Finite Group Databasebeta

← C2×#21461197 #30610908

Subgroups — C2×#21461197 #30610908

Order 15616.

1606 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519834C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊#6923C211yesmaximal, characteristic, complemented
3#21461197C218yesmaximal, direct product factor
4C2×#16840616C212yesmaximal, complemented
5C2×#16840614C212yesmaximal, complemented
6C2×#16840275C211yesmaximal, characteristic
7C23×#92420—21no—
8C22×C22⋊#6923—21no—
9C22×C4×#6923—21no—
10C2×#14519834—21no—
11C2×#14519834—22no—
12C2×#14519834C2212yes—
13C2×#14519834C2214yescomplemented
14C4×C22⋊#6923—22no—
15C4×C22⋊#6923C2214yescomplemented
16C2×(C2×#6923)⋊C4C2212yes—
17#16840616C2218yescomplemented
18#16840614C2218yescomplemented
19C22×#11758226C411yescharacteristic, complemented
20C23×C4×D61C2211yescharacteristic
21#16840275—22no—
22#16840275—22no—
23#16840275C414yescomplemented
24#16840275C2214yes—
25C2×#14519762—21no—
26C2×#14519753C2211yescharacteristic
27C2×#14519752C2211yescharacteristic
28C2×#14519751C411yescharacteristic, complemented
29C2×#14519747—21no—
30C2×#14519739—21no—
31C2×#14519737—21no—
32C2×#14519735C411yescharacteristic, complemented
33C2×C22⋊Q8×C61C411yescharacteristic, complemented
34C23×#6923—22no—
35C22×#92420—22no—
36C22×#92420—24no—
37C22×#92420—44no—
38C2×C22⋊#6923—28no—
39C2×C22⋊#6923—24no—
40C2×C4×#6923—24no—
41C2×C4×#6923—44no—
42#14519834—22no—
43#14519834—24no—
44#14519834C2314yes—
45(C2×#6923)⋊C4—22no—
46(C2×#6923)⋊C4C2314yes—
47C24×D61D411yescharacteristic
48C2×#11758226—22no—
49C2×#11758226—44no—
50C2×#11758226C2×C414yescomplemented
51C22×D244—21no—
52C22×D244—21no—
53C22×D244C2×C411yescharacteristic
54C22×C22⋊D61—21no—
55C23×Dic61D411yescharacteristic
56C22×C4×D61—21no—
57C22×C4×D61—44no—
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2×C412yes—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D412yes—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D412yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2312yes—
66#14519762—24no—
67#14519762—24no—
68#14519753—21no—
69#14519753—21no—
70#14519753C2×C412yes—
71#14519753C2312yes—
72#14519752—21no—
73#14519752—21no—
74#14519752C2×C412yes—
75#14519752C2312yes—
76#14519751—22no—
77#14519751C2×C414yescomplemented
78#14519747—24no—
79#14519747—24no—
80#14519739—24no—
81#14519739—24no—
82#14519737—24no—
83#14519737—24no—
84#14519735—22no—
85#14519735C2×C414yescomplemented
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×#11758196—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×C4×Dic61—21no—
93C22×Q8×C61C2×C411yescharacteristic
94C23×C4×C61C2×C411yescharacteristic
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61C2×C414yescomplemented
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2×C411yescharacteristic
99C2×C22⋊C4×C61—21no—
100C24×C61C22⋊C411yescharacteristic
101C23×D61—24no—
102C23×D61—24no—
103C23×D61—21no—
104C23×D61—42no—
105C23×D61C22⋊C411yescharacteristic
106C23×D61C22⋊C412yes—
107C23×D61C2×D412yes—
108C23×D61C2×D411yescharacteristic
109C22×#6923—28no—
110C22×#6923—24no—
111C22×#6923—44no—
112C22×#6923—44no—
113C2×Q8×C61—21no—
114C2×Q8×C61—42no—
115C2×Q8×C61C22×C412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—42no—
119C22×C4×C61C22⋊C411yescharacteristic
120C22×C4×C61C22⋊C412yes—
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C412yes—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—44no—
125C22×Dic61—24no—
126C22×Dic61—21no—
127C22×Dic61C22⋊C412yes—
128C22×Dic61C22⋊C411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D412yes—
131#11758226—48no—
132#11758226—48no—
133C2×D244—24no—
134C2×D244—22no—
135C2×D244—22no—
136C2×D244—21no—
137C2×D244—42no—
138C2×D244—44no—
139C2×D244—44no—
140C2×D244C22×C412yes—
141C2×C4×D61—24no—
142C2×C4×D61—24no—
143C2×C4×D61—24no—
144C2×C4×D61—44no—
145C2×C4×D61—44no—
146C2×C4×D61—44no—
147C2×C4×D61—44no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×C4×D61—44no—
151C2×C4×D61C22⋊C412yes—
152C2×C4×D61C22⋊C412yes—
153C2×C4×D61C22⋊C412yes—
154C2×C4×D61C22×C412yes—
155C2×C4×D61C22×C411yescharacteristic
156C2×C4×D61C2×D412yes—
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2411yescharacteristic
161C22⋊#6923—216no—
162C2×#92420—24no—
163C2×#92420—44no—
164C2×#92420—48no—
165C4×#6923—48no—
166C4⋊C4×C61—24no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61C22×C412yes—
169C22⋊C4×C61—24no—
170#11758198—24no—
171#11758198—22no—
172#11758198—22no—
173#11758198—24no—
174#11758198—22no—
175#11758198—22no—
176C4⋊Dic61—21no—
177C4⋊Dic61C22×C412yes—
178#11758196—22no—
179#11758196—22no—
180C4×Dic61—24no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61#10414yes—
184C23×C61C2×C22⋊C412yes—
185C23×C61C2×C22⋊C411yescharacteristic
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—22no—
193C22×D61—24no—
194C22×D61—24no—
195C22×D61—22no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—416no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61C2×C22⋊C412yes—
203C22×D61C2×C22⋊C411yescharacteristic
204C22×D61#13514yes—
205C22×D61C22×D411yescharacteristic
206C2×#6923—28no—
207C2×#6923—44no—
208C2×#6923—416no—
209C2×#6923—44no—
210Q8×C61—42no—
211Q8×C61—42no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C23×C411yescharacteristic
223C22⋊D61—48no—
224C2×Dic61—22no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—22no—
228C2×Dic61—42no—
229C2×Dic61#9914yes—
230C2×Dic61C2×C22⋊C411yescharacteristic
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C22×D411yescharacteristic
233D244—42no—
234D244—42no—
235D244—48no—
236D244—44no—
237D244—44no—
238C4×D61—24no—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—42no—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—42no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—48no—
249#92420—44no—
250C2×#1501—611nomaximal, Sylow
251C22×C61—22no—
252C22×C61—22no—
253C22×C61—28no—
254C22×C61C2×#9912yes—
255C22×C61#41612yes—
256C22×C61C2×#10414yes—
257C22×C61C2×#10412yes—
258C22×C61C22×C22⋊C411yescharacteristic
259C2×D61—21no—
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—28no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—48no—
268C2×D61—48no—
269C2×D61—44no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—84no—
274C2×D61#41612yes—
275C2×D61C2×#13512yes—
276#6923—48no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—22no—
280C4×C61—42no—
281C4×C61—42no—
282Dic61—21no—
283Dic61—21no—
284Dic61—42no—
285Dic61C2×#9914yes—
286C2×#525—611no—
287C22×C22⋊C8—611no—
288C2×C4×C22⋊C4—611no—
289#1501—618no—
290C2×#328—612no—
291C2×#327—612no—
292C2×C61—24no—
293C2×C61#150114yescomplemented
294C2×C61C22×#9911yescharacteristic
295C2×C61C2×#41611yescharacteristic
296C2×C61C22×#10411yescharacteristic
297D61—22no—
298D61—22no—
299D61—42no—
300D61—82no—
301C22×#41—611no—
302C24×C4—611no—
303C23×C8—1221no—
304#525—614no—
305#525—614no—
306#525—1222no—
307#525—1222no—
308C2×#123—1221no—
309C2×C22⋊Q8—611no—
310C2×C22⋊Q8#692311yescharacteristic, complemented
311C2×#121—611no—
312C2×#121—611no—
313C2×#120—1221no—
314C2×C4×D4—1221no—
315C2×C4×D4—1221no—
316C2×#117—611no—
317C22×C22⋊C4—1221no—
318C22×C42—1221no—
319C2×C22⋊C8—612no—
320C2×C22⋊C8—614no—
321C2×C22⋊C8—1221no—
322C2×C22⋊C8—1222no—
323C4×C22⋊C4—614no—
324C4×C22⋊C4—1222no—
325C2×#95—612no—
326#328—618no—
327#327—618no—
328C61C2×#150111yescharacteristic, complemented, Sylow
329C25—611no—
330C2×#41—614no—
331C2×#41—1222no—
332C2×#41—2444no—
333C22×Q8C2×#692311yescharacteristic
334C22×D4—611no—
335C22×D4—1221no—
336C22×D4—1221no—
337C22×D4—1221no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—612no—
343C23×C4—612no—
344C23×C4—612no—
345C23×C4—611no—
346C23×C4—612no—
347C23×C4—1221no—
348C23×C4—1222no—
349C23×C4—2444no—
350C23×C4C2×#692311yescharacteristic
351C22×C8—1224no—
352C22×C8—1222no—
353C22×C8—2444no—
354#123—1224no—
355#123—1224no—
356C22⋊Q8—22no—
357C22⋊Q8—612no—
358C22⋊Q8—612no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8C2×#692314yescomplemented
362#121—614no—
363#121—612no—
364#121—612no—
365#121—1221no—
366#121—1221no—
367#121—1222no—
368#120—1224no—
369#120—1224no—
370C4×D4—1224no—
371C4×D4—1224no—
372C4×D4—1224no—
373C4×D4—1224no—
374#117—614no—
375#117—1222no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4C2×#692311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1224no—
385C2×C22⋊C4—1228no—
386C2×C22⋊C4—1221no—
387C2×C42—1221no—
388C2×C42—1224no—
389C2×C42—2444no—
390C22⋊C8—614no—
391C22⋊C8—1222no—
392C22⋊C8—1224no—
393#95—614no—
394#95—1222no—
395C24—611no—
396C24—611no—
397C24—612no—
398C24—612no—
399C24—1224no—
400C24—1224no—
401C24—1221no—
402C24—2442no—
403C24C22⋊#692311yescharacteristic
404#41—2448no—
405#41—2448no—
406C2×Q8—21no—
407C2×Q8—42no—
408C2×Q8C22×#692312yes—
409C2×D4—612no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1224no—
413C2×D4—1224no—
414C2×D4—1221no—
415C2×D4—2442no—
416C2×D4—2444no—
417C2×D4—2444no—
418C2×D4—2444no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—611no—
426C22×C4—612no—
427C22×C4—612no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—1224no—
437C22×C4—1224no—
438C22×C4—1224no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4—1228no—
442C22×C4—1221no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2444no—
446C22×C4—2444no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2444no—
450C22×C4—2444no—
451C22×C4—2444no—
452C22×C4C22⋊#692311yescharacteristic
453C22×C4C22⋊#692312yes—
454C22×C4C22×#692311yescharacteristic
455C22×C4C22×#692312yes—
456C2×C8—1224no—
457C2×C8—2448no—
458C2×C8—2444no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—612no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—1221no—
465C4⋊C4C22×#692312yes—
466C22⋊C4—24no—
467C22⋊C4—1222no—
468C22⋊C4—1222no—
469C22⋊C4—1224no—
470C22⋊C4—1222no—
471C22⋊C4—1222no—
472C22⋊C4—12216no—
473C22⋊C4—1224no—
474C42—1224no—
475C42—2448no—
476C23—22no—
477C23—22no—
478C23—614no—
479C23—611no—
480C23—611no—
481C23—612no—
482C23—1224no—
483C23—1224no—
484C23—1222no—
485C23—1222no—
486C23—1222no—
487C23—1222no—
488C23—1224no—
489C23—1224no—
490C23—1222no—
491C23—1222no—
492C23—2442no—
493C23—2442no—
494C23—24416no—
495C23—2442no—
496C23—2442no—
497C23—2442no—
498C23#1451984214yes—
499C23C2×C22⋊#692312yes—
500C23C2×C22⋊#692311yescharacteristic
501Q8—42no—
502Q8—42no—
503D4—2442no—
504D4—2442no—
505D4—2444no—
506D4—2444no—
507D4—2448no—
508D4—2448no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—614no—
518C2×C4—612no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—1222no—
528C2×C4—1228no—
529C2×C4—1222no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2448no—
537C2×C4—2444no—
538C2×C4—2444no—
539C2×C4—24416no—
540C2×C4—2442no—
541C2×C4C2×C22⋊#692312yes—
542C2×C4C2×C22⋊#692311yescharacteristic
543C2×C4C23×#692311yescharacteristic
544C8—2444no—
545C22—22no—
546C22—22no—
547C22—28no—
548C22—612no—
549C22—612no—
550C22—1221no—
551C22—1221no—
552C22—1221no—
553C22—1221no—
554C22—1228no—
555C22—1222no—
556C22—1222no—
557C22—1222no—
558C22—4884no—
559C22—2448no—
560C22—2448no—
561C22—2442no—
562C22—2442no—
563C22—2442no—
564C22—2444no—
565C22D61⋊#9912yes—
566C22#1684075212yes—
567C22C2×#1451984214yes—
568C22C2×#1451984212yes—
569C22C22×C22⋊#692311yescharacteristic
570C4—22no—
571C4—21no—
572C4—21no—
573C4—42no—
574C4—42no—
575C4—614no—
576C4—1221no—
577C4—1221no—
578C4—2448no—
579C4—2442no—
580C2—24no—
581C2—1222no—
582C2—1222no—
583C2—4882no—
584C2—2442no—
585C2#2146119714yesdirect product factor
586C2C2×D61⋊#9911yescharacteristic
587C2C2×#1684075211yescharacteristic
588C2C22×#1451984211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#16840275C2
cosocleC2×C4×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61C2×#1501
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14519834C2semidirect
2. C2×C4×C22⋊#6923C2semidirect
3. #21461197C2direct (N × H)
4. C2×#16840616C2semidirect
5. C2×#16840614C2semidirect
13. C2×#14519834C22semidirect
15. C4×C22⋊#6923C22semidirect
17. #16840616C22semidirect
18. #16840614C22semidirect
19. C22×#11758226C4semidirect
23. #16840275C4semidirect
28. C2×#14519751C4semidirect
32. C2×#14519735C4semidirect
33. C2×C22⋊Q8×C61C4semidirect
50. C2×#11758226C2×C4semidirect
77. #14519751C2×C4semidirect
85. #14519735C2×C4semidirect
96. C22⋊Q8×C61C2×C4semidirect
293. C2×C61#1501semidirect
310. C2×C22⋊Q8#6923semidirect
328. C61C2×#1501semidirect
361. C22⋊Q8C2×#6923semidirect
585. C2#21461197direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198342
C2×C4×C22⋊#69232
#214611972
C2×#168406162
C2×#168406142
C2×#168402752
C2×#150161