Finite Group Databasebeta

← C2×#21455686 #30547611

Subgroups — C2×#21455686 #30547611

Order 15616.

2094 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840393C211yesmaximal, characteristic, complemented
2C2×#16840243C211yesmaximal, characteristic
3#21455686C2116yesmaximal, direct product factor
4C2×#16840072C212yesmaximal, complemented
5C2×#16840046C212yesmaximal, complemented
6C2×#16840061C212yesmaximal, complemented
7C2×#16840051C212yesmaximal, complemented
8C2×#16839802C211yesmaximal, characteristic, complemented
9C2×#16839788C211yesmaximal, characteristic, complemented
10C2×#16839761C211yesmaximal, characteristic
11C2×#16839759C211yesmaximal, characteristic
12C22×#106×C61C211yesmaximal, characteristic, complemented
13C22×#11758222C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15#16840393—24no—
16#16840393C2218yescomplemented
17C2×#14519776—21no—
18C2×#14519776C2212yes—
19C2×#14519768—21no—
20C2×#14519768C2212yes—
21C2×#14519756—22no—
22#16840243—24no—
23#16840243C2218yes—
24C2×#14519742C2212yes—
25C2×#14519739C2212yes—
26C2×C4×C22⋊D61—22no—
27C2×#14519736C2211yescharacteristic
28C2×#14519736C2211yescharacteristic
29C2×#14519735—21no—
30C2×C4⋊C4×D61—21no—
31#16840072C22116yescomplemented
32#16840061C22116yescomplemented
33#16840051C22116yescomplemented
34#16840046C22116yescomplemented
35#16839802C2218yescomplemented
36C2×#14519710C2212yes—
37C2×#14519708C2212yes—
38#16839788C2218yescomplemented
39C2×#14519706C2212yes—
40C2×#14519706C2212yes—
41#16839761C2218yes—
42#16839759C2218yes—
43C2×#14519702C2211yescharacteristic
44C2×#14519702C2211yescharacteristic
45C2×#14519700C2212yes—
46C2×#14519699C2211yescharacteristic
47C2×#14519699C2211yescharacteristic
48C2×#14519697C2212yes—
49C23×C8×C61C2211yescharacteristic
50C22×C4⋊C4×C61C2211yescharacteristic
51C22×C4⋊C4×C61C2211yescharacteristic
52C2×#106×C61C2212yes—
53C2×#106×C61C2212yes—
54C2×#106×C61C2218yescomplemented
55C2×#11758222—22no—
56C2×#11758222—24no—
57C2×#11758222Q818yescomplemented
58C2×#11758222C2314yes—
59C22×Dic122—22no—
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—22no—
65C2×#11758194—24no—
66C2×#11758194C2314yes—
67C22×#92418C2312yes—
68#14519776—28no—
69#14519776—24no—
70#14519776C2318yes—
71#14519768—28no—
72#14519768—24no—
73#14519768C2318yes—
74#14519756—216no—
75#14519742—24no—
76#14519742C2318yes—
77#14519739—24no—
78#14519739C2318yes—
79C4×C22⋊D61—216no—
80#14519736—22no—
81#14519736D414yes—
82#14519736C2314yes—
83#14519736C2314yes—
84#14519735—28no—
85C2×C4⋊Dic61—22no—
86C2×C4⋊Dic61D411yescharacteristic
87C2×C4⋊Dic61D411yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—22no—
91C2×#11758196—22no—
92C2×#11758196—22no—
93C4⋊C4×D61—28no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61D412yes—
96C2×#11758198—22no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—22no—
99C2×C4×Dic61C2312yes—
100#14519710C2318yes—
101#14519708C2318yes—
102#14519706C2318yes—
103#14519706C2318yes—
104#14519702C2314yes—
105#14519702C2314yes—
106#14519700C2318yes—
107#14519699C2314yes—
108#14519699C2314yes—
109#14519697C2318yes—
110C23×C4×C61—21no—
111C23×C4×C61—21no—
112C23×C4×C61C2311yescharacteristic
113C22×C8×C61—24no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2314yes—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—22no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—22no—
121C2×C4⋊C4×C61C2312yes—
122C2×C4⋊C4×C61C2314yes—
123C2×C4⋊C4×C61C2312yes—
124C2×C4⋊C4×C61C2314yes—
125#106×C61C2318yes—
126#106×C61C2318yes—
127C24×C61C2×D411yescharacteristic
128C23×D61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—24no—
132C22×C4×C61—24no—
133C22×C4×C61—82no—
134C22×C4×C61—82no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×Q814yes—
138C22×C4×C61#4114yes—
139C22×C4×C61C2411yescharacteristic
140C22×C4×C61C2412yes—
141C2×C22⋊D61—28no—
142C2×C22⋊D61—24no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61C2×D412yes—
147C22×Dic61C2×D412yes—
148#11758222—216no—
149#11758222—28no—
150#11758222C2×Q8116yescomplemented
151C2×D244—21no—
152C2×D244—22no—
153C2×D244C2×Q814yes—
154C2×D244C2412yes—
155C2×C4×D61—28no—
156C2×C4×D61—24no—
157C2×C4×D61—28no—
158C2×C4×D61—84no—
159C2×Dic122—21no—
160C2×Dic122—22no—
161C2×Dic122—28no—
162C2×Dic122—84no—
163C2×Dic122C2×Q814yes—
164C2×Dic122C2412yes—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—28no—
168C2×C8×C61C2412yes—
169C2×C8×C61C2412yes—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61C2414yes—
175C4⋊C4×C61C2414yes—
176C22⋊Dic61—24no—
177C22⋊Dic61C2×D418yes—
178#11758198—28no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—28no—
182C4⋊Dic61C2×D414yes—
183C4⋊Dic61C2×D414yes—
184C4⋊Dic61C2412yes—
185C4⋊Dic61C2412yes—
186#11758196—28no—
187#11758196—28no—
188#11758196—28no—
189C4×Dic61—24no—
190C4×Dic61—28no—
191C4×Dic61C2×D414yes—
192C4×Dic61C2414yes—
193#11758194—28no—
194C2×#92418—24no—
195C2×#92418C2414yes—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61C22⋊Q814yes—
199C23×C61C22×D412yes—
200C23×C61C22×D411yescharacteristic
201C22×D61—24no—
202C22×D61—22no—
203C22×D61—42no—
204C22×D61—42no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—44no—
208C2×C4×C61—44no—
209C2×C4×C61—84no—
210C2×C4×C61—84no—
211C2×C4×C61C22⋊Q814yes—
212C2×C4×C61C22⋊Q818yes—
213C2×C4×C61C22×D412yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×Q812yes—
216C2×C4×C61C22×Q814yes—
217C2×C4×C61C2×#4112yes—
218C2×C4×C61C2×#4114yes—
219C2×C4×C61C2511yescharacteristic
220C22⋊D61—216no—
221C2×Dic61—24no—
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61C22×D414yes—
230C2×Dic61C22×D414yes—
231D244—24no—
232D244—22no—
233D244C22×Q814yes—
234C4×D61—216no—
235C4×D61—88no—
236Dic122—24no—
237Dic122—22no—
238Dic122—88no—
239Dic122C22×Q814yes—
240C8×C61—22no—
241C8×C61—22no—
242#92418—24no—
243C2×#3084—611nomaximal, Sylow
244C22×C61—28no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61C2×C22⋊Q814yes—
248C22×C61C2×C22⋊Q812yes—
249C22×C61#57712yes—
250C22×C61#57712yes—
251C22×C61C23×D411yescharacteristic
252C2×D61—24no—
253C2×D61—48no—
254C2×D61—42no—
255C2×D61—42no—
256C4×C61—44no—
257C4×C61—44no—
258C4×C61C2×C22⋊Q812yes—
259C4×C61C2×C22⋊Q814yes—
260C4×C61C23×Q811yescharacteristic
261C4×C61C22×#4111yescharacteristic
262Dic61—24no—
263Dic61—24no—
264Dic61—24no—
265Dic61—42no—
266Dic61—42no—
267C2×#544—611no—
268C2×#520—611no—
269#3084—6116no—
270C2×#479—612no—
271C2×#475—612no—
272C2×#478—612no—
273C2×#476—612no—
274C2×#428—611no—
275C22×#106D6111yescharacteristic, complemented
276C2×#424—611no—
277C2×#417—611no—
278C2×#416—611no—
279C2×C61—24no—
280C2×C61#308414yescomplemented
281C2×C61C22×C22⋊Q811yescharacteristic
282C2×C61C2×#57711yescharacteristic
283C2×C61C2×#57711yescharacteristic
284D61—44no—
285C22×#41—611no—
286C22×M16—611no—
287C23×C8C2×D6111yescharacteristic
288#544—618no—
289#544—1224no—
290C2×C4⋊Q8—612no—
291C2×C4⋊Q8—1221no—
292C2×#125—612no—
293C2×#125—1221no—
294C2×C22⋊Q8—1222no—
295#520—618no—
296#520—1224no—
297C2×C4×Q8—612no—
298C2×C4×D4—612no—
299C2×C4×D4—1222no—
300C2×#117—611no—
301C2×#117—611no—
302C2×#117—1221no—
303C22×C4⋊C4—1221no—
304C22×C4⋊C4C2×D6111yescharacteristic
305C22×C4⋊C4C2×D6111yescharacteristic
306#479—6116no—
307#478—6116no—
308#476—6116no—
309#475—6116no—
310#428—618no—
311C2×#107—612no—
312C2×#106—612no—
313C2×#106C2×D6112yes—
314C2×#106C2×D6112yes—
315C2×#106C2×D6118yescomplemented
316#424—618no—
317C2×C4⋊C8—612no—
318C2×C4⋊C8—612no—
319#417—618no—
320#416—618no—
321C2×#103—611no—
322C2×#103—611no—
323C2×#103—612no—
324C2×#102—611no—
325C2×#102—611no—
326C2×#102—612no—
327C61C2×#308411yescharacteristic, complemented, Sylow
328C2×#41—618no—
329C2×#41—614no—
330C2×#41—1224no—
331C2×#41—1222no—
332C22×Q8—611no—
333C22×Q8—1222no—
334C22×D4—611no—
335C22×D4—1221no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—1222no—
339C23×C4—1221no—
340C23×C4C22×D6111yescharacteristic
341C2×M16—614no—
342C2×M16—1224no—
343C22×C8—24no—
344C22×C8—612no—
345C22×C8C22×D6111yescharacteristic
346C22×C8C22×D6111yescharacteristic
347C22×C8C22×D6114yes—
348C4⋊Q8—618no—
349C4⋊Q8—1224no—
350C4⋊Q8—1228no—
351#125—618no—
352#125—1224no—
353#125—1228no—
354C22⋊Q8—12216no—
355C4×Q8—618no—
356C4×Q8—1224no—
357C4×D4—618no—
358C4×D4—1224no—
359C4×D4—12216no—
360#117—614no—
361#117—614no—
362#117—614no—
363#117—1228no—
364#117—1222no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1228no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4C22×D6112yes—
379C2×C4⋊C4C22×D6114yes—
380C2×C4⋊C4C22×D6112yes—
381C2×C4⋊C4C22×D6114yes—
382C2×C22⋊C4—612no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1221no—
385C2×C42—612no—
386C2×C42—1222no—
387C2×C42—1221no—
388#107—618no—
389#106—618no—
390#106C22×D6118yes—
391#106C22×D6118yes—
392C4⋊C8—618no—
393C4⋊C8—618no—
394#103—614no—
395#103—614no—
396#103—618no—
397#102—614no—
398#102—614no—
399#102—618no—
400C24—1221no—
401C24D4×D6111yescharacteristic
402#41—6116no—
403#41—1228no—
404#41—12216no—
405C2×Q8—614no—
406C2×Q8—612no—
407C2×Q8—1228no—
408C2×Q8—1221no—
409C2×Q8—1222no—
410C2×Q8—4884no—
411C2×D4—614no—
412C2×D4—612no—
413C2×D4—1224no—
414C2×D4—1228no—
415C2×D4—1221no—
416C2×D4—1222no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—1228no—
426C22×C4—1221no—
427C22×C4—1224no—
428C22×C4—1228no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—4884no—
432C22×C4#1175822214yes—
433C22×C4D4×D6112yes—
434C22×C4D4×D6111yescharacteristic
435C22×C4Q8×D6114yes—
436C22×C4C23×D6112yes—
437C22×C4C23×D6111yescharacteristic
438M16—1228no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—28no—
442C2×C8—614no—
443C2×C8—1224no—
444C2×C8C23×D6112yes—
445C2×C8C23×D6112yes—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—614no—
451C4⋊C4—614no—
452C4⋊C4—612no—
453C4⋊C4—612no—
454C4⋊C4—1228no—
455C4⋊C4—1228no—
456C4⋊C4—1228no—
457C4⋊C4—1228no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4C23×D6114yes—
461C4⋊C4C23×D6114yes—
462C22⋊C4—618no—
463C22⋊C4—1228no—
464C22⋊C4—1224no—
465C42—614no—
466C42—614no—
467C42—1228no—
468C42—1224no—
469C23—22no—
470C23—22no—
471C23—1224no—
472C23—1222no—
473C23—2442no—
474C23—2442no—
475C23#1451975514yes—
476C23C2×D4×D6112yes—
477C23C2×D4×D6111yescharacteristic
478Q8—614no—
479Q8—1224no—
480Q8—1222no—
481Q8—4888no—
482D4—614no—
483D4—12216no—
484D4—1224no—
485D4—1222no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—614no—
493C2×C4—614no—
494C2×C4—1224no—
495C2×C4—1224no—
496C2×C4—1224no—
497C2×C4—1222no—
498C2×C4—12216no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—4888no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4#1451975514yes—
505C2×C4#1451975518yes—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C2×D4×D6112yes—
508C2×C4C2×Q8×D6112yes—
509C2×C4C2×Q8×D6114yes—
510C2×C4C2×#1175822212yes—
511C2×C4C2×#1175822214yes—
512C2×C4C24×D6111yescharacteristic
513C8—22no—
514C8—22no—
515C8—1224no—
516C22—28no—
517C22—22no—
518C22—22no—
519C22—1224no—
520C22—2448no—
521C22—2442no—
522C22—2442no—
523C22C2×#1451975514yes—
524C22C2×#1451975512yes—
525C22#1684056812yes—
526C22#1684056812yes—
527C22C22×D4×D6111yescharacteristic
528C4—44no—
529C4—44no—
530C4—1224no—
531C4—1224no—
532C4—1224no—
533C4—2442no—
534C4—2442no—
535C4C2×#1451975514yes—
536C4C2×#1451975512yes—
537C4C22×Q8×D6111yescharacteristic
538C4C22×#1175822211yescharacteristic
539C2—24no—
540C2—2444no—
541C2#2145568614yesdirect product factor
542C2C22×#1451975511yescharacteristic
543C2C2×#1684056811yescharacteristic
544C2C2×#1684056811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC22×#106×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC22×#106D61
hypocenterC61C2×#3084
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
1. C2×#16840393C2semidirect
3. #21455686C2direct (N × H)
4. C2×#16840072C2semidirect
5. C2×#16840046C2semidirect
6. C2×#16840061C2semidirect
7. C2×#16840051C2semidirect
8. C2×#16839802C2semidirect
9. C2×#16839788C2semidirect
12. C22×#106×C61C2semidirect
16. #16840393C22semidirect
31. #16840072C22semidirect
32. #16840061C22semidirect
33. #16840051C22semidirect
34. #16840046C22semidirect
35. #16839802C22semidirect
38. #16839788C22semidirect
54. C2×#106×C61C22semidirect
57. C2×#11758222Q8semidirect
150. #11758222C2×Q8semidirect
275. C22×#106D61semidirect
280. C2×C61#3084semidirect
315. C2×#106C2×D61semidirect
327. C61C2×#3084semidirect
541. C2#21455686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403932
C2×#168402432
#214556862
C2×#168400722
C2×#168400462
C2×#168400612
C2×#168400512
C2×#168398022
C2×#168397882
C2×#168397612
C2×#168397592
C22×#106×C612
C2×#308461