Finite Group Databasebeta

← C2×#21456998 #30552091

Subgroups — C2×#21456998 #30552091

Order 15616.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840296C211yesmaximal, characteristic
2#21456998C2116yesmaximal, direct product factor
3C2×#16840161C212yesmaximal, complemented
4C2×#16840119C212yesmaximal, complemented
5C2×#16840103C212yesmaximal, complemented
6C2×#16840086C212yesmaximal
7C2×#16840054C211yesmaximal, characteristic, complemented
8C2×#16840044C211yesmaximal, characteristic, complemented
9C2×#16840049C211yesmaximal, characteristic, complemented
10C2×#16840039C211yesmaximal, characteristic, complemented
11C2×#16839815C211yesmaximal, characteristic, complemented
12C2×C61×#533C211yesmaximal, 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#16840103C22116yescomplemented
28#16840086C22116yes—
29#16840054C2218yescomplemented
30#16840049C2218yescomplemented
31#16840044C2218yescomplemented
32#16840039C2218yescomplemented
33#16839815C2218yescomplemented
34C2×#14519710C2212yes—
35C2×#14519710C2212yes—
36C2×#14519708C2212yes—
37C2×#14519708C2212yes—
38C2×C4⋊#92418C2211yescharacteristic
39C2×C4⋊#92418C2211yescharacteristic
40C2×#14519700C2212yes—
41C2×#14519700C2212yes—
42C2×#14519697C2212yes—
43C2×#14519697C2212yes—
44C2×C22⋊#92418C2212yes—
45C2×C4×#92418C2212yes—
46C61×#533—24no—
47C61×#533C2218yescomplemented
48C2×C4⋊Q8×C61C2211yescharacteristic
49C2×C4⋊Q8×C61C2211yescharacteristic
50C2×#125×C61C2212yes—
51C2×C22⋊Q8×C61—21no—
52C2×C22⋊Q8×C61—21no—
53C2×C4×Q8×C61—21no—
54C2×C4×Q8×C61—21no—
55C2×#117×C61C2212yes—
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⋊#92418C2314yes—
96C4⋊#92418C2314yes—
97#14519700C2318yes—
98#14519700C2318yes—
99#14519697C2318yes—
100#14519697C2318yes—
101C22⋊#92418C2318yes—
102C4×#92418C2318yes—
103C22×Q8×C61—21no—
104C22×Q8×C61—21no—
105C23×C4×C61D411yescharacteristic
106C23×C4×C61D411yescharacteristic
107C23×C4×C61C2311yescharacteristic
108C4⋊Q8×C61—22no—
109C4⋊Q8×C61—22no—
110C4⋊Q8×C61C2314yes—
111C4⋊Q8×C61C2314yes—
112#125×C61—24no—
113#125×C61C2318yes—
114C22⋊Q8×C61—28no—
115C22⋊Q8×C61—28no—
116C4×Q8×C61—28no—
117C4×Q8×C61—28no—
118#117×C61—24no—
119#117×C61C2318yes—
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×C22⋊C4×C61—22no—
127C2×C42×C61—22no—
128C2×C42×C61D414yes—
129C2×C42×C61D411yescharacteristic
130C2×C42×C61D411yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2314yes—
133C2×C42×C61C2311yescharacteristic
134C24×C61C2×D411yescharacteristic
135C23×D61—21no—
136C2×Q8×C61—24no—
137C2×Q8×C61—24no—
138C2×Q8×C61—44no—
139C2×Q8×C61—44no—
140C22×C4×C61—24no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—42no—
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×C61C2×D411yescharacteristic
154C22×C4×C61C2412yes—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—24no—
157C2×C22⋊D61—44no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244C2412yes—
165C2×C4×D61—24no—
166C2×C4×D61—44no—
167C2×Dic122—22no—
168C2×Dic122—22no—
169C2×Dic122—21no—
170C2×Dic122C2412yes—
171C4⋊C4×C61—28no—
172C4⋊C4×C61—28no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61C2414yes—
178C4⋊C4×C61C2414yes—
179C22⋊C4×C61—28no—
180C42×C61—28no—
181C42×C61C2×D414yes—
182C42×C61C2×D414yes—
183C42×C61C2×D412yes—
184C42×C61C2×D412yes—
185C42×C61C2412yes—
186C42×C61C2412yes—
187C22⋊Dic61—24no—
188#11758198—24no—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C2412yes—
195C4⋊Dic61C2412yes—
196#11758196—24no—
197#11758196—24no—
198#11758196—24no—
199#11758196—24no—
200C4×Dic61—24no—
201C2×#92418—24no—
202C2×#92418—24no—
203C2×#92418C2414yes—
204C2×#92418C2414yes—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61C22×D412yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—22no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214Q8×C61—48no—
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—28no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—44no—
227C2×C4×C61—44no—
228C2×C4×C61—44no—
229C2×C4×C61#13514yes—
230C2×C4×C61C22×D412yes—
231C2×C4×C61C22×D412yes—
232C2×C4×C61C22×D412yes—
233C2×C4×C61C22×D412yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D412yes—
236C2×C4×C612−1+412yes—
237C2×C4×C612−1+412yes—
238C2×C4×C61C2511yescharacteristic
239C22⋊D61—48no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246D244—22no—
247D244—22no—
248D244—42no—
249C4×D61—48no—
250Dic122—22no—
251Dic122—22no—
252Dic122—42no—
253#92418—24no—
254#92418—24no—
255C2×#3233—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—44no—
260C22×C61#53612yes—
261C22×C61C2×#13512yes—
262C22×C61#57512yes—
263C22×C61C23×D411yescharacteristic
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C2×D61—84no—
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—44no—
273C4×C61—44no—
274C4×C61#53612yes—
275C4×C61C2×#13514yes—
276C4×C61C2×2−1+411yescharacteristic
277C4×C61C2×2−1+411yescharacteristic
278Dic61—42no—
279Dic61—42no—
280Dic61—42no—
281C2×#533D6111yescharacteristic, complemented
282C2×#529—611no—
283#3233—6116no—
284C2×#499—612no—
285C2×#488—612no—
286C2×#485—612no—
287C2×#482—612no—
288C2×#477—611no—
289C2×#475—611no—
290C2×#476—611no—
291C2×#474—611no—
292C2×#431—611no—
293C2×C61—42no—
294C2×C61#323314yescomplemented
295C2×C61C2×#53611yescharacteristic
296C2×C61C22×#13511yescharacteristic
297C2×C61C2×#57511yescharacteristic
298D61—82no—
299#533—24no—
300#533C2×D6118yescomplemented
301C2×C4⋊Q8C2×D6111yescharacteristic
302C2×C4⋊Q8C2×D6111yescharacteristic
303#529—618no—
304#529—1224no—
305C2×#126—1221no—
306C2×#125—611no—
307C2×#125C2×D6112yes—
308C2×#124—611no—
309C2×#123—1221no—
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8—611no—
313C2×#121—611no—
314C2×C4×Q8—21no—
315C2×C4×Q8—21no—
316C2×C4×Q8—611no—
317C2×C4×D4—611no—
318C2×C4×D4—1221no—
319C2×#117—1221no—
320C2×#117C2×D6112yes—
321C22×C42C2×D6111yescharacteristic
322#499—6116no—
323#488—6116no—
324#485—6116no—
325#482—6116no—
326#477—618no—
327#476—618no—
328#475—618no—
329#474—618no—
330#431—618no—
331C2×#107—612no—
332C2×#107—612no—
333C2×#106—612no—
334C2×#106—612no—
335C2×C4⋊C8—611no—
336C2×C4⋊C8—611no—
337C2×#103—612no—
338C2×#103—612no—
339C2×#102—612no—
340C2×#102—612no—
341C2×C22⋊C8—612no—
342C2×C4×C8—612no—
343C61C2×#323311yescharacteristic, complemented, Sylow
344C22×Q8—21no—
345C22×Q8—21no—
346C22×Q8—611no—
347C22×D4—611no—
348C22×D4—1221no—
349C23×C4—1221no—
350C23×C4C22⋊D6111yescharacteristic
351C23×C4C22⋊D6111yescharacteristic
352C23×C4C22×D6111yescharacteristic
353C22×C8—612no—
354C22×C8—612no—
355C4⋊Q8—22no—
356C4⋊Q8—22no—
357C4⋊Q8C22×D6114yes—
358C4⋊Q8C22×D6114yes—
359#126—1228no—
360#125—24no—
361#125—614no—
362#125—1222no—
363#125C22×D6118yes—
364#124—614no—
365#124—1222no—
366#123—1228no—
367C22⋊Q8—28no—
368C22⋊Q8—28no—
369C22⋊Q8—614no—
370C22⋊Q8—1222no—
371#121—614no—
372#121—1222no—
373C4×Q8—28no—
374C4×Q8—28no—
375C4×Q8—614no—
376C4×Q8—1222no—
377C4×D4—614no—
378C4×D4—1228no—
379C4×D4—1222no—
380#117—24no—
381#117—1228no—
382#117C22×D6118yes—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6112yes—
394C2×C4⋊C4C22×D6112yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—22no—
402C2×C42—1221no—
403C2×C42C22⋊D6111yescharacteristic
404C2×C42C22⋊D6111yescharacteristic
405C2×C42C22⋊D6114yes—
406C2×C42C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408C2×C42C22×D6114yes—
409#107—618no—
410#107—618no—
411#106—618no—
412#106—618no—
413C4⋊C8—614no—
414C4⋊C8—614no—
415#103—618no—
416#103—618no—
417#102—618no—
418#102—618no—
419C22⋊C8—618no—
420C4×C8—618no—
421C24—1221no—
422C24C2×C22⋊D6111yescharacteristic
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—612no—
428C2×Q8—1221no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×D4—612no—
432C2×D4—1224no—
433C2×D4—1221no—
434C2×D4—1222no—
435C2×D4—1222no—
436C2×D4—2444no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—42no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1224no—
447C22×C4—2444no—
448C22×C4C2×C22⋊D6112yes—
449C22×C4C2×C22⋊D6114yes—
450C22×C4C2×C22⋊D6111yescharacteristic
451C22×C4C2×C22⋊D6111yescharacteristic
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6114yes—
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C23×D6112yes—
457C22×C4C23×D6111yescharacteristic
458C2×C8—614no—
459C2×C8—614no—
460C2×C8—1224no—
461C2×C8—1224no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—612no—
469C4⋊C4—612no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1224no—
473C4⋊C4—1224no—
474C4⋊C4—1224no—
475C4⋊C4—1224no—
476C4⋊C4C23×D6114yes—
477C4⋊C4C23×D6114yes—
478C22⋊C4—28no—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C22⋊C4—1224no—
484C42—28no—
485C42—1224no—
486C42C2×C22⋊D6114yes—
487C42C2×C22⋊D6112yes—
488C42C2×C22⋊D6114yes—
489C42C2×C22⋊D6112yes—
490C42C23×D6112yes—
491C42C23×D6112yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—1222no—
496C23—2442no—
497C23—2442no—
498C23—2442no—
499C23C22×C22⋊D6112yes—
500C23C22×C22⋊D6111yescharacteristic
501Q8—48no—
502Q8—48no—
503Q8—1222no—
504Q8—1222no—
505Q8—2442no—
506D4—1222no—
507D4—1222no—
508D4—2448no—
509D4—2442no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2448no—
530C2×C4#1451979714yes—
531C2×C4C22×C22⋊D6112yes—
532C2×C4C22×C22⋊D6112yes—
533C2×C4C22×C22⋊D6112yes—
534C2×C4C22×C22⋊D6112yes—
535C2×C4C22×C22⋊D6112yes—
536C2×C4C22×C22⋊D6111yescharacteristic
537C2×C4#1451982512yes—
538C2×C4#1451982512yes—
539C2×C4C24×D6111yescharacteristic
540C8—1224no—
541C8—1224no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—44no—
546C22—4884no—
547C22—2442no—
548C22—2442no—
549C22—2442no—
550C22#1684033612yes—
551C22C2×#1451979712yes—
552C22#1684056012yes—
553C22C23×C22⋊D6111yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—44no—
559C4—44no—
560C4—2442no—
561C4—2442no—
562C4—2442no—
563C4#1684033612yes—
564C4C2×#1451979714yes—
565C4C2×#1451982511yescharacteristic
566C4C2×#1451982511yescharacteristic
567C2—42no—
568C2—4882no—
569C2#2145699814yesdirect product factor
570C2C2×#1684033611yescharacteristic
571C2C22×#1451979711yescharacteristic
572C2C2×#1684056011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#533C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenterC2×#533D61
hypocenterC61C2×#3233
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. #21456998C2direct (N × H)
3. C2×#16840161C2semidirect
4. C2×#16840119C2semidirect
5. C2×#16840103C2semidirect
7. C2×#16840054C2semidirect
8. C2×#16840044C2semidirect
9. C2×#16840049C2semidirect
10. C2×#16840039C2semidirect
11. C2×#16839815C2semidirect
12. C2×C61×#533C2semidirect
25. #16840161C22semidirect
26. #16840119C22semidirect
27. #16840103C22semidirect
29. #16840054C22semidirect
30. #16840049C22semidirect
31. #16840044C22semidirect
32. #16840039C22semidirect
33. #16839815C22semidirect
47. C61×#533C22semidirect
281. C2×#533D61semidirect
294. C2×C61#3233semidirect
300. #533C2×D61semidirect
343. C61C2×#3233semidirect
569. C2#21456998direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402962
#214569982
C2×#168401612
C2×#168401192
C2×#168401032
C2×#168400862
C2×#168400542
C2×#168400442
C2×#168400492
C2×#168400392
C2×#168398152
C2×C61×#5332
C2×#323361