Finite Group Databasebeta

← C2×#21454413 #30543679

Subgroups — C2×#21454413 #30543679

Order 15616.

2766 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519785C211yesmaximal, characteristic
2C2×#16840243C211yesmaximal, characteristic
3C2×#16840242C211yesmaximal, characteristic
4#21454413C2116yesmaximal, direct product factor
5C2×#16839804C211yesmaximal, characteristic, complemented
6C2×#16839803C212yesmaximal, complemented
7C2×#16839801C212yesmaximal, complemented
8C2×#16839799C212yesmaximal, complemented
9C2×#16839798C211yesmaximal, characteristic, complemented
10C22×#14519708C211yesmaximal, characteristic, complemented
11C2×#106×D61C212yesmaximal, complemented
12C2×C61×#428C211yesmaximal, characteristic, complemented
13C22×#11758222C2211yescharacteristic
14C2×#14519785C4116yescomplemented
15C2×#14519785C2218yes—
16C22×#11758189C2212yes—
17C23×#92418C2211yescharacteristic
18C22×C8×D61C2212yes—
19#16840243—24no—
20#16840243C2218yes—
21#16840242—24no—
22#16840242C2218yes—
23C2×C4×Dic122—21no—
24C2×#14519742—21no—
25C2×C4×D244—21no—
26C2×#14519739—21no—
27C2×C4×C22⋊D61—22no—
28C2×#14519737—22no—
29C2×#14519736C2211yescharacteristic
30C2×#14519735C2212yes—
31C2×#14519735C2212yes—
32C22×C4⋊Dic61C2211yescharacteristic
33C2×C4⋊C4×D61C2212yes—
34C2×C4⋊C4×D61C2212yes—
35#16839804C2218yescomplemented
36#16839803C22116yescomplemented
37#16839801C22116yescomplemented
38#16839799C22116yescomplemented
39#16839798C2218yescomplemented
40C2×#14519711C2212yes—
41C2×#14519710C2211yescharacteristic
42C2×#14519710C2211yescharacteristic
43C2×#14519709C2212yes—
44C2×#14519708C2211yescharacteristic
45C2×#14519708C2212yes—
46C2×#14519708C2212yes—
47C2×#14519708C2211yescharacteristic
48C2×#14519708C2218yescomplemented
49#106×D61C22116yescomplemented
50C22×M16×C61C2211yescharacteristic
51C2×#117×C61C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C61×#428C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×#106×C61C2212yes—
56C2×#11758222C2×C418yes—
57C2×#11758222D414yes—
58C2×#11758222Q818yes—
59C2×#11758222C2314yes—
60C22×Dic122D411yescharacteristic
61C22×D244D411yescharacteristic
62C22×C22⋊D61D412yes—
63C23×Dic61—21no—
64C22×C4×D61—22no—
65C22×C4×D61—22no—
66C22×C4×D61C2312yes—
67#14519785C2×C4164yescomplemented
68C2×#11758189C2×C4116yescomplemented
69C2×#11758189C2318yes—
70C22×#92418C2×C418yescomplemented
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2314yes—
74C2×C8×D61C2×C4116yescomplemented
75C2×C8×D61C2318yes—
76C4×Dic122—28no—
77#14519742—28no—
78C4×D244—28no—
79#14519739—28no—
80C4×C22⋊D61—216no—
81#14519737—216no—
82#14519736—22no—
83#14519736C2314yes—
84#14519735—24no—
85#14519735—24no—
86#14519735C2318yes—
87#14519735C2318yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—22no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2312yes—
92C2×C4⋊Dic61C2314yes—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×#11758196—22no—
95C2×#11758196—22no—
96C4⋊C4×D61—24no—
97C4⋊C4×D61—24no—
98C4⋊C4×D61C2318yes—
99C4⋊C4×D61C2318yes—
100C2×C22⋊Dic61—21no—
101C2×#11758198—22no—
102C2×#11758198—22no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—22no—
105C2×C4×Dic61—22no—
106#14519711C2318yes—
107#14519710C2314yes—
108#14519710C2314yes—
109#14519709C2318yes—
110#14519708C2314yes—
111#14519708C2318yes—
112#14519708C2318yes—
113#14519708C2314yes—
114C23×C4×C61—21no—
115C23×C4×C61C2311yescharacteristic
116C2×M16×C61C2×C418yescomplemented
117C2×M16×C61C2314yes—
118C22×C8×C61C2312yes—
119#117×C61—22no—
120#117×C61C2314yes—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—22no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2312yes—
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2314yes—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61—21no—
129#107×C61C2318yes—
130#106×C61C2318yes—
131C24×C61C2×D411yescharacteristic
132C23×D61C2×D412yes—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—24no—
137C22×C4×C61—82no—
138C22×C4×C61C22×C414yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×Q814yes—
142C22×C4×C61C2411yescharacteristic
143C22×C4×C61C2412yes—
144C2×C22⋊D61C4⋊C4116yes—
145C2×C22⋊D61C2×D418yes—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—24no—
150C22×Dic61—82no—
151C22×Dic61C2×D412yes—
152#11758222C4⋊C4132yes—
153#11758222C22×C4116yes—
154#11758222C2×Q8116yes—
155C2×D244C4⋊C418yes—
156C2×D244C2×D414yes—
157C2×C4×D61—28no—
158C2×C4×D61—28no—
159C2×C4×D61—84no—
160C2×C4×D61—84no—
161C2×C4×D61C22×C418yes—
162C2×C4×D61C2×D414yes—
163C2×C4×D61C2×Q818yes—
164C2×C4×D61C2414yes—
165C2×Dic122C4⋊C418yes—
166C2×Dic122C2×D414yes—
167M16×C61C22×C4116yescomplemented
168C2×C8×C61C22×C418yes—
169C2×C8×C61C2414yes—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2412yes—
175C4⋊C4×C61C2414yes—
176C4⋊C4×C61C2412yes—
177C22⋊C4×C61—24no—
178C42×C61—24no—
179C22⋊Dic61—24no—
180#11758198—28no—
181#11758198—28no—
182C4⋊Dic61—21no—
183C4⋊Dic61—22no—
184C4⋊Dic61—24no—
185C4⋊Dic61—21no—
186C4⋊Dic61C2414yes—
187C4⋊Dic61C2412yes—
188C4⋊Dic61C2412yes—
189#11758196—28no—
190#11758196—28no—
191C4×Dic61—24no—
192C4×Dic61—28no—
193C4×Dic61—28no—
194C2×#92418C22×C414yes—
195C2×#92418C22×C414yes—
196C2×#92418C22×C4116yescomplemented
197C2×#92418C2412yes—
198C2×#92418C2412yes—
199#11758189C22×C4132yescomplemented
200C8×D61C22×C4132yescomplemented
201C23×C61—22no—
202C23×C61—22no—
203C23×C61C2×C4⋊C414yes—
204C23×C61C22×D412yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—24no—
207C22×D61—24no—
208C22×D61C2×C4⋊C418yes—
209C22×D61C22×D414yes—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—42no—
214C2×C4×C61—44no—
215C2×C4×C61—42no—
216C2×C4×C61—84no—
217C2×C4×C61C2×C4⋊C414yes—
218C2×C4×C61C2×C4⋊C418yes—
219C2×C4×C61C23×C412yes—
220C2×C4×C61C23×C414yes—
221C2×C4×C61C22×D412yes—
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C22×Q812yes—
224C2×C4×C61C22×Q814yes—
225C2×C4×C61C2511yescharacteristic
226C22⋊D61C2×C4⋊C4132yes—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—24no—
230C2×Dic61—42no—
231C2×Dic61—44no—
232C2×Dic61—42no—
233C2×Dic61—84no—
234C2×Dic61C2×C4⋊C418yes—
235C2×Dic61C22×D414yes—
236D244C2×C4⋊C4116yes—
237C4×D61—88no—
238C4×D61—88no—
239C4×D61C2×C4⋊C4116yes—
240C4×D61C23×C418yes—
241C4×D61C22×Q818yes—
242Dic122C2×C4⋊C4116yes—
243C8×C61C23×C418yes—
244#92418C23×C414yes—
245#92418C23×C414yes—
246C2×#2912—611nomaximal, Sylow
247C22×C61—28no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61C22×C4⋊C414yes—
251C22×C61C22×C4⋊C412yes—
252C22×C61#57712yes—
253C22×C61#57712yes—
254C22×C61C23×D411yescharacteristic
255C2×D61—216no—
256C2×D61—24no—
257C2×D61—24no—
258C2×D61C22×C4⋊C418yes—
259C4×C61—42no—
260C4×C61—44no—
261C4×C61—42no—
262C4×C61C22×C4⋊C414yes—
263C4×C61C22×C4⋊C412yes—
264C4×C61C24×C411yescharacteristic
265C4×C61C23×Q811yescharacteristic
266Dic61—42no—
267Dic61—44no—
268Dic61—42no—
269Dic61C22×C4⋊C418yes—
270C22×#131—611no—
271C2×#520—611no—
272C2×#520—611no—
273#2912—6116no—
274C2×#428—612no—
275C2×#428—612no—
276C2×#428—611no—
277C2×#428D6111yescharacteristic, complemented
278C2×#427—612no—
279C2×#427—611no—
280C22×#106—612no—
281C22×#106—611no—
282C2×C61—24no—
283C2×C61#291214yescomplemented
284C2×C61C23×C4⋊C411yescharacteristic
285C2×C61C2×#57711yescharacteristic
286C2×C61C2×#57711yescharacteristic
287D61—28no—
288C22×#41—611no—
289C2×#131—6116no—
290C2×#131—618no—
291C22×M16—612no—
292C22×M16C2×D6111yescharacteristic
293C23×C8—611no—
294C23×C8—612no—
295#520—618no—
296#520—618no—
297#520—1224no—
298#520—1224no—
299C2×C4×Q8—1221no—
300C2×C4×Q8—1221no—
301C2×C4×D4—1221no—
302C2×C4×D4—1222no—
303C2×C4×D4—1221no—
304C2×C4×D4—1222no—
305C2×#117—611no—
306C2×#117—612no—
307C2×#117—612no—
308C2×#117C2×D6111yescharacteristic
309C22×C4⋊C4—611no—
310C22×C4⋊C4—612no—
311C22×C4⋊C4—612no—
312C22×C4⋊C4C2×D6111yescharacteristic
313#428—618no—
314#428—6116no—
315#428—6116no—
316#428C2×D6118yescomplemented
317#427—618no—
318#427—6116no—
319C2×#107—612no—
320C2×#107—611no—
321C2×#107—611no—
322C2×#107C2×D6112yes—
323C2×#106—612no—
324C2×#106—611no—
325C2×#106—612no—
326C2×#106—612no—
327C2×#106—618no—
328C2×#106—611no—
329C2×#106—6116no—
330C2×#106C2×D6112yes—
331C61C2×#291211yescharacteristic, complemented, Sylow
332C2×#41—618no—
333C2×#41—614no—
334C2×#41—618no—
335C2×#41—614no—
336C22×Q8—611no—
337C22×D4—612no—
338C22×D4—611no—
339C23×C4—21no—
340C23×C4—612no—
341C23×C4—1222no—
342C23×C4—1222no—
343C23×C4—1221no—
344C23×C4C22×D6111yescharacteristic
345#131—6164no—
346C2×M16—6116no—
347C2×M16—618no—
348C2×M16C4×D6118yescomplemented
349C2×M16C22×D6114yes—
350C22×C8—618no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8—614no—
354C22×C8—6116no—
355C22×C8—618no—
356C22×C8C22×D6112yes—
357C4×Q8—1228no—
358C4×Q8—1228no—
359C4×D4—1228no—
360C4×D4—12216no—
361C4×D4—1228no—
362C4×D4—12216no—
363#117—22no—
364#117—614no—
365#117—618no—
366#117—618no—
367#117—1224no—
368#117—1224no—
369#117—1222no—
370#117C22×D6114yes—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—618no—
378C2×C4⋊C4—618no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4—1224no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6112yes—
387C2×C4⋊C4C22×D6114yes—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1222no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1221no—
393C2×C42—21no—
394C2×C42—1222no—
395C2×C42—1222no—
396C2×C42—1221no—
397#107—618no—
398#107—614no—
399#107—614no—
400#107C22×D6118yes—
401#106—618no—
402#106—618no—
403#106—614no—
404#106—618no—
405#106—614no—
406#106C22×D6118yes—
407C24—612no—
408C24D4×D6111yescharacteristic
409#41—6116no—
410#41—6116no—
411#41—6132no—
412C2×Q8—614no—
413C2×Q8—618no—
414C2×D4—618no—
415C2×D4—6116no—
416C2×D4—614no—
417C2×D4—618no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—82no—
423C22×C4—612no—
424C22×C4—618no—
425C22×C4—614no—
426C22×C4—618no—
427C22×C4—614no—
428C22×C4—1228no—
429C22×C4—1228no—
430C22×C4—1224no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—1221no—
434C22×C4—4884no—
435C22×C4—4884no—
436C22×C4—4882no—
437C22×C4C2×C4×D6114yes—
438C22×C4D4×D6112yes—
439C22×C4D4×D6111yescharacteristic
440C22×C4Q8×D6114yes—
441C22×C4C23×D6112yes—
442C22×C4C23×D6111yescharacteristic
443M16—6132no—
444M16C2×C4×D61116yescomplemented
445C2×C8—614no—
446C2×C8—614no—
447C2×C8—6116no—
448C2×C8—612no—
449C2×C8—612no—
450C2×C8—6132no—
451C2×C8C2×C4×D6118yes—
452C2×C8C23×D6114yes—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—612no—
458C4⋊C4—614no—
459C4⋊C4—612no—
460C4⋊C4—1228no—
461C4⋊C4—1228no—
462C4⋊C4—1221no—
463C4⋊C4—1222no—
464C4⋊C4—1224no—
465C4⋊C4—1221no—
466C4⋊C4C23×D6114yes—
467C4⋊C4C23×D6112yes—
468C4⋊C4C23×D6112yes—
469C22⋊C4—24no—
470C22⋊C4—1228no—
471C22⋊C4—1228no—
472C22⋊C4—1224no—
473C42—24no—
474C42—1228no—
475C42—1228no—
476C42—1224no—
477C23—22no—
478C23—22no—
479C23—618no—
480C23—614no—
481C23—1224no—
482C23—1224no—
483C23C4⋊C4×D6114yes—
484C23C2×D4×D6112yes—
485C23C2×D4×D6111yescharacteristic
486Q8—6116no—
487D4—6132no—
488D4—6116no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—84no—
496C2×C4—618no—
497C2×C4—614no—
498C2×C4—6116no—
499C2×C4—618no—
500C2×C4—618no—
501C2×C4—1222no—
502C2×C4—1224no—
503C2×C4—1222no—
504C2×C4—4888no—
505C2×C4—4888no—
506C2×C4—4884no—
507C2×C4—2442no—
508C2×C4—2444no—
509C2×C4—2442no—
510C2×C4C4⋊C4×D6114yes—
511C2×C4C4⋊C4×D6118yes—
512C2×C4C22×C4×D6112yes—
513C2×C4C22×C4×D6114yes—
514C2×C4C2×D4×D6112yes—
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4C2×Q8×D6112yes—
517C2×C4C2×Q8×D6114yes—
518C2×C4C24×D6111yescharacteristic
519C8—614no—
520C8—614no—
521C8C22×C4×D6118yes—
522C22—28no—
523C22—22no—
524C22—22no—
525C22—618no—
526C22—12216no—
527C22—1224no—
528C22—1224no—
529C22C2×C4⋊C4×D6114yes—
530C22C2×C4⋊C4×D6112yes—
531C22#1684057112yes—
532C22#1684057212yes—
533C22C22×D4×D6111yescharacteristic
534C4—42no—
535C4—42no—
536C4—44no—
537C4—618no—
538C4—2444no—
539C4—2442no—
540C4—2442no—
541C4C2×C4⋊C4×D6112yes—
542C4C2×C4⋊C4×D6114yes—
543C4C23×C4×D6111yescharacteristic
544C4C22×Q8×D6111yescharacteristic
545C2—24no—
546C2—1228no—
547C2#2145441314yesdirect product factor
548C2C22×C4⋊C4×D6111yescharacteristic
549C2C2×#1684057111yescharacteristic
550C2C2×#1684057211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC4×C61C24×C4
FrattiniC2×C4C24×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC2×C4×C61C25
hypercenterC2×#428D61
hypocenterC61C2×#2912
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #21454413C2direct (N × H)
5. C2×#16839804C2semidirect
6. C2×#16839803C2semidirect
7. C2×#16839801C2semidirect
8. C2×#16839799C2semidirect
9. C2×#16839798C2semidirect
10. C22×#14519708C2semidirect
11. C2×#106×D61C2semidirect
12. C2×C61×#428C2semidirect
14. C2×#14519785C4semidirect
35. #16839804C22semidirect
36. #16839803C22semidirect
37. #16839801C22semidirect
38. #16839799C22semidirect
39. #16839798C22semidirect
48. C2×#14519708C22semidirect
49. #106×D61C22semidirect
53. C61×#428C22semidirect
67. #14519785C2×C4semidirect
68. C2×#11758189C2×C4semidirect
70. C22×#92418C2×C4semidirect
74. C2×C8×D61C2×C4semidirect
116. C2×M16×C61C2×C4semidirect
167. M16×C61C22×C4semidirect
196. C2×#92418C22×C4semidirect
199. #11758189C22×C4semidirect
200. C8×D61C22×C4semidirect
277. C2×#428D61semidirect
283. C2×C61#2912semidirect
316. #428C2×D61semidirect
331. C61C2×#2912semidirect
348. C2×M16C4×D61semidirect
444. M16C2×C4×D61semidirect
547. C2#21454413direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197852
C2×#168402432
C2×#168402422
#214544132
C2×#168398042
C2×#168398032
C2×#168398012
C2×#168397992
C2×#168397982
C22×#145197082
C2×#106×D612
C2×C61×#4282
C2×#291261