Finite Group Databasebeta

← C2×#21456987 #30552043

Subgroups — C2×#21456987 #30552043

Order 15616.

1822 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×#16840296C211yesmaximal, characteristic
2#21456987C2116yesmaximal, direct product factor
3C2×#16840161C212yesmaximal, complemented
4C2×#16840119C212yesmaximal, complemented
5C2×#16840097C211yesmaximal, characteristic, complemented
6C2×#16840080C211yesmaximal, characteristic
7C2×#16840092C211yesmaximal, characteristic, complemented
8C2×#16840075C211yesmaximal, characteristic
9C2×#16840066C212yesmaximal, complemented
10C2×#16840059C212yesmaximal, complemented
11C2×#16839815C211yesmaximal, characteristic, complemented
12C22×#125×C61C211yesmaximal, characteristic, complemented
13#16840296—24no—
14#16840296C2218yes—
15C2×#14519771—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×#14519760—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519751C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4×C22⋊D61—21no—
24C2×#14519734—21no—
25#16840161C22116yescomplemented
26#16840119C22116yescomplemented
27#16840097C2218yescomplemented
28#16840092C2218yescomplemented
29#16840080C2218yes—
30#16840075C2218yes—
31#16840066C22116yescomplemented
32#16840059C22116yescomplemented
33#16839815C2218yescomplemented
34C2×#14519710C2212yes—
35C2×#14519710C2212yes—
36C2×#14519708C2212yes—
37C2×#14519708C2212yes—
38C2×C4⋊#92418C2212yes—
39C2×#14519700C2212yes—
40C2×#14519700C2212yes—
41C2×#14519697C2212yes—
42C2×#14519697C2212yes—
43C2×C22⋊#92418C2211yescharacteristic
44C2×C22⋊#92418C2211yescharacteristic
45C2×C4×#92418C2212yes—
46C2×#125×C61—21no—
47C2×#125×C61—21no—
48C2×#125×C61—24no—
49C2×#125×C61C2212yes—
50C2×#125×C61C2212yes—
51C2×#125×C61C2218yescomplemented
52C22×C4⋊C4×C61—21no—
53C22×C4⋊C4×C61—21no—
54C22×C4⋊C4×C61C2211yescharacteristic
55C22×C4⋊C4×C61C2211yescharacteristic
56C22×C42×C61C2211yescharacteristic
57C22×Dic122C2311yescharacteristic
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61C22×#92418C2312yes—
62C22×#92418C2312yes—
63#14519771—28no—
64#14519769—22no—
65#14519769C2314yes—
66#14519765—22no—
67#14519765C2314yes—
68#14519760—28no—
69#14519756—22no—
70#14519756C2314yes—
71#14519751—22no—
72#14519751C2314yes—
73C4×Dic122—22no—
74C4×Dic122C2314yes—
75C4×D244—22no—
76C4×D244C2314yes—
77C4×C22⋊D61—28no—
78#14519734—28no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91#14519710C2318yes—
92#14519710C2318yes—
93#14519708C2318yes—
94#14519708C2318yes—
95C4⋊#92418C2318yes—
96#14519700C2318yes—
97#14519700C2318yes—
98#14519697C2318yes—
99#14519697C2318yes—
100C22⋊#92418C2314yes—
101C22⋊#92418C2314yes—
102C4×#92418C2318yes—
103C23×C4×C61—21no—
104C23×C4×C61—21no—
105C23×C4×C61D411yescharacteristic
106C23×C4×C61D411yescharacteristic
107C23×C4×C61C2311yescharacteristic
108#125×C61—24no—
109#125×C61—24no—
110#125×C61—28no—
111#125×C61—28no—
112#125×C61C2318yes—
113#125×C61C2318yes—
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61—22no—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61—22no—
118C2×C4⋊C4×C61—28no—
119C2×C4⋊C4×C61—28no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61C2312yes—
125C2×C4⋊C4×C61C2312yes—
126C2×C4⋊C4×C61C2314yes—
127C2×C4⋊C4×C61C2314yes—
128C2×C42×C61D411yescharacteristic
129C2×C42×C61D411yescharacteristic
130C2×C42×C61D414yes—
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2311yescharacteristic
133C2×C42×C61C2314yes—
134C24×C61C2×D411yescharacteristic
135C23×D61—21no—
136C22×C4×C61—24no—
137C22×C4×C61—24no—
138C22×C4×C61—24no—
139C22×C4×C61—24no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—42no—
143C22×C4×C61—44no—
144C22×C4×C61—44no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D414yes—
152C22×C4×C61C2×D414yes—
153C22×C4×C61C2412yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—24no—
156C2×C22⋊D61—44no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C2×D244—24no—
161C2×D244—21no—
162C2×D244C2412yes—
163C2×C4×D61—24no—
164C2×C4×D61—44no—
165C2×Dic122—24no—
166C2×Dic122—21no—
167C2×Dic122C2412yes—
168C4⋊C4×C61—28no—
169C4⋊C4×C61—28no—
170C4⋊C4×C61—28no—
171C4⋊C4×C61—28no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C2414yes—
177C4⋊C4×C61C2414yes—
178C42×C61C2×D414yes—
179C42×C61C2×D412yes—
180C42×C61C2×D412yes—
181C42×C61C2×D414yes—
182C42×C61C2412yes—
183C42×C61C2412yes—
184C22⋊Dic61—24no—
185#11758198—24no—
186#11758198—24no—
187#11758198—24no—
188#11758198—24no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C2412yes—
192C4⋊Dic61C2412yes—
193#11758196—24no—
194#11758196—24no—
195#11758196—24no—
196#11758196—24no—
197C4×Dic61—24no—
198C2×#92418—24no—
199C2×#92418—24no—
200C2×#92418C2414yes—
201C2×#92418C2414yes—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61#13514yes—
205C23×C61C22×D412yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C2×C4×C61—22no—
212C2×C4×C61—28no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—44no—
222C2×C4×C61—48no—
223C2×C4×C61—48no—
224C2×C4×C61—44no—
225C2×C4×C61—44no—
226C2×C4×C61C22×D412yes—
227C2×C4×C61C22×D412yes—
228C2×C4×C61C22×D412yes—
229C2×C4×C61C22×D412yes—
230C2×C4×C61C22×D412yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C612−1+414yes—
233C2×C4×C61C2511yescharacteristic
234C22⋊D61—48no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241D244—24no—
242D244—42no—
243C4×D61—48no—
244Dic122—24no—
245Dic122—42no—
246#92418—24no—
247#92418—24no—
248C2×#3232—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—28no—
251C22×C61—22no—
252C22×C61#53612yes—
253C22×C61C2×#13514yes—
254C22×C61C2×#13512yes—
255C22×C61#57512yes—
256C22×C61C23×D411yescharacteristic
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—84no—
261C4×C61—22no—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—22no—
266C4×C61—44no—
267C4×C61—44no—
268C4×C61#53612yes—
269C4×C61C2×2−1+412yes—
270Dic61—42no—
271Dic61—42no—
272Dic61—42no—
273C2×#529—611no—
274C22×#125D6111yescharacteristic, complemented
275#3232—6116no—
276C2×#499—612no—
277C2×#488—612no—
278C2×#484—611no—
279C2×#481—611no—
280C2×#483—611no—
281C2×#480—611no—
282C2×#479—612no—
283C2×#478—612no—
284C2×#431—611no—
285C2×C61—24no—
286C2×C61#323214yescomplemented
287C2×C61C2×#53611yescharacteristic
288C2×C61C22×#13511yescharacteristic
289C2×C61C2×#57511yescharacteristic
290D61—82no—
291#529—618no—
292#529—1224no—
293C2×#126—1221no—
294C2×#125—21no—
295C2×#125—21no—
296C2×#125—24no—
297C2×#125—611no—
298C2×#125C2×D6112yes—
299C2×#125C2×D6112yes—
300C2×#125C2×D6118yescomplemented
301C2×#124—611no—
302C2×#123—1221no—
303C2×C22⋊Q8—611no—
304C2×#121—611no—
305C2×C4×Q8—611no—
306C2×C4×D4—611no—
307C2×C4×D4—1221no—
308C2×#117—1221no—
309C22×C4⋊C4—21no—
310C22×C4⋊C4—21no—
311C22×C4⋊C4C2×D6111yescharacteristic
312C22×C4⋊C4C2×D6111yescharacteristic
313C22×C42C2×D6111yescharacteristic
314#499—6116no—
315#488—6116no—
316#484—618no—
317#483—618no—
318#481—618no—
319#480—618no—
320#479—6116no—
321#478—6116no—
322#431—618no—
323C2×#107—612no—
324C2×#107—612no—
325C2×#106—612no—
326C2×#106—612no—
327C2×C4⋊C8—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#102—612no—
331C2×#102—612no—
332C2×C22⋊C8—611no—
333C2×C22⋊C8—611no—
334C2×C4×C8—612no—
335C61C2×#323211yescharacteristic, complemented, Sylow
336C22×Q8—611no—
337C22×D4—611no—
338C22×D4—1221no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—1221no—
342C23×C4C22⋊D6111yescharacteristic
343C23×C4C22⋊D6111yescharacteristic
344C23×C4C22×D6111yescharacteristic
345C22×C8—612no—
346C22×C8—612no—
347#126—1228no—
348#125—24no—
349#125—24no—
350#125—28no—
351#125—28no—
352#125—614no—
353#125—1222no—
354#125C22×D6118yes—
355#125C22×D6118yes—
356#124—614no—
357#124—1222no—
358#123—1228no—
359C22⋊Q8—614no—
360C22⋊Q8—1222no—
361#121—614no—
362#121—1222no—
363C4×Q8—614no—
364C4×Q8—1222no—
365C4×D4—614no—
366C4×D4—1228no—
367C4×D4—1222no—
368#117—1228no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—28no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22×D6112yes—
386C2×C4⋊C4C22×D6112yes—
387C2×C4⋊C4C22×D6114yes—
388C2×C4⋊C4C22×D6114yes—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C22⋊C4—1221no—
393C2×C22⋊C4—1221no—
394C2×C42—1221no—
395C2×C42C22⋊D6111yescharacteristic
396C2×C42C22⋊D6111yescharacteristic
397C2×C42C22⋊D6114yes—
398C2×C42C22×D6111yescharacteristic
399C2×C42C22×D6111yescharacteristic
400C2×C42C22×D6114yes—
401#107—618no—
402#107—618no—
403#106—618no—
404#106—618no—
405C4⋊C8—618no—
406#103—618no—
407#103—618no—
408#102—618no—
409#102—618no—
410C22⋊C8—614no—
411C22⋊C8—614no—
412C4×C8—618no—
413C24—1221no—
414C24C2×C22⋊D6111yescharacteristic
415C2×Q8—612no—
416C2×Q8—1221no—
417C2×Q8—1224no—
418C2×D4—612no—
419C2×D4—1224no—
420C2×D4—1221no—
421C2×D4—1224no—
422C2×D4—2444no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1224no—
436C22×C4—2444no—
437C22×C4C2×C22⋊D6112yes—
438C22×C4C2×C22⋊D6114yes—
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6114yes—
445C22×C4C23×D6112yes—
446C22×C4C23×D6111yescharacteristic
447C2×C8—614no—
448C2×C8—614no—
449C2×C8—1224no—
450C2×C8—1224no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—612no—
460C4⋊C4—612no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—1224no—
466C4⋊C4—1224no—
467C4⋊C4C23×D6114yes—
468C4⋊C4C23×D6114yes—
469C22⋊C4—1224no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C42—1224no—
475C42C2×C22⋊D6114yes—
476C42C2×C22⋊D6112yes—
477C42C2×C22⋊D6112yes—
478C42C2×C22⋊D6114yes—
479C42C23×D6112yes—
480C42C23×D6112yes—
481C23—22no—
482C23—22no—
483C23—1222no—
484C23—2442no—
485C23—2442no—
486C23—2442no—
487C23#1451979714yes—
488C23C22×C22⋊D6112yes—
489C23C22×C22⋊D6111yescharacteristic
490Q8—1224no—
491Q8—2442no—
492D4—1224no—
493D4—2448no—
494D4—2442no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—2448no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4C22×C22⋊D6112yes—
518C2×C4C22×C22⋊D6112yes—
519C2×C4C22×C22⋊D6112yes—
520C2×C4C22×C22⋊D6112yes—
521C2×C4C22×C22⋊D6112yes—
522C2×C4C22×C22⋊D6111yescharacteristic
523C2×C4#1451982514yes—
524C2×C4C24×D6111yescharacteristic
525C8—1224no—
526C8—1224no—
527C22—22no—
528C22—28no—
529C22—22no—
530C22—4884no—
531C22—2442no—
532C22—2442no—
533C22—2442no—
534C22#1684033612yes—
535C22C2×#1451979714yes—
536C22C2×#1451979712yes—
537C22#1684056012yes—
538C22C23×C22⋊D6111yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—22no—
544C4—44no—
545C4—44no—
546C4—2442no—
547C4—2442no—
548C4—2442no—
549C4#1684033612yes—
550C4C2×#1451982512yes—
551C2—24no—
552C2—4882no—
553C2#2145698714yesdirect product factor
554C2C2×#1684033611yescharacteristic
555C2C22×#1451979711yescharacteristic
556C2C2×#1684056011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC22×#125×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC22×#125D61
hypocenterC61C2×#3232
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #21456987C2direct (N × H)
3. C2×#16840161C2semidirect
4. C2×#16840119C2semidirect
5. C2×#16840097C2semidirect
7. C2×#16840092C2semidirect
9. C2×#16840066C2semidirect
10. C2×#16840059C2semidirect
11. C2×#16839815C2semidirect
12. C22×#125×C61C2semidirect
25. #16840161C22semidirect
26. #16840119C22semidirect
27. #16840097C22semidirect
28. #16840092C22semidirect
31. #16840066C22semidirect
32. #16840059C22semidirect
33. #16839815C22semidirect
51. C2×#125×C61C22semidirect
274. C22×#125D61semidirect
286. C2×C61#3232semidirect
300. C2×#125C2×D61semidirect
335. C61C2×#3232semidirect
553. C2#21456987direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402962
#214569872
C2×#168401612
C2×#168401192
C2×#168400972
C2×#168400802
C2×#168400922
C2×#168400752
C2×#168400662
C2×#168400592
C2×#168398152
C22×#125×C612
C2×#323261