Finite Group Databasebeta

← C2×#21457032 #30552164

Subgroups — C2×#21457032 #30552164

Order 15616.

1758 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2C2×#16840316C211yesmaximal, characteristic
3#21457032C2116yesmaximal, direct product factor
4C2×#16840169C212yesmaximal, complemented
5C2×#16840129C212yesmaximal, complemented
6C2×#16840106C212yesmaximal
7C2×#16840102C212yesmaximal
8C2×#16840058C211yesmaximal, characteristic, complemented
9C2×#16840057C211yesmaximal, characteristic, complemented
10C2×#16840044C211yesmaximal, characteristic, complemented
11C2×#16839818C211yesmaximal, characteristic
12C2×C61×#475C211yesmaximal, characteristic, complemented
13#16840317—24no—
14#16840317C2218yes—
15#16840316—24no—
16#16840316C2218yes—
17C2×#14519777C2211yescharacteristic
18C2×#14519776C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519768C2212yes—
21C2×#14519767—21no—
22C2×#14519757—21no—
23C2×#14519757C2211yescharacteristic
24C2×#14519755—21no—
25C2×#14519753—21no—
26C2×#14519753C2211yescharacteristic
27C2×Q8×Dic61—21no—
28C2×C4×Dic122—21no—
29C2×C4×Dic122C2211yescharacteristic
30C2×#14519742—21no—
31C2×#14519735C2212yes—
32C2×#14519734—21no—
33C2×C42×D61C2211yescharacteristic
34#16840169C22116yescomplemented
35#16840129C22116yescomplemented
36#16840106C22116yes—
37#16840102C22116yes—
38#16840058C2218yescomplemented
39#16840057C2218yescomplemented
40#16840044C2218yescomplemented
41#16839818C2218yes—
42C2×#14519711C2212yes—
43C2×#14519710C2212yes—
44C2×#14519708C2212yes—
45C2×C4⋊#92418C2211yescharacteristic
46C2×#14519702C2212yes—
47C2×#14519701C2212yes—
48C2×#14519700C2212yes—
49C2×#14519689C2212yes—
50C2×#14519687C2212yes—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#475C2218yescomplemented
54C2×#106×C61C2212yes—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2212yes—
57C22×Dic122—21no—
58C22×Dic122—21no—
59C22×Dic122C2311yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×#92418C2312yes—
64#14519777—22no—
65#14519777C2314yes—
66#14519776—22no—
67#14519776C2314yes—
68#14519775—22no—
69#14519775C2314yes—
70#14519768—24no—
71#14519768C2318yes—
72#14519767—28no—
73#14519757—28no—
74#14519757—22no—
75#14519757C2314yes—
76#14519755—28no—
77#14519753—28no—
78#14519753—22no—
79#14519753C2314yes—
80Q8×Dic61—28no—
81C4×Dic122—22no—
82C4×Dic122—28no—
83C4×Dic122C2314yes—
84#14519742—28no—
85#14519735—24no—
86#14519735C2318yes—
87#14519734—28no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—22no—
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—22no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—22no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C42×D61D414yes—
108C42×D61C2314yes—
109#14519711C2318yes—
110#14519710C2318yes—
111#14519708C2318yes—
112C4⋊#92418C2314yes—
113#14519702C2318yes—
114#14519701C2318yes—
115#14519700C2318yes—
116#14519689C2318yes—
117#14519687C2318yes—
118C22×Q8×C61—21no—
119C22×Q8×C61C2311yescharacteristic
120C22×C8×C61C2312yes—
121C4⋊Q8×C61—22no—
122C4⋊Q8×C61C2314yes—
123C4×Q8×C61—22no—
124C4×Q8×C61C2314yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2312yes—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#106×C61C2318yes—
132C4⋊C8×C61C2314yes—
133#103×C61C2318yes—
134C23×D61C2×D411yescharacteristic
135C2×Q8×C61—21no—
136C2×Q8×C61—24no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61—44no—
140C2×Q8×C61C2412yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—22no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×C4×D61—24no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—42no—
157C2×C4×D61C2×D414yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D414yes—
160C2×C4×D61C2412yes—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—24no—
164C2×Dic122—21no—
165C2×Dic122—24no—
166C2×Dic122—44no—
167C2×Dic122—44no—
168C2×Dic122C2412yes—
169C2×C8×C61—24no—
170C2×C8×C61C2414yes—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C2414yes—
176C4⋊C4×C61C2412yes—
177C42×C61—24no—
178C42×C61C2×D412yes—
179C42×C61C2412yes—
180#11758198—24no—
181#11758198—24no—
182#11758198—28no—
183C4⋊Dic61—24no—
184C4⋊Dic61—22no—
185C4⋊Dic61—21no—
186C4⋊Dic61—24no—
187C4⋊Dic61—28no—
188C4⋊Dic61C2414yes—
189C4⋊Dic61C2412yes—
190#11758196—24no—
191#11758196—24no—
192#11758196—24no—
193#11758196—28no—
194#11758196—24no—
195#11758196—24no—
196C4×Dic61—24no—
197C4×Dic61—28no—
198C4×Dic61C2×D412yes—
199C4×Dic61C2×D414yes—
200C4×Dic61C2×D414yes—
201C4×Dic61C2412yes—
202C2×#92418—24no—
203C2×#92418C2414yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61C22×D412yes—
209Q8×C61—22no—
210Q8×C61—22no—
211Q8×C61—42no—
212Q8×C61—48no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—42no—
218C2×C4×C61—44no—
219C2×C4×C61—42no—
220C2×C4×C61#13714yes—
221C2×C4×C61C22×D412yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×D412yes—
224C2×C4×C612−1+412yes—
225C2×C4×C612−1+412yes—
226C2×C4×C61C2511yescharacteristic
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—24no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—44no—
236C2×Dic61C22×D412yes—
237C2×Dic61C22×D412yes—
238C2×Dic61C22×D412yes—
239C4×D61—24no—
240C4×D61—28no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—44no—
244Dic122—22no—
245Dic122—22no—
246Dic122—48no—
247Dic122—48no—
248Dic122—42no—
249C8×C61—24no—
250#92418—24no—
251C2×#3237—611nomaximal, Sylow
252C22×C61#53612yes—
253C22×C61C2×#13712yes—
254C22×C61#57512yes—
255C22×C61C23×D411yescharacteristic
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—44no—
260C4×C61—22no—
261C4×C61—42no—
262C4×C61—44no—
263C4×C61—42no—
264C4×C61#53612yes—
265C4×C61C2×#13714yes—
266C4×C61C2×2−1+411yescharacteristic
267C4×C61C2×2−1+411yescharacteristic
268Dic61—22no—
269Dic61—22no—
270Dic61—22no—
271Dic61—42no—
272Dic61—42no—
273Dic61—44no—
274C2×#533—611no—
275C2×#533—611no—
276#3237—6116no—
277C2×#501—612no—
278C2×#491—612no—
279C2×#485—612no—
280C2×#485—612no—
281C2×#477—611no—
282C2×#477—611no—
283C2×#475—611no—
284C2×#475D6111yescharacteristic, complemented
285C2×#432—611no—
286C2×C61#323714yescomplemented
287C2×C61C2×#53611yescharacteristic
288C2×C61C22×#13711yescharacteristic
289C2×C61C2×#57511yescharacteristic
290D61—42no—
291#533—618no—
292#533—618no—
293#533—1224no—
294#533—1224no—
295C2×C4⋊Q8—611no—
296C2×C4⋊Q8—611no—
297C2×C4⋊Q8—611no—
298C2×C4⋊Q8C2×D6111yescharacteristic
299C2×#125—612no—
300C2×#125—1221no—
301C2×C22⋊Q8—611no—
302C2×C22⋊Q8—611no—
303C2×C22⋊Q8—1221no—
304C2×C22⋊Q8—1221no—
305C2×C22⋊Q8—1221no—
306C2×C4×Q8—611no—
307C2×C4×Q8—1221no—
308C2×C4×Q8—1221no—
309C2×C4×Q8—1221no—
310C2×C4×Q8C2×D6111yescharacteristic
311C2×#117—612no—
312C2×#117—1221no—
313C22×C42—611no—
314#501—6116no—
315#491—6116no—
316#485—6116no—
317#485—6116no—
318#477—618no—
319#477—618no—
320#475—618no—
321#475C2×D6118yescomplemented
322#432—618no—
323C2×#107—612no—
324C2×#107—612no—
325C2×#106—612no—
326C2×#106C2×D6112yes—
327C2×C4⋊C8—611no—
328C2×C4⋊C8C2×D6111yescharacteristic
329C2×#103—612no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#103C2×D6112yes—
333C2×C22⋊C8—612no—
334C2×#97—612no—
335C61C2×#323711yescharacteristic, complemented, Sylow
336C22×Q8—21no—
337C22×Q8—611no—
338C22×Q8—1221no—
339C22×Q8—1221no—
340C22×Q8C22×D6111yescharacteristic
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C22×C8—612no—
345C22×C8C22×D6112yes—
346C4⋊Q8—22no—
347C4⋊Q8—614no—
348C4⋊Q8—614no—
349C4⋊Q8—614no—
350C4⋊Q8—1222no—
351C4⋊Q8—1222no—
352C4⋊Q8—1222no—
353C4⋊Q8C22×D6114yes—
354#125—618no—
355#125—1224no—
356#125—1228no—
357C22⋊Q8—614no—
358C22⋊Q8—614no—
359C22⋊Q8—1222no—
360C22⋊Q8—1228no—
361C22⋊Q8—1228no—
362C22⋊Q8—1222no—
363C22⋊Q8—1228no—
364C4×Q8—22no—
365C4×Q8—614no—
366C4×Q8—1228no—
367C4×Q8—1228no—
368C4×Q8—1228no—
369C4×Q8—1222no—
370C4×Q8C22×D6114yes—
371#117—618no—
372#117—1228no—
373#117—1224no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C22×D6112yes—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—1222no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C42—21no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—614no—
397C2×C42—614no—
398C2×C42—1222no—
399C2×C42—1221no—
400C2×C42C22×D6111yescharacteristic
401#107—618no—
402#107—618no—
403#106—618no—
404#106C22×D6118yes—
405C4⋊C8—614no—
406C4⋊C8C22×D6114yes—
407#103—618no—
408#103—618no—
409#103—618no—
410#103C22×D6118yes—
411C22⋊C8—618no—
412#97—618no—
413C24—611no—
414C2×Q8—21no—
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—44no—
419C2×Q8—612no—
420C2×Q8—1224no—
421C2×Q8—1224no—
422C2×Q8—1221no—
423C2×Q8—1222no—
424C2×Q8—1222no—
425C2×Q8—2444no—
426C2×Q8—2444no—
427C2×Q8C23×D6112yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—614no—
436C22×C4—612no—
437C22×C4—614no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1222no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—2442no—
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448C2×C8—24no—
449C2×C8—614no—
450C2×C8—1224no—
451C2×C8C23×D6114yes—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—614no—
457C4⋊C4—612no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—1224no—
461C4⋊C4—1228no—
462C4⋊C4—1224no—
463C4⋊C4—1224no—
464C4⋊C4—1228no—
465C4⋊C4—1222no—
466C4⋊C4—1221no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4C23×D6114yes—
470C4⋊C4C23×D6112yes—
471C22⋊C4—1228no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C42—24no—
475C42—614no—
476C42—614no—
477C42—612no—
478C42—612no—
479C42—1228no—
480C42—1224no—
481C42D4×D6112yes—
482C42C23×D6112yes—
483C23—612no—
484C23—1222no—
485C23—1222no—
486C23—1222no—
487C23C2×D4×D6111yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—48no—
491Q8—42no—
492Q8—1222no—
493Q8—1222no—
494Q8—2448no—
495Q8—2448no—
496Q8—2442no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1228no—
511C2×C4—1224no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1224no—
516C2×C4—1222no—
517C2×C4—2444no—
518C2×C4—2444no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4#1451980614yes—
522C2×C4C2×D4×D6112yes—
523C2×C4C2×D4×D6112yes—
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4#1451982412yes—
526C2×C4#1451982512yes—
527C2×C4C24×D6111yescharacteristic
528C8—24no—
529C8—1224no—
530C22—1222no—
531C22—1222no—
532C22—1222no—
533C22—2444no—
534C22#1684033512yes—
535C22C2×#1451980612yes—
536C22#1684056112yes—
537C22C22×D4×D6111yescharacteristic
538C4—22no—
539C4—44no—
540C4—42no—
541C4—42no—
542C4—1222no—
543C4—1222no—
544C4—1222no—
545C4—2444no—
546C4—2442no—
547C4—2442no—
548C4#1684033512yes—
549C4C2×#1451980614yes—
550C4C2×#1451982411yescharacteristic
551C4C2×#1451982511yescharacteristic
552C2—2442no—
553C2#2145703214yesdirect product factor
554C2C2×#1684033511yescharacteristic
555C2C22×#1451980611yescharacteristic
556C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#475C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#475D61
hypocenterC61C2×#3237
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #21457032C2direct (N × H)
4. C2×#16840169C2semidirect
5. C2×#16840129C2semidirect
8. C2×#16840058C2semidirect
9. C2×#16840057C2semidirect
10. C2×#16840044C2semidirect
12. C2×C61×#475C2semidirect
34. #16840169C22semidirect
35. #16840129C22semidirect
38. #16840058C22semidirect
39. #16840057C22semidirect
40. #16840044C22semidirect
53. C61×#475C22semidirect
284. C2×#475D61semidirect
286. C2×C61#3237semidirect
321. #475C2×D61semidirect
335. C61C2×#3237semidirect
553. C2#21457032direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168403162
#214570322
C2×#168401692
C2×#168401292
C2×#168401062
C2×#168401022
C2×#168400582
C2×#168400572
C2×#168400442
C2×#168398182
C2×C61×#4752
C2×#323761