Finite Group Databasebeta

← C2×#21461217 #30610929

Subgroups — C2×#21461217 #30610929

Order 15616.

1554 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840734C211yesmaximal, characteristic
2C2×C4×C22⋊#6923C211yesmaximal, characteristic
3#21461217C218yesmaximal, direct product factor
4C2×#16840616C212yesmaximal, complemented
5C2×#16840615C212yesmaximal, complemented
6C2×C22⋊Q8×D61C211yesmaximal, characteristic
7C22×#11758217—21no—
8C22×C22⋊#6923—21no—
9C22×C4×#6923—21no—
10#16840734—22no—
11#16840734C2214yes—
12C2×#14519834—21no—
13C2×#14519834C2212yes—
14C4×C22⋊#6923—22no—
15C4×C22⋊#6923C2214yes—
16C2×(C2×#6923)⋊C4C2212yes—
17#16840616C2218yescomplemented
18#16840615C2218yescomplemented
19C22×Q8×D61C411yescharacteristic, complemented
20C23×C4×D61C2211yescharacteristic
21C2×#14519757C2211yescharacteristic
22C2×#14519756C411yescharacteristic, complemented
23C2×#14519755—21no—
24C2×#14519754—21no—
25C2×#14519753C2211yescharacteristic
26C22⋊Q8×D61—22no—
27C22⋊Q8×D61—22no—
28C22⋊Q8×D61C414yescomplemented
29C22⋊Q8×D61C2214yes—
30C2×C4⋊C4×D61—21no—
31C2×C4⋊C4×D61C411yescharacteristic, complemented
32C2×C22⋊C4×D61—21no—
33C2×C22⋊Q8×C61C411yescharacteristic, complemented
34C23×#6923—22no—
35C2×#11758217—24no—
36C2×#11758217—44no—
37C22×#92420—22no—
38C2×C22⋊#6923—28no—
39C2×C22⋊#6923—24no—
40C2×C4×#6923—24no—
41C2×C4×#6923—44no—
42#14519834—24no—
43#14519834—22no—
44#14519834C2314yes—
45(C2×#6923)⋊C4—22no—
46(C2×#6923)⋊C4C2314yes—
47C24×D61D411yescharacteristic
48C22×Dic122—21no—
49C22×Dic122C2×C411yescharacteristic
50C2×Q8×D61—22no—
51C2×Q8×D61—44no—
52C2×Q8×D61C2×C414yescomplemented
53C23×Dic61D411yescharacteristic
54C22×C4×D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—44no—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61C2×C412yes—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D412yes—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D412yes—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2312yes—
65#14519757—21no—
66#14519757—21no—
67#14519757C2×C412yes—
68#14519757C2312yes—
69#14519756—22no—
70#14519756C2×C414yescomplemented
71#14519755—24no—
72#14519755—24no—
73#14519754—24no—
74#14519754—24no—
75#14519753—21no—
76#14519753—21no—
77#14519753C2×C412yes—
78#14519753C2312yes—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2×C411yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C4⋊C4×D61—24no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61—24no—
87C4⋊C4×D61C2×C414yescomplemented
88C2×C22⋊Dic61—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C22⋊C4×D61—24no—
92C22⋊C4×D61—24no—
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—42no—
104C23×D61C22⋊C412yes—
105C23×D61C22⋊C411yescharacteristic
106C23×D61C2×D412yes—
107C23×D61C2×D411yescharacteristic
108C22×#6923—28no—
109C22×#6923—24no—
110C22×#6923—44no—
111C22×#6923—44no—
112C2×Q8×C61—21no—
113C2×Q8×C61—42no—
114C2×Q8×C61C22×C412yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—42no—
118C22×C4×C61C22⋊C411yescharacteristic
119C22×C4×C61C22⋊C412yes—
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C412yes—
122C22×Dic61—24no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61C22⋊C412yes—
126C22×Dic61C22⋊C411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D412yes—
129Q8×D61—48no—
130Q8×D61—48no—
131C2×C4×D61—24no—
132C2×C4×D61—24no—
133C2×C4×D61—24no—
134C2×C4×D61—24no—
135C2×C4×D61—44no—
136C2×C4×D61—44no—
137C2×C4×D61—44no—
138C2×C4×D61—44no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×C4×D61—44no—
142C2×C4×D61—44no—
143C2×C4×D61C22⋊C412yes—
144C2×C4×D61C22⋊C412yes—
145C2×C4×D61C22⋊C412yes—
146C2×C4×D61C22×C412yes—
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D412yes—
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—24no—
155C2×Dic122—44no—
156C2×Dic122—42no—
157C2×Dic122C22×C412yes—
158C22⋊#6923—216no—
159#11758217—48no—
160C2×#92420—24no—
161C2×#92420—44no—
162C4×#6923—48no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C22×C412yes—
166C22⋊C4×C61—24no—
167C22⋊Dic61—24no—
168#11758198—22no—
169#11758198—22no—
170#11758198—22no—
171#11758198—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61C22×C412yes—
176#11758196—24no—
177#11758196—22no—
178#11758196—24no—
179#11758196—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61C2×C22⋊C412yes—
184C23×C61C2×C22⋊C411yescharacteristic
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—24no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—416no—
196C22×D61#10414yes—
197C22×D61C2×C22⋊C411yescharacteristic
198C22×D61C2×C22⋊C412yes—
199C22×D61#13714yes—
200C22×D61C22×D411yescharacteristic
201C2×#6923—28no—
202C2×#6923—44no—
203C2×#6923—416no—
204C2×#6923—44no—
205Q8×C61—42no—
206Q8×C61—42no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61C2×C22⋊C412yes—
216C2×C4×C61C2×C22⋊C411yescharacteristic
217C2×C4×C61C23×C411yescharacteristic
218C2×Dic61—22no—
219C2×Dic61—24no—
220C2×Dic61—24no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61#9914yes—
226C2×Dic61C2×C22⋊C411yescharacteristic
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C22×D411yescharacteristic
229C4×D61—24no—
230C4×D61—24no—
231C4×D61—24no—
232C4×D61—24no—
233C4×D61—42no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61—42no—
237C4×D61—41no—
238C4×D61—41no—
239C4×D61—48no—
240C4×D61—48no—
241Dic122—42no—
242Dic122—48no—
243Dic122—42no—
244#92420—44no—
245C2×#1508—611nomaximal, Sylow
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—44no—
250C22×C61C2×#9912yes—
251C22×C61#41912yes—
252C22×C61C2×#10412yes—
253C22×C61C22×C22⋊C411yescharacteristic
254C2×D61—22no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—22no—
258C2×D61—21no—
259C2×D61—21no—
260C2×D61—28no—
261C2×D61—28no—
262C2×D61—48no—
263C2×D61—48no—
264C2×D61#41912yes—
265C2×D61C2×#10414yes—
266C2×D61C2×#13712yes—
267#6923—48no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—22no—
271C4×C61—42no—
272C4×C61—42no—
273Dic61—21no—
274Dic61—21no—
275Dic61—42no—
276Dic61—42no—
277Dic61C2×#9914yes—
278C22×C22⋊Q8—611no—
279C2×#407—611no—
280C2×C4×C22⋊C4—611no—
281#1508—618no—
282C2×#328—612no—
283C2×#328—612no—
284C2×C61—42no—
285C2×C61#150814yescomplemented
286C2×C61C22×#9911yescharacteristic
287C2×C61C2×#41911yescharacteristic
288C2×C61C22×#10411yescharacteristic
289D61—22no—
290D61—22no—
291D61—24no—
292C23×Q8—611no—
293C24×C4—611no—
294C22×M16—1221no—
295C2×C22⋊Q8—611no—
296C2×C22⋊Q8—611no—
297C2×C22⋊Q8—614no—
298C2×C22⋊Q8—614no—
299C2×C22⋊Q8—611no—
300C2×C22⋊Q8—1221no—
301C2×C22⋊Q8—1221no—
302C2×C22⋊Q8—1222no—
303C2×C22⋊Q8—1222no—
304C2×C22⋊Q8#692311yescharacteristic, complemented
305C22×C4⋊C4—611no—
306C22×C4⋊C4—1221no—
307C22×C22⋊C4—1221no—
308C22×C22⋊C4—1221no—
309C22×C42—1221no—
310#407—614no—
311#407—1222no—
312C2×C22⋊C8—612no—
313C2×C22⋊C8—1221no—
314C4×C22⋊C4—614no—
315C4×C22⋊C4—1222no—
316C2×#95—612no—
317#328—618no—
318#328—618no—
319C61C2×#150811yescharacteristic, complemented, Sylow
320C25—611no—
321C22×Q8—614no—
322C22×Q8—611no—
323C22×Q8—1221no—
324C22×Q8—1222no—
325C22×Q8—2444no—
326C22×Q8C2×#692311yescharacteristic
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—612no—
332C23×C4—612no—
333C23×C4—612no—
334C23×C4—611no—
335C23×C4—612no—
336C23×C4—1221no—
337C23×C4—1221no—
338C23×C4—1222no—
339C23×C4—2444no—
340C23×C4C2×#692311yescharacteristic
341C2×M16—1224no—
342C2×M16—2444no—
343C22×C8—1222no—
344C22⋊Q8—22no—
345C22⋊Q8—612no—
346C22⋊Q8—612no—
347C22⋊Q8—614no—
348C22⋊Q8—612no—
349C22⋊Q8—612no—
350C22⋊Q8—1224no—
351C22⋊Q8—1224no—
352C22⋊Q8—1224no—
353C22⋊Q8—1224no—
354C22⋊Q8—1221no—
355C22⋊Q8—1221no—
356C22⋊Q8—1221no—
357C22⋊Q8—1221no—
358C22⋊Q8—1222no—
359C22⋊Q8C2×#692314yescomplemented
360C2×C4⋊C4—21no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—614no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—1224no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C2×#692311yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1224no—
373C2×C22⋊C4—1224no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1228no—
378C2×C22⋊C4—1221no—
379C2×C42—1224no—
380C2×C42—2444no—
381C22⋊C8—614no—
382C22⋊C8—1222no—
383C22⋊C8—1224no—
384#95—614no—
385#95—1222no—
386C24—611no—
387C24—611no—
388C24—612no—
389C24—612no—
390C24—1224no—
391C24—1224no—
392C24—2442no—
393C24C22⋊#692311yescharacteristic
394C2×Q8—21no—
395C2×Q8—42no—
396C2×Q8—612no—
397C2×Q8—1224no—
398C2×Q8—1221no—
399C2×Q8—2442no—
400C2×Q8—2444no—
401C2×Q8—2448no—
402C2×Q8—2448no—
403C2×Q8C22×#692312yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—611no—
408C22×C4—612no—
409C22×C4—612no—
410C22×C4—611no—
411C22×C4—612no—
412C22×C4—612no—
413C22×C4—612no—
414C22×C4—612no—
415C22×C4—611no—
416C22×C4—612no—
417C22×C4—611no—
418C22×C4—612no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—1224no—
422C22×C4—1224no—
423C22×C4—1224no—
424C22×C4—1224no—
425C22×C4—1224no—
426C22×C4—1224no—
427C22×C4—1228no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4—2444no—
433C22×C4—2444no—
434C22×C4—2444no—
435C22×C4—2444no—
436C22×C4—2444no—
437C22×C4—2444no—
438C22×C4—2444no—
439C22×C4—2444no—
440C22×C4C22⋊#692311yescharacteristic
441C22×C4C22⋊#692312yes—
442C22×C4C22×#692311yescharacteristic
443C22×C4C22×#692312yes—
444M16—2448no—
445C2×C8—1224no—
446C2×C8—2444no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—612no—
450C4⋊C4—1224no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1221no—
456C4⋊C4—1224no—
457C4⋊C4C22×#692312yes—
458C22⋊C4—24no—
459C22⋊C4—1222no—
460C22⋊C4—1222no—
461C22⋊C4—1222no—
462C22⋊C4—1222no—
463C22⋊C4—12216no—
464C22⋊C4—1224no—
465C42—2448no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—614no—
470C23—614no—
471C23—611no—
472C23—612no—
473C23—611no—
474C23—1224no—
475C23—1224no—
476C23—1224no—
477C23—1224no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—2442no—
483C23—2442no—
484C23—24416no—
485C23C2×C22⋊#692312yes—
486C23C2×C22⋊#692311yescharacteristic
487Q8—42no—
488Q8—42no—
489Q8—2442no—
490Q8—2442no—
491Q8—2448no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—614no—
501C2×C4—612no—
502C2×C4—611no—
503C2×C4—611no—
504C2×C4—1224no—
505C2×C4—1224no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1222no—
511C2×C4—1228no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2448no—
521C2×C4—2448no—
522C2×C4—2444no—
523C2×C4—2444no—
524C2×C4—24416no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4C2×C22⋊#692312yes—
528C2×C4C2×C22⋊#692311yescharacteristic
529C2×C4C23×#692311yescharacteristic
530C8—2444no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—44no—
535C22—612no—
536C22—612no—
537C22—614no—
538C22—1221no—
539C22—1222no—
540C22—1221no—
541C22—1222no—
542C22—1221no—
543C22—1221no—
544C22—1228no—
545C22—1228no—
546C22—2448no—
547C22—2448no—
548C22D61⋊#9912yes—
549C22#1684075512yes—
550C22D61⋊#10412yes—
551C22C22×C22⋊#692311yescharacteristic
552C4—22no—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—42no—
557C4—614no—
558C4—1221no—
559C4—1221no—
560C4—2448no—
561C4—2442no—
562C4—2442no—
563C2—42no—
564C2—1222no—
565C2—1222no—
566C2—1224no—
567C2#2146121714yesdirect product factor
568C2C2×D61⋊#9911yescharacteristic
569C2C2×#1684075511yescharacteristic
570C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC2×C4×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61C2×#1508
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
3. #21461217C2direct (N × H)
4. C2×#16840616C2semidirect
5. C2×#16840615C2semidirect
17. #16840616C22semidirect
18. #16840615C22semidirect
19. C22×Q8×D61C4semidirect
22. C2×#14519756C4semidirect
28. C22⋊Q8×D61C4semidirect
31. C2×C4⋊C4×D61C4semidirect
33. C2×C22⋊Q8×C61C4semidirect
52. C2×Q8×D61C2×C4semidirect
70. #14519756C2×C4semidirect
87. C4⋊C4×D61C2×C4semidirect
96. C22⋊Q8×C61C2×C4semidirect
285. C2×C61#1508semidirect
304. C2×C22⋊Q8#6923semidirect
319. C61C2×#1508semidirect
359. C22⋊Q8C2×#6923semidirect
567. C2#21461217direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407342
C2×C4×C22⋊#69232
#214612172
C2×#168406162
C2×#168406152
C2×C22⋊Q8×D612
C2×#150861