Finite Group Databasebeta

← C2×#21456522 #30549930

Subgroups — C2×#21456522 #30549930

Order 15616.

1998 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840296C211yesmaximal, characteristic
2#21456522C2116yesmaximal, direct product factor
3C2×#16840168C211yesmaximal, characteristic, complemented
4C2×#16840154C211yesmaximal, characteristic, complemented
5C2×#16840130C211yesmaximal, characteristic, complemented
6C2×#16840124C211yesmaximal, characteristic, complemented
7C2×#16840001C212yesmaximal, complemented
8C2×#16839993C212yesmaximal
9C2×#16839867C212yesmaximal, complemented
10C2×#16839860C212yesmaximal, complemented
11C22×#14519688C211yesmaximal, characteristic, complemented
12C2×C61×#533C211yesmaximal, characteristic, complemented
13C22×#11758202C2212yes—
14C22×#11758201C2212yes—
15C23×#92418C2212yes—
16#16840296—24no—
17#16840296C2218yes—
18C2×#14519771—21no—
19C2×#14519769C2211yescharacteristic
20C2×#14519765C2211yescharacteristic
21C2×#14519760—21no—
22C2×#14519756C2211yescharacteristic
23C2×#14519751C2211yescharacteristic
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26C2×C4×C22⋊D61—21no—
27C2×#14519734—21no—
28#16840168C2218yescomplemented
29#16840154C2218yescomplemented
30#16840130C2218yescomplemented
31#16840124C2218yescomplemented
32#16840001C22116yescomplemented
33#16839993C22116yes—
34#16839867C22116yescomplemented
35#16839860C22116yescomplemented
36C2×#14519710C2212yes—
37C2×#14519708C2212yes—
38C2×#14519701C2212yes—
39C2×#14519701C2212yes—
40C2×#14519700C2212yes—
41C2×#14519697C2212yes—
42C2×#14519688C2212yes—
43C2×#14519688C2211yescharacteristic
44C2×#14519688C2211yescharacteristic
45C2×#14519688C2218yescomplemented
46C61×#533—24no—
47C61×#533C2218yescomplemented
48C2×C4⋊Q8×C61C2211yescharacteristic
49C2×C4⋊Q8×C61C2211yescharacteristic
50C2×#125×C61—21no—
51C2×C22⋊Q8×C61—21no—
52C2×C22⋊Q8×C61C2212yes—
53C2×C4×Q8×C61—21no—
54C2×C4×Q8×C61C2212yes—
55C2×#117×C61—21no—
56C22×C42×C61C2211yescharacteristic
57C22×Dic122C2311yescharacteristic
58C22×D244C2311yescharacteristic
59C22×C22⋊D61—21no—
60C22×C4×D61—21no—
61C2×#11758202—28no—
62C2×#11758202C2318yes—
63C2×#11758201—28no—
64C2×#11758201C2318yes—
65C22×#92418—28no—
66C22×#92418C2312yes—
67C22×#92418C2318yes—
68C22×#92418C2312yes—
69#14519771—28no—
70#14519769—22no—
71#14519769C2314yes—
72#14519765—22no—
73#14519765C2314yes—
74#14519760—28no—
75#14519756—22no—
76#14519756C2314yes—
77#14519751—22no—
78#14519751C2314yes—
79C4×Dic122—22no—
80C4×Dic122C2314yes—
81C4×D244—22no—
82C4×D244C2314yes—
83C4×C22⋊D61—28no—
84#14519734—28no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×C22⋊Dic61—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97#14519710C2318yes—
98#14519708C2318yes—
99#14519701C2318yes—
100#14519701C2318yes—
101#14519700C2318yes—
102#14519697C2318yes—
103#14519688C2318yes—
104#14519688C2314yes—
105#14519688C2314yes—
106C22×Q8×C61—21no—
107C22×Q8×C61C2312yes—
108C23×C4×C61D411yescharacteristic
109C23×C4×C61D411yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C4⋊Q8×C61—22no—
112C4⋊Q8×C61—22no—
113C4⋊Q8×C61C2314yes—
114C4⋊Q8×C61C2314yes—
115#125×C61—28no—
116C22⋊Q8×C61—24no—
117C22⋊Q8×C61—28no—
118C22⋊Q8×C61C2318yes—
119C4×Q8×C61—24no—
120C4×Q8×C61—28no—
121C4×Q8×C61C2318yes—
122#117×C61—28no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—22no—
127C2×C4⋊C4×C61—22no—
128C2×C4⋊C4×C61C2312yes—
129C2×C22⋊C4×C61—22no—
130C2×C42×C61—22no—
131C2×C42×C61D414yes—
132C2×C42×C61D411yescharacteristic
133C2×C42×C61D411yescharacteristic
134C2×C42×C61C2311yescharacteristic
135C2×C42×C61C2314yes—
136C2×C42×C61C2311yescharacteristic
137C24×C61C2×D411yescharacteristic
138C23×D61—21no—
139C2×Q8×C61—22no—
140C2×Q8×C61—24no—
141C2×Q8×C61—24no—
142C2×Q8×C61—24no—
143C2×Q8×C61—44no—
144C2×Q8×C61C2414yes—
145C22×C4×C61—24no—
146C22×C4×C61—24no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—42no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D414yes—
157C22×C4×C61C2×D414yes—
158C22×C4×C61C2412yes—
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—24no—
161C2×C22⋊D61—44no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—21no—
165C2×D244—24no—
166C2×D244—21no—
167C2×D244C2412yes—
168C2×C4×D61—24no—
169C2×C4×D61—44no—
170C2×Dic122—24no—
171C2×Dic122—21no—
172C2×Dic122C2412yes—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—28no—
177C4⋊C4×C61—28no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61C2414yes—
180C22⋊C4×C61—28no—
181C42×C61—28no—
182C42×C61C2×D412yes—
183C42×C61C2×D414yes—
184C42×C61C2×D414yes—
185C42×C61C2×D412yes—
186C42×C61C2412yes—
187C42×C61C2412yes—
188C22⋊Dic61—24no—
189#11758202—216no—
190#11758201—216no—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194#11758198—24no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C2412yes—
198C4⋊Dic61C2412yes—
199#11758196—24no—
200#11758196—24no—
201#11758196—24no—
202#11758196—24no—
203C4×Dic61—24no—
204C2×#92418—24no—
205C2×#92418—216no—
206C2×#92418#4118yes—
207C2×#92418C2414yes—
208C2×#92418C2414yes—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61#13714yes—
212C23×C61C22×D412yes—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218Q8×C61—24no—
219Q8×C61—24no—
220Q8×C61—48no—
221Q8×C61—44no—
222C2×C4×C61—22no—
223C2×C4×C61—28no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—44no—
233C2×C4×C61—44no—
234C2×C4×C61—44no—
235C2×C4×C61C22×D412yes—
236C2×C4×C61C22×D412yes—
237C2×C4×C61C22×D412yes—
238C2×C4×C61C22×D412yes—
239C2×C4×C61C22×D412yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4114yes—
242C2×C4×C61C2511yescharacteristic
243C22⋊D61—48no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250D244—24no—
251D244—42no—
252C4×D61—48no—
253Dic122—24no—
254Dic122—42no—
255#92418—24no—
256#92418C2×#4118yes—
257C2×#3181—611nomaximal, Sylow
258C22×C61—28no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61#53212yes—
262C22×C61C2×#13714yes—
263C22×C61C2×#13712yes—
264C22×C61#57512yes—
265C22×C61C23×D411yescharacteristic
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—84no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—44no—
276C4×C61—44no—
277C4×C61#53212yes—
278C4×C61C22×#4112yes—
279Dic61—42no—
280Dic61—42no—
281Dic61—42no—
282C2×#533D6111yescharacteristic, complemented
283C2×#529—611no—
284#3181—6116no—
285C2×#501—611no—
286C2×#497—611no—
287C2×#491—611no—
288C2×#489—611no—
289C2×#467—612no—
290C2×#465—612no—
291C2×#441—612no—
292C2×#440—612no—
293C22×#97—611no—
294C2×C61—24no—
295C2×C61#318114yescomplemented
296C2×C61C2×#53211yescharacteristic
297C2×C61C22×#13711yescharacteristic
298C2×C61C2×#57511yescharacteristic
299D61—82no—
300C22×Q16—612no—
301C22×SD16—612no—
302C23×C8—612no—
303#533—24no—
304#533C2×D6118yescomplemented
305C2×C4⋊Q8C2×D6111yescharacteristic
306C2×C4⋊Q8C2×D6111yescharacteristic
307#529—618no—
308#529—1224no—
309C2×#126—1221no—
310C2×#125—21no—
311C2×#125—611no—
312C2×#124—611no—
313C2×#123—1221no—
314C2×C22⋊Q8—21no—
315C2×C22⋊Q8—611no—
316C2×C22⋊Q8C2×D6112yes—
317C2×#121—611no—
318C2×C4×Q8—21no—
319C2×C4×Q8—611no—
320C2×C4×Q8C2×D6112yes—
321C2×C4×D4—611no—
322C2×C4×D4—1221no—
323C2×#117—21no—
324C2×#117—1221no—
325C22×C42C2×D6111yescharacteristic
326#501—618no—
327#497—618no—
328#491—618no—
329#489—618no—
330#467—6116no—
331#465—6116no—
332#441—6116no—
333#440—6116no—
334C2×#107—612no—
335C2×#106—612no—
336C2×#103—612no—
337C2×#103—612no—
338C2×#103—612no—
339C2×#102—612no—
340C2×#97—611no—
341C2×#97—611no—
342C2×#97—612no—
343C2×#97—618no—
344C61C2×#318111yescharacteristic, complemented, Sylow
345C22×Q8—21no—
346C22×Q8—611no—
347C22×Q8C22×D6112yes—
348C22×D4—611no—
349C22×D4—1221no—
350C23×C4—1221no—
351C23×C4C22⋊D6111yescharacteristic
352C23×C4C22⋊D6111yescharacteristic
353C23×C4C22×D6111yescharacteristic
354C2×Q16—618no—
355C2×Q16—1228no—
356C2×SD16—618no—
357C2×SD16—1228no—
358C22×C8—612no—
359C22×C8—612no—
360C22×C8—618no—
361C22×C8—1228no—
362C4⋊Q8—22no—
363C4⋊Q8—22no—
364C4⋊Q8C22×D6114yes—
365C4⋊Q8C22×D6114yes—
366#126—1228no—
367#125—28no—
368#125—614no—
369#125—1222no—
370#124—614no—
371#124—1222no—
372#123—1228no—
373C22⋊Q8—28no—
374C22⋊Q8—24no—
375C22⋊Q8—614no—
376C22⋊Q8—1222no—
377C22⋊Q8C22×D6118yes—
378#121—614no—
379#121—1222no—
380C4×Q8—28no—
381C4×Q8—24no—
382C4×Q8—614no—
383C4×Q8—1222no—
384C4×Q8C22×D6118yes—
385C4×D4—614no—
386C4×D4—1228no—
387C4×D4—1222no—
388#117—28no—
389#117—1228no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C42—22no—
409C2×C42—1221no—
410C2×C42C22⋊D6111yescharacteristic
411C2×C42C22⋊D6111yescharacteristic
412C2×C42C22⋊D6114yes—
413C2×C42C22×D6111yescharacteristic
414C2×C42C22×D6111yescharacteristic
415C2×C42C22×D6114yes—
416#107—618no—
417#106—618no—
418#103—618no—
419#103—618no—
420#103—618no—
421#102—618no—
422#97—618no—
423#97—614no—
424#97—614no—
425C24—1221no—
426C24C2×C22⋊D6111yescharacteristic
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—44no—
432C2×Q8—612no—
433C2×Q8—1221no—
434C2×Q8—1224no—
435C2×Q8C23×D6114yes—
436C2×D4—612no—
437C2×D4—1224no—
438C2×D4—1221no—
439C2×D4—1224no—
440C2×D4—2444no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—42no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1224no—
450C22×C4—2444no—
451C22×C4C2×C22⋊D6112yes—
452C22×C4C2×C22⋊D6114yes—
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6114yes—
459C22×C4C23×D6112yes—
460C22×C4C23×D6111yescharacteristic
461Q16—12216no—
462SD16—12216no—
463C2×C8—618no—
464C2×C8—614no—
465C2×C8—614no—
466C2×C8—12216no—
467C2×C8—1224no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—22no—
474C4⋊C4—612no—
475C4⋊C4—612no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C4⋊C4C23×D6114yes—
483C22⋊C4—28no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4—1224no—
488C22⋊C4—1224no—
489C42—28no—
490C42—1224no—
491C42C2×C22⋊D6114yes—
492C42C2×C22⋊D6112yes—
493C42C2×C22⋊D6114yes—
494C42C2×C22⋊D6112yes—
495C42C23×D6112yes—
496C42C23×D6112yes—
497C23—22no—
498C23—22no—
499C23—1222no—
500C23—2442no—
501C23—2442no—
502C23—2442no—
503C23#1451980814yes—
504C23C22×C22⋊D6112yes—
505C23C22×C22⋊D6111yescharacteristic
506Q8—24no—
507Q8—24no—
508Q8—48no—
509Q8—44no—
510Q8—1224no—
511Q8—2442no—
512D4—1224no—
513D4—2448no—
514D4—2442no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—2448no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4C22×C22⋊D6112yes—
536C2×C4C22×C22⋊D6112yes—
537C2×C4C22×C22⋊D6112yes—
538C2×C4C22×C22⋊D6112yes—
539C2×C4C22×C22⋊D6112yes—
540C2×C4C22×C22⋊D6111yescharacteristic
541C2×C4#41×D6114yes—
542C2×C4C24×D6111yescharacteristic
543C8—618no—
544C8—1224no—
545C22—22no—
546C22—28no—
547C22—22no—
548C22—4884no—
549C22—2442no—
550C22—2442no—
551C22—2442no—
552C22#1684030812yes—
553C22C2×#1451980814yes—
554C22C2×#1451980812yes—
555C22#1684056012yes—
556C22C23×C22⋊D6111yescharacteristic
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—44no—
563C4—44no—
564C4—2442no—
565C4—2442no—
566C4—2442no—
567C4#1684030812yes—
568C4C2×#41×D6112yes—
569C2—24no—
570C2—4882no—
571C2#2145652214yesdirect product factor
572C2C2×#1684030811yescharacteristic
573C2C22×#1451980811yescharacteristic
574C2C2×#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×#3181
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. #21456522C2direct (N × H)
3. C2×#16840168C2semidirect
4. C2×#16840154C2semidirect
5. C2×#16840130C2semidirect
6. C2×#16840124C2semidirect
7. C2×#16840001C2semidirect
9. C2×#16839867C2semidirect
10. C2×#16839860C2semidirect
11. C22×#14519688C2semidirect
12. C2×C61×#533C2semidirect
28. #16840168C22semidirect
29. #16840154C22semidirect
30. #16840130C22semidirect
31. #16840124C22semidirect
32. #16840001C22semidirect
34. #16839867C22semidirect
35. #16839860C22semidirect
45. C2×#14519688C22semidirect
47. C61×#533C22semidirect
282. C2×#533D61semidirect
295. C2×C61#3181semidirect
304. #533C2×D61semidirect
344. C61C2×#3181semidirect
571. C2#21456522direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402962
#214565222
C2×#168401682
C2×#168401542
C2×#168401302
C2×#168401242
C2×#168400012
C2×#168399932
C2×#168398672
C2×#168398602
C22×#145196882
C2×C61×#5332
C2×#318161