Finite Group Databasebeta

← #29834294

Subgroups — #29834294

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×#16840203C211yesmaximal, characteristic, complemented
2#21456243C211yesmaximal, characteristic
3C2×Q16×Dic61C211yesmaximal, characteristic, complemented
4#21447883C212yesmaximal, complemented
5#21447879C212yesmaximal, complemented
6#21447867C212yesmaximal, complemented
7#21447857C212yesmaximal, complemented
8#21447574C211yesmaximal, characteristic, complemented
9#21447532C211yesmaximal, characteristic, complemented
10#21447531C211yesmaximal, characteristic, complemented
11C2×C61×#510C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C2×#14519797—21no—
14#16840308—21no—
15C2×Q8×Dic61—21no—
16#16840203—24no—
17C2×#14519723C2212yes—
18C2×#14519720C2212yes—
19C2×#14519717C2211yescharacteristic
20#16840147C2211yescharacteristic
21#16840145C2211yescharacteristic
22#16840143—21no—
23#16840140C2211yescharacteristic
24#16840137C2211yescharacteristic
25#16840133—21no—
26Q16×Dic61—21no—
27Q16×Dic61—21no—
28Q16×Dic61C2212yes—
29Q16×Dic61C2212yes—
30C2×#14519711C2211yescharacteristic
31C2×#14519701—21no—
32C2×C8×Dic61C2211yescharacteristic
33#16839580C2212yes—
34#16839580C2212yes—
35#16839567C2212yes—
36#16839566C2212yes—
37#16839564C2212yes—
38#16839563C2212yes—
39C2×#135×C61C2211yescharacteristic
40C22×Q16×C61C2211yescharacteristic
41C61×#510—24no—
42C2×Q32×C61C2212yes—
43C2×SD32×C61C2212yes—
44C2×M32×C61C2211yescharacteristic
45C2×#11758222—21no—
46#14519799—22no—
47#14519799—22no—
48#14519797—44no—
49C2×Dic244C2311yescharacteristic
50C2×#11758202—21no—
51C2×#11758190—21no—
52C2×#11758200—21no—
53C2×#11758201—21no—
54C2×D488C2311yescharacteristic
55C2×#11758199—21no—
56C22×#92418D411yescharacteristic
57#14519775—21no—
58#14519773—21no—
59#14519766—21no—
60#14519764—21no—
61#14519749—41no—
62#14519749—41no—
63Q8×Dic61—22no—
64Q8×Dic61—22no—
65Q8×Dic61—41no—
66Q8×Dic61—41no—
67C4×C22⋊D61—41no—
68C4×C22⋊D61—41no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—41no—
71C2×C4×Dic61—41no—
72C2×C4×Dic61D411yescharacteristic
73#14519723—24no—
74#14519720—24no—
75#14519717—22no—
76C2×#11758185C2312yes—
77#14519711—21no—
78#14519711C2312yes—
79#14519701—22no—
80#14519701—22no—
81C8×Dic61D411yescharacteristic
82C8×Dic61D411yescharacteristic
83C8×Dic61C2311yescharacteristic
84C8×Dic61C2311yescharacteristic
85C2×#41×C61—21no—
86C22×Q8×C61—21no—
87#135×C61—22no—
88#135×C61—22no—
89C2×Q16×C61—21no—
90C2×Q16×C61—22no—
91C2×Q16×C61D414yescomplemented
92C2×Q16×C61C2311yescharacteristic
93C2×Q16×C61C2312yes—
94C2×SD16×C61—21no—
95C2×D8×C61C2311yescharacteristic
96C22×C8×C61C2311yescharacteristic
97Q32×C61—24no—
98SD32×C61—24no—
99M32×C61—22no—
100C2×C16×C61C2312yes—
101#41×C61—42no—
102#41×C61—42no—
103C2×Q8×C61—24no—
104C2×Q8×C61—22no—
105C2×Q8×C61—21no—
106C2×Q8×C61—41no—
107C2×Q8×C61—42no—
108C2×D4×C61—21no—
109C2×D4×C61—41no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—41no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—41no—
116C22×Dic61—41no—
117C22×Dic61D811yescharacteristic
118C22×Dic61D811yescharacteristic
119#11758222—42no—
120#11758222—42no—
121C2×D244—21no—
122C2×C4×D61—41no—
123C2×Dic122—21no—
124Q16×C61—24no—
125Q16×C61—21no—
126Q16×C61—22no—
127Q16×C61—21no—
128Q16×C61C2×D414yes—
129SD16×C61—42no—
130D8×C61—21no—
131D8×C61—21no—
132C2×C8×C61C2×D411yescharacteristic
133C2×C8×C61C2×D412yes—
134C2×C8×C61#4112yes—
135C2×C8×C61C2411yescharacteristic
136C22⋊Dic61—41no—
137C22⋊Dic61—41no—
138#11758202—42no—
139#11758201—42no—
140#11758200—42no—
141#11758199—42no—
142#11758198—41no—
143#11758198—41no—
144C4⋊Dic61—22no—
145C4⋊Dic61—41no—
146C4⋊Dic61—42no—
147C4⋊Dic61—42no—
148#11758196—41no—
149#11758196—41no—
150C4×Dic61—42no—
151C4×Dic61—42no—
152C4×Dic61D811yescharacteristic
153C4×Dic61D811yescharacteristic
154C4×Dic61C2×D411yescharacteristic
155C4×Dic61C2×D411yescharacteristic
156C2×#92418—22no—
157C2×#92418C2×D411yescharacteristic
158C2×#92418C2×D411yescharacteristic
159Dic244—21no—
160Dic244—21no—
161D488—21no—
162D488—21no—
163#11758190—42no—
164C16×C61—22no—
165#11758185—22no—
166C23×C61—41no—
167C23×C61C2×D811yescharacteristic
168C22×D61—41no—
169Q8×C61—24no—
170Q8×C61—42no—
171Q8×C61—44no—
172Q8×C61—41no—
173Q8×C61—41no—
174D4×C61—41no—
175D4×C61—41no—
176D4×C61—82no—
177C2×C4×C61—42no—
178C2×C4×C61—44no—
179C2×C4×C61—41no—
180C2×C4×C61—82no—
181C2×C4×C61#12112yes—
182C2×C4×C61C2×D811yescharacteristic
183C2×C4×C61AGL(1,ℤ/8)12yes—
184C2×C4×C61C22×D411yescharacteristic
185C22⋊D61—82no—
186C22⋊D61—82no—
187C22⋊D61—82no—
188C2×Dic61—22no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—82no—
193C2×Dic61C2×D811yescharacteristic
194C2×Dic61C2×D811yescharacteristic
195C2×Dic61C2×D811yescharacteristic
196C2×Dic61C2×D811yescharacteristic
197D244—41no—
198D244—41no—
199C4×D61—82no—
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—44no—
219C4×C61—81no—
220C4×C61#45912yes—
221C4×C61C2×#12111yescharacteristic
222C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
223Dic61—21no—
224Dic61—21no—
225Dic61—21no—
226Dic61—82no—
227Dic61—81no—
228Dic61#51012yescomplemented
229C2×#510—611no—
230C2×#510D6111yescharacteristic, complemented
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#2139—611no—
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×Q16C2×D6111yescharacteristic
250#532—1221no—
251C2×C4×Q8—1221no—
252#510—24no—
253#510—1224no—
254C2×Q32—612no—
255C2×Q32C2×D6112yes—
256C2×SD32—612no—
257C2×SD32C2×D6112yes—
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×#107—611no—
271C2×#103—1221no—
272C2×C4×C8—611no—
273C4⋊C16—612no—
274C4⋊C16—612no—
275#358—612no—
276#358—612no—
277#357—612no—
278#357—612no—
279C61#3358111yescharacteristic, complemented, Sylow
280C2×#41—21no—
281C2×#41—1221no—
282C22×Q8—21no—
283#135—22no—
284#135—22no—
285#135—1222no—
286#135—1222no—
287#135—2444no—
288C2×Q16—21no—
289C2×Q16—22no—
290C2×Q16—611no—
291C2×Q16—1221no—
292C2×Q16C22⋊D6114yescomplemented
293C2×Q16C22×D6111yescharacteristic
294C2×Q16C22×D6112yes—
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—1221no—
317C2×C4⋊C4—2441no—
318C2×C42—611no—
319C2×C42—2441no—
320Q32—24no—
321Q32—1224no—
322SD32—24no—
323SD32—1224no—
324M32—22no—
325M32—1222no—
326C2×C16—612no—
327C2×C16C22×D6112yes—
328#107—612no—
329#107—1221no—
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—22no—
341C2×Q8—21no—
342C2×Q8—24no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—1221no—
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—22no—
361Q16—21no—
362Q16—24no—
363Q16—1221no—
364Q16—1221no—
365Q16—2442no—
366Q16C2×C22⋊D6114yes—
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×C8D4×D6111yescharacteristic
380C2×C8#1175822412yes—
381C2×C8C2×C22⋊D6112yes—
382C2×C8C23×D6111yescharacteristic
383C4⋊C4—1222no—
384C4⋊C4—2442no—
385C4⋊C4—2442no—
386C4⋊C4—2441no—
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—24no—
405Q8—44no—
406Q8—42no—
407Q8—41no—
408Q8—41no—
409Q8—2441no—
410Q8—2441no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—4882no—
415D4—4882no—
416D4—4882no—
417D4—2441no—
418D4—2441no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—82no—
423C2×C4—611no—
424C2×C4—611no—
425C2×C4—611no—
426C2×C4—611no—
427C2×C4—1222no—
428C2×C4—4882no—
429C2×C4—4882no—
430C2×C4—2441no—
431C2×C4—2442no—
432C2×C4—2441no—
433C2×C4#1451974912yes—
434C2×C4D8×D6111yescharacteristic
435C2×C4#1451980112yes—
436C2×C4C2×D4×D6111yescharacteristic
437C8—1221no—
438C8—1221no—
439C8#1451974912yes—
440C8C22×C22⋊D6111yescharacteristic
441C8C2×#1175822411yescharacteristic
442C22—21no—
443C22—21no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—4881no—
448C22—4881no—
449C22—4881no—
450C22#1683995512yes—
451C22C2×D8×D6111yescharacteristic
452C4—44no—
453C4—81no—
454C4—612no—
455C4—1221no—
456C4—1221no—
457C4—1221no—
458C4—4882no—
459C4—4881no—
460C4#1683995512yes—
461C4C2×#1451974911yescharacteristic
462C4C2×#1451980111yescharacteristic
463C2—22no—
464C2—161no—
465C2—9761no—
466C2C2×#1683995511yescharacteristic
467C2#510×D6111yescharacteristic
468C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#510C2
socleC22×C61C22×D8
prime radical#21456243C2
hypercenterC2×#510D61
hypocenterC61#33581
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
3. C2×Q16×Dic61C2semidirect
4. #21447883C2semidirect
5. #21447879C2semidirect
6. #21447867C2semidirect
7. #21447857C2semidirect
8. #21447574C2semidirect
9. #21447532C2semidirect
10. #21447531C2semidirect
11. C2×C61×#510C2semidirect
91. C2×Q16×C61D4semidirect
228. Dic61#510semidirect
230. C2×#510D61semidirect
279. C61#33581semidirect
292. C2×Q16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
#214562432
C2×Q16×Dic612
#214478832
#214478792
#214478672
#214478572
#214475742
#214475322
#214475312
C2×C61×#5102
#3358161