Finite Group Databasebeta

← #29695662

Subgroups — #29695662

Order 15616.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840416C211yesmaximal, characteristic
2C2×#16839839C211yesmaximal, characteristic, complemented
3C2×#16839760C212yesmaximal, complemented
4#21448719C211yesmaximal, characteristic, complemented
5#21446741C211yesmaximal, characteristic, complemented
6#21446620C211yesmaximal, characteristic, complemented
7#21446251C212yesmaximal, complemented
8#21446125C212yesmaximal, complemented
9#21445756C212yesmaximal, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758224C2212yes—
13C22×#11758189C2212yes—
14#16840416—24no—
15#16840416C418yescomplemented
16C2×#14519777C2211yescharacteristic
17C2×#14519761—22no—
18C2×#14519757C2212yes—
19C2×#14519751—21no—
20C2×C4×D244C2211yescharacteristic
21C2×#14519739—21no—
22C22×C4⋊Dic61—21no—
23C2×C4⋊C4×D61C2212yes—
24#16839839—24no—
25#16839760—28no—
26C2×#14519702C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519689C2212yes—
33C2×C4×#92418C2211yescharacteristic
34#16839658—21no—
35#16839633—22no—
36#16839619C2212yes—
37C4×C4⋊Dic61C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C2×#11758224—24no—
46C2×#11758224—24no—
47C2×#11758224—24no—
48C2×#11758224—42no—
49C2×#11758224—42no—
50C2×#11758224C2×C418yescomplemented
51C22×Dic122C2312yes—
52C22×D244D411yescharacteristic
53C22×C22⋊D61—22no—
54C23×Dic61—22no—
55C22×C4×D61—22no—
56C22×C4×D61C2312yes—
57C2×#11758189—24no—
58C2×#11758189—24no—
59C2×#11758189—42no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519777—22no—
63#14519777C2×C414yescomplemented
64#14519761—216no—
65#14519757—24no—
66#14519757C2×C418yescomplemented
67#14519751—28no—
68C4×D244C2×C414yes—
69C4×D244D414yes—
70#14519739—44no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—28no—
75C2×C4⋊Dic61D411yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—22no—
79C4⋊C4×D61—24no—
80C4⋊C4×D61—42no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198D412yes—
84C2×C4×Dic61—21no—
85#14519702—24no—
86#14519702—42no—
87#14519700—24no—
88#14519700—42no—
89#14519698—22no—
90#14519698D414yescomplemented
91#14519689—24no—
92C4×#92418—22no—
93#14519684—21no—
94#14519684—22no—
95C22×D4×C61C2311yescharacteristic
96C23×C4×C61—21no—
97C22×C8×C61C2312yes—
98C4×D4×C61—22no—
99C4×D4×C61C2×C414yescomplemented
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—22no—
107#102×C61—24no—
108C24×C61—21no—
109C23×D61C2×D412yes—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61C22×C412yes—
114C2×D4×C61C2×D412yes—
115C2×D4×C61#4112yes—
116C22×C4×C61—21no—
117C22×C4×C61—24no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2411yescharacteristic
124C2×C22⋊D61—28no—
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61—28no—
130C22×Dic61—21no—
131C22×Dic61—44no—
132C22×Dic61—44no—
133C22×Dic61C2×D412yes—
134#11758224—216no—
135#11758224—48no—
136#11758224—48no—
137#11758224—48no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244C22⋊C414yes—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—44no—
146C2×C4×D61—44no—
147C2×C4×D61—82no—
148C2×C4×D61C22×C414yes—
149C2×C4×D61C2×D414yes—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122C22×C414yes—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—81no—
164C4⋊C4×C61C22×C412yes—
165C22⋊C4×C61—24no—
166C42×C61C22×C412yes—
167C42×C61C2×D412yes—
168#11758198—28no—
169#11758198—44no—
170#11758198C22⋊C418yes—
171C4⋊Dic61—24no—
172C4⋊Dic61—24no—
173C4⋊Dic61—21no—
174C4⋊Dic61—28no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61C22⋊C414yes—
178C4⋊Dic61C2×D412yes—
179#11758196—44no—
180C4×Dic61—42no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418C2×D412yes—
187#11758189—48no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C22×D411yescharacteristic
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61C2×C22⋊C414yes—
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—22no—
206D4×C61C4×D414yescomplemented
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—48no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C412yes—
217C2×C4×C61C2×C22⋊C412yes—
218C2×C4×C61AGL(1,ℤ/8)12yes—
219C2×C4×C61#13712yes—
220C2×C4×C61C23×C411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C22×D411yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C22⋊D61—416no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—24no—
230C2×Dic61—22no—
231C2×Dic61—416no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61C2×C22⋊C414yes—
240D244—28no—
241C4×D61—24no—
242C4×D61—24no—
243C4×D61—24no—
244C4×D61—24no—
245C4×D61—88no—
246Dic122—24no—
247Dic122—24no—
248Dic122—42no—
249Dic122—42no—
250C8×C61—42no—
251#92418—22no—
252#92418—41no—
253#22431—611nomaximal, Sylow
254C22×C61—24no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61D4211yescharacteristic
263C22×C61#54611yescharacteristic
264C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C61C2×#13711yescharacteristic
266C22×C61#57711yescharacteristic
267C22×C61#57811yescharacteristic
268C2×D61—24no—
269C2×D61—24no—
270C2×D61—24no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C4×C61—22no—
276C4×C61—42no—
277C4×C61—82no—
278C4×C61#41714yescomplemented
279C4×C61C2×C4×D411yescharacteristic
280C4×C61#51811yescharacteristic
281C4×C61D4211yescharacteristic
282C4×C61#54611yescharacteristic
283Dic61—24no—
284Dic61—44no—
285Dic61—42no—
286Dic61—81no—
287C2×#547—611no—
288C2×#436—611no—
289C2×#417—612no—
290D4×C4⋊C4—611no—
291#1981—611no—
292#1956—611no—
293#1894—612no—
294#1872—612no—
295#1802D6111yescharacteristic, complemented
296#1788—612no—
297C4×#102—611no—
298C2×C61—44no—
299C2×C61D4×C22⋊C411yescharacteristic
300C2×C61C2×#41711yescharacteristic
301C2×C61#289811yescharacteristic
302C2×C61#327211yescharacteristic
303C2×C61#327611yescharacteristic
304C2×C61#333011yescharacteristic
305C2×C61#333111yescharacteristic
306D61—44no—
307C22×#41—612no—
308C22×M16—612no—
309#547—618no—
310#547—1224no—
311C2×C4⋊Q8—611no—
312C2×#123—1222no—
313C2×C22⋊Q8—612no—
314C2×#121—1221no—
315C2×C4×D4—611no—
316C2×C4×D4—1221no—
317C2×C4×D4C2×D6111yescharacteristic
318C22×C4⋊C4—612no—
319C22×C4⋊C4—1221no—
320#436—1224no—
321C2×C4⋊C8C2×D6111yescharacteristic
322#417—1228no—
323C2×#103—612no—
324C2×#103—612no—
325C2×#103—612no—
326C2×#103—612no—
327C2×#102—611no—
328C2×#102—611no—
329C2×#102C2×D6112yes—
330C2×C22⋊C8—612no—
331C2×C4×C8—611no—
332#389—1221no—
333#385—1222no—
334#384C2×D6111yescharacteristic
335#380—612no—
336C4×C4⋊C4—611no—
337#340—611no—
338#340—611no—
339#340C2×D6112yes—
340C61#2243111yescharacteristic, complemented, Sylow
341C2×#41—618no—
342C2×#41—1224no—
343C2×#41—1224no—
344C2×#41—1224no—
345C2×#41—2442no—
346C2×#41—2442no—
347C22×Q8—612no—
348C22×D4—611no—
349C22×D4—1222no—
350C22×D4C22×D6111yescharacteristic
351C23×C4—21no—
352C23×C4—612no—
353C23×C4—1222no—
354C23×C4—1222no—
355C2×M16—1224no—
356C2×M16—1224no—
357C2×M16—2442no—
358C22×C8—611no—
359C22×C8—611no—
360C22×C8C22×D6112yes—
361C4⋊Q8—614no—
362C4⋊Q8—1222no—
363#123—12216no—
364C22⋊Q8—618no—
365C22⋊Q8—1224no—
366#121—1228no—
367C4×D4—22no—
368C4×D4—614no—
369C4×D4—614no—
370C4×D4—2444no—
371C4×D4C4×D6114yescomplemented
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1228no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—2442no—
384C2×C4⋊C4C22×D6112yes—
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—612no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C42—1221no—
391C2×C42C22×D6111yescharacteristic
392C4⋊C8—22no—
393#103—1224no—
394#103—1224no—
395#103—2442no—
396#103—2442no—
397#102—24no—
398#102—614no—
399#102—1222no—
400C22⋊C8—1224no—
401C4×C8—1222no—
402#95—1222no—
403#95—1221no—
404C24—21no—
405C24—612no—
406#41—12216no—
407#41—2448no—
408#41—2448no—
409#41—2448no—
410C2×Q8—614no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—614no—
420C2×D4—1228no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—2444no—
424C2×D4—2444no—
425C2×D4C2×C4×D6112yes—
426C2×D4D4×D6112yes—
427C2×D4#1175822612yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—612no—
434C22×C4—614no—
435C22×C4—614no—
436C22×C4—1228no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1221no—
442C22×C4—1222no—
443C22×C4—1221no—
444C22×C4—4882no—
445C22×C4—2444no—
446C22×C4—2444no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4D4×D6111yescharacteristic
450C22×C4D4×D6111yescharacteristic
451C22×C4C23×D6111yescharacteristic
452M16—2448no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—612no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—614no—
468C4⋊C4—612no—
469C4⋊C4—1228no—
470C4⋊C4—1221no—
471C4⋊C4—1224no—
472C4⋊C4—1224no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—2444no—
476C4⋊C4C2×C4×D6112yes—
477C22⋊C4—24no—
478C22⋊C4—618no—
479C22⋊C4—1228no—
480C22⋊C4—2444no—
481C42—2442no—
482C42C2×C4×D6112yes—
483C42D4×D6112yes—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—614no—
492C23—1222no—
493C23—1222no—
494C23—1222no—
495C23—1222no—
496C23—1222no—
497C23—1222no—
498C23C2×D4×D6111yescharacteristic
499Q8—1224no—
500Q8—1224no—
501Q8—2442no—
502Q8—2442no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4—1228no—
507D4—24416no—
508D4#1451973914yescomplemented
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—614no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1222no—
527C2×C4—4888no—
528C2×C4—24416no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4C22⋊C4×D6112yes—
537C2×C4C22⋊C4×D6112yes—
538C2×C4#1451980112yes—
539C2×C4#1451980612yes—
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C2×#1175822611yescharacteristic
544C2×C4#1451982311yescharacteristic
545C2×C4#1451982311yescharacteristic
546C8—42no—
547C8—1222no—
548C8—2441no—
549C22—24no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—1224no—
557C22—1224no—
558C22—1224no—
559C22—2442no—
560C22—2442no—
561C22—2442no—
562C22—2442no—
563C22C2×C22⋊C4×D6111yescharacteristic
564C22#1684039611yescharacteristic
565C22#1684040311yescharacteristic
566C22C2×#1451980111yescharacteristic
567C22C2×#1451980611yescharacteristic
568C22#1684057211yescharacteristic
569C22#1684057811yescharacteristic
570C4—22no—
571C4—42no—
572C4—82no—
573C4—1224no—
574C4—4881no—
575C4—2442no—
576C4—2444no—
577C4#1683976014yescomplemented
578C4C2×#1451973911yescharacteristic
579C4#1684023611yescharacteristic
580C4#1684039611yescharacteristic
581C4#1684040311yescharacteristic
582C2—44no—
583C2—2444no—
584C2#2144863911yescharacteristic
585C2C2×#1683976011yescharacteristic
586C2#2145435011yescharacteristic
587C2#2145741411yescharacteristic
588C2#2145746811yescharacteristic
589C2#2145812111yescharacteristic
590C2#2145813211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC2×#16840416C2
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#22431
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16839839C2semidirect
3. C2×#16839760C2semidirect
4. #21448719C2semidirect
5. #21446741C2semidirect
6. #21446620C2semidirect
7. #21446251C2semidirect
8. #21446125C2semidirect
9. #21445756C2semidirect
10. C4×#14519698C2semidirect
11. C61×#1802C2semidirect
15. #16840416C4semidirect
50. C2×#11758224C2×C4semidirect
63. #14519777C2×C4semidirect
66. #14519757C2×C4semidirect
90. #14519698D4semidirect
99. C4×D4×C61C2×C4semidirect
206. D4×C61C4×D4semidirect
278. C4×C61#417semidirect
295. #1802D61semidirect
340. C61#22431semidirect
371. C4×D4C4×D61semidirect
508. D4#14519739semidirect
577. C4#16839760semidirect

Maximal subgroups

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

subgroupindex
C2×#168404162
C2×#168398392
C2×#168397602
#214487192
#214467412
#214466202
#214462512
#214461252
#214457562
C4×#145196982
C61×#18022
#2243161