Finite Group Databasebeta

← C22⋊C4×#11758200 #29695842

Subgroups — C22⋊C4×#11758200 #29695842

Order 15616.

1614 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758200C211yesmaximal, characteristic
2C2×C4×#11758200C212yesmaximal, complemented
3C22⋊C4×Dic122C211yesmaximal, characteristic, complemented
4#21446672C212yesmaximal, complemented
5#21446177C212yesmaximal, complemented
6#21446153C212yesmaximal, complemented
7#21445842C211yesmaximal, characteristic, complemented
8#21445753C211yesmaximal, characteristic, complemented
9#21445750C211yesmaximal, characteristic
10C22⋊C4×#92418C211yesmaximal, characteristic, complemented
11D4×C22⋊C4×C61C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758200—28no—
14C22×#11758200C418yescomplemented
15C22×#11758200C2211yescharacteristic
16C22×#11758200C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758200C2211yescharacteristic
19C23×#92418C2211yescharacteristic
20C2×C4×Dic122C2212yescomplemented
21C4×#11758200—28no—
22C2×#14519708C2212yescomplemented
23C2×#14519700C2212yescomplemented
24C2×#14519700C2212yescomplemented
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27C2×C22⋊#92418C2211yescharacteristic
28C2×C4×#92418C2212yescomplemented
29#16839668C2212yescomplemented
30#16839666—22no—
31#16839622C2211yescharacteristic
32#16839620—21no—
33C22⋊C4×Dic61—21no—
34#16839524C2212yescomplemented
35#16839517C2212yescomplemented
36C23×D4×C61C2211yescharacteristic
37C2×C4×D4×C61C2212yes—
38C22×C22⋊C4×C61—21no—
39C61×#390C2212yes—
40C61×#386—22no—
41C61×#380C2211yescharacteristic
42C61×#379—21no—
43C4×C22⋊C4×C61C2211yescharacteristic
44C22×Dic122—24no—
45C22×Dic122—21no—
46C22×Dic122—22no—
47C22×Dic122—42no—
48C22×Dic122C2×C414yescomplemented
49C22×Dic122C2311yescharacteristic
50C22×Dic122C2311yescharacteristic
51C23×Dic61—21no—
52C2×#11758200—24no—
53C2×#11758200—24no—
54C2×#11758200—232no—
55C2×#11758200—24no—
56C2×#11758200C2×C414yes—
57C2×#11758200C2×C414yes—
58C2×#11758200C2×C414yescomplemented
59C2×#11758200C2×C414yescomplemented
60C2×#11758200D418yes—
61C22×#92418—24no—
62C22×#92418C2×C414yescomplemented
63C22×#92418C2312yes—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C4×Dic122—24no—
67C4×Dic122—42no—
68C2×C4⋊Dic61C2312yes—
69C2×#11758196—22no—
70C2×C22⋊Dic61—21no—
71C2×C4×Dic61—22no—
72#14519708—24no—
73#14519700—24no—
74#14519700—42no—
75#14519698—24no—
76#14519698—42no—
77C22⋊#92418—22no—
78C4×#92418—24no—
79#14519684—22no—
80#14519684—22no—
81#14519684—21no—
82C25×C61—21no—
83C22×D4×C61—24no—
84C22×D4×C61—21no—
85C22×D4×C61—22no—
86C22×D4×C61—42no—
87C22×D4×C61C2×C414yes—
88C22×D4×C61C2311yescharacteristic
89C22×D4×C61C2311yescharacteristic
90C23×C4×C61—22no—
91C23×C4×C61C2311yescharacteristic
92C4×D4×C61—24no—
93C4×D4×C61—42no—
94C2×C4⋊C4×C61C2312yes—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—44no—
100C2×C22⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61D411yescharacteristic
102C2×C42×C61C2312yes—
103#95×C61—21no—
104#95×C61D412yes—
105C24×C61—21no—
106C24×C61—21no—
107C24×C61—24no—
108C24×C61—44no—
109C24×C61—42no—
110C24×C61—42no—
111C24×C61—82no—
112C24×C61C2×D411yescharacteristic
113C2×D4×C61—28no—
114C2×D4×C61—24no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—44no—
121C2×D4×C61—48no—
122C2×D4×C61—44no—
123C2×D4×C61C22×C412yes—
124C2×D4×C61C22×C412yes—
125C2×D4×C61C2×D414yes—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—44no—
130C22×C4×C61—44no—
131C22×C4×C61—82no—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D412yes—
136C22×C4×C61#4112yes—
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—24no—
140C22×Dic61—21no—
141C22×Dic61—22no—
142C22×Dic61—44no—
143C2×Dic122—21no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—28no—
148C2×Dic122—24no—
149C2×Dic122—21no—
150C2×Dic122—44no—
151C2×Dic122—48no—
152C2×Dic122—44no—
153C2×Dic122C22×C412yes—
154C2×Dic122C22×C412yes—
155C2×Dic122C2×D414yes—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—44no—
160C22⋊C4×C61—42no—
161C22⋊C4×C61—44no—
162C22⋊C4×C61—82no—
163C22⋊C4×C61SD1614yescomplemented
164C42×C61—22no—
165C42×C61—22no—
166C22⋊Dic61—42no—
167#11758200—28no—
168#11758200—28no—
169#11758200—28no—
170#11758200C22⋊C4116yesdirect product factor
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173#11758196—44no—
174C4×Dic61—44no—
175C2×#92418—22no—
176C2×#92418—28no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C2×#92418C22×C412yes—
181C2×#92418C22×C412yes—
182C2×#92418C2×D414yes—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—24no—
187C23×C61—22no—
188C23×C61—48no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61—48no—
197C23×C61—84no—
198C23×C61—88no—
199C23×C61—88no—
200C23×C61C4×D412yes—
201C23×C61C2×SD1612yes—
202C23×C61#13512yes—
203C23×C61C22×D411yescharacteristic
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—22no—
207D4×C61—24no—
208D4×C61—48no—
209D4×C61—42no—
210D4×C61C2×C22⋊C414yes—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—88no—
224C2×C4×C61C4×D412yes—
225C2×C4×C61C2×SD1614yescomplemented
226C2×C4×C61C23×C411yescharacteristic
227C2×C4×C61C22×D412yes—
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)12yes—
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—22no—
233C2×Dic61—48no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238Dic122—22no—
239Dic122—24no—
240Dic122—22no—
241Dic122—22no—
242Dic122—48no—
243Dic122—42no—
244Dic122C2×C22⋊C414yescomplemented
245#92418—22no—
246#92418—42no—
247#92418C2×C22⋊C414yescomplemented
248C22⋊C4×SD16—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—24no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—816no—
262C22×C61—82no—
263C22×C61—82no—
264C22×C61C4×SD1614yescomplemented
265C22×C61C2×C4×D411yescharacteristic
266C22×C61D4212yes—
267C22×C61C22×SD1611yescharacteristic
268C22×C61C2×#13511yescharacteristic
269C22×C61#57712yes—
270C4×C61—22no—
271C4×C61—24no—
272C4×C61—42no—
273C4×C61—82no—
274C4×C61C22×C22⋊C411yescharacteristic
275C4×C61#51811yescharacteristic
276C4×C61#54612yes—
277Dic61—24no—
278Dic61—42no—
279Dic61—82no—
280C23×SD16—611no—
281C2×C4×SD16—612no—
282Q8×C22⋊C4—611no—
283D4×C22⋊C4D6111yescharacteristic
284#1968—612no—
285#1880—612no—
286#1877—612no—
287#1807—611no—
288#1787—611no—
289#1786—611no—
290C8×C22⋊C4—611no—
291C2×C61—24no—
292C2×C61—44no—
293C2×C61—82no—
294C2×C61D4×C22⋊C411yescharacteristic
295C2×C61C2×C4×SD1611yescharacteristic
296C2×C61#294911yescharacteristic
297C2×C61D4×SD1612yes—
298C2×C61#328212yes—
299C23×Q8—611no—
300C23×D4C2×D6111yescharacteristic
301C22×SD16—611no—
302C22×SD16—618no—
303C22×SD16—611no—
304C22×SD16—611no—
305C22×SD16—611no—
306C22×SD16—1228no—
307C23×C8—611no—
308C2×C4×Q8—612no—
309C2×C4×D4C2×D6112yes—
310C22×C22⋊C4—21no—
311C4×SD16—1228no—
312C2×#106—612no—
313C2×#103—612no—
314C2×#103—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×C22⋊C8—611no—
318C2×C4×C8—612no—
319#391—612no—
320#391—1222no—
321#390C2×D6112yes—
322#386—22no—
323#380—611no—
324#380—1221no—
325#380C2×D6111yescharacteristic
326#379—21no—
327C4×C22⋊C4—1221no—
328C4×C22⋊C4C2×D6111yescharacteristic
329#340—612no—
330#336—612no—
331C61C22⋊C4×SD1611yescharacteristic, complemented, Sylow
332C25—21no—
333C22×Q8—614no—
334C22×Q8—611no—
335C22×Q8—611no—
336C22×Q8—1224no—
337C22×Q8—1222no—
338C22×Q8—1221no—
339C22×Q8—2442no—
340C22×D4—24no—
341C22×D4—22no—
342C22×D4—21no—
343C22×D4—42no—
344C22×D4C4×D6114yes—
345C22×D4C22×D6111yescharacteristic
346C22×D4C22×D6111yescharacteristic
347C23×C4—22no—
348C23×C4—1221no—
349C23×C4C22×D6111yescharacteristic
350C2×SD16—614no—
351C2×SD16—618no—
352C2×SD16—614no—
353C2×SD16—614no—
354C2×SD16—614no—
355C2×SD16—1224no—
356C2×SD16—12232no—
357C2×SD16—1224no—
358C2×SD16—1224no—
359C22×C8—611no—
360C22×C8—614no—
361C22×C8—611no—
362C22×C8—612no—
363C22×C8—1224no—
364C4×Q8—1224no—
365C4×Q8—2442no—
366C4×D4—24no—
367C4×D4—42no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4C22×D6112yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—44no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4C22⋊D6111yescharacteristic
378C2×C22⋊C4C22⋊D6111yescharacteristic
379C2×C42—1222no—
380C2×C42C22×D6112yes—
381#106—1224no—
382#103—1224no—
383#103—2442no—
384#102—1224no—
385#102—2442no—
386C22⋊C8—1222no—
387C4×C8—1224no—
388#95—21no—
389#95—1222no—
390#95—1221no—
391#95—1222no—
392#95C22⋊D6112yes—
393C24—21no—
394C24—21no—
395C24—24no—
396C24—44no—
397C24—42no—
398C24—42no—
399C24—82no—
400C24C2×C22⋊D6111yescharacteristic
401C2×Q8—612no—
402C2×Q8—612no—
403C2×Q8—614no—
404C2×Q8—1228no—
405C2×Q8—1221no—
406C2×Q8—1221no—
407C2×Q8—1222no—
408C2×Q8—1224no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—2444no—
412C2×Q8—2444no—
413C2×Q8—2448no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4C2×C4×D6112yes—
425C2×D4C2×C4×D6112yes—
426C2×D4D4×D6114yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—82no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—2444no—
438C22×C4C2×C4×D6112yes—
439C22×C4#1175822212yes—
440C22×C4C2×C22⋊D6112yes—
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6112yes—
443C22×C4C23×D6111yescharacteristic
444SD16—6116no—
445SD16—1228no—
446SD16—1228no—
447SD16—1228no—
448C2×C8—612no—
449C2×C8—614no—
450C2×C8—612no—
451C2×C8—1228no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—2444no—
461C22⋊C4—22no—
462C22⋊C4—44no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C22⋊C4—2442no—
467C22⋊C4#1175820014yesdirect product factor
468C42—22no—
469C42—22no—
470C42—2444no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—48no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—48no—
484C23—42no—
485C23—84no—
486C23—88no—
487C23—88no—
488C23C4×C22⋊D6112yes—
489C23C2×#1175820012yes—
490C23#1451979712yes—
491C23C22×C22⋊D6111yescharacteristic
492Q8—614no—
493Q8—1222no—
494Q8—1224no—
495Q8—1222no—
496Q8—1222no—
497Q8—2448no—
498Q8—2442no—
499D4—22no—
500D4—22no—
501D4—22no—
502D4—24no—
503D4—48no—
504D4—42no—
505D4C22⋊C4×D6114yes—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—1222no—
520C2×C4—1224no—
521C2×C4—1222no—
522C2×C4—2448no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4C4×C22⋊D6112yes—
528C2×C4C2×#1175820014yescomplemented
529C2×C4C22×C4×D6111yescharacteristic
530C2×C4C2×D4×D6112yes—
531C2×C4C2×#1175822211yescharacteristic
532C2×C4#1451982212yes—
533C8—614no—
534C8—1222no—
535C8—2442no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—22no—
540C22—24no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—816no—
549C22—82no—
550C22—82no—
551C22C4×#1175820014yescomplemented
552C22C2×C4×C22⋊D6111yescharacteristic
553C22D4×C22⋊D6112yes—
554C22C22×#1175820011yescharacteristic
555C22C2×#1451979711yescharacteristic
556C22#1684057312yes—
557C4—24no—
558C4—22no—
559C4—42no—
560C4—82no—
561C4—1224no—
562C4—4882no—
563C4—2442no—
564C4C2×C22⋊C4×D6111yescharacteristic
565C4#1684023511yescharacteristic
566C4#1684040212yes—
567C2—24no—
568C2—44no—
569C2—82no—
570C2C22⋊C4×C22⋊D6111yescharacteristic
571C2C2×C4×#1175820011yescharacteristic
572C2#2145467611yescharacteristic
573C2D4×#1175820012yes—
574C2#2145752212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingD4×C22⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
cosocleC22×C4×C61C24
hypercenterD4×C22⋊C4D61
hypocenterC61C22⋊C4×SD16
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ D4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758200C2semidirect
3. C22⋊C4×Dic122C2semidirect
4. #21446672C2semidirect
5. #21446177C2semidirect
6. #21446153C2semidirect
7. #21445842C2semidirect
8. #21445753C2semidirect
10. C22⋊C4×#92418C2semidirect
14. C22×#11758200C4semidirect
20. C2×C4×Dic122C22semidirect
22. C2×#14519708C22semidirect
23. C2×#14519700C22semidirect
24. C2×#14519700C22semidirect
28. C2×C4×#92418C22semidirect
29. #16839668C22semidirect
34. #16839524C22semidirect
35. #16839517C22semidirect
48. C22×Dic122C2×C4semidirect
58. C2×#11758200C2×C4semidirect
59. C2×#11758200C2×C4semidirect
62. C22×#92418C2×C4semidirect
163. C22⋊C4×C61SD16semidirect
170. #11758200C22⋊C4direct (N × H)
225. C2×C4×C61C2×SD16semidirect
244. Dic122C2×C22⋊C4semidirect
247. #92418C2×C22⋊C4semidirect
264. C22×C61C4×SD16semidirect
331. C61C22⋊C4×SD16semidirect
467. C22⋊C4#11758200direct (N × H)
528. C2×C4C2×#11758200semidirect
551. C22C4×#11758200semidirect

Maximal subgroups

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

subgroupindex
C23×#117582002
C2×C4×#117582002
C22⋊C4×Dic1222
#214466722
#214461772
#214461532
#214458422
#214457532
#214457502
C22⋊C4×#924182
D4×C22⋊C4×C612
C22⋊C4×SD1661