Finite Group Databasebeta

← #29696851

Subgroups — #29696851

Order 15616.

1366 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840434C211yesmaximal, characteristic
2#21448776C211yesmaximal, characteristic, complemented
3#21446745C212yesmaximal, complemented
4#21446745C211yesmaximal, characteristic, complemented
5#21446643C212yesmaximal, complemented
6#21446625C211yesmaximal, characteristic, complemented
7#21445830C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9#21445645C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic, complemented
11C2×C61×#446C211yesmaximal, characteristic, complemented
12C22×#11758226—21no—
13#16840434—24no—
14#16840434C418yescomplemented
15C2×#14519774C2211yescharacteristic
16C2×#14519774C2212yes—
17C2×#14519752—21no—
18C2×#14519752—22no—
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2212yes—
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28#16839669—21no—
29#16839648—22no—
30#16839632C2211yescharacteristic
31#16839628C2212yes—
32C4⋊C4×Dic61—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34C4×C4⋊Dic61C2212yes—
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839525C2212yes—
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#446—24no—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2212yes—
42C2×#97×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C2×#11758226—28no—
45C2×#11758226—44no—
46C2×#11758226—44no—
47C22×D244—22no—
48C22×D244—21no—
49C22×D244—22no—
50C22×D244—21no—
51C22×D244C2311yescharacteristic
52C22×D244C2312yes—
53C22×C4×D61—22no—
54C22×C4×D61—21no—
55#14519774—22no—
56#14519774—24no—
57#14519774C2×C414yes—
58#14519774C2×C418yes—
59#14519752—216no—
60#14519752—28no—
61C4×D244—22no—
62C4×D244—24no—
63C4×D244C2×C414yes—
64C4×D244C2×C418yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2312yes—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758198—22no—
72C2×#11758198—21no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61—22no—
75#14519699—22no—
76#14519699—24no—
77#14519699—22no—
78#14519699—24no—
79#14519699—42no—
80#14519699—42no—
81#14519699—42no—
82#14519684—21no—
83#14519684—22no—
84#14519684—22no—
85#14519684—22no—
86C22×Q8×C61D411yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C22×C8×C61C2312yes—
89C22×C8×C61C2311yescharacteristic
90C4×Q8×C61C2×C414yes—
91C4×Q8×C61D414yes—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D412yes—
94C2×C42×C61C2311yescharacteristic
95C2×C42×C61C2312yes—
96C4⋊C8×C61—22no—
97C4⋊C8×C61—24no—
98#97×C61—24no—
99C4×C8×C61—22no—
100C23×D61—22no—
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61C22⋊C414yes—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—22no—
114#11758226—432no—
115C2×D244—21no—
116C2×D244—22no—
117C2×D244—21no—
118C2×D244—22no—
119C2×D244—21no—
120C2×D244—22no—
121C2×D244—21no—
122C2×D244—22no—
123C2×D244—28no—
124C2×D244—24no—
125C2×D244—21no—
126C2×D244—22no—
127C2×D244—24no—
128C2×D244—28no—
129C2×D244—44no—
130C2×D244—44no—
131C2×D244—42no—
132C2×D244—42no—
133C2×D244—82no—
134C2×D244—81no—
135C2×D244—82no—
136C2×D244—81no—
137C2×D244C22×C412yes—
138C2×D244C22×C414yes—
139C2×C4×D61—28no—
140C2×C4×D61—24no—
141C2×C4×D61—44no—
142C2×C4×D61—44no—
143C2×C4×D61—42no—
144C2×C4×D61—42no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—22no—
149C2×C8×C61—21no—
150C2×C8×C61—22no—
151C2×C8×C61—21no—
152C2×C8×C61—22no—
153C2×C8×C61#4112yescomplemented
154C4⋊C4×C61C22⋊C418yes—
155C4⋊C4×C61C22⋊C414yes—
156C42×C61C22×C414yes—
157C42×C61C22×C412yes—
158C42×C61C2×D412yes—
159C42×C61C2×D414yes—
160#11758198—24no—
161#11758198—28no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—42no—
169C4⋊Dic61—81no—
170C4⋊Dic61C22×C412yes—
171C4⋊Dic61#4112yescomplemented
172C4×Dic61—42no—
173C4×Dic61—82no—
174C23×C61C22×D411yescharacteristic
175C22×D61—24no—
176C22×D61—22no—
177C22×D61—42no—
178C22×D61—41no—
179C22×D61—42no—
180C22×D61—41no—
181C22×D61—42no—
182C22×D61—41no—
183C22×D61—42no—
184C22×D61—41no—
185C22×D61—41no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—41no—
189Q8×C61—28no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—21no—
196C2×C4×C61C2×C22⋊C412yes—
197C2×C4×C61C2×C22⋊C414yes—
198C2×C4×C61C2×C22⋊C414yes—
199C2×C4×C61C2×C22⋊C412yes—
200C2×C4×C61AGL(1,ℤ/8)12yes—
201C2×C4×C61AGL(1,ℤ/8)12yes—
202C2×C4×C61C23×C411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61H(5,𝔽2)11yescharacteristic
207C2×C4×C612−1+411yescharacteristic
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220D244—22no—
221D244—24no—
222D244—22no—
223D244—24no—
224D244—416no—
225D244—41no—
226D244—42no—
227D244—41no—
228D244—42no—
229D244—48no—
230D244—88no—
231D244—84no—
232C4×D61—416no—
233C4×D61—48no—
234C8×C61—22no—
235C8×C61—41no—
236C8×C61—42no—
237#22545—611nomaximal, Sylow
238C22×C61C22×C22⋊C411yescharacteristic
239C22×C61#54811yescharacteristic
240C22×C61#55011yescharacteristic
241C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
243C22×C61#57611yescharacteristic
244C22×C61#57711yescharacteristic
245C2×D61—44no—
246C2×D61—42no—
247C2×D61—44no—
248C2×D61—42no—
249C2×D61—44no—
250C2×D61—42no—
251C2×D61—82no—
252C2×D61—81no—
253C2×D61—82no—
254C2×D61—81no—
255C2×D61—82no—
256C2×D61—81no—
257C2×D61—82no—
258C2×D61—81no—
259C4×C61—24no—
260C4×C61—24no—
261C4×C61—22no—
262C4×C61#41814yes—
263C4×C61C4×#4111yescharacteristic
264C4×C61#52011yescharacteristic
265C4×C61#54811yescharacteristic
266C4×C61#55011yescharacteristic
267Dic61—42no—
268Dic61—42no—
269Dic61—81no—
270Dic61—82no—
271C2×#550—611no—
272C2×#446D6111yescharacteristic, complemented
273#2355—611no—
274#1982—611no—
275#1982—612no—
276#1961—612no—
277#1957—611no—
278#1804—612no—
279#1802—611no—
280#1762—612no—
281C4×#102—611no—
282C2×C61#234111yescharacteristic
283C2×C61C2×#41811yescharacteristic
284C2×C61#289711yescharacteristic
285C2×C61#336611yescharacteristic
286C2×C61#336811yescharacteristic
287C2×C61#339711yescharacteristic
288C2×C61#339911yescharacteristic
289D61—84no—
290D61—82no—
291C22×#41—1221no—
292#550—618no—
293#550—1224no—
294C2×#127—611no—
295C2×#127—612no—
296C2×#121—1221no—
297C2×#121—1222no—
298C2×C4×Q8C2×D6111yescharacteristic
299C2×C4×D4—611no—
300C2×C4×D4—612no—
301#446—24no—
302C2×C4⋊C8C2×D6111yescharacteristic
303C2×C4⋊C8C2×D6112yes—
304C2×#102—612no—
305C2×#102—611no—
306C2×#102—611no—
307C2×#102—612no—
308C2×#102—612no—
309C2×#102—612no—
310C2×#102—612no—
311C2×#97C2×D6112yes—
312C2×C4×C8C2×D6111yescharacteristic
313#391—1221no—
314#387—1222no—
315#384—611no—
316#382—612no—
317C4×C4⋊C4—611no—
318C4×C4⋊C4—612no—
319C4×C4⋊C4—1221no—
320#340—612no—
321#340—611no—
322#340—611no—
323C61#2254511yescharacteristic, complemented, Sylow
324C2×#41—1228no—
325C2×#41—2444no—
326C2×#41—2444no—
327C22×Q8D24411yescharacteristic
328C22×D4—611no—
329C22×D4—612no—
330C22×D4—1222no—
331C22×D4—1221no—
332C22×D4—1222no—
333C22×D4—1221no—
334C23×C4—1222no—
335C23×C4—1221no—
336C22×C8C22×D6111yescharacteristic
337C22×C8C22×D6112yes—
338C22×C8C22×D6111yescharacteristic
339#127—614no—
340#127—618no—
341#127—1222no—
342#127—1224no—
343#121—12216no—
344#121—1228no—
345C4×Q8C4×D6114yes—
346C4×Q8C22⋊D6114yes—
347C4×D4—614no—
348C4×D4—618no—
349C4×D4—1222no—
350C4×D4—1224no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—611no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4D24411yescharacteristic
358C2×C4⋊C4D24412yes—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1221no—
361C2×C42—1222no—
362C2×C42—1221no—
363C2×C42C22×D6112yes—
364C2×C42C22×D6111yescharacteristic
365C4⋊C8—22no—
366C4⋊C8—24no—
367#102—1222no—
368#102—1224no—
369#102—1224no—
370#102—1222no—
371#102—2442no—
372#102—2442no—
373#102—2442no—
374#97—24no—
375C4×C8—22no—
376#95—1221no—
377#95—1222no—
378#95—1222no—
379#95—1222no—
380C24—1222no—
381C24—1221no—
382#41—24432no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8#1175819814yes—
386C2×D4—612no—
387C2×D4—614no—
388C2×D4—1221no—
389C2×D4—1222no—
390C2×D4—1221no—
391C2×D4—1222no—
392C2×D4—1222no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1222no—
396C2×D4—1221no—
397C2×D4—1222no—
398C2×D4—1224no—
399C2×D4—1228no—
400C2×D4—1228no—
401C2×D4—1224no—
402C2×D4—4882no—
403C2×D4—4881no—
404C2×D4—4882no—
405C2×D4—4881no—
406C2×D4—2444no—
407C2×D4—2444no—
408C2×D4—2442no—
409C2×D4—2442no—
410C22×C4—1221no—
411C22×C4—1221no—
412C22×C4—1228no—
413C22×C4—1224no—
414C22×C4—1221no—
415C22×C4—1222no—
416C22×C4—2444no—
417C22×C4—2444no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4C2×D24412yes—
421C22×C4C2×D24412yes—
422C22×C4C2×D24411yescharacteristic
423C22×C4C2×D24411yescharacteristic
424C22×C4C23×D6111yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8#1175822612yescomplemented
434C4⋊C4—612no—
435C4⋊C4—612no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—1221no—
439C4⋊C4—1222no—
440C4⋊C4—1221no—
441C4⋊C4—1222no—
442C4⋊C4—4881no—
443C4⋊C4—2442no—
444C4⋊C4#1175819818yes—
445C4⋊C4#1175819814yes—
446C22⋊C4—1224no—
447C22⋊C4—1228no—
448C42—4882no—
449C42—2442no—
450C42C2×C4×D6114yes—
451C42C2×C4×D6112yes—
452C42C2×C22⋊D6112yes—
453C42C2×C22⋊D6114yes—
454C23—1224no—
455C23—1222no—
456C23—2442no—
457C23—2441no—
458C23—2442no—
459C23—2441no—
460C23—2441no—
461C23—2442no—
462C23—2442no—
463C23—2441no—
464C23—2442no—
465C23—2441no—
466C23—2442no—
467C23—2441no—
468C23C22×D24411yescharacteristic
469Q8—28no—
470D4—1222no—
471D4—1224no—
472D4—1222no—
473D4—1224no—
474D4—4888no—
475D4—4884no—
476D4—24416no—
477D4—2448no—
478D4—2441no—
479D4—2442no—
480D4—2441no—
481D4—2442no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—24416no—
491C2×C4—2441no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2448no—
502C2×C4C2×#1175819812yes—
503C2×C4C2×#1175819814yes—
504C2×C4C2×#1175819814yes—
505C2×C4C2×#1175819812yes—
506C2×C4#1451980512yes—
507C2×C4#1451980512yes—
508C2×C4C22×C4×D6111yescharacteristic
509C2×C4C22×C22⋊D6111yescharacteristic
510C2×C4#41×D6111yescharacteristic
511C2×C4C2×#1175822611yescharacteristic
512C2×C4#1451982311yescharacteristic
513C2×C4#1451982511yescharacteristic
514C8—22no—
515C8—41no—
516C8—42no—
517C22—4882no—
518C22—4881no—
519C22—4882no—
520C22—4881no—
521C22—4882no—
522C22—4881no—
523C22—4882no—
524C22—4881no—
525C22—2444no—
526C22—2442no—
527C22—2444no—
528C22—2442no—
529C22—2444no—
530C22—2442no—
531C22C22×#1175819811yescharacteristic
532C22#1684042411yescharacteristic
533C22#1684043411yescharacteristic
534C22C2×#1451980511yescharacteristic
535C22C2×#1451980511yescharacteristic
536C22#1684056711yescharacteristic
537C22#1684057411yescharacteristic
538C4—24no—
539C4—22no—
540C4—24no—
541C4—4881no—
542C4—4882no—
543C4—2442no—
544C4—2442no—
545C4#1683976814yes—
546C4C4×#1175822611yescharacteristic
547C4#1684024411yescharacteristic
548C4#1684042311yescharacteristic
549C4#1684043111yescharacteristic
550C2—4884no—
551C2—4882no—
552C2#2144865711yescharacteristic
553C2C2×#1683976811yescharacteristic
554C2#2145434611yescharacteristic
555C2#2145852811yescharacteristic
556C2#2145855011yescharacteristic
557C2#2145884111yescharacteristic
558C2#2145886311yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#446C2
socleC23×C61C22×D4
prime radicalC2×#16840434C2
cosocleC22×C4×C61C24
hypercenterC2×#446D61
hypocenterC61#22545
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #21448776C2semidirect
3. #21446745C2semidirect
4. #21446745C2semidirect
5. #21446643C2semidirect
6. #21446625C2semidirect
7. #21445830C2semidirect
8. #21445820C2semidirect
9. #21445645C2semidirect
10. C4×#14519699C2semidirect
11. C2×C61×#446C2semidirect
14. #16840434C4semidirect
153. C2×C8×C61#41semidirect
171. C4⋊Dic61#41semidirect
272. C2×#446D61semidirect
323. C61#22545semidirect
433. C2×C8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168404342
#214487762
#214467452
#214466432
#214466252
#214458302
#214458202
#214456452
C4×#145196992
C2×C61×#4462
#2254561