Finite Group Databasebeta

← #29833879

Subgroups — #29833879

Order 15616.

1150 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×#16840564C211yesmaximal, characteristic, complemented
2C2×#16840200C211yesmaximal, characteristic, complemented
3C2×#16840192C211yesmaximal, characteristic, complemented
4#21455491C211yesmaximal, characteristic, complemented
5#21447762C212yesmaximal, complemented
6#21447749C212yesmaximal, complemented
7#21447745C212yesmaximal, complemented
8#21447732C212yesmaximal, complemented
9C2×#16839562C211yesmaximal, characteristic, complemented
10#21447434C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C2×#14519823—21no—
13#16840564—24no—
14#16840564—24no—
15C2×#14519804—22no—
16C2×#14519802—21no—
17C2×#14519800C2212yes—
18C2×#14519799C2211yescharacteristic, complemented
19C22×D488C2211yescharacteristic
20C2×D8×D61C2212yes—
21C2×#14519783C2211yescharacteristic
22#16840343—21no—
23#16840200—24no—
24#16840192—24no—
25C2×#14519723C2211yescharacteristic, complemented
26C2×#14519722C2212yescomplemented
27C2×#14519721C2211yescharacteristic, complemented
28C2×#14519720C2211yescharacteristic, complemented
29C2×D976C2212yescomplemented
30C2×#14519718C2211yescharacteristic, complemented
31C22×#11758185C2211yescharacteristic, complemented
32#16840009—22no—
33#16840004C2212yescomplemented
34#16839999C2212yescomplemented
35#16839761—21no—
36#16839562C2212yescomplemented
37#16839562C2212yescomplemented
38#16839541C2212yescomplemented
39#16839541C2212yescomplemented
40C2×#135×C61C2211yescharacteristic
41C2×M32×C61C2211yescharacteristic, complemented
42C2×#107×C61C2211yescharacteristic, complemented
43C61×#358C2212yescomplemented
44C61×#357C2212yescomplemented
45#14519823—28no—
46#14519823—44no—
47C2×#11758226—22no—
48C2×#11758222—21no—
49C2×#11758222—41no—
50C2×#11758222D411yescharacteristic
51C22×D244—21no—
52C22×D244—41no—
53C2×D4×D61—22no—
54C2×D4×D61—42no—
55#14519804—216no—
56#14519802—28no—
57#14519800—24no—
58#14519800—24no—
59#14519799—22no—
60#14519799—22no—
61C2×Dic244C2311yescharacteristic
62C2×#11758190—21no—
63C2×#11758201—22no—
64C2×D488—21no—
65C2×D488—22no—
66C2×D488D414yescomplemented
67C2×D488C2311yescharacteristic
68C2×D488C2312yes—
69C2×#11758199—22no—
70D8×D61—24no—
71D8×D61—24no—
72#14519783—22no—
73#14519783—22no—
74C2×#11758194D411yescharacteristic
75C2×#11758189D412yes—
76C2×C8×D61C2312yes—
77#14519757—22no—
78#14519757—41no—
79#14519752—22no—
80#14519752—41no—
81#14519751—42no—
82#14519723—22no—
83#14519722—24no—
84#14519721—22no—
85#14519720—22no—
86D976—24no—
87#14519718—22no—
88C2×#11758185—22no—
89C2×#11758185C2311yescharacteristic
90C2×#11758185C2311yescharacteristic
91#14519700—22no—
92#14519697—22no—
93C2×#41×C61—21no—
94#135×C61—22no—
95#135×C61D414yescomplemented
96C2×Q16×C61C2311yescharacteristic
97C2×SD16×C61—21no—
98C2×D8×C61C2311yescharacteristic
99C22×C8×C61C2311yescharacteristic
100C2×C4⋊C4×C61—21no—
101M32×C61—22no—
102C2×C16×C61C2312yes—
103#107×C61—21no—
104#107×C61C2312yes—
105C23×D61—41no—
106C23×D61—42no—
107#41×C61—24no—
108#41×C61—42no—
109C2×Q8×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—41no—
112C22×C4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C2×C22⋊D61—41no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61D812yes—
118#11758226—28no—
119#11758226—44no—
120D4×D61—28no—
121D4×D61—44no—
122D4×D61—416no—
123#11758222—22no—
124#11758222—22no—
125#11758222—48no—
126#11758222—42no—
127#11758222—42no—
128C2×D244—24no—
129C2×D244—22no—
130C2×D244—21no—
131C2×D244—48no—
132C2×D244—42no—
133C2×D244—41no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244D811yescharacteristic
137C2×C4×D61—42no—
138C2×C4×D61—41no—
139C2×C4×D61C2×D412yes—
140C2×Dic122—21no—
141C2×Dic122D811yescharacteristic
142Q16×C61—21no—
143Q16×C61C2×D412yes—
144SD16×C61—24no—
145D8×C61—21no—
146D8×C61C2×D412yes—
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2×D412yes—
149C2×C8×C61C2×D412yes—
150C2×C8×C61C2411yescharacteristic
151C4⋊C4×C61—22no—
152C4⋊C4×C61—41no—
153#11758201—28no—
154#11758199—28no—
155#11758198—42no—
156#11758198—42no—
157C4⋊Dic61—42no—
158#11758194—22no—
159C2×#92418C2×D412yes—
160Dic244—21no—
161Dic244—21no—
162D488—24no—
163D488—22no—
164D488—21no—
165D488—21no—
166D488C2×D414yes—
167#11758190—42no—
168#11758189—24no—
169C8×D61—22no—
170C8×D61—22no—
171C16×C61—22no—
172#11758185—21no—
173#11758185—21no—
174C23×C61—41no—
175C23×C61C2×D811yescharacteristic
176C22×D61—48no—
177C22×D61—42no—
178C22×D61—44no—
179C22×D61—41no—
180C22×D61—162no—
181C22×D61—162no—
182C22×D61—161no—
183C22×D61—161no—
184C22×D61C2×D812yes—
185Q8×C61—22no—
186Q8×C61—41no—
187D4×C61—22no—
188D4×C61—44no—
189D4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—44no—
192C2×C4×C61—42no—
193C2×C4×C61—161no—
194C2×C4×C61#12012yes—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61AGL(1,ℤ/8)12yes—
197C2×C4×C61C22×D411yescharacteristic
198C22⋊D61—24no—
199C22⋊D61—48no—
200C22⋊D61—82no—
201C2×Dic61—41no—
202C2×Dic61C2×D812yes—
203D244—24no—
204D244—22no—
205D244—48no—
206D244—41no—
207D244—42no—
208D244—41no—
209D244—44no—
210D244—162no—
211C4×D61—22no—
212C4×D61—22no—
213C4×D61—48no—
214C4×D61—82no—
215Dic122—22no—
216Dic122—41no—
217Dic122—41no—
218C8×C61#12012yes—
219C8×C61C22×D411yescharacteristic
220C8×C61C22×D411yescharacteristic
221#92418—22no—
222#33526—611nomaximal, Sylow
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—42no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61#44712yes—
229C22×C61C22×D811yescharacteristic
230C2×D61—22no—
231C2×D61—22no—
232C2×D61—22no—
233C2×D61—44no—
234C2×D61—82no—
235C2×D61—82no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—168no—
240C2×D61—164no—
241C4×C61—42no—
242C4×C61—82no—
243C4×C61#44712yes—
244C4×C61C2×#12011yescharacteristic
245C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
246Dic61—22no—
247Dic61—81no—
248C2×#576—611no—
249C2×#509—611no—
250C2×#508—611no—
251#3061—611no—
252#2189—612no—
253#2187—612no—
254#2186—612no—
255#2184—612no—
256#2139D6111yescharacteristic, complemented
257C2×#357—611no—
258#2113—611no—
259C2×C61—22no—
260C2×C61—82no—
261C2×C61C2×#44711yescharacteristic
262C2×C61#341511yescharacteristic
263C2×C61#341611yescharacteristic
264D61—42no—
265D61—84no—
266D61—161no—
267C2×H(5,𝔽2)—1221no—
268#576—1224no—
269#576—1224no—
270C2×AGL(1,ℤ/8)—1222no—
271C2×AGL(1,ℤ/8)—1221no—
272C2×#135—612no—
273C2×#135—611no—
274C2×#135C2×D6111yescharacteristic
275C22×D8—612no—
276C22×D8—611no—
277C2×#131—611no—
278#537—1221no—
279#509—1224no—
280#508—1224no—
281C2×Q32—611no—
282C2×SD32—611no—
283C2×SD32—611no—
284C2×SD32—612no—
285C2×D16—611no—
286C2×D16—612no—
287C2×M32C2×D6111yescharacteristic, complemented
288C22×C16—611no—
289#468—1222no—
290#467—612no—
291#466—612no—
292C2×#107C2×D6111yescharacteristic, complemented
293#417—1221no—
294#358C2×D6112yescomplemented
295#357—612no—
296#357—612no—
297#357C2×D6112yescomplemented
298C22⋊C16—612no—
299C22⋊C16—612no—
300C61#3352611yescharacteristic, complemented, Sylow
301H(5,𝔽2)—1228no—
302H(5,𝔽2)—2444no—
303C2×#41—21no—
304C2×#41—611no—
305C2×#41—1222no—
306C2×#41—1221no—
307C2×#41—2441no—
308C22×D4—1222no—
309C22×D4—1221no—
310C22×D4—2441no—
311C22×D4—2442no—
312AGL(1,ℤ/8)—12216no—
313AGL(1,ℤ/8)—1228no—
314#135—22no—
315#135—1224no—
316#135—1224no—
317#135—1222no—
318#135—1222no—
319#135D24414yescomplemented
320C2×Q16—611no—
321C2×Q16C22×D6111yescharacteristic
322C2×SD16—21no—
323C2×SD16—1222no—
324C2×SD16—1221no—
325C2×D8—614no—
326C2×D8—612no—
327C2×D8—611no—
328C2×D8—1222no—
329C2×D8—1224no—
330C2×D8—1224no—
331C2×D8—1222no—
332C2×D8—1221no—
333C2×D8C22×D6111yescharacteristic
334#131—1222no—
335#131—1222no—
336C2×M16—611no—
337C2×M16—612no—
338C22×C8—612no—
339C22×C8C22×D6111yescharacteristic
340C22⋊Q8—1222no—
341C22⋊Q8—2441no—
342#121—1222no—
343#121—2441no—
344#121—2442no—
345C2×C4⋊C4—21no—
346Q32—1222no—
347SD32—1224no—
348SD32—1222no—
349SD32—1222no—
350D16—1224no—
351D16—1222no—
352M32—22no—
353C2×C16—611no—
354C2×C16—611no—
355C2×C16—1222no—
356C2×C16C22×D6112yes—
357#107—21no—
358#107C22×D6112yes—
359#103—1222no—
360#102—1222no—
361C24—2442no—
362C24—2441no—
363#41—24no—
364#41—42no—
365#41—1222no—
366#41—1222no—
367#41—1228no—
368#41—2448no—
369#41—2444no—
370#41—2442no—
371#41—2442no—
372C2×Q8—21no—
373C2×Q8—611no—
374C2×Q8—1221no—
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—612no—
378C2×D4—611no—
379C2×D4—1228no—
380C2×D4—1222no—
381C2×D4—1221no—
382C2×D4—1224no—
383C2×D4—2448no—
384C2×D4—2442no—
385C2×D4—2442no—
386C2×D4—2442no—
387C2×D4—2444no—
388C2×D4—24416no—
389C2×D4—2441no—
390C2×D4—2441no—
391C2×D4—2442no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—612no—
395C22×C4—2442no—
396C22×C4—2441no—
397C22×C4D4×D6111yescharacteristic
398Q16—21no—
399Q16—1221no—
400Q16—1221no—
401Q16C2×D24412yes—
402SD16—24no—
403SD16—1228no—
404SD16—2442no—
405D8—21no—
406D8—614no—
407D8—1228no—
408D8—1221no—
409D8—1222no—
410D8—1221no—
411D8—1224no—
412D8C2×D24412yes—
413M16—1222no—
414M16—1224no—
415C2×C8—612no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8C2×D24412yes—
419C2×C8D4×D6112yes—
420C2×C8D4×D6111yescharacteristic
421C2×C8C23×D6111yescharacteristic
422C4⋊C4—22no—
423C4⋊C4—41no—
424C4⋊C4—2442no—
425C22⋊C4—2442no—
426C22⋊C4—2442no—
427C16—22no—
428C16—1221no—
429C16—1221no—
430C23—41no—
431C23—612no—
432C23—9761no—
433C23—9761no—
434C23—9762no—
435C23—9762no—
436C23—2448no—
437C23—2444no—
438C23—2441no—
439C23—2442no—
440C23D8×D6111yescharacteristic
441Q8—22no—
442Q8—41no—
443Q8—1222no—
444Q8—2441no—
445Q8—2441no—
446D4—22no—
447D4—44no—
448D4—41no—
449D4—1224no—
450D4—1224no—
451D4—1222no—
452D4—9762no—
453D4—4882no—
454D4—2448no—
455D4—2448no—
456D4—2441no—
457D4—2441no—
458D4—2442no—
459D4—2444no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—161no—
464C2×C4—612no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—4882no—
468C2×C4—2448no—
469C2×C4—2441no—
470C2×C4#1451974712yes—
471C2×C4D8×D6111yescharacteristic
472C2×C4#1451980412yes—
473C2×C4C2×D4×D6111yescharacteristic
474C8—1222no—
475C8#1451974712yes—
476C8C2×D4×D6111yescharacteristic
477C8C22×D24411yescharacteristic
478C22—21no—
479C22—21no—
480C22—42no—
481C22—81no—
482C22—81no—
483C22—1222no—
484C22—1222no—
485C22—1222no—
486C22—9764no—
487C22—9768no—
488C22—4881no—
489C22—4881no—
490C22—4881no—
491C22—4882no—
492C22—4882no—
493C22—2444no—
494C22#1683989712yes—
495C22C2×D8×D6111yescharacteristic
496C4—42no—
497C4—82no—
498C4—1222no—
499C4—4881no—
500C4#1683989712yes—
501C4C2×#1451974711yescharacteristic
502C4C2×#1451980411yescharacteristic
503C2—22no—
504C2—82no—
505C2—9761no—
506C2—4884no—
507C2—2442no—
508C2C2×#1683989711yescharacteristic
509C2#2145895911yescharacteristic
510C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
hypercenter#2139D61
hypocenterC61#33526
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
1. C2×#16840564C2semidirect
2. C2×#16840200C2semidirect
3. C2×#16840192C2semidirect
4. #21455491C2semidirect
5. #21447762C2semidirect
6. #21447749C2semidirect
7. #21447745C2semidirect
8. #21447732C2semidirect
9. C2×#16839562C2semidirect
10. #21447434C2semidirect
11. C61×#2139C2semidirect
18. C2×#14519799C22semidirect
25. C2×#14519723C22semidirect
26. C2×#14519722C22semidirect
27. C2×#14519721C22semidirect
28. C2×#14519720C22semidirect
29. C2×D976C22semidirect
30. C2×#14519718C22semidirect
31. C22×#11758185C22semidirect
33. #16840004C22semidirect
34. #16839999C22semidirect
36. #16839562C22semidirect
37. #16839562C22semidirect
38. #16839541C22semidirect
39. #16839541C22semidirect
41. C2×M32×C61C22semidirect
42. C2×#107×C61C22semidirect
43. C61×#358C22semidirect
44. C61×#357C22semidirect
66. C2×D488D4semidirect
95. #135×C61D4semidirect
256. #2139D61semidirect
287. C2×M32C2×D61semidirect
292. C2×#107C2×D61semidirect
294. #358C2×D61semidirect
297. #357C2×D61semidirect
300. C61#33526semidirect
319. #135D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168405642
C2×#168402002
C2×#168401922
#214554912
#214477622
#214477492
#214477452
#214477322
C2×#168395622
#214474342
C61×#21392
#3352661