Finite Group Databasebeta

← #29696474

Subgroups — #29696474

Order 15616.

1246 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic
2D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
3#21446780C211yesmaximal, characteristic, complemented
4#21446766C211yesmaximal, characteristic, complemented
5#21446257C212yesmaximal, complemented
6#21446134C212yesmaximal, complemented
7#21445915C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9C4×#14519699C211yesmaximal, characteristic, complemented
10#21445539C212yesmaximal, complemented
11C2×C61×#436C211yesmaximal, characteristic, complemented
12#16840477—24no—
13#16840477C418yescomplemented
14C2×#14519777C2211yescharacteristic
15C2×#14519763—22no—
16C2×#14519751C2212yes—
17C2×#14519748—21no—
18C2×C4×D244C2211yescharacteristic
19C2×D4×Dic61—21no—
20C2×#14519736C2212yes—
21C2×#14519735—21no—
22C22×C4⋊Dic61C2212yes—
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2212yes—
26#16839661—21no—
27#16839637—22no—
28#16839632C2211yescharacteristic
29#16839622C2212yes—
30C4×C4⋊Dic61C2211yescharacteristic
31#16839525C2211yescharacteristic
32#16839525C2211yescharacteristic
33#16839525C2212yes—
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38C22×M16×C61C2212yes—
39C2×C4×D4×C61C2211yescharacteristic
40C61×#436—24no—
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C22×Dic122—21no—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C23×Dic61—22no—
48C22×C4×D61—21no—
49#14519777—22no—
50#14519777C2×C414yescomplemented
51#14519763—216no—
52#14519751—24no—
53#14519751C2×C418yescomplemented
54#14519748—28no—
55C4×D244—22no—
56C4×D244C2×C414yescomplemented
57D4×Dic61—44no—
58#14519736—24no—
59#14519736C2×C418yescomplemented
60#14519735—28no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—24no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—42no—
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C22⋊Dic61—22no—
73C2×C22⋊Dic61—22no—
74C2×#11758198—22no—
75C2×#11758198—21no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—21no—
78#14519699—22no—
79#14519699—22no—
80#14519699—42no—
81#14519684—21no—
82#14519684—22no—
83C22×D4×C61D411yescharacteristic
84C23×C4×C61C2312yes—
85C2×M16×C61—24no—
86C2×M16×C61—24no—
87C2×M16×C61—42no—
88C22×C8×C61C2312yes—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C4×D4×C61C2×C414yes—
92C4×D4×C61D414yes—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C22⋊C4×C61D412yes—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—22no—
97C22⋊C8×C61—24no—
98C4×C8×C61—22no—
99C24×C61C2×D412yes—
100C23×D61—21no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61C22⋊C414yes—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61C22×C414yes—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D414yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2411yescharacteristic
114C2×C22⋊D61—28no—
115C2×C22⋊D61—82no—
116C2×C22⋊D61—82no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—21no—
122C22×Dic61—44no—
123C22×Dic61—44no—
124C22×Dic61—82no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244C22×C412yes—
131C2×C4×D61—24no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×Dic122—24no—
135C2×Dic122—81no—
136C2×Dic122—81no—
137M16×C61—48no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—21no—
144C2×C8×C61—22no—
145C2×C8×C61—21no—
146C2×C8×C61#4112yes—
147C4⋊C4×C61C22⋊C414yes—
148C22⋊C4×C61C22⋊C418yes—
149C42×C61C22×C412yes—
150C42×C61C2×D412yes—
151C22⋊Dic61—28no—
152C22⋊Dic61—44no—
153#11758198—28no—
154#11758198—24no—
155C4⋊Dic61—24no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—44no—
164C4⋊Dic61—42no—
165C4⋊Dic61—81no—
166C4⋊Dic61C22×C412yes—
167C4⋊Dic61C22×C414yes—
168C4⋊Dic61#4112yes—
169C4×Dic61—28no—
170C4×Dic61—42no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61C2×C22⋊C414yes—
178C23×C61C22×D411yescharacteristic
179C22×D61—22no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—41no—
186D4×C61—28no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—24no—
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—24no—
195C2×C4×C61C2×C22⋊C414yes—
196C2×C4×C61C2×C22⋊C412yes—
197C2×C4×C61C2×C22⋊C412yes—
198C2×C4×C61AGL(1,ℤ/8)12yes—
199C2×C4×C61#13712yes—
200C2×C4×C61C23×C411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61H(5,𝔽2)11yescharacteristic
205C2×C4×C612−1+411yescharacteristic
206C22⋊D61—88no—
207C2×Dic61—22no—
208C2×Dic61—24no—
209C2×Dic61—22no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—41no—
222C2×Dic61—88no—
223D244—22no—
224D244—22no—
225D244—41no—
226D244—41no—
227C4×D61—48no—
228Dic122—84no—
229C8×C61—22no—
230C8×C61—42no—
231C8×C61—41no—
232#22503—611nomaximal, Sylow
233C22×C61—24no—
234C22×C61—24no—
235C22×C61—24no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61C22×C22⋊C411yescharacteristic
241C22×C61#54611yescharacteristic
242C22×C61#54711yescharacteristic
243C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
244C22×C61C2×#13711yescharacteristic
245C22×C61#57611yescharacteristic
246C22×C61#57711yescharacteristic
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—81no—
251C2×D61—81no—
252C2×D61—81no—
253C2×D61—81no—
254C4×C61—24no—
255C4×C61—22no—
256C4×C61#41714yes—
257C4×C61C4×#4111yescharacteristic
258C4×C61#52011yescharacteristic
259C4×C61#54611yescharacteristic
260C4×C61#54711yescharacteristic
261Dic61—42no—
262Dic61—44no—
263Dic61—42no—
264Dic61—81no—
265Dic61—82no—
266C2×#555—611no—
267C2×#436D6111yescharacteristic, complemented
268D4×C4⋊C4—611no—
269#1990—611no—
270#1987—611no—
271#1895—612no—
272#1874—612no—
273#1825—612no—
274#1802—611no—
275C4×#102—611no—
276#1736—612no—
277C2×C61—44no—
278C2×C61#233911yescharacteristic
279C2×C61C2×#41711yescharacteristic
280C2×C61#289711yescharacteristic
281C2×C61#332511yescharacteristic
282C2×C61#332911yescharacteristic
283C2×C61#335411yescharacteristic
284C2×C61#335711yescharacteristic
285D61—82no—
286C22×M16C2×D6112yes—
287#555—618no—
288#555—1224no—
289C2×C4⋊Q8—611no—
290C2×#124—1222no—
291C2×#121—612no—
292C2×#121—1221no—
293C2×C4×D4—611no—
294C2×C4×D4—1221no—
295C2×C4×D4C2×D6111yescharacteristic
296C2×#117—612no—
297C2×#117—1221no—
298C22×C4⋊C4—612no—
299#436—24no—
300C2×C4⋊C8C2×D6111yescharacteristic
301C2×#102—611no—
302C2×#102—611no—
303C2×#102—612no—
304C2×C22⋊C8C2×D6112yes—
305C2×C4×C8C2×D6111yescharacteristic
306#389—1221no—
307#385—1222no—
308#384—611no—
309#380—612no—
310C4×C4⋊C4—611no—
311#340—612no—
312#340—611no—
313#340—612no—
314#340—612no—
315#340—612no—
316#340—612no—
317#340—611no—
318C61#2250311yescharacteristic, complemented, Sylow
319C22×Q8—1221no—
320C22×D4—611no—
321C22×D4—1222no—
322C22×D4D24411yescharacteristic
323C23×C4—1221no—
324C23×C4—1222no—
325C23×C4C22×D6112yes—
326C2×M16—24no—
327C2×M16—24no—
328C2×M16—42no—
329C22×C8C22×D6112yes—
330C22×C8C22×D6111yescharacteristic
331C22×C8C22×D6111yescharacteristic
332C4⋊Q8—614no—
333C4⋊Q8—1222no—
334#124—12216no—
335#121—618no—
336#121—1228no—
337#121—1224no—
338C4×D4—614no—
339C4×D4—1222no—
340C4×D4—2444no—
341C4×D4C4×D6114yes—
342C4×D4C22⋊D6114yes—
343#117—618no—
344#117—1224no—
345#117—1228no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—2442no—
357C2×C4⋊C4D24411yescharacteristic
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1221no—
361C2×C22⋊C4—1222no—
362C2×C22⋊C4D24412yes—
363C2×C42—1222no—
364C2×C42—1221no—
365C2×C42C22×D6111yescharacteristic
366C4⋊C8—22no—
367#102—1222no—
368#102—1222no—
369#102—2442no—
370C22⋊C8—24no—
371C4×C8—22no—
372#95—1221no—
373#95—1222no—
374C24—1221no—
375C24C2×D24412yes—
376C2×Q8—1224no—
377C2×Q8—4881no—
378C2×Q8—4881no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—612no—
382C2×D4—1228no—
383C2×D4—1221no—
384C2×D4—1221no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—4882no—
389C2×D4—4882no—
390C2×D4#1175819814yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1222no—
398C22×C4—1224no—
399C22×C4—1221no—
400C22×C4—1222no—
401C22×C4—4882no—
402C22×C4—2444no—
403C22×C4—2444no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4C2×C4×D6114yes—
407C22×C4C2×D24412yes—
408C22×C4C2×D24411yescharacteristic
409C22×C4C2×D24411yescharacteristic
410C22×C4C2×C22⋊D6114yes—
411C22×C4C23×D6111yescharacteristic
412M16—48no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8#1175822412yes—
422C4⋊C4—612no—
423C4⋊C4—614no—
424C4⋊C4—612no—
425C4⋊C4—1221no—
426C4⋊C4—1222no—
427C4⋊C4—1221no—
428C4⋊C4—1224no—
429C4⋊C4—1222no—
430C4⋊C4—1221no—
431C4⋊C4—1222no—
432C4⋊C4—1221no—
433C4⋊C4—4881no—
434C4⋊C4—2444no—
435C4⋊C4—2442no—
436C4⋊C4#1175819814yes—
437C22⋊C4—1228no—
438C22⋊C4—1228no—
439C22⋊C4—1224no—
440C22⋊C4—2444no—
441C22⋊C4#1175819818yes—
442C42—1228no—
443C42—2442no—
444C42C2×C4×D6112yes—
445C42C2×C22⋊D6112yes—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—1222no—
453C23—2441no—
454C23—2441no—
455C23—2441no—
456C23—2441no—
457C23—2441no—
458C23—2441no—
459C23C2×#1175819814yes—
460C23C22×D24411yescharacteristic
461Q8—4884no—
462D4—28no—
463D4—1222no—
464D4—1222no—
465D4—4888no—
466D4—2441no—
467D4—2441no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—1222no—
477C2×C4—1224no—
478C2×C4—1222no—
479C2×C4—4888no—
480C2×C4—2441no—
481C2×C4—2442no—
482C2×C4—2442no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2448no—
493C2×C4C2×#1175819814yes—
494C2×C4C2×#1175819812yes—
495C2×C4C2×#1175819812yes—
496C2×C4#1451980512yes—
497C2×C4#1451981012yes—
498C2×C4C22×C4×D6111yescharacteristic
499C2×C4C22×C22⋊D6111yescharacteristic
500C2×C4#41×D6111yescharacteristic
501C2×C4C2×#1175822411yescharacteristic
502C2×C4#1451982211yescharacteristic
503C2×C4#1451982411yescharacteristic
504C8—22no—
505C8—42no—
506C8—41no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—4881no—
515C22—4881no—
516C22—4881no—
517C22—4881no—
518C22—2442no—
519C22—2442no—
520C22—2442no—
521C22C22×#1175819811yescharacteristic
522C22#1684040711yescharacteristic
523C22#1684041611yescharacteristic
524C22C2×#1451980511yescharacteristic
525C22C2×#1451981011yescharacteristic
526C22#1684056711yescharacteristic
527C22#1684057411yescharacteristic
528C4—22no—
529C4—24no—
530C4—4882no—
531C4—4881no—
532C4—2444no—
533C4—2442no—
534C4—2442no—
535C4#1683976414yes—
536C4C4×#1175822411yescharacteristic
537C4#1684024111yescharacteristic
538C4#1684040611yescharacteristic
539C4#1684041111yescharacteristic
540C2—44no—
541C2—4882no—
542C2#2144864411yescharacteristic
543C2C2×#1683976411yescharacteristic
544C2#2145434611yescharacteristic
545C2#2145806211yescharacteristic
546C2#2145811411yescharacteristic
547C2#2145841511yescharacteristic
548C2#2145844611yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#436C2
socleC23×C61C22×D4
prime radicalC2×#16840477C2
cosocleC22×C4×C61C24
hypercenterC2×#436D61
hypocenterC61#22503
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
2. D4×C4⋊Dic61C2semidirect
3. #21446780C2semidirect
4. #21446766C2semidirect
5. #21446257C2semidirect
6. #21446134C2semidirect
7. #21445915C2semidirect
8. #21445820C2semidirect
9. C4×#14519699C2semidirect
10. #21445539C2semidirect
11. C2×C61×#436C2semidirect
13. #16840477C4semidirect
50. #14519777C2×C4semidirect
53. #14519751C2×C4semidirect
56. C4×D244C2×C4semidirect
59. #14519736C2×C4semidirect
267. C2×#436D61semidirect
318. C61#22503semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
D4×C4⋊Dic612
#214467802
#214467662
#214462572
#214461342
#214459152
#214458202
C4×#145196992
#214455392
C2×C61×#4362
#2250361