Finite Group Databasebeta

← #29834296

Subgroups — #29834296

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840205C211yesmaximal, characteristic, complemented
2C2×#16840202C211yesmaximal, characteristic, complemented
3#21456243C211yesmaximal, characteristic
4C2×#16839856C211yesmaximal, characteristic, complemented
5#21447884C212yesmaximal, complemented
6#21447880C212yesmaximal, complemented
7#21447868C212yesmaximal, complemented
8#21447858C212yesmaximal, complemented
9#21447574C211yesmaximal, characteristic, complemented
10#21447529C211yesmaximal, characteristic, complemented
11C61×#2139C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519797—21no—
14C22×Dic244C2211yescharacteristic
15#16840308—21no—
16C2×#14519742—21no—
17#16840205—24no—
18#16840202—24no—
19C2×Dic488C2212yes—
20C2×#14519723C2212yes—
21C2×#14519722C2212yes—
22C2×#14519721C2212yes—
23C2×#14519717C2211yescharacteristic
24#16840147C2211yescharacteristic
25#16840145C2211yescharacteristic
26#16840143—21no—
27#16840140C2211yescharacteristic
28#16840137C2211yescharacteristic
29#16840133—21no—
30#16839856—21no—
31#16839856—21no—
32#16839856C2212yes—
33#16839856C2212yes—
34C2×#14519700—21no—
35C2×C8×Dic61C2211yescharacteristic
36#16839580C2212yes—
37#16839580C2212yes—
38#16839565C2212yes—
39#16839562C2212yes—
40C2×#135×C61C2211yescharacteristic
41C2×M32×C61C2211yescharacteristic
42C2×#107×C61C2211yescharacteristic
43C61×#358C2212yes—
44C61×#357C2212yes—
45C2×#11758222—21no—
46C22×Dic122—21no—
47#14519799—22no—
48#14519799—22no—
49#14519797—44no—
50C2×Dic244—21no—
51C2×Dic244—22no—
52C2×Dic244D414yescomplemented
53C2×Dic244C2311yescharacteristic
54C2×Dic244C2312yes—
55C2×#11758202—21no—
56C2×#11758190—21no—
57C2×#11758200—21no—
58C2×#11758201—21no—
59C2×D488C2311yescharacteristic
60C2×#11758199—21no—
61C22×#92418D411yescharacteristic
62#14519775—21no—
63#14519773—21no—
64#14519766—21no—
65#14519764—21no—
66#14519749—41no—
67#14519749—41no—
68#14519742—22no—
69#14519742—22no—
70#14519742—41no—
71#14519742—41no—
72C4×C22⋊D61—41no—
73C4×C22⋊D61—41no—
74C2×#11758196—41no—
75C2×C4×Dic61—41no—
76C2×C4×Dic61D411yescharacteristic
77Dic488—24no—
78#14519723—24no—
79#14519722—24no—
80#14519721—24no—
81#14519717—22no—
82C2×#11758185C2312yes—
83#14519700—22no—
84#14519700—22no—
85C8×Dic61D411yescharacteristic
86C8×Dic61D411yescharacteristic
87C8×Dic61C2311yescharacteristic
88C8×Dic61C2311yescharacteristic
89C2×#41×C61—21no—
90#135×C61—22no—
91#135×C61—22no—
92C2×Q16×C61C2311yescharacteristic
93C2×SD16×C61—21no—
94C2×D8×C61C2311yescharacteristic
95C22×C8×C61C2311yescharacteristic
96C2×C4⋊C4×C61—21no—
97M32×C61—22no—
98C2×C16×C61C2312yes—
99#107×C61—21no—
100#107×C61C2312yes—
101#41×C61—42no—
102#41×C61—42no—
103C2×Q8×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C2×C22⋊D61—41no—
110C2×C22⋊D61—41no—
111C2×C22⋊D61—41no—
112C22×Dic61—41no—
113C22×Dic61D811yescharacteristic
114C22×Dic61D811yescharacteristic
115#11758222—42no—
116#11758222—42no—
117C2×D244—21no—
118C2×C4×D61—41no—
119C2×Dic122—24no—
120C2×Dic122—22no—
121C2×Dic122—21no—
122C2×Dic122—41no—
123C2×Dic122—42no—
124Q16×C61—21no—
125Q16×C61—21no—
126SD16×C61—42no—
127D8×C61—21no—
128D8×C61—21no—
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D412yes—
131C2×C8×C61#4112yes—
132C2×C8×C61C2411yescharacteristic
133C4⋊C4×C61—22no—
134C4⋊C4×C61—41no—
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137#11758202—42no—
138#11758201—42no—
139#11758200—42no—
140#11758199—42no—
141#11758198—41no—
142#11758198—41no—
143#11758196—42no—
144#11758196—42no—
145#11758196—41no—
146#11758196—41no—
147C4×Dic61—42no—
148C4×Dic61—42no—
149C4×Dic61D811yescharacteristic
150C4×Dic61D811yescharacteristic
151C4×Dic61C2×D411yescharacteristic
152C4×Dic61C2×D411yescharacteristic
153C2×#92418—22no—
154C2×#92418C2×D411yescharacteristic
155C2×#92418C2×D411yescharacteristic
156Dic244—24no—
157Dic244—21no—
158Dic244—22no—
159Dic244—21no—
160Dic244C2×D414yes—
161D488—21no—
162D488—21no—
163#11758190—42no—
164C16×C61—22no—
165#11758185—22no—
166C23×C61—41no—
167C23×C61C2×D811yescharacteristic
168C22×D61—41no—
169Q8×C61—41no—
170Q8×C61—41no—
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—82no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61—82no—
177C2×C4×C61—82no—
178C2×C4×C61#12112yes—
179C2×C4×C61C2×D811yescharacteristic
180C2×C4×C61AGL(1,ℤ/8)12yes—
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—82no—
183C22⋊D61—82no—
184C22⋊D61—82no—
185C2×Dic61—22no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—44no—
189C2×Dic61—41no—
190C2×Dic61C2×D811yescharacteristic
191C2×Dic61C2×D811yescharacteristic
192C2×Dic61C2×D811yescharacteristic
193C2×Dic61C2×D811yescharacteristic
194D244—41no—
195D244—41no—
196C4×D61—82no—
197Dic122—24no—
198Dic122—42no—
199Dic122—44no—
200Dic122—41no—
201Dic122—41no—
202C8×C61#12112yes—
203C8×C61C22×D411yescharacteristic
204C8×C61C2×#4111yescharacteristic
205#92418—21no—
206#92418—21no—
207#33581—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—81no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61#45912yes—
214C22×C61C22×D811yescharacteristic
215C2×D61—81no—
216C2×D61—81no—
217C2×D61—81no—
218C4×C61—82no—
219C4×C61—81no—
220C4×C61#45912yes—
221C4×C61C2×#12111yescharacteristic
222C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
223Dic61—21no—
224Dic61—21no—
225Dic61—21no—
226Dic61—44no—
227Dic61—81no—
228Dic61#51012yescomplemented
229C2×#510—611no—
230C2×#510—611no—
231#3147—611no—
232C2×C4×Q16—611no—
233#2211—612no—
234#2211—612no—
235#2209—612no—
236#2207—612no—
237#2150—611no—
238#2139—611no—
239#2139D6111yescharacteristic, complemented
240C2×C61—22no—
241C2×C61—161no—
242C2×C61C2×#45911yescharacteristic
243C2×C61C2×#51011yescharacteristic
244C2×C61#341411yescharacteristic
245D61—161no—
246C2×#135—611no—
247C2×#135—1221no—
248C2×#135C2×D6111yescharacteristic
249C22×Q16—611no—
250#532—1221no—
251C2×C4×Q8—1221no—
252#510—1224no—
253#510—1224no—
254C2×Q32—612no—
255C2×Q32—612no—
256C2×SD32—612no—
257C2×SD32—612no—
258C2×M32—611no—
259C2×M32C2×D6111yescharacteristic
260#495—611no—
261#495—611no—
262#495—1221no—
263#494—611no—
264#493—611no—
265#492—1221no—
266C4×Q16—612no—
267C4×Q16—612no—
268C4×Q16—1221no—
269C4×Q16—1221no—
270C2×#107C2×D6111yescharacteristic
271C2×#103—1221no—
272C2×C4×C8—611no—
273C4⋊C16—612no—
274C4⋊C16—612no—
275#358—612no—
276#358C2×D6112yes—
277#357—612no—
278#357C2×D6112yes—
279C61#3358111yescharacteristic, complemented, Sylow
280C2×#41—21no—
281C2×#41—1221no—
282C22×Q8—1221no—
283#135—22no—
284#135—22no—
285#135—1222no—
286#135—1222no—
287#135—2444no—
288C2×Q16—614no—
289C2×Q16—612no—
290C2×Q16—611no—
291C2×Q16—1221no—
292C2×Q16—1222no—
293C2×Q16—1221no—
294C2×Q16C22×D6111yescharacteristic
295C2×SD16—21no—
296C2×SD16—1221no—
297C2×SD16—1221no—
298C2×SD16—1221no—
299C2×D8—611no—
300C2×D8—1221no—
301C2×D8C22×D6111yescharacteristic
302C22×C8—611no—
303C22×C8C22×D6111yescharacteristic
304C4⋊Q8—1221no—
305#127—1221no—
306#124—1221no—
307#124—1221no—
308#121—2441no—
309#121—2441no—
310C4×Q8—1222no—
311C4×Q8—1222no—
312C4×Q8—2441no—
313C4×Q8—2441no—
314C4×D4—2441no—
315C4×D4—2441no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—2441no—
318C2×C42—611no—
319C2×C42—2441no—
320Q32—1224no—
321Q32—1224no—
322SD32—1224no—
323SD32—1224no—
324M32—22no—
325M32—1222no—
326C2×C16—612no—
327C2×C16C22×D6112yes—
328#107—21no—
329#107C22×D6112yes—
330#103—1222no—
331#103—1222no—
332C4×C8—611no—
333C4×C8—611no—
334C4×C8—611no—
335C4×C8—611no—
336#41—42no—
337#41—42no—
338#41—2442no—
339#41—2442no—
340C2×Q8—21no—
341C2×Q8—1222no—
342C2×Q8—1221no—
343C2×Q8—1224no—
344C2×Q8—2441no—
345C2×Q8—2442no—
346C2×D4—21no—
347C2×D4—41no—
348C2×D4—1221no—
349C2×D4—2441no—
350C2×D4—2441no—
351C2×D4—2441no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—611no—
355C22×C4—611no—
356C22×C4—2441no—
357C22×C4—2441no—
358C22×C4D4×D6111yescharacteristic
359Q16—21no—
360Q16—21no—
361Q16—614no—
362Q16—1221no—
363Q16—1222no—
364Q16—1221no—
365Q16—1224no—
366Q16—2442no—
367SD16—42no—
368SD16—2442no—
369SD16—2442no—
370SD16—2442no—
371D8—21no—
372D8—21no—
373D8—1221no—
374D8—1221no—
375D8—2442no—
376C2×C8—611no—
377C2×C8—611no—
378C2×C8—1222no—
379C2×C8#1175822212yes—
380C2×C8D4×D6112yes—
381C2×C8D4×D6111yescharacteristic
382C2×C8C23×D6111yescharacteristic
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—2442no—
386C4⋊C4—2442no—
387C4⋊C4—2441no—
388C4⋊C4—2441no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C22⋊C4—2441no—
392C22⋊C4—2441no—
393C42—611no—
394C42—611no—
395C42—611no—
396C42—611no—
397C42—2442no—
398C42—2442no—
399C16—22no—
400C16—1222no—
401C23—41no—
402C23—2441no—
403C23D8×D6111yescharacteristic
404Q8—41no—
405Q8—41no—
406Q8—1224no—
407Q8—2441no—
408Q8—2441no—
409Q8—2444no—
410Q8—2442no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—4882no—
415D4—4882no—
416D4—4882no—
417D4—2441no—
418D4—2441no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—611no—
424C2×C4—611no—
425C2×C4—611no—
426C2×C4—611no—
427C2×C4—1222no—
428C2×C4—4882no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2444no—
432C2×C4—2442no—
433C2×C4#1451975012yes—
434C2×C4D8×D6111yescharacteristic
435C2×C4#1451980412yes—
436C2×C4C2×D4×D6111yescharacteristic
437C8—1221no—
438C8—1221no—
439C8#1451975012yes—
440C8C2×D4×D6111yescharacteristic
441C8C2×#1175822211yescharacteristic
442C22—21no—
443C22—21no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—4881no—
448C22—4881no—
449C22—4881no—
450C22#1683995612yes—
451C22C2×D8×D6111yescharacteristic
452C4—82no—
453C4—81no—
454C4—612no—
455C4—1221no—
456C4—1221no—
457C4—1221no—
458C4—4881no—
459C4—2444no—
460C4#1683995612yes—
461C4C2×#1451975011yescharacteristic
462C4C2×#1451980411yescharacteristic
463C2—22no—
464C2—161no—
465C2—9761no—
466C2C2×#1683995611yescharacteristic
467C2#510×D6111yescharacteristic
468C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2139C2
socleC22×C61C22×D8
prime radical#21456243C2
hypercenter#2139D61
hypocenterC61#33581
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×#16840205C2semidirect
2. C2×#16840202C2semidirect
4. C2×#16839856C2semidirect
5. #21447884C2semidirect
6. #21447880C2semidirect
7. #21447868C2semidirect
8. #21447858C2semidirect
9. #21447574C2semidirect
10. #21447529C2semidirect
11. C61×#2139C2semidirect
52. C2×Dic244D4semidirect
228. Dic61#510semidirect
239. #2139D61semidirect
279. C61#33581semidirect

Maximal subgroups

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

subgroupindex
C2×#168402052
C2×#168402022
#214562432
C2×#168398562
#214478842
#214478802
#214478682
#214478582
#214475742
#214475292
C61×#21392
#3358161