Finite Group Databasebeta

← #29716395

Subgroups — #29716395

Order 15616.

1146 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×#16840469C211yesmaximal, characteristic, complemented
2C2×#16840052C211yesmaximal, characteristic, complemented
3#21450741C211yesmaximal, characteristic, complemented
4#21447155C212yesmaximal, complemented
5#21446963C212yesmaximal, complemented
6#21446780C211yesmaximal, characteristic, complemented
7#21446755C211yesmaximal, characteristic, complemented
8#21446169C212yesmaximal
9#21446078C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C4×D8×C61C211yesmaximal, characteristic
12#16840469—24no—
13#16840469—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519768C2212yes—
16C2×#14519756—21no—
17C2×#14519756—21no—
18C2×#14519754—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61C2212yes—
21C22×C4⋊Dic61—21no—
22#16840052—24no—
23C2×#14519708C2212yes—
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519698C2212yes—
26C2×#14519698C2212yes—
27#16839685—22no—
28#16839665C2212yes—
29#16839658C2212yes—
30#16839632C2211yescharacteristic
31#16839622—21no—
32#16839525C2212yes—
33#16839525C2212yes—
34#16839524C2212yes—
35#16839516C2212yes—
36#16839516C2212yes—
37C22×D8×C61C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C4×D8×C61—24no—
41C2×#107×C61C2211yescharacteristic
42C2×#102×C61C2211yescharacteristic
43C2×#102×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122D411yescharacteristic
47C23×Dic61—22no—
48C22×#92418C2312yes—
49#14519777—22no—
50#14519777D414yes—
51#14519768—24no—
52#14519768—24no—
53#14519756—28no—
54#14519756—44no—
55#14519754—216no—
56C4×Dic122—22no—
57C4×Dic122—22no—
58D4×Dic61—24no—
59D4×Dic61—24no—
60C2×C4⋊Dic61—28no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2312yes—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196D412yes—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×C4×Dic61C2312yes—
75#14519708—24no—
76C4⋊#92418—22no—
77#14519698—24no—
78#14519698—24no—
79#14519684—22no—
80C22×D4×C61C2311yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—21no—
83C23×C4×C61—21no—
84C2×D8×C61—22no—
85C2×D8×C61—22no—
86C2×D8×C61Q814yescomplemented
87C22×C8×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61D414yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#107×C61—22no—
99#102×C61—22no—
100#102×C61—22no—
101C4×C8×C61—22no—
102C24×C61—21no—
103C24×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C2×D412yes—
113C2×D4×C61C2×Q812yes—
114C2×D4×C61C2×Q812yes—
115C2×D4×C61#4112yes—
116C22×C4×C61—21no—
117C22×C4×C61—24no—
118C22×C4×C61—21no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—22no—
128C22×Dic61—28no—
129C22×Dic61—22no—
130C22×Dic61—82no—
131C22×Dic61—82no—
132C22×Dic61C2×D412yes—
133C22×Dic61C2×D412yes—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—24no—
138C2×Dic122—81no—
139C2×Dic122—81no—
140D8×C61—28no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61C2×Q812yes—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D412yes—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—42no—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155C22⋊Dic61—28no—
156C22⋊Dic61—44no—
157C22⋊Dic61—82no—
158C4⋊Dic61—28no—
159C4⋊Dic61—22no—
160C4⋊Dic61—24no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—42no—
165C4⋊Dic61—81no—
166C4⋊Dic61C2×D414yes—
167#11758196—28no—
168#11758196—24no—
169#11758196—44no—
170C4×Dic61—22no—
171C4×Dic61—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61C22×D411yescharacteristic
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—22no—
193D4×C61—22no—
194D4×C61—22no—
195D4×C61—41no—
196D4×C61—41no—
197D4×C61C22⋊Q814yescomplemented
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—48no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—84no—
208C2×C4×C61#12012yes—
209C2×C4×C61#12012yes—
210C2×C4×C61#13512yes—
211C2×C4×C61AGL(1,ℤ/8)12yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×Q811yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C612−1+411yescharacteristic
217C2×C4×C612−1+411yescharacteristic
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—24no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—88no—
231Dic122—44no—
232Dic122—84no—
233C8×C61—22no—
234C8×C61—41no—
235#92418—42no—
236#24327—611nomaximal, Sylow
237C22×C61—24no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61C2×#12011yescharacteristic
252C22×C61#54811yescharacteristic
253C22×C61D4×Q811yescharacteristic
254C22×C61C2×#13511yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57611yescharacteristic
257C22×C61#57711yescharacteristic
258C4×C61—22no—
259C4×C61—42no—
260C4×C61—81no—
261C4×C61#44914yes—
262C4×C61C2×C22⋊Q811yescharacteristic
263C4×C61#53611yescharacteristic
264C4×C61#54811yescharacteristic
265C4×C61D4×Q811yescharacteristic
266Dic61—42no—
267Dic61—44no—
268Dic61—42no—
269Dic61—82no—
270C2×#554—611no—
271C2×#476—611no—
272C2×C4×D8D6111yescharacteristic
273#2538—611no—
274#2059—612no—
275#2025—612no—
276#1990—611no—
277#1984—611no—
278#1879—612no—
279#1864—612no—
280#1802—611no—
281C2×C61—44no—
282C2×C61—82no—
283C2×C61#240911yescharacteristic
284C2×C61C2×#44911yescharacteristic
285C2×C61#301211yescharacteristic
286C2×C61#336011yescharacteristic
287C2×C61#336611yescharacteristic
288C2×C61#338011yescharacteristic
289C2×C61#338411yescharacteristic
290C22×D8C2×D6111yescharacteristic
291#554—1224no—
292#554—1224no—
293C2×C4⋊Q8—611no—
294C2×#125—612no—
295C2×C22⋊Q8—1222no—
296C2×C22⋊Q8—1221no—
297C2×C22⋊Q8—1221no—
298C2×C4×Q8—611no—
299C2×C4×D4—612no—
300C2×C4×D4C2×D6111yescharacteristic
301C2×C4×D4C2×D6111yescharacteristic
302C22×C4⋊C4—1221no—
303#476—1224no—
304C4×D8—24no—
305C2×#107C2×D6111yescharacteristic
306C2×#106—612no—
307C2×C4⋊C8—611no—
308C2×#102—612no—
309C2×#102—612no—
310C2×#102C2×D6111yescharacteristic
311C2×#102C2×D6111yescharacteristic
312C2×C4×C8C2×D6111yescharacteristic
313#396—1222no—
314#390—612no—
315#389—612no—
316#384—611no—
317#380—1221no—
318#340—612no—
319#340—612no—
320#340—612no—
321#336—612no—
322#336—612no—
323C61#2432711yescharacteristic, complemented, Sylow
324C22×Q8—611no—
325C22×Q8—1221no—
326C22×D4C22×D6111yescharacteristic
327C22×D4C22×D6111yescharacteristic
328C23×C4—21no—
329C23×C4—21no—
330C23×C4—1222no—
331C2×D8—22no—
332C2×D8—22no—
333C2×D8Dic12214yescomplemented
334C22×C8—612no—
335C22×C8C22×D6111yescharacteristic
336C22×C8C22×D6111yescharacteristic
337C4⋊Q8—614no—
338C4⋊Q8—1222no—
339#125—1224no—
340#125—1224no—
341C22⋊Q8—12216no—
342C22⋊Q8—1228no—
343C22⋊Q8—2444no—
344C4×Q8—1222no—
345C4×Q8—1222no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—1224no—
350C4×D4—1224no—
351C4×D4C22⋊D6114yes—
352C2×C4⋊C4—612no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1228no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4C22×D6111yescharacteristic
364C2×C4⋊C4C22×D6111yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1222no—
370C2×C42—612no—
371C2×C42C22×D6111yescharacteristic
372#107—22no—
373#106—1224no—
374C4⋊C8—1222no—
375#102—22no—
376#102—22no—
377#102—1224no—
378#102—1224no—
379C4×C8—22no—
380#95—1222no—
381C24—21no—
382C24—21no—
383C2×Q8—1224no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—4881no—
388C2×Q8—4881no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4C2×Dic12212yes—
398C2×D4C2×Dic12212yes—
399C2×D4#1175822212yes—
400C2×D4C2×C22⋊D6112yes—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—1228no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—4882no—
415C22×C4—4882no—
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419D8—28no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—1222no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8C2×Dic12212yes—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—614no—
433C4⋊C4—1228no—
434C4⋊C4—1228no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—1224no—
438C4⋊C4—1224no—
439C4⋊C4—1222no—
440C4⋊C4—1222no—
441C4⋊C4—4881no—
442C4⋊C4—2444no—
443C4⋊C4—2442no—
444C4⋊C4C2×C22⋊D6112yes—
445C22⋊C4—24no—
446C22⋊C4—42no—
447C22⋊C4—1228no—
448C22⋊C4—4882no—
449C22⋊C4—2444no—
450C42—1222no—
451C42—1222no—
452C42D4×D6112yes—
453C42C2×C22⋊D6112yes—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23C2×D4×D6111yescharacteristic
469Q8—4884no—
470Q8—2444no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—22no—
475D4—22no—
476D4—41no—
477D4—41no—
478D4#1451975614yescomplemented
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—48no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—84no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1224no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—4888no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451974512yes—
503C2×C4#1451974512yes—
504C2×C4#1451979612yes—
505C2×C4#1451980112yes—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C22×C22⋊D6111yescharacteristic
508C2×C4C22×Dic12211yescharacteristic
509C2×C4C2×#1175822211yescharacteristic
510C2×C4#1451982411yescharacteristic
511C2×C4#1451982411yescharacteristic
512C8—22no—
513C8—41no—
514C8—2442no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22C2×#1451974511yescharacteristic
530C22#1684042111yescharacteristic
531C22D4×Dic12211yescharacteristic
532C22C2×#1451979611yescharacteristic
533C22C2×#1451980111yescharacteristic
534C22#1684056411yescharacteristic
535C22#1684056911yescharacteristic
536C4—22no—
537C4—42no—
538C4—81no—
539C4—4882no—
540C4—2442no—
541C4—2444no—
542C4—2442no—
543C4#1683990414yes—
544C4C2×#1451975611yescharacteristic
545C4#1684033311yescharacteristic
546C4#1684042111yescharacteristic
547C4D4×Dic12211yescharacteristic
548C2—44no—
549C2—82no—
550C2#2144918411yescharacteristic
551C2C2×#1683990411yescharacteristic
552C2#2145511911yescharacteristic
553C2#2145846611yescharacteristic
554C2#2145852411yescharacteristic
555C2#2145868411yescharacteristic
556C2#2145871611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×D8×C61C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×C4×D8D61
hypocenterC61#24327
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16840469C2semidirect
2. C2×#16840052C2semidirect
3. #21450741C2semidirect
4. #21447155C2semidirect
5. #21446963C2semidirect
6. #21446780C2semidirect
7. #21446755C2semidirect
9. #21446078C2semidirect
10. #21445819C2semidirect
86. C2×D8×C61Q8semidirect
197. D4×C61C22⋊Q8semidirect
323. C61#24327semidirect
333. C2×D8Dic122semidirect
478. D4#14519756semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
C2×#168400522
#214507412
#214471552
#214469632
#214467802
#214467552
#214461692
#214460782
#214458192
C2×C4×D8×C612
#2432761