Finite Group Databasebeta

← #29719017

Subgroups — #29719017

Order 15616.

1064 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839900C211yesmaximal, characteristic
2C22×#14519702C211yesmaximal, characteristic, complemented
3#21450291C211yesmaximal, characteristic
4#21448651C211yesmaximal, characteristic, complemented
5#21447231C212yesmaximal, complemented
6#21447135C212yesmaximal, complemented
7#21446171C212yesmaximal, complemented
8#21445997C211yesmaximal, characteristic, complemented
9#21445903C211yesmaximal, characteristic, complemented
10#21445823C212yesmaximal, complemented
11C61×#1893C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758200C2211yescharacteristic
14C22×#11758200C2211yescharacteristic
15C2×#14519758—21no—
16C2×#14519758—21no—
17C2×#14519742C2212yes—
18C22×C4⋊Dic61C2211yescharacteristic
19#16839900—24no—
20C2×#14519706C2212yes—
21C2×#14519702—24no—
22C2×#14519702C2212yes—
23C2×#14519702C2212yes—
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27#16839698—22no—
28#16839667C2212yes—
29#16839666—22no—
30#16839665C2212yes—
31#16839631C2212yes—
32#16839620—21no—
33C22⋊C4×Dic61—21no—
34C4×C22⋊Dic61C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38C23×C8×C61C2211yescharacteristic
39C2×#121×C61C2211yescharacteristic
40C2×#102×C61C2212yes—
41C61×#380C2211yescharacteristic
42C61×#340C2212yes—
43C22×Dic122—24no—
44C22×Dic122—21no—
45C22×Dic122—22no—
46C22×Dic122—42no—
47C22×Dic122D414yes—
48C22×Dic122C2311yescharacteristic
49C22×Dic122C2311yescharacteristic
50C23×Dic61—21no—
51C23×Dic61—21no—
52C2×#11758200—22no—
53C2×#11758200—22no—
54C2×#11758200—24no—
55C2×#11758200—42no—
56C22×#92418C2311yescharacteristic
57C22×#92418C2311yescharacteristic
58#14519758—44no—
59#14519758—44no—
60#14519742—24no—
61#14519742—42no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61—22no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2312yes—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C2×#11758196—22no—
69C2×C22⋊Dic61—22no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61D411yescharacteristic
72C2×C22⋊Dic61D411yescharacteristic
73C2×C4×Dic61—22no—
74C2×C4×Dic61C2312yes—
75#14519706—42no—
76#14519702—24no—
77#14519702—24no—
78#14519700—22no—
79#14519700—22no—
80C22⋊#92418—22no—
81#14519684—22no—
82#14519684—22no—
83#14519684D412yes—
84C22×D4×C61—21no—
85C22×D4×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61—22no—
88C22×C8×C61—42no—
89C22×C8×C61C2312yes—
90#121×C61—22no—
91#121×C61—22no—
92C2×C4⋊C4×C61C2312yes—
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96#102×C61—42no—
97#95×C61—21no—
98C24×C61—21no—
99C24×C61C2×D411yescharacteristic
100C2×D4×C61—21no—
101C2×D4×C61—21no—
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61—42no—
105C2×D4×C61—42no—
106C2×D4×C61—42no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D412yes—
113C22×C4×C61#4112yes—
114C22×C4×C61C2411yescharacteristic
115C22×Dic61—21no—
116C22×Dic61—24no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—44no—
120C22×Dic61—42no—
121C22×Dic61—42no—
122C22×Dic61—42no—
123C22×Dic61C2×D412yes—
124C22×Dic61C2×D412yes—
125C2×Dic122—21no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—28no—
130C2×Dic122—24no—
131C2×Dic122—21no—
132C2×Dic122—44no—
133C2×Dic122—48no—
134C2×Dic122—44no—
135C2×Dic122C2×D412yes—
136C2×Dic122C2×D412yes—
137C2×Dic122#4114yes—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—44no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C22⋊C4×C61—42no—
147C22⋊C4×C61—42no—
148C22⋊Dic61—22no—
149C22⋊Dic61—44no—
150C22⋊Dic61—44no—
151C22⋊Dic61SD1614yescomplemented
152#11758200—44no—
153#11758200—44no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—42no—
157#11758196—44no—
158#11758196—44no—
159#11758196—44no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C4×Dic61—44no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—41no—
170C23×C61—42no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—42no—
174C23×C61#12112yes—
175C23×C61C2×SD1612yes—
176C23×C61AGL(1,ℤ/8)12yes—
177C23×C61C22×D411yescharacteristic
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—42no—
181D4×C61—84no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61#12112yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4112yes—
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61H(5,𝔽2)12yes—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—24no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—48no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—84no—
211C2×Dic61C2×SD1614yes—
212Dic122—22no—
213Dic122—24no—
214Dic122—22no—
215Dic122—22no—
216Dic122—48no—
217Dic122—42no—
218Dic122#12314yescomplemented
219C8×C61—42no—
220#92418—41no—
221#92418—41no—
222#24528—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—24no—
225C22×C61—22no—
226C22×C61—24no—
227C22×C61—81no—
228C22×C61—81no—
229C22×C61—82no—
230C22×C61—81no—
231C22×C61—82no—
232C22×C61#46014yes—
233C22×C61C2×#12111yescharacteristic
234C22×C61#54612yes—
235C22×C61C22×SD1611yescharacteristic
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C61#57812yes—
238C4×C61—22no—
239C4×C61—81no—
240C4×C61—82no—
241C4×C61C2×#12311yescharacteristic
242C4×C61#53411yescharacteristic
243C4×C61#55112yes—
244Dic61—24no—
245Dic61—24no—
246Dic61—42no—
247Dic61—42no—
248Dic61—82no—
249C2×#448—611no—
250C22×#103—611no—
251#2501—611no—
252Q8×C22⋊C4—611no—
253#2072—612no—
254#2055—612no—
255#1893D6111yescharacteristic
256#1880—612no—
257#1850—611no—
258#1823—611no—
259#1803—612no—
260C2×C61—24no—
261C2×C61—161no—
262C2×C61#241211yescharacteristic
263C2×C61C2×#46011yescharacteristic
264C2×C61#304411yescharacteristic
265C2×C61#329812yes—
266C2×C61#330812yes—
267C23×Q8—611no—
268C22×SD16—611no—
269C22×SD16—611no—
270C23×C8C2×D6111yescharacteristic
271C2×#123—1221no—
272C2×#123—1221no—
273C2×#121C2×D6111yescharacteristic
274C2×C4×Q8—612no—
275C22×C4⋊C4—611no—
276#448—1224no—
277C2×C4⋊C8—612no—
278C2×#103—612no—
279C2×#103—612no—
280C2×#103—611no—
281C2×#103—611no—
282C2×#103—1224no—
283C2×#102C2×D6112yes—
284C2×C22⋊C8—611no—
285#399—1222no—
286#391—612no—
287#391—1222no—
288#390—612no—
289#384—612no—
290#380—1221no—
291#380C2×D6111yescharacteristic
292C4×C22⋊C4—611no—
293C4×C22⋊C4—1221no—
294#340—612no—
295#340—612no—
296#340C2×D6112yes—
297#336—612no—
298C61#2452811yescharacteristic, complemented, Sylow
299C22×Q8—614no—
300C22×Q8—611no—
301C22×Q8—611no—
302C22×Q8—1224no—
303C22×Q8—1222no—
304C22×Q8—1221no—
305C22×Q8—2442no—
306C22×D4—21no—
307C22×D4C22×D6111yescharacteristic
308C23×C4—1221no—
309C23×C4—1221no—
310C23×C4C22×D6111yescharacteristic
311C2×SD16—1222no—
312C2×SD16—1222no—
313C2×SD16—1224no—
314C2×SD16—2442no—
315C22×C8—22no—
316C22×C8—42no—
317C22×C8—611no—
318C22×C8—611no—
319C22×C8C22×D6112yes—
320#123—2444no—
321#123—2444no—
322#121—22no—
323#121—22no—
324C4×Q8—1224no—
325C4×Q8—2442no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—1222no—
328C2×C4⋊C4—1222no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4C22×D6112yes—
334C2×C4⋊C4C22×D6111yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—611no—
338C2×C22⋊C4—611no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C42—612no—
342C2×C42—1222no—
343C4⋊C8—2442no—
344#103—1222no—
345#103—1222no—
346#103—1224no—
347#103—1224no—
348#102—42no—
349C22⋊C8—1222no—
350#95—21no—
351#95—612no—
352#95—1222no—
353#95—1222no—
354C24—21no—
355C24D4×D6111yescharacteristic
356C2×Q8—612no—
357C2×Q8—612no—
358C2×Q8—614no—
359C2×Q8—1228no—
360C2×Q8—1221no—
361C2×Q8—1221no—
362C2×Q8—1222no—
363C2×Q8—1224no—
364C2×Q8—1222no—
365C2×Q8—1222no—
366C2×Q8—2444no—
367C2×Q8—2444no—
368C2×Q8—2448no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—612no—
381C22×C4—612no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1224no—
385C22×C4—1222no—
386C22×C4—2444no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4#1175822212yes—
391C22×C4D4×D6112yes—
392C22×C4D4×D6111yescharacteristic
393C22×C4C23×D6111yescharacteristic
394SD16—2444no—
395SD16—2444no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—44no—
400C2×C8—1221no—
401C2×C8—1221no—
402C2×C8—1222no—
403C2×C8—1222no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—2444no—
411C4⋊C4—2444no—
412C4⋊C4—2444no—
413C4⋊C4—2442no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—614no—
417C22⋊C4—1222no—
418C22⋊C4—2444no—
419C22⋊C4—2444no—
420C42—1222no—
421C42—1222no—
422C42—2444no—
423C23—22no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23#1451975012yes—
431C23SD16×D6112yes—
432C23#1451980112yes—
433C23C2×D4×D6111yescharacteristic
434Q8—614no—
435Q8—1222no—
436Q8—1224no—
437Q8—1222no—
438Q8—1222no—
439Q8—2448no—
440Q8—2442no—
441D4—41no—
442D4—41no—
443D4—42no—
444D4—84no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—614no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1224no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—4884no—
465C2×C4—2448no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4#1451975012yes—
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4#41×D6112yes—
473C2×C4C2×#1175822211yescharacteristic
474C2×C4#1451982212yes—
475C8—42no—
476C8—2441no—
477C8—2441no—
478C22—22no—
479C22—24no—
480C22—24no—
481C22—22no—
482C22—81no—
483C22—82no—
484C22—81no—
485C22—82no—
486C22—81no—
487C22#1683996114yes—
488C22C2×#1451975011yescharacteristic
489C22#1684040112yes—
490C22C2×SD16×D6111yescharacteristic
491C22C2×#1451980111yescharacteristic
492C22#1684057612yes—
493C4—22no—
494C4—81no—
495C4—82no—
496C4—1224no—
497C4—1224no—
498C4—4882no—
499C4—2442no—
500C4—2442no—
501C4#123×D6111yescharacteristic
502C4#1684032211yescharacteristic
503C4#1684044112yes—
504C2—24no—
505C2—161no—
506C2#2144921811yescharacteristic
507C2C2×#1683996111yescharacteristic
508C2#2145535211yescharacteristic
509C2#2145767912yes—
510C2#2145781612yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1893C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#1893D61
hypocenterC61#24528
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. C22×#14519702C2semidirect
4. #21448651C2semidirect
5. #21447231C2semidirect
6. #21447135C2semidirect
7. #21446171C2semidirect
8. #21445997C2semidirect
9. #21445903C2semidirect
10. #21445823C2semidirect
151. C22⋊Dic61SD16semidirect
218. Dic122#123semidirect
298. C61#24528semidirect

Maximal subgroups

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

subgroupindex
C2×#168399002
C22×#145197022
#214502912
#214486512
#214472312
#214471352
#214461712
#214459972
#214459032
#214458232
C61×#18932
#2452861