Finite Group Databasebeta

← #29719012

Subgroups — #29719012

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×#16839903C211yesmaximal, characteristic
2C22×#14519702C211yesmaximal, characteristic, complemented
3#21450290C211yesmaximal, characteristic
4C22⋊C4×Dic122C211yesmaximal, characteristic, complemented
5#21447229C212yesmaximal, complemented
6#21447137C212yesmaximal, complemented
7#21446247C211yesmaximal, characteristic
8#21446173C212yesmaximal, complemented
9#21445906C211yesmaximal, characteristic, complemented
10#21445825C212yesmaximal, complemented
11C61×#1850C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×#11758190C2211yescharacteristic
14C22×#11758190C2211yescharacteristic
15C2×#14519761—21no—
16C2×#14519761—21no—
17C2×#14519751C2211yescharacteristic
18C2×C4×Dic122C2212yes—
19C22×C4⋊Dic61C2211yescharacteristic
20#16839903—24no—
21C2×#14519702—24no—
22C2×#14519702C2212yes—
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2212yes—
25C2×#14519702C2211yescharacteristic
26C2×#14519699C2212yes—
27#16839699—22no—
28#16839668C2212yes—
29#16839666—22no—
30#16839664C2212yes—
31#16839632C2212yes—
32#16839622C2211yescharacteristic
33#16839620—21no—
34C22⋊C4×Dic61—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839525C2212yes—
38C23×C8×C61C2211yescharacteristic
39C2×C4⋊C8×C61C2212yes—
40C2×C22⋊C8×C61C2211yescharacteristic
41C4×C22⋊C4×C61C2211yescharacteristic
42C61×#336C2212yes—
43C22×Dic122—24no—
44C22×Dic122—21no—
45C22×Dic122—22no—
46C22×Dic122—42no—
47C22×Dic122D414yes—
48C22×Dic122C2311yescharacteristic
49C22×Dic122C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C23×Dic61—21no—
53C23×Dic61—21no—
54C2×#11758190—22no—
55C2×#11758190—22no—
56C2×#11758190—24no—
57C2×#11758190—42no—
58#14519761—44no—
59#14519761—44no—
60#14519751—22no—
61#14519751—22no—
62C4×Dic122—24no—
63C4×Dic122—42no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—22no—
73C2×C22⋊Dic61—21no—
74C2×#11758198—21no—
75C2×#11758198—22no—
76C2×#11758198—22no—
77C2×C4×Dic61—22no—
78#14519702—24no—
79#14519702—24no—
80#14519702—22no—
81#14519702—22no—
82#14519699—42no—
83#14519684—22no—
84#14519684—22no—
85#14519684—21no—
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61—22no—
88C22×C8×C61—42no—
89C22×C8×C61C2312yes—
90C22×C8×C61C2311yescharacteristic
91C22×C8×C61C2311yescharacteristic
92C2×C22⋊C4×C61D411yescharacteristic
93C2×C22⋊C4×C61D411yescharacteristic
94C2×C42×C61C2312yes—
95C4⋊C8×C61—42no—
96C22⋊C8×C61—22no—
97#95×C61D412yes—
98C24×C61C2×D411yescharacteristic
99C23×D61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2×D412yes—
106C22×C4×C61#4112yes—
107C22×C4×C61C2411yescharacteristic
108C2×C22⋊D61—42no—
109C2×C22⋊D61—42no—
110C2×C22⋊D61—42no—
111C22×Dic61—21no—
112C22×Dic61—24no—
113C22×Dic61—21no—
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—22no—
117C22×Dic61—44no—
118C22×Dic61—42no—
119C22×Dic61—42no—
120C22×Dic61—42no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—22no—
124C2×D244—22no—
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—21no—
139C2×C8×C61—21no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—44no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61SD1614yescomplemented
148C42×C61—22no—
149C42×C61—22no—
150C22⋊Dic61—42no—
151#11758198—44no—
152#11758198—42no—
153#11758198—44no—
154C4⋊Dic61—22no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61—44no—
161C4⋊Dic61—42no—
162C4⋊Dic61—44no—
163#11758196—44no—
164C4×Dic61—44no—
165#11758190—44no—
166#11758190—44no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61#12112yes—
170C23×C61C2×SD1612yes—
171C23×C61AGL(1,ℤ/8)12yes—
172C23×C61C22×D411yescharacteristic
173C22×D61—41no—
174C22×D61—42no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—42no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—24no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61#12112yes—
188C2×C4×C61C2×SD1614yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4112yes—
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61H(5,𝔽2)12yes—
193C22⋊D61—84no—
194C2×Dic61—22no—
195C2×Dic61—24no—
196C2×Dic61—22no—
197C2×Dic61—48no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—84no—
209D244—41no—
210D244—41no—
211D244—42no—
212Dic122—22no—
213Dic122—24no—
214Dic122—22no—
215Dic122—22no—
216Dic122—48no—
217Dic122—42no—
218Dic122#12314yescomplemented
219C8×C61—41no—
220C8×C61—41no—
221C8×C61—42no—
222#24528—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—24no—
225C22×C61—22no—
226C22×C61—24no—
227C22×C61#46014yes—
228C22×C61C2×#12111yescharacteristic
229C22×C61#54612yes—
230C22×C61C22×SD1611yescharacteristic
231C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C61#57812yes—
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—82no—
236C2×D61—81no—
237C2×D61—82no—
238C4×C61—22no—
239C4×C61—24no—
240C4×C61—42no—
241C4×C61C2×#12311yescharacteristic
242C4×C61#53411yescharacteristic
243C4×C61#55112yes—
244Dic61—24no—
245Dic61—42no—
246Dic61—82no—
247Dic61—81no—
248Dic61—82no—
249C2×#448—611no—
250C22×#103—611no—
251#2501—611no—
252Q8×C22⋊C4—611no—
253#2072—612no—
254#2055—612no—
255#1893—611no—
256#1880—612no—
257#1850D6111yescharacteristic, complemented
258#1823—611no—
259#1803—612no—
260C2×C61—24no—
261C2×C61#241211yescharacteristic
262C2×C61C2×#46011yescharacteristic
263C2×C61#304411yescharacteristic
264C2×C61#329812yes—
265C2×C61#330812yes—
266D61—161no—
267C23×Q8—611no—
268C22×SD16—611no—
269C22×SD16—611no—
270C23×C8C2×D6111yescharacteristic
271C2×#123—1221no—
272C2×#123—1221no—
273C2×#121—611no—
274C2×C4×Q8—612no—
275C22×C4⋊C4—611no—
276#448—1224no—
277C2×C4⋊C8C2×D6112yes—
278C2×#103—612no—
279C2×#103—611no—
280C2×#103—612no—
281C2×#103—611no—
282C2×#103—1224no—
283C2×#102—612no—
284C2×C22⋊C8C2×D6111yescharacteristic
285#399—1222no—
286#391—612no—
287#391—1222no—
288#390—612no—
289#384—612no—
290#380—611no—
291#380—1221no—
292C4×C22⋊C4—1221no—
293C4×C22⋊C4C2×D6111yescharacteristic
294#340—612no—
295#340—612no—
296#340—612no—
297#336C2×D6112yes—
298C61#2452811yescharacteristic, complemented, Sylow
299C22×Q8—614no—
300C22×Q8—611no—
301C22×Q8—611no—
302C22×Q8—1224no—
303C22×Q8—1221no—
304C22×Q8—1222no—
305C22×Q8—2442no—
306C22×D4—611no—
307C22×D4—1221no—
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×C8C22×D6112yes—
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6111yescharacteristic
320#123—2444no—
321#123—2444no—
322#121—1222no—
323#121—1222no—
324C4×Q8—1224no—
325C4×Q8—2442no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1222no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1222no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1222no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4D24411yescharacteristic
340C2×C22⋊C4D24411yescharacteristic
341C2×C42—1222no—
342C2×C42C22×D6112yes—
343C4⋊C8—42no—
344#103—1224no—
345#103—1224no—
346#103—1222no—
347#103—1222no—
348#102—2442no—
349C22⋊C8—22no—
350#95—1222no—
351#95—1221no—
352#95—1222no—
353#95D24412yes—
354C24—1221no—
355C24C2×D24411yescharacteristic
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—1221no—
370C2×D4—1222no—
371C2×D4—1221no—
372C2×D4—1222no—
373C2×D4—2442no—
374C2×D4—2442no—
375C2×D4—2442no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1224no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1221no—
384C22×C4—2444no—
385C22×C4—2442no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4C2×D24412yes—
389C22×C4C2×D24411yescharacteristic
390C22×C4C2×D24412yes—
391C22×C4D4×D6112yes—
392C22×C4#1175822612yes—
393C22×C4C23×D6111yescharacteristic
394SD16—2444no—
395SD16—2444no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—44no—
404C4⋊C4—1222no—
405C4⋊C4—1221no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4—1221no—
409C4⋊C4—1222no—
410C4⋊C4—2444no—
411C4⋊C4—2444no—
412C4⋊C4—2442no—
413C4⋊C4—2444no—
414C22⋊C4—22no—
415C22⋊C4—2444no—
416C22⋊C4—2442no—
417C22⋊C4—2442no—
418C22⋊C4—2444no—
419C22⋊C4#1175819014yescomplemented
420C42—22no—
421C42—22no—
422C42—2444no—
423C23—22no—
424C23—22no—
425C23—2441no—
426C23—2441no—
427C23—2441no—
428C23—2442no—
429C23—2442no—
430C23#1451975212yes—
431C23C2×#1175819012yes—
432C23#1451980512yes—
433C23C22×D24411yescharacteristic
434Q8—614no—
435Q8—1222no—
436Q8—1224no—
437Q8—1222no—
438Q8—1222no—
439Q8—2448no—
440Q8—2442no—
441D4—4884no—
442D4—2441no—
443D4—2441no—
444D4—2442no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—1222no—
455C2×C4—1224no—
456C2×C4—1222no—
457C2×C4—4884no—
458C2×C4—2448no—
459C2×C4—2441no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2441no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2441no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4#1451975212yes—
470C2×C4C2×#1175819014yes—
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4#41×D6112yes—
473C2×C4C2×#1175822611yescharacteristic
474C2×C4#1451982212yes—
475C8—41no—
476C8—42no—
477C8—41no—
478C22—24no—
479C22—24no—
480C22—22no—
481C22—22no—
482C22—4882no—
483C22—4881no—
484C22—4882no—
485C22—4881no—
486C22—4881no—
487C22#1683996314yes—
488C22C2×#1451975211yescharacteristic
489C22#1684040712yes—
490C22C22×#1175819011yescharacteristic
491C22C2×#1451980511yescharacteristic
492C22#1684057912yes—
493C4—24no—
494C4—22no—
495C4—42no—
496C4—1224no—
497C4—4882no—
498C4—4881no—
499C4—4882no—
500C4—2442no—
501C4#123×D6111yescharacteristic
502C4#1684032211yescharacteristic
503C4#1684043912yes—
504C2—24no—
505C2—9761no—
506C2#2144921311yescharacteristic
507C2C2×#1683996311yescharacteristic
508C2#2145535011yescharacteristic
509C2#2145768512yes—
510C2#2145781212yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1850D61
hypocenterC61#24528
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
2. C22×#14519702C2semidirect
4. C22⋊C4×Dic122C2semidirect
5. #21447229C2semidirect
6. #21447137C2semidirect
8. #21446173C2semidirect
9. #21445906C2semidirect
10. #21445825C2semidirect
11. C61×#1850C2semidirect
147. C22⋊C4×C61SD16semidirect
218. Dic122#123semidirect
257. #1850D61semidirect
298. C61#24528semidirect
419. C22⋊C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168399032
C22×#145197022
#214502902
C22⋊C4×Dic1222
#214472292
#214471372
#214462472
#214461732
#214459062
#214458252
C61×#18502
#2452861