Finite Group Databasebeta

← #29834854

Subgroups — #29834854

Order 15616.

588 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455462C211yesmaximal, characteristic
2#21454416C211yesmaximal, characteristic, complemented
3#21448013C211yesmaximal, characteristic, complemented
4#21448010C211yesmaximal, characteristic, complemented
5#21448008C211yesmaximal, characteristic, complemented
6#21448006C211yesmaximal, characteristic, complemented
7#21448002C211yesmaximal, characteristic, complemented
8#21448000C211yesmaximal, characteristic, complemented
9#21447998C211yesmaximal, characteristic, complemented
10#21447606C211yesmaximal, characteristic, complemented
11#21447600C211yesmaximal, characteristic, complemented
12#21447533C211yesmaximal, characteristic, complemented
13#357×D61C211yesmaximal, characteristic, complemented
14#21447431C211yesmaximal, characteristic, complemented
15C61×#2234C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18#16840328—21no—
19#16840242—21no—
20C2×#14519716C2211yescharacteristic
21C2×C16×D61C2211yescharacteristic
22#16840014—21no—
23#16840013—21no—
24#16839999C2211yescharacteristic
25#16839998C2211yescharacteristic
26#16839995C2211yescharacteristic
27#16839805—21no—
28#16839801—21no—
29#16839801—21no—
30#16839799C2211yescharacteristic
31#16839799C2211yescharacteristic
32#16839799C2211yescharacteristic, complemented
33C2×#14519711C2211yescharacteristic, complemented
34#107×D61C2211yescharacteristic
35#107×D61C2211yescharacteristic, complemented
36#16839765—21no—
37#16839589C2211yescharacteristic, complemented
38#16839589C2211yescharacteristic, complemented
39#16839589C2211yescharacteristic, complemented
40#16839588C2211yescharacteristic, complemented
41#16839588C2211yescharacteristic, complemented
42#16839587C2211yescharacteristic, complemented
43#16839564C2211yescharacteristic
44#16839564C2211yescharacteristic
45#16839563C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic, complemented
50#16839541C2211yescharacteristic, complemented
51#16839541C2211yescharacteristic, complemented
52C61×#466C2211yescharacteristic
53C61×#427C2211yescharacteristic, complemented
54C61×#367C2211yescharacteristic, complemented
55C61×#366C2211yescharacteristic, complemented
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic, complemented
59C2×#11758222D411yescharacteristic
60C2×D4×D61—21no—
61C2×D488C2311yescharacteristic
62C2×#11758199—21no—
63D8×D61—22no—
64D8×D61—22no—
65#14519783—22no—
66#14519783—22no—
67C2×#11758194D411yescharacteristic
68C2×#11758189D411yescharacteristic
69C2×#11758189D411yescharacteristic
70C2×C8×D61C2311yescharacteristic
71C2×C8×D61C2311yescharacteristic
72#14519773—41no—
73#14519763—41no—
74#14519752—21no—
75#14519751—21no—
76#14519748—21no—
77#14519747—41no—
78#14519745—41no—
79C4×Dic122—41no—
80C4×D244—41no—
81#14519737—41no—
82#14519737—41no—
83#14519735—21no—
84#14519735—21no—
85#14519735—21no—
86C2×C4⋊Dic61—21no—
87C4⋊C4×D61—21no—
88C4⋊C4×D61—21no—
89#14519716—22no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92C16×D61—22no—
93#14519711—21no—
94#14519711C2311yescharacteristic
95#14519711C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519710—21no—
98#14519710—21no—
99#14519708—21no—
100#14519708—21no—
101#14519708—21no—
102#14519699—21no—
103#14519698—21no—
104#14519697—21no—
105C2×D8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107#121×C61—21no—
108#117×C61—21no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#107×C61C2311yescharacteristic
114#106×C61—21no—
115#102×C61—21no—
116C23×D61—41no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C2×C22⋊D61D811yescharacteristic
124C2×C22⋊D61D811yescharacteristic
125C22×Dic61—41no—
126D4×D61—42no—
127D4×D61—42no—
128#11758222—22no—
129#11758222—22no—
130C2×D244—21no—
131C2×D244—41no—
132C2×D244D811yescharacteristic
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137C2×Dic122D811yescharacteristic
138D8×C61—21no—
139D8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2411yescharacteristic
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C22⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C42×C61—41no—
150C22⋊Dic61—41no—
151#11758199—42no—
152#11758198—41no—
153#11758198—41no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—41no—
161#11758196—41no—
162#11758196—41no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166#11758194—22no—
167C2×#92418C2×D411yescharacteristic
168C2×#92418C2×D411yescharacteristic
169D488—21no—
170D488—21no—
171#11758189—22no—
172#11758189—22no—
173C8×D61—21no—
174C8×D61—21no—
175C8×D61#4112yes—
176C8×D61#4112yes—
177C16×C61—21no—
178C16×C61—21no—
179#11758185—21no—
180#11758185—21no—
181C23×C61—41no—
182C23×C61C2×D811yescharacteristic
183C22×D61—41no—
184C22×D61—82no—
185C22×D61—161no—
186C22×D61—161no—
187C22×D61C2×D811yescharacteristic
188C22×D61C2×D811yescharacteristic
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—161no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C22⋊D61—22no—
200C22⋊D61—22no—
201C22⋊D61—82no—
202C22⋊D61—161no—
203C22⋊D61—161no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—161no—
208C2×Dic61C2×D811yescharacteristic
209C2×Dic61C2×D811yescharacteristic
210D244—22no—
211D244—41no—
212D244—41no—
213D244—161no—
214C4×D61—21no—
215C4×D61—21no—
216C4×D61—161no—
217C4×D61—161no—
218C4×D61#12312yes—
219C4×D61#13712yes—
220Dic122—22no—
221C8×C61—21no—
222C8×C61C2×#4111yescharacteristic
223C8×C61C2×#4111yescharacteristic
224#92418—21no—
225#92418#12312yes—
226#33650—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61C22×D811yescharacteristic
234C2×D61—21no—
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—21no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—164no—
243C2×D61#48012yes—
244C4×C61—21no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61—81no—
248C4×C61C2×#12311yescharacteristic
249C4×C61C2×#13711yescharacteristic
250Dic61—21no—
251Dic61—81no—
252Dic61—81no—
253Dic61—81no—
254Dic61#48012yes—
255#3058—611no—
256#2913—611no—
257#2234—611no—
258#2234—611no—
259#2234—611no—
260#2234D6111yescharacteristic, complemented
261#2233—611no—
262#2233—611no—
263#2232—611no—
264#2232—611no—
265#2159—611no—
266#2158—611no—
267#2140—611no—
268C2×#357—611no—
269#2113—611no—
270C2×C61—41no—
271C2×C61—161no—
272C2×C61C2×#48011yescharacteristic
273C2×C61#341511yescharacteristic
274C2×C61#341611yescharacteristic
275D61—22no—
276D61—41no—
277D61—161no—
278C22×D8—611no—
279C2×#131—611no—
280#535—1221no—
281#520—1221no—
282C2×M32—611no—
283C22×C16—611no—
284#469—1221no—
285#469—1221no—
286#466—611no—
287#466—611no—
288#466—611no—
289#466C2×D6111yescharacteristic
290#428—1221no—
291#428—1221no—
292#428—1221no—
293#427—611no—
294#427—611no—
295#427—611no—
296#427C2×D6111yescharacteristic, complemented
297C2×#107—611no—
298C2×#107—611no—
299C2×#107—611no—
300#418—1221no—
301#367—611no—
302#367—611no—
303#367—611no—
304#367C2×D6111yescharacteristic, complemented
305#366—611no—
306#366—611no—
307#366—611no—
308#366C2×D6111yescharacteristic, complemented
309#357—611no—
310#357—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315#357C2×D6111yescharacteristic
316#357C2×D6111yescharacteristic
317C22⋊C16—611no—
318C22⋊C16—611no—
319C22⋊C16—611no—
320C22⋊C16C2×D6111yescharacteristic, complemented
321C61#3365011yescharacteristic, complemented, Sylow
322C2×#41—611no—
323C22×D4—1221no—
324C2×D8—611no—
325C2×D8—1222no—
326C2×D8—1222no—
327C2×D8—1221no—
328C2×D8C22×D6111yescharacteristic
329#131—1222no—
330#131—1222no—
331C2×M16—611no—
332C2×M16—611no—
333C2×M16—611no—
334C22×C8—611no—
335C22×C8—611no—
336C22×C8C22×D6111yescharacteristic
337#127—2441no—
338#124—2441no—
339#121—21no—
340#121—1221no—
341#121—1221no—
342#121—1221no—
343#120—2441no—
344#120—2441no—
345C4×Q8—2441no—
346C4×D4—2441no—
347C4×D4—2441no—
348C4×D4—2441no—
349#117—21no—
350#117—1221no—
351#117—1221no—
352#117—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4—1221no—
355C2×C4⋊C4—1221no—
356M32—1222no—
357C2×C16—611no—
358C2×C16—611no—
359C2×C16—1222no—
360C2×C16C22×D6111yescharacteristic
361C2×C16C22×D6111yescharacteristic
362#107—611no—
363#107—611no—
364#107—611no—
365#107—1221no—
366#107—1221no—
367#107—1221no—
368#107C22×D6111yescharacteristic
369#107C22×D6111yescharacteristic
370#107C22×D6111yescharacteristic
371#106—21no—
372#106—1221no—
373#106—1221no—
374#106—1221no—
375#102—21no—
376#102—1221no—
377#102—1221no—
378#102—1221no—
379C24—2441no—
380#41—1222no—
381#41—1222no—
382C2×Q8—611no—
383C2×D4—21no—
384C2×D4—41no—
385C2×D4—611no—
386C2×D4—611no—
387C2×D4—611no—
388C2×D4—1221no—
389C2×D4—2441no—
390C2×D4—2442no—
391C2×D4—2442no—
392C2×D4—2441no—
393C2×D4—2441no—
394C2×D4—2441no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—2441no—
398C22×C4—2441no—
399C22×C4—2441no—
400C22×C4D4×D6111yescharacteristic
401D8—21no—
402D8—21no—
403D8—1221no—
404D8—1221no—
405D8—2442no—
406M16—1222no—
407M16—1222no—
408M16—1222no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—611no—
412C2×C8—612no—
413C2×C8—612no—
414C2×C8—611no—
415C2×C8—1221no—
416C2×C8—1221no—
417C2×C8D4×D6111yescharacteristic
418C2×C8C23×D6111yescharacteristic
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—2441no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C42—41no—
437C42—2441no—
438C42—2441no—
439C42—2441no—
440C16—21no—
441C16—21no—
442C16—1221no—
443C16—1221no—
444C23—41no—
445C23—611no—
446C23—611no—
447C23—9761no—
448C23—9761no—
449C23—4882no—
450C23—2441no—
451C23D8×D6111yescharacteristic
452Q8—1222no—
453D4—41no—
454D4—41no—
455D4—161no—
456D4—1222no—
457D4—1222no—
458D4—1222no—
459D4—9761no—
460D4—9761no—
461D4—9761no—
462D4—4882no—
463D4—2441no—
464D4—2441no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—611no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—611no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—9761no—
477C2×C4—9761no—
478C2×C4—9761no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4D8×D6111yescharacteristic
483C2×C4C2×D4×D6111yescharacteristic
484C8—21no—
485C8—612no—
486C8—1221no—
487C8#41×D6111yescharacteristic
488C8#41×D6111yescharacteristic
489C22—21no—
490C22—21no—
491C22—21no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—612no—
496C22—1221no—
497C22—1221no—
498C22—1221no—
499C22—1221no—
500C22—1221no—
501C22—9764no—
502C22—4881no—
503C22—4881no—
504C22—4881no—
505C22C2×D8×D6111yescharacteristic
506C4—21no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—612no—
511C4—1221no—
512C4—4881no—
513C4—4881no—
514C4—4881no—
515C4#123×D6111yescharacteristic
516C4#137×D6111yescharacteristic
517C2—41no—
518C2—161no—
519C2—1222no—
520C2—9761no—
521C2—2441no—
522C2#480×D6111yescharacteristic
523C2#2145895711yescharacteristic
524C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2234C2
socleC22×C61C22×D8
prime radical#21455462C2
hypercenter#2234D61
hypocenterC61#33650
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
2. #21454416C2semidirect
3. #21448013C2semidirect
4. #21448010C2semidirect
5. #21448008C2semidirect
6. #21448006C2semidirect
7. #21448002C2semidirect
8. #21448000C2semidirect
9. #21447998C2semidirect
10. #21447606C2semidirect
11. #21447600C2semidirect
12. #21447533C2semidirect
13. #357×D61C2semidirect
14. #21447431C2semidirect
15. C61×#2234C2semidirect
32. #16839799C22semidirect
33. C2×#14519711C22semidirect
35. #107×D61C22semidirect
37. #16839589C22semidirect
38. #16839589C22semidirect
39. #16839589C22semidirect
40. #16839588C22semidirect
41. #16839588C22semidirect
42. #16839587C22semidirect
49. C22⋊#11758185C22semidirect
50. #16839541C22semidirect
51. #16839541C22semidirect
53. C61×#427C22semidirect
54. C61×#367C22semidirect
55. C61×#366C22semidirect
58. C61×C22⋊C16C22semidirect
260. #2234D61semidirect
296. #427C2×D61semidirect
304. #367C2×D61semidirect
308. #366C2×D61semidirect
320. C22⋊C16C2×D61semidirect
321. C61#33650semidirect

Maximal subgroups

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

subgroupindex
#214554622
#214544162
#214480132
#214480102
#214480082
#214480062
#214480022
#214480002
#214479982
#214476062
#214476002
#214475332
#357×D612
#214474312
C61×#22342
#3365061