Finite Group Databasebeta

← C2×#21457004 #30552113

Subgroups — C2×#21457004 #30552113

Order 15616.

2050 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840310C211yesmaximal, characteristic
2C2×#125×D61C211yesmaximal, characteristic, complemented
3#21457004C2116yesmaximal, direct product factor
4C2×#16840172C211yesmaximal, characteristic, complemented
5C2×#16840171C211yesmaximal, characteristic, complemented
6C2×#16840126C211yesmaximal, characteristic, complemented
7C2×#16840079C212yesmaximal, complemented
8C2×#16840074C212yesmaximal, complemented
9C2×#16840064C212yesmaximal, complemented
10C2×#16840060C212yesmaximal, complemented
11C2×#16839817C211yesmaximal, characteristic, complemented
12C2×C61×#490C211yesmaximal, characteristic, complemented
13C22×#11758224—21no—
14#16840310—24no—
15#16840310C2218yes—
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519768C2211yescharacteristic
20C2×#14519767—21no—
21C2×#14519767—21no—
22#125×D61—24no—
23#125×D61C2218yescomplemented
24C2×#14519764—21no—
25C2×#14519748—21no—
26C2×#14519748C2212yes—
27C2×D4×Dic61—21no—
28C2×D4×Dic61C2212yes—
29C2×C4⋊C4×D61—21no—
30C2×C4⋊C4×D61—21no—
31C2×C4⋊C4×D61C2211yescharacteristic
32C2×C4⋊C4×D61C2211yescharacteristic
33C2×C42×D61C2211yescharacteristic
34#16840172C2218yescomplemented
35#16840171C2218yescomplemented
36#16840126C2218yescomplemented
37#16840079C22116yescomplemented
38#16840074C22116yescomplemented
39#16840064C22116yescomplemented
40#16840060C22116yescomplemented
41#16839817C2218yescomplemented
42C2×#14519711C2212yes—
43C2×#14519710C2212yes—
44C2×#14519709C2212yes—
45C2×#14519708C2212yes—
46C2×#14519706C2212yes—
47C2×#14519698C2212yes—
48C2×#14519698C2212yes—
49C2×#14519689C2212yes—
50C2×#14519688C2211yescharacteristic
51C2×#127×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#490C2218yescomplemented
54C2×#102×C61C2212yes—
55C2×#102×C61C2212yes—
56C2×#97×C61C2211yescharacteristic
57C2×#11758224—28no—
58C2×#11758224—48no—
59C22×Dic122—21no—
60C22×C22⋊D61—22no—
61C23×Dic61—22no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C22×#92418C2312yes—
68#14519777—22no—
69#14519777C2314yes—
70#14519769—22no—
71#14519769C2314yes—
72#14519768—22no—
73#14519768—22no—
74#14519768C2314yes—
75#14519768C2314yes—
76#14519767—28no—
77#14519767—28no—
78#14519764—28no—
79#14519748—28no—
80#14519748—24no—
81#14519748C2318yes—
82D4×Dic61—28no—
83D4×Dic61—24no—
84D4×Dic61C2318yes—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2312yes—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C4⋊C4×D61—28no—
98C4⋊C4×D61—28no—
99C4⋊C4×D61—22no—
100C4⋊C4×D61—22no—
101C4⋊C4×D61C2314yes—
102C4⋊C4×D61C2314yes—
103C2×C22⋊Dic61—22no—
104C2×C22⋊Dic61—22no—
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C42×D61D414yes—
109C42×D61C2314yes—
110#14519711C2318yes—
111#14519710C2318yes—
112#14519709C2318yes—
113#14519708C2318yes—
114#14519706C2318yes—
115#14519698C2318yes—
116#14519698C2318yes—
117#14519689C2318yes—
118#14519688C2314yes—
119C22×D4×C61—21no—
120C22×D4×C61—21no—
121C22×D4×C61C2312yes—
122C22×C8×C61C2312yes—
123#127×C61—22no—
124#127×C61C2314yes—
125#125×C61—22no—
126#125×C61C2314yes—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132#102×C61C2318yes—
133#102×C61C2318yes—
134#97×C61C2314yes—
135C24×C61—22no—
136C23×D61C2×D411yescharacteristic
137C2×D4×C61—24no—
138C2×D4×C61—22no—
139C2×D4×C61—24no—
140C2×D4×C61—28no—
141C2×D4×C61—44no—
142C2×D4×C61—44no—
143C2×D4×C61C2414yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—28no—
150C2×C22⋊D61—48no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61—28no—
155C22×Dic61—48no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159#11758224—432no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—42no—
165C2×C4×D61—44no—
166C2×C4×D61—44no—
167C2×C4×D61C2×D414yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D414yes—
170C2×C4×D61C2412yes—
171C2×Dic122—24no—
172C2×Dic122—44no—
173C2×C8×C61—24no—
174C2×C8×C61C2414yes—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2412yes—
180C4⋊C4×C61C2412yes—
181C42×C61C2×D412yes—
182C42×C61C2412yes—
183C22⋊Dic61—28no—
184C22⋊Dic61—28no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—24no—
188C4⋊Dic61—22no—
189C4⋊Dic61—24no—
190C4⋊Dic61—24no—
191C4⋊Dic61C2412yes—
192C4⋊Dic61C2412yes—
193C4⋊Dic61C2414yes—
194#11758196—24no—
195#11758196—24no—
196#11758196—24no—
197#11758196—24no—
198#11758196—24no—
199#11758196—24no—
200C4×Dic61C2×D412yes—
201C4×Dic61C2×D414yes—
202C4×Dic61C2×D414yes—
203C4×Dic61C2412yes—
204C2×#92418—24no—
205C2×#92418C2414yes—
206C23×C61—24no—
207C23×C61—44no—
208C23×C61—44no—
209C23×C61—44no—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—22no—
213C22×D61AGL(1,ℤ/8)14yes—
214C22×D61C22×D412yes—
215D4×C61—28no—
216D4×C61—48no—
217D4×C61—48no—
218D4×C61—44no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61C22×D412yes—
226C2×C4×C61C22×D412yes—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C612−1+414yes—
229C2×C4×C61C2511yescharacteristic
230C22⋊D61—416no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—416no—
238C2×Dic61—44no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61C22×D412yes—
242C2×Dic61C22×D412yes—
243C2×Dic61C22×D412yes—
244C4×D61—28no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—44no—
248C4×D61—48no—
249C4×D61—48no—
250Dic122—48no—
251C8×C61—24no—
252#92418—24no—
253C2×#3234—611nomaximal, Sylow
254C22×C61—44no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—88no—
258C22×C61#53612yes—
259C22×C61C2×AGL(1,ℤ/8)12yes—
260C22×C61#57512yes—
261C22×C61C23×D411yescharacteristic
262C2×D61—22no—
263C2×D61—28no—
264C2×D61—22no—
265C2×D61C2×AGL(1,ℤ/8)14yes—
266C4×C61—22no—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61#53612yes—
271C4×C61C2×2−1+412yes—
272Dic61—22no—
273Dic61—22no—
274Dic61—22no—
275Dic61—44no—
276Dic61—42no—
277Dic61—42no—
278C2×#532—611no—
279C22×#125—611no—
280#3234—6116no—
281C2×#501—611no—
282C2×#501—611no—
283C2×#490—611no—
284C2×#490D6111yescharacteristic, complemented
285C2×#481—612no—
286C2×#480—612no—
287C2×#478—612no—
288C2×#478—612no—
289C2×#432—611no—
290C2×C61—84no—
291C2×C61#323414yescomplemented
292C2×C61C2×#53611yescharacteristic
293C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
294C2×C61C2×#57511yescharacteristic
295D61—24no—
296C22×#41—1221no—
297#532—618no—
298#532—1224no—
299C2×C4⋊Q8—611no—
300C2×#127C2×D6111yescharacteristic
301C2×#125—618no—
302C2×#125—611no—
303C2×#125—611no—
304C2×#125—611no—
305C2×#125—1221no—
306C2×#125—1221no—
307C2×#125—1224no—
308C2×#125C2×D6111yescharacteristic
309C2×#124—1221no—
310C2×#121—612no—
311C2×#121—1221no—
312C2×C4×D4—612no—
313C2×C4×D4—1221no—
314C22×C4⋊C4—611no—
315C22×C4⋊C4—611no—
316C22×C4⋊C4—1221no—
317C22×C4⋊C4—1221no—
318C22×C42—611no—
319#501—618no—
320#501—618no—
321#490—618no—
322#490C2×D6118yescomplemented
323#481—6116no—
324#480—6116no—
325#478—6116no—
326#478—6116no—
327#432—618no—
328C2×#107—612no—
329C2×#107—612no—
330C2×#106—612no—
331C2×#106—612no—
332C2×C4⋊C8—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102C2×D6112yes—
336C2×#102C2×D6112yes—
337C2×C22⋊C8—612no—
338C2×#97—611no—
339C2×#97C2×D6111yescharacteristic
340C61C2×#323411yescharacteristic, complemented, Sylow
341C2×#41—1228no—
342C2×#41—2448no—
343C22×Q8—1221no—
344C22×D4—21no—
345C22×D4—21no—
346C22×D4—1222no—
347C22×D4C22×D6112yes—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—1221no—
352C23×C4—1221no—
353C23×C4—1222no—
354C22×C8—612no—
355C22×C8C22×D6112yes—
356C4⋊Q8—614no—
357C4⋊Q8—1222no—
358#127—22no—
359#127C22×D6114yes—
360#125—22no—
361#125—614no—
362#125—614no—
363#125—614no—
364#125—1228no—
365#125—1228no—
366#125—1222no—
367#125—1222no—
368#125—1222no—
369#125C22×D6114yes—
370#124—1228no—
371#121—618no—
372#121—1224no—
373#121—1228no—
374C4×D4—618no—
375C4×D4—1224no—
376C4×D4—1228no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—612no—
382C2×C4⋊C4—614no—
383C2×C4⋊C4—614no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4—1228no—
388C2×C4⋊C4—1221no—
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⋊C4—1221no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—614no—
405C2×C42—614no—
406C2×C42C22×D6111yescharacteristic
407#107—618no—
408#107—618no—
409#106—618no—
410#106—618no—
411C4⋊C8—618no—
412#102—618no—
413#102—618no—
414#102C22×D6118yes—
415#102C22×D6118yes—
416C22⋊C8—618no—
417#97—614no—
418#97C22×D6114yes—
419C24—22no—
420C24—611no—
421#41—24432no—
422C2×Q8—1224no—
423C2×Q8—2444no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—28no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—1228no—
431C2×D4—2448no—
432C2×D4C23×D6114yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—614no—
440C22×C4—612no—
441C22×C4—614no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1224no—
445C22×C4—1228no—
446C22×C4—1224no—
447C22×C4—1222no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—2442no—
451C22×C4—2444no—
452C22×C4—2444no—
453C22×C4—2448no—
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457C2×C8—24no—
458C2×C8—614no—
459C2×C8—1224no—
460C2×C8C23×D6114yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—612no—
466C4⋊C4—612no—
467C4⋊C4—614no—
468C4⋊C4—1224no—
469C4⋊C4—1224no—
470C4⋊C4—1224no—
471C4⋊C4—1224no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1222no—
477C4⋊C4—1224no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4C23×D6112yes—
481C4⋊C4C23×D6112yes—
482C22⋊C4—1228no—
483C22⋊C4—1228no—
484C42—614no—
485C42—612no—
486C42—614no—
487C42—612no—
488C42D4×D6112yes—
489C42C23×D6112yes—
490C23—24no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—614no—
495C23—612no—
496C23—1222no—
497C23—1222no—
498C23C2×D4×D6111yescharacteristic
499Q8—2448no—
500D4—28no—
501D4—48no—
502D4—48no—
503D4—44no—
504D4—24416no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—1228no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1224no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—2444no—
524C2×C4—2448no—
525C2×C4—2448no—
526C2×C4—24416no—
527C2×C4—2442no—
528C2×C4—2444no—
529C2×C4—2442no—
530C2×C4C2×D4×D6112yes—
531C2×C4C2×D4×D6112yes—
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4#1451982414yes—
534C2×C4C24×D6111yescharacteristic
535C8—24no—
536C8—1224no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—88no—
541C22—614no—
542C22—1222no—
543C22—1228no—
544C22—1222no—
545C22#1684033212yes—
546C22AGL(1,ℤ/8)×D6112yes—
547C22#1684055812yes—
548C22C22×D4×D6111yescharacteristic
549C4—22no—
550C4—22no—
551C4—42no—
552C4—42no—
553C4—1222no—
554C4—1222no—
555C4—1222no—
556C4—2442no—
557C4—2444no—
558C4—2442no—
559C4#1684033212yes—
560C4C2×#1451982412yes—
561C2—84no—
562C2—1224no—
563C2#2145700414yesdirect product factor
564C2C2×#1684033211yescharacteristic
565C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
566C2C2×#1684055811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#490C2
socleC23×C61C22×D4
prime radicalC2×#16840310C2
hypercenterC2×#490D61
hypocenterC61C2×#3234
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
2. C2×#125×D61C2semidirect
3. #21457004C2direct (N × H)
4. C2×#16840172C2semidirect
5. C2×#16840171C2semidirect
6. C2×#16840126C2semidirect
7. C2×#16840079C2semidirect
8. C2×#16840074C2semidirect
9. C2×#16840064C2semidirect
10. C2×#16840060C2semidirect
11. C2×#16839817C2semidirect
12. C2×C61×#490C2semidirect
23. #125×D61C22semidirect
34. #16840172C22semidirect
35. #16840171C22semidirect
36. #16840126C22semidirect
37. #16840079C22semidirect
38. #16840074C22semidirect
39. #16840064C22semidirect
40. #16840060C22semidirect
41. #16839817C22semidirect
53. C61×#490C22semidirect
284. C2×#490D61semidirect
291. C2×C61#3234semidirect
322. #490C2×D61semidirect
340. C61C2×#3234semidirect
563. C2#21457004direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403102
C2×#125×D612
#214570042
C2×#168401722
C2×#168401712
C2×#168401262
C2×#168400792
C2×#168400742
C2×#168400642
C2×#168400602
C2×#168398172
C2×C61×#4902
C2×#323461