—
360C4⋊Q8—1224no—
361C4⋊Q8—1228no—
362C4⋊Q8—1228no—
363C4⋊Q8—1222no—
364C4⋊Q8C22×D6114yes—
365#126—1228no—
366#125—22no—
367#125C22×D6114yes—
368#124—614no—
369#124—1222no—
370#123—1228no—
371C22⋊Q8—614no—
372C22⋊Q8—1222no—
373#121—614no—
374#121—1222no—
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×D4—614no—
378C4×D4—1222no—
379C4×D4—1228no—
380#117—1228no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—618no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1228no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6112yes—
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
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#107C22×D6118yes—
410#106—618no—
411#106C22×D6118yes—
412C4⋊C8—618no—
413#103—618no—
414#103—618no—
415#102—618no—
416#102—618no—
417C22⋊C8—618no—
418#97—614no—
419#97C22×D6114yes—
420C24—611no—
421C24—1221no—
422C2×Q8—24no—
423C2×Q8—44no—
424C2×Q8—612no—
425C2×Q8—1228no—
426C2×Q8—1221no—
427C2×Q8—1224no—
428C2×Q8—1224no—
429C2×Q8—2444no—
430C2×Q8—2448no—
431C2×Q8—24432no—
432C2×D4—612no—
433C2×D4—1224no—
434C2×D4—1221no—
435C2×D4—1224no—
436C2×D4—2444no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—614no—
444C22×C4—612no—
445C22×C4—614no—
446C22×C4—612no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1228no—
451C22×C4—1224no—
452C22×C4—1221no—
453C22×C4—1222no—
454C22×C4—2444no—
455C22×C4—2448no—
456C22×C4D4×D6111yescharacteristic
457C22×C4D4×D6111yescharacteristic
458C22×C4C23×D6111yescharacteristic
459C2×C8—24no—
460C2×C8—614no—
461C2×C8—1224no—
462C2×C8C23×D6114yes—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—614no—
470C4⋊C4—1228no—
471C4⋊C4—1228no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C4⋊C4—1222no—
475C4⋊C4—1224no—
476C4⋊C4C23×D6114yes—
477C4⋊C4C23×D6112yes—
478C4⋊C4C23×D6112yes—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—1224no—
484C42—614no—
485C42—612no—
486C42—614no—
487C42—612no—
488C42—1224no—
489C42D4×D6112yes—
490C42C23×D6112yes—
491C23—614no—
492C23—612no—
493C23—1222no—
494C23—1222no—
495C23—1222no—
496C23—2442no—
497C23—2442no—
498C23—2442no—
499C23C2×D4×D6111yescharacteristic
500Q8—48no—
501Q8—1224no—
502Q8—2448no—
503Q8—24416no—
504Q8—2442no—
505D4—1224no—
506D4—2448no—
507D4—2442no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—614no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—1228no—
521C2×C4—1228no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1222no—
527C2×C4—1224no—
528C2×C4—2448no—
529C2×C4—24416no—
530C2×C4—2444no—
531C2×C4—2442no—
532C2×C4C2×D4×D6112yes—
533C2×C4C2×D4×D6112yes—
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4#1451982514yes—
536C2×C4C24×D6111yescharacteristic
537C8—24no—
538C8—1224no—
539C22—614no—
540C22—1222no—
541C22—1228no—
542C22—1222no—
543C22—4884no—
544C22—2442no—
545C22—2442no—
546C22—2442no—
547C22#1684033712yes—
548C22AGL(1,ℤ/8)×D6112yes—
549C22#137×D6112yes—
550C22C22×D4×D6111yescharacteristic
551C4—22no—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—614no—
557C4—1222no—
558C4—1222no—
559C4—2444no—
560C4—2442no—
561C4#1684033712yes—
562C4C2×#1451982512yes—
563C2—1224no—
564C2—4882no—
565C2#2145704114yesdirect product factor
566C2C2×#1684033711yescharacteristic
567C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
568C2C2×#137×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenterC2×#501D61
hypocenterC61C2×#3238
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D61C2semidirect
3. #21457041C2direct (N × H)
4. C2×#16840172C2semidirect
5. C2×#16840124C2semidirect
6. C2×#16840122C2semidirect
7. C2×#16840094C2semidirect
9. C2×#16840056C2semidirect
10. C2×#16840051C2semidirect
11. C2×#16839817C2semidirect
12. C2×C61×#501C2semidirect
19. C4⋊Q8×D61C22semidirect
34. #16840172C22semidirect
35. #16840124C22semidirect
36. #16840122C22semidirect
37. #16840094C22semidirect
39. #16840056C22semidirect
40. #16840051C22semidirect
41. #16839817C22semidirect
53. C61×#501C22semidirect
283. C2×#501D61semidirect
291. C2×C61#3238semidirect
321. #501C2×D61semidirect
341. C61C2×#3238semidirect
565. C2#21457041direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D612
C2×#168402972
#214570412
C2×#168401722
C2×#168401242
C2×#168401222
C2×#168400942
C2×#168400772
C2×#168400562
C2×#168400512
C2×#168398172
C2×C61×#5012
C2×#323861