Finite Group Databasebeta

← C2×#21462083 #30623919

Subgroups — C2×#21462083 #30623919

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840748C211yesmaximal, characteristic
2C2×#16840741C211yesmaximal, characteristic
3#21462083C218yesmaximal, direct product factor
4C2×#16840665C214yesmaximal, complemented
5C2×#16840338C211yesmaximal, characteristic
6C2×D61⋊M16—21no—
7C2×#14519856—21no—
8#16840748—22no—
9#16840748C2214yes—
10C2×#14519838—21no—
11C2×#14519837C2212yes—
12#16840741—22no—
13#16840741C2214yes—
14C2×#14519836—21no—
15C2×#14519835C2212yes—
16#16840665C22116yescomplemented
17C22×#11758222C2211yescharacteristic
18C23×Dic122C411yescharacteristic, complemented
19#16840338—22no—
20#16840338—22no—
21#16840338C414yescomplemented
22#16840338C2214yes—
23C2×#14519777—21no—
24C2×#14519765—21no—
25C2×#14519761C412yescomplemented
26C2×#14519761C2212yes—
27C2×#14519757—21no—
28C2×#14519757—21no—
29C2×#117×C61C411yescharacteristic, complemented
30C2×#11758213—41no—
31D61⋊M16—24no—
32D61⋊M16—42no—
33C2×#11758217—22no—
34C2×D61⋊C8—41no—
35#14519856—24no—
36#14519856—42no—
37C2×C4⋊#6923—41no—
38C2×C22⋊#6923—22no—
39C2×C4×#6923—41no—
40#14519838—24no—
41#14519837—22no—
42#14519837C2314yes—
43#14519836—24no—
44#14519835—22no—
45#14519835C2314yes—
46C2×#11758222—41no—
47C2×#11758222—41no—
48C2×#11758222—42no—
49C2×#11758222C2×C412yes—
50C2×#11758222D412yes—
51C2×#11758222D412yes—
52C2×#11758222C2312yes—
53C22×Dic122—21no—
54C22×Dic122—21no—
55C22×Dic122—22no—
56C22×Dic122—42no—
57C22×Dic122—41no—
58C22×Dic122—41no—
59C22×Dic122C2×C414yescomplemented
60C22×Dic122D411yescharacteristic
61C22×D244D411yescharacteristic
62C22×C22⋊D61C2×C411yescharacteristic
63C22×C22⋊D61C2311yescharacteristic
64C23×Dic61C2×C412yes—
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67#14519777—24no—
68#14519777—24no—
69#14519765—24no—
70#14519765—24no—
71#14519761—24no—
72#14519761—22no—
73#14519761—22no—
74#14519761C2×C414yes—
75#14519761C2×C418yescomplemented
76#14519761C2314yes—
77#14519757—24no—
78#14519757—24no—
79#14519757—24no—
80#14519757—24no—
81C2×C4⋊Dic61—22no—
82C2×C4⋊Dic61—22no—
83C2×#11758198—22no—
84C2×#11758198—22no—
85C23×C4×C61C2×C411yescharacteristic
86#117×C61—22no—
87#117×C61C2×C414yescomplemented
88C2×C4⋊C4×C61—21no—
89C2×C22⋊C4×C61C2×C412yes—
90C2×C42×C61—21no—
91C24×C61C22×C411yescharacteristic
92C23×D61C22⋊C411yescharacteristic
93C23×D61C2×D411yescharacteristic
94C22×#6923—42no—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61C22⋊C411yescharacteristic
100C22×C4×C61C22⋊C412yes—
101C22×C4×C61C22⋊C411yescharacteristic
102C22×C4×C61C22×C412yes—
103C2×C22⋊D61—41no—
104C2×C22⋊D61—41no—
105C2×C22⋊D61—41no—
106C2×C22⋊D61—41no—
107C2×C22⋊D61C22⋊C412yes—
108C2×C22⋊D61C22×C412yes—
109C2×C22⋊D61C22×C411yescharacteristic
110C2×C22⋊D61C2×D411yescharacteristic
111C2×C22⋊D61C2×D411yescharacteristic
112C2×C22⋊D61C2411yescharacteristic
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61—42no—
116C22×Dic61—42no—
117C22×Dic61C22⋊C411yescharacteristic
118C22×Dic61C22×C414yes—
119C22×Dic61C2×D411yescharacteristic
120#11758222—42no—
121#11758222—42no—
122#11758222—42no—
123#11758222—42no—
124#11758222—42no—
125#11758222—42no—
126#11758222—42no—
127#11758222—42no—
128C2×D244—41no—
129C2×D244—41no—
130C2×D244C22⋊C412yes—
131C2×D244C2×D412yes—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—22no—
135C2×C4×D61—21no—
136C2×C4×D61—22no—
137C2×C4×D61—21no—
138C2×C4×D61C22⋊C412yes—
139C2×C4×D61C22⋊C412yes—
140C2×C4×D61C2×D412yes—
141C2×C4×D61C2×D412yes—
142C2×Dic122—24no—
143C2×Dic122—24no—
144C2×Dic122—41no—
145C2×Dic122—41no—
146C2×Dic122—42no—
147C2×Dic122—42no—
148C2×Dic122—44no—
149C2×Dic122—44no—
150C2×Dic122—42no—
151C2×Dic122—41no—
152C2×Dic122—41no—
153C2×Dic122—44no—
154C2×Dic122—44no—
155C2×Dic122C22⋊C412yes—
156C2×Dic122C2×D412yes—
157C22⋊#6923—24no—
158C22⋊#6923—42no—
159#11758217—24no—
160#11758217—42no—
161C2×#92420—42no—
162C4⋊#6923—44no—
163C4×#6923—44no—
164#11758213—44no—
165D61⋊C8—44no—
166C4⋊C4×C61—24no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61C22×C414yes—
169C42×C61—24no—
170#11758198—24no—
171#11758198—28no—
172#11758198—24no—
173C4⋊Dic61—24no—
174C4⋊Dic61—28no—
175C4⋊Dic61—24no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—42no—
179C23×C61C2×C22⋊C411yescharacteristic
180C23×C61C2×C22⋊C411yescharacteristic
181C23×C61C23×C411yescharacteristic
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—21no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61C2×C22⋊C411yescharacteristic
189C22×D61C2×C22⋊C412yes—
190C22×D61C22×D411yescharacteristic
191C2×#6923—44no—
192C2×#6923—84no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—24no—
200C2×C4×C61—44no—
201C2×C4×C61#9912yes—
202C2×C4×C61#9911yescharacteristic
203C2×C4×C61#9911yescharacteristic
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C2×C22⋊C412yes—
206C22⋊D61—42no—
207C22⋊D61—41no—
208C22⋊D61—42no—
209C22⋊D61—41no—
210C22⋊D61—41no—
211C22⋊D61—41no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—24no—
217C2×Dic61—44no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220C2×Dic61—44no—
221C2×Dic61C2×C22⋊C411yescharacteristic
222C2×Dic61C2×C22⋊C412yes—
223C2×Dic61C22×D411yescharacteristic
224D244—42no—
225D244—42no—
226C4×D61—22no—
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—22no—
230C4×D61—22no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—22no—
234Dic122—42no—
235Dic122—42no—
236Dic122—48no—
237Dic122—44no—
238Dic122—44no—
239#92420—44no—
240C2×#2123—611nomaximal, Sylow
241C22×C61—22no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61C2×#9911yescharacteristic
249C22×C61C2×#9911yescharacteristic
250C22×C61C22×C22⋊C411yescharacteristic
251C2×D61—21no—
252C2×D61—22no—
253C2×D61—21no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—81no—
260C2×D61—81no—
261#6923—84no—
262C4×C61—22no—
263C4×C61—44no—
264C4×C61C2×#9912yes—
265C4×C61C2×#9912yes—
266Dic61—21no—
267Dic61—21no—
268Dic61—22no—
269Dic61—44no—
270C2×#536—611no—
271C2×#413—611no—
272C2×#410—611no—
273#2123—618no—
274C2×#352—614no—
275C2×C61—21no—
276C2×C61—21no—
277C2×C61—41no—
278C2×C61—41no—
279C2×C61#212312yescomplemented
280C2×C61C22×#9911yescharacteristic
281D61—41no—
282D61—41no—
283D61—42no—
284C22×#41—611no—
285C23×Q8—611no—
286C22×M16—1221no—
287#536—614no—
288#536—614no—
289#536—1222no—
290#536—1222no—
291C2×C4⋊Q8—1221no—
292C2×#124—1221no—
293C2×#123—612no—
294C2×#123—612no—
295C2×C22⋊Q8—1221no—
296C2×C22⋊Q8—1221no—
297C2×#117—1221no—
298C2×#117#692311yescharacteristic, complemented
299#413—614no—
300#413—1222no—
301C2×#101—1221no—
302C2×#100—612no—
303#410—614no—
304#410—1222no—
305C2×#99—612no—
306C2×#99—1221no—
307#352—6116no—
308C61C2×#212311yescharacteristic, complemented, Sylow
309C2×#41—612no—
310C2×#41—612no—
311C2×#41—612no—
312C2×#41—612no—
313C2×#41—2441no—
314C2×#41—2441no—
315C2×#41—2442no—
316C22×Q8—611no—
317C22×Q8—614no—
318C22×Q8—1221no—
319C22×Q8—1222no—
320C22×Q8—1221no—
321C22×Q8—2442no—
322C22×Q8—2441no—
323C22×Q8—2441no—
324C22×D4—611no—
325C22×D4—611no—
326C22×D4—611no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—612no—
330C23×C4C2×#692311yescharacteristic
331C2×M16—1222no—
332C2×M16—1224no—
333C2×M16—2442no—
334C2×M16—2441no—
335C22×C8—2441no—
336C4⋊Q8—1224no—
337C4⋊Q8—1224no—
338#124—1224no—
339#124—1224no—
340#123—614no—
341#123—614no—
342#123—618no—
343#123—1222no—
344#123—1222no—
345#123—1224no—
346C22⋊Q8—1224no—
347C22⋊Q8—1224no—
348C22⋊Q8—1224no—
349C22⋊Q8—1224no—
350#117—22no—
351#117—1224no—
352#117—2442no—
353#117C2×#692314yescomplemented
354C2×C4⋊C4—21no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—2441no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4C2×#692312yes—
362C2×C42—21no—
363C2×C42—2441no—
364#101—1224no—
365#100—614no—
366#100—1222no—
367#99—614no—
368#99—1222no—
369#99—1224no—
370C24—611no—
371C24—611no—
372C24C22×#692311yescharacteristic
373#41—2442no—
374#41—2442no—
375#41—2442no—
376#41—2442no—
377#41—2442no—
378#41—2442no—
379#41—2442no—
380#41—2442no—
381C2×Q8—612no—
382C2×Q8—612no—
383C2×Q8—1224no—
384C2×Q8—1224no—
385C2×Q8—2441no—
386C2×Q8—2441no—
387C2×Q8—2444no—
388C2×Q8—2444no—
389C2×Q8—2442no—
390C2×Q8—2442no—
391C2×Q8—2444no—
392C2×Q8—2444no—
393C2×Q8—2442no—
394C2×Q8—2441no—
395C2×Q8—2441no—
396C2×D4—611no—
397C2×D4—611no—
398C2×D4—612no—
399C2×D4—611no—
400C2×D4—611no—
401C2×D4—612no—
402C2×D4—612no—
403C2×D4—612no—
404C2×D4—2441no—
405C2×D4—2441no—
406C2×D4—2441no—
407C2×D4—2441no—
408C2×D4—2441no—
409C2×D4—2441no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—612no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—612no—
420C22×C4—614no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—2442no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4C22⋊#692311yescharacteristic
433C22×C4C22⋊#692312yes—
434C22×C4C22⋊#692311yescharacteristic
435C22×C4C22×#692312yes—
436M16—1224no—
437M16—2444no—
438M16—2442no—
439C2×C8—2444no—
440C2×C8—2442no—
441C4⋊C4—24no—
442C4⋊C4—1224no—
443C4⋊C4—1228no—
444C4⋊C4—1224no—
445C4⋊C4—2444no—
446C22⋊C4—22no—
447C22⋊C4—1224no—
448C22⋊C4—1228no—
449C22⋊C4—1224no—
450C22⋊C4—1224no—
451C22⋊C4—2442no—
452C22⋊C4C22×#692314yes—
453C42—24no—
454C42—2444no—
455C23—21no—
456C23—21no—
457C23—42no—
458C23—611no—
459C23—612no—
460C23—611no—
461C23—1221no—
462C23—1221no—
463C23—1221no—
464C23—1221no—
465C23—2442no—
466C23—2442no—
467C23C2×C22⋊#692311yescharacteristic
468C23C2×C22⋊#692311yescharacteristic
469C23C23×#692311yescharacteristic
470Q8—2442no—
471Q8—2442no—
472Q8—2448no—
473Q8—2444no—
474Q8—2444no—
475D4—2442no—
476D4—2442no—
477D4—2441no—
478D4—2441no—
479D4—2441no—
480D4—2442no—
481D4—2441no—
482D4—2442no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—44no—
491C2×C4—611no—
492C2×C4—612no—
493C2×C4—611no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1224no—
507C2×C4—4884no—
508C2×C4—2444no—
509C2×C4—2444no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2444no—
513C2×C4#1451983612yes—
514C2×C4#1451983611yescharacteristic
515C2×C4#1451983611yescharacteristic
516C2×C4C2×C22⋊#692312yes—
517C2×C4C2×C22⋊#692312yes—
518C8—2444no—
519C22—22no—
520C22—21no—
521C22—21no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—1221no—
527C22—1222no—
528C22—1221no—
529C22—4881no—
530C22—4881no—
531C22—2441no—
532C22—2441no—
533C22—2442no—
534C22—2442no—
535C22—2442no—
536C22C2×#1451983611yescharacteristic
537C22C2×#1451983611yescharacteristic
538C22C22×C22⋊#692311yescharacteristic
539C4—22no—
540C4—44no—
541C4—1221no—
542C4—1221no—
543C4—1222no—
544C4—4884no—
545C4—2444no—
546C4C2×#1451983612yes—
547C4C2×#1451983612yes—
548C2—21no—
549C2—21no—
550C2—41no—
551C2—41no—
552C2—2441no—
553C2—2441no—
554C2—2442no—
555C2#2146208312yesdirect product factor
556C2C22×#1451983611yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC2×C22⋊D61C24
hypercenterC2×#117#6923
hypocenterC61C2×#2123
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21462083C2direct (N × H)
4. C2×#16840665C2semidirect
16. #16840665C22semidirect
18. C23×Dic122C4semidirect
21. #16840338C4semidirect
25. C2×#14519761C4semidirect
29. C2×#117×C61C4semidirect
59. C22×Dic122C2×C4semidirect
75. #14519761C2×C4semidirect
87. #117×C61C2×C4semidirect
279. C2×C61#2123semidirect
298. C2×#117#6923semidirect
308. C61C2×#2123semidirect
353. #117C2×#6923semidirect
555. C2#21462083direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407482
C2×#168407412
#214620832
C2×#168406652
C2×#168403382
C2×#212361