Finite Group Databasebeta

← #29716393

Subgroups — #29716393

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×#16840473C211yesmaximal, characteristic, complemented
2C2×#16840049C211yesmaximal, characteristic, complemented
3C2×C4×D488C211yesmaximal, characteristic
4#21450742C211yesmaximal, characteristic, complemented
5#21447160C212yesmaximal, complemented
6#21446964C212yesmaximal, complemented
7#21446755C211yesmaximal, characteristic, complemented
8#21446168C212yesmaximal
9#21446081C212yesmaximal, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1990C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13#16840473—24no—
14#16840473—24no—
15C2×#14519768C2212yes—
16C2×#14519757—21no—
17C2×#14519757—21no—
18C2×#14519753—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519739C2212yes—
23C2×C4⋊C4×D61—21no—
24#16840049—24no—
25C4×D488—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519708C2212yes—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2212yes—
32C2×#14519697C2212yes—
33#16839684—22no—
34#16839663C2212yes—
35#16839661C2212yes—
36#16839619—21no—
37#16839524C2212yes—
38#16839516C2212yes—
39#16839516C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C61×#384C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C22×Dic122D411yescharacteristic
46C22×D244C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C2×D488—22no—
52C2×D488—22no—
53C2×D488Q814yescomplemented
54C22×#92418C2312yes—
55#14519768—24no—
56#14519768—24no—
57#14519757—28no—
58#14519757—44no—
59#14519753—216no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62C4×D244—22no—
63C4×D244—22no—
64C4×D244—22no—
65C4×D244D414yes—
66#14519739—24no—
67#14519739—24no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—22no—
74C2×#11758196D412yes—
75C4⋊C4×D61—28no—
76C2×#11758198—21no—
77C2×#11758198—22no—
78C2×#11758198—21no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×C4×Dic61C2312yes—
82#14519711—22no—
83#14519708—24no—
84C4⋊#92418—22no—
85#14519699—22no—
86#14519699—22no—
87#14519697—24no—
88#14519697—24no—
89#14519684—22no—
90C22×Q8×C61—21no—
91C22×C8×C61C2311yescharacteristic
92C22×C8×C61C2311yescharacteristic
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61D414yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101C4×C8×C61—22no—
102C23×D61—21no—
103C23×D61—21no—
104C2×Q8×C61—24no—
105C2×Q8×C61—81no—
106C2×Q8×C61—81no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61C2×D412yes—
115C22×Dic61C2×D412yes—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244C2×D412yes—
125C2×D244C2×Q812yes—
126C2×D244C2×Q812yes—
127C2×D244#4112yes—
128C2×C4×D61—24no—
129C2×C4×D61—28no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×C4×D61—82no—
136C2×C4×D61—82no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61C2×Q812yes—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—81no—
152C4⋊C4×C61C2×D414yes—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155#11758198—28no—
156#11758198—24no—
157#11758198—44no—
158#11758198—42no—
159#11758198—82no—
160C4⋊Dic61—28no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—44no—
166C4⋊Dic61C2×D412yes—
167#11758196—28no—
168#11758196—24no—
169C4×Dic61—22no—
170C4×Dic61—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174D488—28no—
175C23×C61C22×D411yescharacteristic
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—22no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190Q8×C61—84no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—24no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61#12012yes—
200C2×C4×C61#12012yes—
201C2×C4×C61#13512yes—
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×Q811yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C2×C4×C612−1+411yescharacteristic
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221D244—22no—
222D244—22no—
223D244—22no—
224D244—22no—
225D244—22no—
226D244—41no—
227D244—41no—
228D244C22⋊Q814yescomplemented
229C4×D61—48no—
230C4×D61—88no—
231C4×D61—84no—
232Dic122—44no—
233C8×C61—22no—
234C8×C61—41no—
235#92418—42no—
236#24327—611nomaximal, Sylow
237C22×C61C2×#12011yescharacteristic
238C22×C61#54811yescharacteristic
239C22×C61D4×Q811yescharacteristic
240C22×C61C2×#13511yescharacteristic
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C61#57611yescharacteristic
243C22×C61#57711yescharacteristic
244C2×D61—24no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61—81no—
255C2×D61—81no—
256C2×D61—81no—
257C2×D61—81no—
258C4×C61—22no—
259C4×C61—44no—
260C4×C61—82no—
261C4×C61#44914yes—
262C4×C61C2×C22⋊Q811yescharacteristic
263C4×C61#53611yescharacteristic
264C4×C61#54811yescharacteristic
265C4×C61D4×Q811yescharacteristic
266Dic61—42no—
267Dic61—42no—
268Dic61—42no—
269Dic61—81no—
270C2×#554—611no—
271C2×#476—611no—
272C2×C4×D8—611no—
273#2538—611no—
274#2059—612no—
275#2025—612no—
276#1990D6111yescharacteristic, complemented
277#1984—611no—
278#1879—612no—
279#1864—612no—
280#1802—611no—
281C2×C61#240911yescharacteristic
282C2×C61C2×#44911yescharacteristic
283C2×C61#301211yescharacteristic
284C2×C61#336011yescharacteristic
285C2×C61#336611yescharacteristic
286C2×C61#338011yescharacteristic
287C2×C61#338411yescharacteristic
288D61—44no—
289D61—82no—
290C22×D8—611no—
291#554—1224no—
292#554—1224no—
293C2×C4⋊Q8C2×D6111yescharacteristic
294C2×#125—612no—
295C2×C22⋊Q8—1221no—
296C2×C22⋊Q8—1221no—
297C2×C22⋊Q8—1222no—
298C2×C4×Q8—611no—
299C2×C4×D4—612no—
300C2×C4×D4—611no—
301C2×C4×D4—611no—
302C22×C4⋊C4—1221no—
303#476—1224no—
304C4×D8—1224no—
305C2×#107—611no—
306C2×#106—612no—
307C2×C4⋊C8—611no—
308C2×#102—611no—
309C2×#102—611no—
310C2×#102—612no—
311C2×#102—612no—
312C2×C4×C8C2×D6111yescharacteristic
313#396—1222no—
314#390—612no—
315#389—612no—
316#384C2×D6111yescharacteristic
317#380—1221no—
318#340—612no—
319#340C2×D6112yes—
320#340C2×D6112yes—
321#336—612no—
322#336—612no—
323C61#2432711yescharacteristic, complemented, Sylow
324C22×Q8—21no—
325C22×Q8—611no—
326C22×D4—611no—
327C22×D4—611no—
328C23×C4—1222no—
329C23×C4—1221no—
330C23×C4—1221no—
331C2×D8—614no—
332C2×D8—1222no—
333C2×D8—1222no—
334C22×C8—612no—
335C22×C8C22×D6111yescharacteristic
336C22×C8C22×D6111yescharacteristic
337C4⋊Q8—22no—
338C4⋊Q8D24414yes—
339#125—1224no—
340#125—1224no—
341C22⋊Q8—1228no—
342C22⋊Q8—12216no—
343C22⋊Q8—2444no—
344C4×Q8—1222no—
345C4×Q8—1222no—
346C4×D4—614no—
347C4×D4—1224no—
348C4×D4—1224no—
349C4×D4—1222no—
350C4×D4—1222no—
351C4×D4—1222no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—1228no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4C22×D6112yes—
364C2×C4⋊C4C22×D6112yes—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4—1221no—
370C2×C42—612no—
371C2×C42C22×D6111yescharacteristic
372#107—1222no—
373#106—1224no—
374C4⋊C8—1222no—
375#102—1224no—
376#102—1222no—
377#102—1222no—
378#102—1224no—
379C4×C8—22no—
380#95—1222no—
381C24—1221no—
382C24—1221no—
383C2×Q8—24no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×D4—612no—
390C2×D4—612no—
391C2×D4—612no—
392C2×D4—612no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1221no—
400C2×D4—1221no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—612no—
404C22×C4—612no—
405C22×C4—1228no—
406C22×C4—1221no—
407C22×C4—1224no—
408C22×C4—1221no—
409C22×C4—4882no—
410C22×C4—4882no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4—2442no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4D4×D6111yescharacteristic
417C22×C4D4×D6111yescharacteristic
418C22×C4C23×D6111yescharacteristic
419D8—1228no—
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×C8Q8×D6112yes—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—42no—
434C4⋊C4—81no—
435C4⋊C4—612no—
436C4⋊C4—1228no—
437C4⋊C4—1228no—
438C4⋊C4—1221no—
439C4⋊C4—1222no—
440C4⋊C4—1224no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—2444no—
444C4⋊C4C2×D24414yes—
445C22⋊C4—1228no—
446C22⋊C4—1224no—
447C22⋊C4—4882no—
448C22⋊C4—2444no—
449C22⋊C4—2442no—
450C42—1222no—
451C42—1222no—
452C42C2×D24412yes—
453C42D4×D6112yes—
454C23—1222no—
455C23—1222no—
456C23—1222no—
457C23—1222no—
458C23—2441no—
459C23—2441no—
460C23—2441no—
461C23—2441no—
462C23—2441no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23C2×D4×D6111yescharacteristic
469Q8—84no—
470Q8—2444no—
471D4—614no—
472D4—1222no—
473D4—1222no—
474D4—1222no—
475D4—1222no—
476D4—1222no—
477D4—2441no—
478D4—2441no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—4888no—
495C2×C4—4884no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2448no—
502C2×C4#1451974712yes—
503C2×C4#1451974712yes—
504C2×C4#1451980012yes—
505C2×C4#1451980412yes—
506C2×C4C2×D4×D6111yescharacteristic
507C2×C4C22×D24411yescharacteristic
508C2×C4C2×Q8×D6111yescharacteristic
509C2×C4#41×D6111yescharacteristic
510C2×C4#1451982411yescharacteristic
511C2×C4#1451982511yescharacteristic
512C8—22no—
513C8—41no—
514C8—2442no—
515C22—1224no—
516C22—4881no—
517C22—4881no—
518C22—4881no—
519C22—4881no—
520C22—2442no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22—2442no—
528C22—2442no—
529C22C2×#1451974711yescharacteristic
530C22#1684042011yescharacteristic
531C22#1684042711yescharacteristic
532C22C2×#1451980011yescharacteristic
533C22C2×#1451980411yescharacteristic
534C22#1684056311yescharacteristic
535C22#1684056811yescharacteristic
536C4—22no—
537C4—44no—
538C4—82no—
539C4—4881no—
540C4—2442no—
541C4—2442no—
542C4—2442no—
543C4#1683991014yes—
544C4C22⋊Q8×D6111yescharacteristic
545C4#1684033511yescharacteristic
546C4#1684042411yescharacteristic
547C4Q8×D24411yescharacteristic
548C2—4882no—
549C2—2444no—
550C2#2144918211yescharacteristic
551C2C2×#1683991011yescharacteristic
552C2#2145512811yescharacteristic
553C2#2145847311yescharacteristic
554C2#2145853111yescharacteristic
555C2#2145868811yescharacteristic
556C2#2145872311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radicalC22×D488C22
hypercenter#1990D61
hypocenterC61#24327
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
1. C2×#16840473C2semidirect
2. C2×#16840049C2semidirect
4. #21450742C2semidirect
5. #21447160C2semidirect
6. #21446964C2semidirect
7. #21446755C2semidirect
9. #21446081C2semidirect
10. #21445816C2semidirect
11. C61×#1990C2semidirect
53. C2×D488Q8semidirect
228. D244C22⋊Q8semidirect
276. #1990D61semidirect
323. C61#24327semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
C2×#168400492
C2×C4×D4882
#214507422
#214471602
#214469642
#214467552
#214461682
#214460812
#214458162
C61×#19902
#2432761