Finite Group Databasebeta

← C2×#21457010 #30552119

Subgroups — C2×#21457010 #30552119

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×#16840313C211yesmaximal, characteristic
2C2×#125×D61C211yesmaximal, characteristic, complemented
3#21457010C2116yesmaximal, direct product factor
4C2×#16840168C211yesmaximal, characteristic, complemented
5C2×#16840128C211yesmaximal, characteristic, complemented
6C2×#16840127C211yesmaximal, characteristic, complemented
7C2×#16840082C212yesmaximal, complemented
8C2×#16840077C212yesmaximal, complemented
9C2×#16840065C212yesmaximal, complemented
10C2×#16840061C212yesmaximal, complemented
11C2×#16839817C211yesmaximal, characteristic, complemented
12C2×C61×#501C211yesmaximal, characteristic, complemented
13C22×#11758226—21no—
14#16840313—24no—
15#16840313C2218yes—
16C2×#14519774C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519768C2211yescharacteristic
20C2×#14519767—21no—
21C2×#14519767—21no—
22#125×D61—24no—
23#125×D61C2218yescomplemented
24C2×#14519766—21no—
25C2×#14519752—21no—
26C2×#14519752C2212yes—
27C2×#14519739—21no—
28C2×#14519739C2212yes—
29C2×C4⋊C4×D61—21no—
30C2×C4⋊C4×D61—21no—
31C2×C4⋊C4×D61C2211yescharacteristic
32C2×C4⋊C4×D61C2211yescharacteristic
33C2×C42×D61C2211yescharacteristic
34#16840168C2218yescomplemented
35#16840128C2218yescomplemented
36#16840127C2218yescomplemented
37#16840082C22116yescomplemented
38#16840077C22116yescomplemented
39#16840065C22116yescomplemented
40#16840061C22116yescomplemented
41#16839817C2218yescomplemented
42C2×#14519710C2212yes—
43C2×#14519708C2212yes—
44C2×#14519706C2212yes—
45C2×#14519699C2212yes—
46C2×#14519699C2212yes—
47C2×#14519697C2212yes—
48C2×#14519697C2212yes—
49C2×#14519689C2212yes—
50C2×#14519688C2211yescharacteristic
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#501C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×#106×C61C2212yes—
56C2×#97×C61C2211yescharacteristic
57C2×#11758226—28no—
58C2×#11758226—48no—
59C22×D244—22no—
60C22×D244—21no—
61C22×D244—21no—
62C22×D244C2312yes—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×C4×D61—22no—
66C22×C4×D61D411yescharacteristic
67C22×C4×D61D411yescharacteristic
68C22×C4×D61C2311yescharacteristic
69C22×#92418C2312yes—
70#14519774—22no—
71#14519774C2314yes—
72#14519769—22no—
73#14519769C2314yes—
74#14519768—22no—
75#14519768—22no—
76#14519768C2314yes—
77#14519768C2314yes—
78#14519767—28no—
79#14519767—28no—
80#14519766—28no—
81#14519752—28no—
82#14519752—24no—
83#14519752C2318yes—
84#14519739—28no—
85#14519739—24no—
86#14519739C2318yes—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
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×#11758198—22no—
104C2×#11758198—22no—
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C42×D61D414yes—
109C42×D61C2314yes—
110#14519710C2318yes—
111#14519708C2318yes—
112#14519706C2318yes—
113#14519699C2318yes—
114#14519699C2318yes—
115#14519697C2318yes—
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—22no—
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—21no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150#11758226—432no—
151C2×D244—28no—
152C2×D244—28no—
153C2×D244—24no—
154C2×D244—24no—
155C2×D244—22no—
156C2×D244—44no—
157C2×D244—44no—
158C2×D244—48no—
159C2×D244C2414yes—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—28no—
165C2×C4×D61—42no—
166C2×C4×D61—48no—
167C2×C4×D61—44no—
168C2×C4×D61—44no—
169C2×C4×D61C2×D414yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×D414yes—
172C2×C4×D61C2412yes—
173C2×C8×C61—24no—
174C2×C8×C61C2414yes—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C2412yes—
182C4⋊C4×C61C2414yes—
183C4⋊C4×C61C2412yes—
184C42×C61C2×D412yes—
185C42×C61C2412yes—
186#11758198—28no—
187#11758198—28no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—24no—
191C4⋊Dic61—24no—
192C4⋊Dic61C2412yes—
193C4⋊Dic61C2412yes—
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×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—24no—
210C22×D61—44no—
211C22×D61—44no—
212C22×D61—44no—
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—22no—
220C2×C4×C61—24no—
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—22no—
233C2×Dic61—22no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61C22×D412yes—
237C2×Dic61C22×D412yes—
238C2×Dic61C22×D412yes—
239D244—28no—
240D244—48no—
241D244—48no—
242D244—416no—
243D244—44no—
244C4×D61—28no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—44no—
248C4×D61—416no—
249C4×D61—48no—
250C4×D61—48no—
251C8×C61—24no—
252#92418—24no—
253C2×#3234—611nomaximal, Sylow
254C22×C61#53612yes—
255C22×C61C2×AGL(1,ℤ/8)12yes—
256C22×C61#57512yes—
257C22×C61C23×D411yescharacteristic
258C2×D61—22no—
259C2×D61—28no—
260C2×D61—22no—
261C2×D61—44no—
262C2×D61—44no—
263C2×D61—44no—
264C2×D61—88no—
265C2×D61C2×AGL(1,ℤ/8)14yes—
266C4×C61—22no—
267C4×C61—22no—
268C4×C61—44no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61#53612yes—
272C4×C61C2×2−1+412yes—
273Dic61—22no—
274Dic61—22no—
275Dic61—22no—
276Dic61—42no—
277Dic61—42no—
278C2×#532—611no—
279C22×#125—611no—
280#3234—6116no—
281C2×#501—611no—
282C2×#501D6111yescharacteristic, complemented
283C2×#490—611no—
284C2×#490—611no—
285C2×#481—612no—
286C2×#480—612no—
287C2×#478—612no—
288C2×#478—612no—
289C2×#432—611no—
290C2×C61#323414yescomplemented
291C2×C61C2×#53611yescharacteristic
292C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
293C2×C61C2×#57511yescharacteristic
294D61—24no—
295D61—84no—
296C22×#41—1221no—
297#532—618no—
298#532—1224no—
299C2×C4⋊Q8C2×D6111yescharacteristic
300C2×#127—611no—
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#501C2×D6118yescomplemented
321#490—618no—
322#490—618no—
323#481—6116no—
324#480—6116no—
325#478—6116no—
326#478—6116no—
327#432—618no—
328C2×#107—612no—
329C2×#107C2×D6112yes—
330C2×#106—612no—
331C2×#106C2×D6112yes—
332C2×C4⋊C8—612no—
333C2×#102—612no—
334C2×#102—612no—
335C2×#102—612no—
336C2×#102—612no—
337C2×C22⋊C8—612no—
338C2×#97—611no—
339C2×#97C2×D6111yescharacteristic
340C61C2×#323411yescharacteristic, complemented, Sylow
341C2×#41—1228no—
342C2×#41—2448no—
343C22×Q8—21no—
344C22×D4—612no—
345C22×D4—1222no—
346C22×D4—1221no—
347C22×D4—1221no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—1222no—
352C23×C4—1221no—
353C23×C4—1221no—
354C22×C8—612no—
355C22×C8C22×D6112yes—
356C4⋊Q8—22no—
357C4⋊Q8C22×D6114yes—
358#127—614no—
359#127—1222no—
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—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
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⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6112yes—
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#107C22×D6118yes—
409#106—618no—
410#106C22×D6118yes—
411C4⋊C8—618no—
412#102—618no—
413#102—618no—
414#102—618no—
415#102—618no—
416C22⋊C8—618no—
417#97—614no—
418#97C22×D6114yes—
419C24—611no—
420C24—1222no—
421#41—24432no—
422C2×Q8—24no—
423C2×Q8—44no—
424C2×D4—614no—
425C2×D4—1228no—
426C2×D4—1224no—
427C2×D4—1222no—
428C2×D4—1224no—
429C2×D4—1228no—
430C2×D4—2444no—
431C2×D4—2444no—
432C2×D4—2448no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—614no—
441C22×C4—612no—
442C22×C4—614no—
443C22×C4—1228no—
444C22×C4—1224no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1224no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—2442no—
451C22×C4—2448no—
452C22×C4—2444no—
453C22×C4—2444no—
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—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—612no—
468C4⋊C4—612no—
469C4⋊C4—1224no—
470C4⋊C4—1224no—
471C4⋊C4—1224no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C4⋊C4—1224no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1224no—
478C4⋊C4—1224no—
479C4⋊C4C23×D6112yes—
480C4⋊C4C23×D6112yes—
481C4⋊C4C23×D6114yes—
482C22⋊C4—1228no—
483C22⋊C4—1228no—
484C42—614no—
485C42—612no—
486C42—614no—
487C42—612no—
488C42D4×D6112yes—
489C42C23×D6112yes—
490C23—614no—
491C23—612no—
492C23—1222no—
493C23—1222no—
494C23—1224no—
495C23—2444no—
496C23—2444no—
497C23—2444no—
498C23C2×D4×D6111yescharacteristic
499Q8—48no—
500D4—1228no—
501D4—2448no—
502D4—2448no—
503D4—24416no—
504D4—2444no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—1228no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—2444no—
525C2×C4—24416no—
526C2×C4—2448no—
527C2×C4—2448no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4C2×D4×D6112yes—
531C2×C4C2×D4×D6112yes—
532C2×C4C2×D4×D6111yescharacteristic
533C2×C4#1451982514yes—
534C2×C4C24×D6111yescharacteristic
535C8—24no—
536C8—1224no—
537C22—614no—
538C22—1222no—
539C22—1228no—
540C22—1222no—
541C22—4888no—
542C22—2444no—
543C22—2444no—
544C22—2444no—
545C22#1684033712yes—
546C22AGL(1,ℤ/8)×D6112yes—
547C22#1684056112yes—
548C22C22×D4×D6111yescharacteristic
549C4—22no—
550C4—22no—
551C4—42no—
552C4—44no—
553C4—42no—
554C4—1222no—
555C4—1222no—
556C4—1222no—
557C4—2442no—
558C4—2442no—
559C4#1684033712yes—
560C4C2×#1451982512yes—
561C2—1224no—
562C2—4884no—
563C2#2145701014yesdirect product factor
564C2C2×#1684033711yescharacteristic
565C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
566C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC2×#16840313C2
hypercenterC2×#501D61
hypocenterC61C2×#3234
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
2. C2×#125×D61C2semidirect
3. #21457010C2direct (N × H)
4. C2×#16840168C2semidirect
5. C2×#16840128C2semidirect
6. C2×#16840127C2semidirect
7. C2×#16840082C2semidirect
8. C2×#16840077C2semidirect
9. C2×#16840065C2semidirect
10. C2×#16840061C2semidirect
11. C2×#16839817C2semidirect
12. C2×C61×#501C2semidirect
23. #125×D61C22semidirect
34. #16840168C22semidirect
35. #16840128C22semidirect
36. #16840127C22semidirect
37. #16840082C22semidirect
38. #16840077C22semidirect
39. #16840065C22semidirect
40. #16840061C22semidirect
41. #16839817C22semidirect
53. C61×#501C22semidirect
282. C2×#501D61semidirect
290. C2×C61#3234semidirect
320. #501C2×D61semidirect
340. C61C2×#3234semidirect
563. C2#21457010direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403132
C2×#125×D612
#214570102
C2×#168401682
C2×#168401282
C2×#168401272
C2×#168400822
C2×#168400772
C2×#168400652
C2×#168400612
C2×#168398172
C2×C61×#5012
C2×#323461