Finite Group Databasebeta

← C2×#21461910 #30615951

Subgroups — C2×#21461910 #30615951

Order 15616.

1206 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519840C211yesmaximal, characteristic
2C22×#14519831C211yesmaximal, characteristic
3C2×(C2×#6923)⋊Q8C211yesmaximal, characteristic
4C2×#16840713C211yesmaximal, characteristic
5#21461910C218yesmaximal, direct product factor
6C2×#16840636C211yesmaximal, characteristic
7C2×#16840635C211yesmaximal, characteristic
8C2×#103×D61C211yesmaximal, characteristic
9C22×D61⋊C8C2211yescharacteristic
10C22×C4⋊#6923—21no—
11C22×C4⋊#6923C2211yescharacteristic
12C22×C4⋊#6923C2211yescharacteristic
13C22×C4×#6923C2211yescharacteristic
14C2×#14519840—24no—
15C2×#14519840C2214yes—
16C2×#14519831—24no—
17C2×#14519831C2214yes—
18(C2×#6923)⋊Q8—22no—
19(C2×#6923)⋊Q8C2214yes—
20#16840713—22no—
21#16840713C2214yes—
22C2×(C2×#6923)⋊C4—21no—
23C2×(C2×#6923)⋊C4—21no—
24C2×(C2×#6923)⋊C4—21no—
25#16840636C2214yes—
26#16840635C2214yes—
27C22×Q8×D61C2211yescharacteristic
28C22×C8×D61C2211yescharacteristic
29C2×C4⋊C4×D61C2211yescharacteristic
30C2×#14519702C411yescharacteristic, complemented
31C2×#14519701C411yescharacteristic, complemented
32C2×#14519700C411yescharacteristic, complemented
33#103×D61C414yescomplemented
34#103×D61C2214yes—
35C2×#103×C61C411yescharacteristic, complemented
36C23×#6923—21no—
37C23×#6923—21no—
38C23×#6923D411yescharacteristic
39C23×#6923D411yescharacteristic
40C2×D61⋊C8—21no—
41C2×D61⋊C8—22no—
42C2×D61⋊C8D414yes—
43C2×D61⋊C8C2312yes—
44C22×#92420—21no—
45C2×C4⋊#6923—22no—
46C2×C4⋊#6923—22no—
47C2×C4⋊#6923—22no—
48C2×C4⋊#6923—22no—
49C2×C4⋊#6923—24no—
50C2×C4⋊#6923—21no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923—44no—
53C2×C4⋊#6923C2312yes—
54C2×C4⋊#6923C2312yes—
55C2×C4×#6923—22no—
56C2×C4×#6923—22no—
57C2×C4×#6923D412yes—
58C2×C4×#6923C2312yes—
59#14519840—28no—
60#14519831—28no—
61(C2×#6923)⋊C4—24no—
62(C2×#6923)⋊C4—24no—
63(C2×#6923)⋊C4—24no—
64C22×Dic122—21no—
65C22×Dic122C2×C411yescharacteristic
66C2×Q8×D61—21no—
67C2×Q8×D61—21no—
68C2×Q8×D61—22no—
69C2×Q8×D61—22no—
70C2×Q8×D61—22no—
71C2×Q8×D61—22no—
72C2×Q8×D61C2×C412yes—
73C2×Q8×D61C2312yes—
74C22×C4×D61—21no—
75C22×C4×D61—21no—
76C22×C4×D61C2311yescharacteristic
77C22×#92418C2×C411yescharacteristic
78C2×C8×D61—22no—
79C2×C8×D61—22no—
80C2×C8×D61C2×C412yes—
81C2×C8×D61C2312yes—
82C2×C4⋊Dic61C2×C411yescharacteristic
83C2×#11758196—21no—
84C4⋊C4×D61—21no—
85C4⋊C4×D61—21no—
86C4⋊C4×D61C2×C412yes—
87C4⋊C4×D61C2312yes—
88#14519702C2×C414yescomplemented
89#14519701C2×C414yescomplemented
90#14519700C2×C414yescomplemented
91C22×Q8×C61C2×C411yescharacteristic
92C22×C8×C61C2×C411yescharacteristic
93C2×C4⋊C4×C61C2×C411yescharacteristic
94#103×C61C2×C414yescomplemented
95C23×D61C2×D411yescharacteristic
96C22×#6923—22no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—22no—
100C22×#6923—22no—
101C22×#6923—22no—
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—42no—
105C22×#6923—42no—
106C22×#6923—42no—
107C22×#6923—42no—
108C22×#6923—42no—
109C22×#6923—42no—
110C22×#6923C2×D412yes—
111C22×#6923C2×D412yes—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61C22×C412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61C22×C411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61#4111yescharacteristic
122Q8×D61—24no—
123Q8×D61—24no—
124Q8×D61—22no—
125Q8×D61—22no—
126Q8×D61—22no—
127Q8×D61—22no—
128Q8×D61—42no—
129Q8×D61—42no—
130Q8×D61—42no—
131Q8×D61—42no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61C22×C411yescharacteristic
141C2×C4×D61C2×D412yes—
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61#4112yes—
144C2×C4×D61#4112yes—
145C2×C4×D61#4112yes—
146C2×C4×D61#4111yescharacteristic
147C2×C4×D61C2411yescharacteristic
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—21no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×Dic122C22×C412yes—
156C2×#92420—24no—
157C2×#92420—22no—
158C4⋊#6923—22no—
159C4⋊#6923—22no—
160C4⋊#6923—22no—
161C4⋊#6923—22no—
162C4⋊#6923—48no—
163C4⋊#6923—42no—
164C4⋊#6923—42no—
165C4×#6923—24no—
166C4×#6923—22no—
167C4×#6923—22no—
168D61⋊C8—24no—
169D61⋊C8—22no—
170D61⋊C8C2×D414yes—
171C2×C8×C61—22no—
172C2×C8×C61C22×C412yes—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C412yes—
175C4⋊Dic61—21no—
176C4⋊Dic61C22×C412yes—
177#11758196—22no—
178#11758196—22no—
179C2×#92418—22no—
180C2×#92418C22×C412yes—
181C8×D61—24no—
182C8×D61—22no—
183C8×D61—22no—
184C23×C61C4×D411yescharacteristic
185C22×D61C4×D411yescharacteristic
186C22×D61#12112yes—
187C22×D61#12412yes—
188C22×D61AGL(1,ℤ/8)12yes—
189C22×D61#13712yes—
190C22×D61#13712yes—
191C22×D61#13712yes—
192C22×D61C22×D411yescharacteristic
193C2×#6923—22no—
194C2×#6923—22no—
195C2×#6923—22no—
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—44no—
200C2×#6923—44no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—84no—
208C2×#6923—84no—
209Q8×C61—22no—
210Q8×C61—22no—
211Q8×C61—42no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61#11712yes—
217C2×C4×C61C4×D412yes—
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C23×C411yescharacteristic
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61C4×D411yescharacteristic
225C2×Dic61#12312yes—
226C2×Dic61#12612yes—
227C2×Dic61C2×#4111yescharacteristic
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—42no—
232C4×D61—44no—
233C4×D61—44no—
234C4×D61#11712yes—
235C4×D61C4×D412yes—
236C4×D61#12112yes—
237C4×D61#12312yes—
238C4×D61#12412yes—
239C4×D61#12612yes—
240C4×D61C22×D411yescharacteristic
241C4×D61C2×#4111yescharacteristic
242C4×D61C2×#4111yescharacteristic
243C4×D61C2×#4111yescharacteristic
244Dic122—22no—
245Dic122—22no—
246Dic122—42no—
247Dic122—42no—
248Dic122—42no—
249Dic122—44no—
250#92420—24no—
251C8×C61—22no—
252#92418—22no—
253C2×#1935—611nomaximal, Sylow
254C22×C61#38812yes—
255C22×C61#44012yes—
256C22×C61#44112yes—
257C22×C61C2×C4×D411yescharacteristic
258C2×D61#38812yes—
259C2×D61#44012yes—
260C2×D61#44112yes—
261C2×D61#46814yes—
262C2×D61#47014yes—
263C2×D61#48914yes—
264C2×D61#49114yes—
265C2×D61C2×#12111yescharacteristic
266C2×D61C2×#12411yescharacteristic
267C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
268C2×D61C2×#13711yescharacteristic
269C2×D61C2×#13711yescharacteristic
270C2×D61C2×#13711yescharacteristic
271#6923—42no—
272#6923—42no—
273#6923—82no—
274#6923—82no—
275C4×C61—42no—
276C4×C61—42no—
277C4×C61#38812yes—
278C4×C61C2×#11711yescharacteristic
279C4×C61C2×C4×D411yescharacteristic
280Dic61—42no—
281Dic61—42no—
282Dic61#38812yes—
283Dic61C2×#12311yescharacteristic
284Dic61C2×#12611yescharacteristic
285C22×#103—611no—
286C22×#103—611no—
287C22×#97—611no—
288C2×#391—611no—
289C2×#389—611no—
290#1935—618no—
291C2×#340—611no—
292C2×#340—611no—
293C2×C61#193514yescomplemented
294C2×C61C2×#38811yescharacteristic
295C2×C61C2×#44011yescharacteristic
296C2×C61C2×#44111yescharacteristic
297D61#193514yescomplemented
298D61C2×#46811yescharacteristic
299D61C2×#47011yescharacteristic
300D61C2×#48911yescharacteristic
301D61C2×#49111yescharacteristic
302C23×Q8—611no—
303C23×C8—611no—
304C23×C8—611no—
305C22×C4⋊C4—611no—
306C22×C4⋊C4—611no—
307C22×C4⋊C4—611no—
308C22×C4⋊C4—1221no—
309C22×C42—611no—
310C2×#103—611no—
311C2×#103—611no—
312C2×#103—611no—
313C2×#103—614no—
314C2×#103—614no—
315C2×#103—614no—
316C2×#103—1224no—
317C2×#103#692311yescharacteristic, complemented
318C2×#97—614no—
319C2×#97—1224no—
320#391—614no—
321#391—1222no—
322#389—614no—
323#389—1222no—
324C2×#95—1221no—
325C2×#95—1221no—
326C2×#95—1221no—
327#340—614no—
328#340—614no—
329C61C2×#193511yescharacteristic, complemented, Sylow
330C22×Q8—612no—
331C22×Q8—612no—
332C22×Q8—611no—
333C22×Q8—1221no—
334C22×Q8—1221no—
335C22×Q8—1221no—
336C22×Q8—1222no—
337C22×Q8—1222no—
338C22×Q8—1222no—
339C22×Q8—1222no—
340C22×Q8C2×#692311yescharacteristic
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—1221no—
345C23×C4—1221no—
346C23×C4—1221no—
347C23×C4—1221no—
348C22×C8—614no—
349C22×C8—612no—
350C22×C8—611no—
351C22×C8—612no—
352C22×C8—612no—
353C22×C8—1222no—
354C22×C8—1222no—
355C22×C8—1221no—
356C22×C8—1222no—
357C22×C8—1221no—
358C22×C8C2×#692311yescharacteristic
359C2×C4⋊C4—611no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—2444no—
375C2×C4⋊C4C2×#692311yescharacteristic
376C2×C42—612no—
377C2×C42—612no—
378C2×C42—1222no—
379C2×C42—1222no—
380#103—614no—
381#103—614no—
382#103—614no—
383#103—1228no—
384#103C2×#692314yescomplemented
385#97—1228no—
386#95—1224no—
387#95—1224no—
388#95—1224no—
389C24—611no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—612no—
394C2×Q8—1222no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8—1224no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—1224no—
401C2×Q8—1222no—
402C2×Q8—1221no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—2442no—
406C2×Q8—2442no—
407C2×Q8—2442no—
408C2×Q8—2442no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C2×Q8C22×#692312yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—611no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—611no—
418C22×C4—612no—
419C22×C4—612no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—2442no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4C22×#692311yescharacteristic
450C2×C8—22no—
451C2×C8—614no—
452C2×C8—612no—
453C2×C8—1222no—
454C2×C8—1224no—
455C2×C8—1222no—
456C2×C8—1224no—
457C2×C8—1222no—
458C2×C8—1224no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8C22×#692312yes—
462C4⋊C4—21no—
463C4⋊C4—612no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1221no—
471C4⋊C4—2442no—
472C4⋊C4—2442no—
473C4⋊C4—2448no—
474C4⋊C4C22×#692312yes—
475C42—1222no—
476C42—1224no—
477C42—1222no—
478C23—612no—
479C23—612no—
480C23—612no—
481C23—612no—
482C23—612no—
483C23—612no—
484C23—611no—
485C23—611no—
486C23D4×#692311yescharacteristic
487Q8—22no—
488Q8—22no—
489Q8—42no—
490Q8—1222no—
491Q8—1222no—
492Q8—2442no—
493Q8—2442no—
494Q8—2444no—
495Q8—2442no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—611no—
503C2×C4—611no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—611no—
510C2×C4—611no—
511C2×C4—612no—
512C2×C4—611no—
513C2×C4—611no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—4884no—
523C2×C4—4884no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2444no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2444no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2444no—
537C2×C4—2444no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4#1451985612yes—
541C2×C4D4×#692311yescharacteristic
542C2×C4D4×#692312yes—
543C2×C4C23×#692311yescharacteristic
544C8—22no—
545C8—1224no—
546C8—1222no—
547C22—611no—
548C22—611no—
549C22—612no—
550C22—611no—
551C22—611no—
552C22—612no—
553C22—611no—
554C22—611no—
555C22—612no—
556C22—614no—
557C22—614no—
558C22—614no—
559C22—614no—
560C22#1684071212yes—
561C22#1684080112yes—
562C22#1684080212yes—
563C22C2×D4×#692311yescharacteristic
564C4—42no—
565C4—42no—
566C4—611no—
567C4—611no—
568C4—612no—
569C4—4882no—
570C4—4882no—
571C4—2442no—
572C4—2442no—
573C4—2442no—
574C4—2442no—
575C4#1684071212yes—
576C4C2×#1451985611yescharacteristic
577C4C2×D4×#692311yescharacteristic
578C2—611no—
579C2—611no—
580C2—611no—
581C2—611no—
582C2—614no—
583C2#2146191014yesdirect product factor
584C2C2×#1684071211yescharacteristic
585C2C2×#1684080111yescharacteristic
586C2C2×#1684080211yescharacteristic
subgroupG/H
centerC23D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#103×C61C4
socleC23×C61C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×#103#6923
hypocenterC61C2×#1935
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
5. #21461910C2direct (N × H)
30. C2×#14519702C4semidirect
31. C2×#14519701C4semidirect
32. C2×#14519700C4semidirect
33. #103×D61C4semidirect
35. C2×#103×C61C4semidirect
88. #14519702C2×C4semidirect
89. #14519701C2×C4semidirect
90. #14519700C2×C4semidirect
94. #103×C61C2×C4semidirect
293. C2×C61#1935semidirect
297. D61#1935semidirect
317. C2×#103#6923semidirect
329. C61C2×#1935semidirect
384. #103C2×#6923semidirect
583. C2#21461910direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198402
C22×#145198312
C2×(C2×#6923)⋊Q82
C2×#168407132
#214619102
C2×#168406362
C2×#168406352
C2×#103×D612
C2×#193561