Finite Group Databasebeta

← #29834290

Subgroups — #29834290

Order 15616.

770 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840199C212yesmaximal, complemented
2#21456248C211yesmaximal, characteristic, complemented
3C4×D8×D61C211yesmaximal, characteristic, complemented
4#21447877C212yesmaximal, complemented
5#21447876C212yesmaximal, complemented
6#21447861C211yesmaximal, characteristic, complemented
7#21447855C211yesmaximal, characteristic, complemented
8#21447854C211yesmaximal, characteristic, complemented
9#21447573C211yesmaximal, characteristic, complemented
10#21447530C212yesmaximal, complemented
11C61×#2208C211yesmaximal, characteristic, complemented
12C2×#14519800—21no—
13C2×#14519800C2212yes—
14C2×D8×D61C2211yescharacteristic
15#16840313—21no—
16C4×D4×D61—21no—
17#16840199—28no—
18C2×#14519720C2212yes—
19C2×D976C2212yes—
20C2×#14519718C2212yes—
21C2×#14519716C2212yes—
22#16840147—21no—
23#16840147C2212yescomplemented
24#16840139C2211yescharacteristic, complemented
25#16840136—21no—
26C4×D488C2211yescharacteristic
27D8×Dic61C2211yescharacteristic
28#16839844C2211yescharacteristic
29C4×#11758199—21no—
30#16839842—21no—
31#107×D61C2211yescharacteristic
32#102×D61—21no—
33C4×C8×D61C2211yescharacteristic
34C4⋊#11758185C2211yescharacteristic, complemented
35#16839580C2212yescomplemented
36#16839566C2212yescomplemented
37#16839564C2212yescomplemented
38#16839562C2212yescomplemented
39C2×SD32×C61C2212yes—
40C61×#494C2211yescharacteristic, complemented
41C4×D8×C61C2211yescharacteristic
42C61×C4⋊C16C2211yescharacteristic, complemented
43C61×#358C2212yescomplemented
44C2×#11758226—22no—
45C2×D4×D61—21no—
46C22×C4×D61—41no—
47#14519800—24no—
48#14519800—24no—
49#14519800—44no—
50C2×#11758201—22no—
51C2×#11758201—22no—
52C2×D488—21no—
53C2×D488C2312yes—
54C2×D488C2311yescharacteristic
55C2×#11758199—21no—
56D8×D61—22no—
57D8×D61D414yescomplemented
58C2×C8×D61D411yescharacteristic
59C2×C8×D61C2311yescharacteristic
60#14519774—21no—
61#14519766—22no—
62#14519752—41no—
63#14519752—41no—
64C4×D244—21no—
65D4×Dic61—21no—
66#14519739—21no—
67#14519739—41no—
68#14519739—41no—
69C4×C22⋊D61—41no—
70#14519737—41no—
71C4⋊C4×D61—21no—
72C22⋊C4×D61—41no—
73C42×D61D411yescharacteristic
74#14519720—24no—
75D976—24no—
76#14519718—24no—
77#14519716—24no—
78C2×#11758185C2312yes—
79#14519711C2311yescharacteristic
80#14519710—21no—
81#14519699—21no—
82#14519698—21no—
83#14519697—21no—
84C4×#92418D411yescharacteristic
85C8×Dic61D411yescharacteristic
86C8×Dic61C2311yescharacteristic
87C2×Q16×C61—21no—
88C2×Q16×C61C2312yes—
89C2×D8×C61C2311yescharacteristic
90C4⋊Q8×C61—21no—
91C4×D4×C61—21no—
92SD32×C61—24no—
93C2×C16×C61C2312yes—
94#107×C61C2311yescharacteristic
95#102×C61—21no—
96C4×C8×C61C2311yescharacteristic
97C23×D61—41no—
98C2×Q8×C61—22no—
99C2×D4×C61—21no—
100C22×C4×C61—41no—
101C2×C22⋊D61—41no—
102C22×Dic61—41no—
103#11758226—44no—
104#11758226—44no—
105D4×D61—24no—
106D4×D61—42no—
107C2×D244—21no—
108C2×D244—22no—
109C2×D244—42no—
110C2×D244—41no—
111C2×D244—41no—
112C2×C4×D61—41no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61—42no—
116C2×C4×D61—84no—
117C2×C4×D61D811yescharacteristic
118C2×C4×D61D811yescharacteristic
119C2×C4×D61C2×D411yescharacteristic
120Q16×C61—22no—
121Q16×C61—22no—
122Q16×C61—42no—
123D8×C61—21no—
124D8×C61C2×D412yes—
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2411yescharacteristic
127C4⋊C4×C61—21no—
128C4⋊C4×C61—41no—
129C4⋊C4×C61—41no—
130C22⋊C4×C61—41no—
131C42×C61C2×D411yescharacteristic
132C22⋊Dic61—41no—
133#11758201—44no—
134#11758201—44no—
135#11758199—24no—
136#11758198—41no—
137#11758198—41no—
138#11758198—42no—
139#11758198—42no—
140C4⋊Dic61—21no—
141#11758196—41no—
142C4×Dic61D811yescharacteristic
143C4×Dic61D811yescharacteristic
144C4×Dic61C2×D411yescharacteristic
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147D488—22no—
148D488—21no—
149D488—22no—
150D488—42no—
151D488C2×D412yes—
152C8×D61—22no—
153C8×D61C2×D412yes—
154C8×D61#4112yes—
155C16×C61—22no—
156#11758185—22no—
157C23×C61—41no—
158C22×D61—41no—
159C22×D61—44no—
160C22×D61—42no—
161C22×D61—82no—
162C22×D61—82no—
163C22×D61C2×D811yescharacteristic
164Q8×C61—42no—
165Q8×C61—42no—
166D4×C61—22no—
167D4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—82no—
171C2×C4×C61C2×D811yescharacteristic
172C2×C4×C61C2×D811yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C22⋊D61—44no—
175C2×Dic61—41no—
176C2×Dic61—82no—
177C2×Dic61C2×D811yescharacteristic
178C2×Dic61C2×D811yescharacteristic
179C2×Dic61C2×D811yescharacteristic
180D244—22no—
181D244—42no—
182D244—41no—
183D244—42no—
184D244—82no—
185D244—84no—
186D244—82no—
187C4×D61—22no—
188C4×D61—41no—
189C4×D61—82no—
190C4×D61—82no—
191C4×D61—82no—
192C4×D61—84no—
193C4×D61#12112yes—
194C4×D61AGL(1,ℤ/8)12yes—
195C8×C61—21no—
196C8×C61C22×D411yescharacteristic
197C8×C61C2×#4111yescharacteristic
198#92418—21no—
199#92418#12112yes—
200#33580—611nomaximal, Sylow
201C22×C61—42no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61C22×D811yescharacteristic
205C2×D61—21no—
206C2×D61—21no—
207C2×D61—42no—
208C2×D61—88no—
209C2×D61—81no—
210C2×D61—81no—
211C2×D61—82no—
212C2×D61—82no—
213C2×D61—82no—
214C2×D61#45912yes—
215C4×C61—21no—
216C4×C61—81no—
217C4×C61—82no—
218C4×C61#50912yescomplemented
219C4×C61C2×#12111yescharacteristic
220C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
221Dic61—21no—
222Dic61—21no—
223Dic61—81no—
224Dic61#45912yes—
225C2×#509—612no—
226#3147—611no—
227C2×C4×D8—611no—
228#2210—612no—
229#2210—612no—
230#2208—611no—
231#2208D6111yescharacteristic, complemented
232#2206—611no—
233#2206—611no—
234#2150—611no—
235#2139—612no—
236C2×C61—82no—
237C2×C61C2×#45911yescharacteristic
238C2×C61C2×#50911yescharacteristic
239C2×C61#341411yescharacteristic
240D61—22no—
241D61—82no—
242D61—162no—
243C2×#135—612no—
244C2×#135—1221no—
245C22×D8—611no—
246#532—1221no—
247C2×C4×D4—1221no—
248#509—1228no—
249C2×SD32—612no—
250C2×SD32C2×D6112yes—
251C2×D16—612no—
252C2×D16—612no—
253C2×M32—612no—
254#495—612no—
255#495—1221no—
256#494C2×D6111yescharacteristic, complemented
257#493—611no—
258#492—1221no—
259C4×D8—611no—
260C4×D8—611no—
261C4×D8—611no—
262C4×D8—1221no—
263C4×D8—1221no—
264C4×D8C2×D6111yescharacteristic
265C2×#107—611no—
266C2×#102—1221no—
267C2×C4×C8—611no—
268C4⋊C16—612no—
269C4⋊C16—611no—
270C4⋊C16C2×D6111yescharacteristic, complemented
271#358—612no—
272#358C2×D6112yescomplemented
273#357—612no—
274#357—612no—
275C61#3358011yescharacteristic, complemented, Sylow
276C2×#41—1222no—
277C22×D4—1221no—
278C23×C4—2441no—
279#135—1224no—
280#135—1224no—
281#135—2444no—
282C2×Q16—21no—
283C2×Q16C22×D6112yes—
284C2×SD16—1222no—
285C2×SD16—1222no—
286C2×D8—614no—
287C2×D8—611no—
288C2×D8—612no—
289C2×D8—1222no—
290C2×D8—1221no—
291C2×D8—1221no—
292C2×D8C22×D6111yescharacteristic
293C22×C8—611no—
294C22×C8—611no—
295C4⋊Q8—21no—
296#127—1221no—
297#124—1222no—
298#121—2441no—
299#121—2441no—
300C4×D4—21no—
301C4×D4—1221no—
302C4×D4—1221no—
303C4×D4—1221no—
304C4×D4—2441no—
305C4×D4—2441no—
306C4×D4—2441no—
307C4×D4—2441no—
308C2×C4⋊C4—1221no—
309C2×C22⋊C4—2441no—
310C2×C42—611no—
311SD32—24no—
312SD32—1224no—
313D16—1224no—
314D16—1224no—
315M32—1224no—
316C2×C16—612no—
317C2×C16C22×D6112yes—
318#107—611no—
319#107—1221no—
320#107C22×D6111yescharacteristic
321#102—21no—
322#102—1221no—
323#102—1221no—
324#102—1221no—
325C4×C8—611no—
326C4×C8—611no—
327C4×C8—611no—
328C4×C8C22×D6111yescharacteristic
329C24—2441no—
330#41—2444no—
331#41—2444no—
332C2×Q8—22no—
333C2×D4—21no—
334C2×D4—1224no—
335C2×D4—1221no—
336C2×D4—1222no—
337C2×D4—2442no—
338C2×D4—2442no—
339C2×D4—2441no—
340C2×D4—2441no—
341C2×D4—2441no—
342C22×C4—41no—
343C22×C4—611no—
344C22×C4—611no—
345C22×C4—611no—
346C22×C4—4884no—
347C22×C4—2441no—
348C22×C4—2442no—
349C22×C4—2441no—
350C22×C4—2441no—
351C22×C4—2441no—
352Q16—22no—
353Q16—22no—
354Q16—42no—
355SD16—2444no—
356SD16—2444no—
357D8—21no—
358D8—612no—
359D8—1224no—
360D8—1222no—
361D8—1221no—
362D8—1222no—
363D8—2442no—
364D8D4×D6112yes—
365C2×C8—611no—
366C2×C8—612no—
367C2×C8—612no—
368C2×C8—611no—
369C2×C8—1222no—
370C2×C8D4×D6111yescharacteristic
371C2×C8C23×D6111yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—1221no—
376C4⋊C4—2441no—
377C22⋊C4—41no—
378C22⋊C4—2441no—
379C22⋊C4—2442no—
380C22⋊C4—2442no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C42—611no—
384C42—611no—
385C42—611no—
386C42D4×D6111yescharacteristic
387C16—22no—
388C16—1222no—
389C23—41no—
390C23—611no—
391C23—4882no—
392C23—4882no—
393C23—2444no—
394C23—2442no—
395C23—2441no—
396Q8—42no—
397Q8—42no—
398D4—22no—
399D4—41no—
400D4—1222no—
401D4—4882no—
402D4—4884no—
403D4—4882no—
404D4—2444no—
405D4—2442no—
406D4—2442no—
407D4—2441no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—611no—
412C2×C4—612no—
413C2×C4—612no—
414C2×C4—611no—
415C2×C4—611no—
416C2×C4—1222no—
417C2×C4—4884no—
418C2×C4—4882no—
419C2×C4—4882no—
420C2×C4—4882no—
421C2×C4—4882no—
422C2×C4—2441no—
423C2×C4—2441no—
424C2×C4D8×D6111yescharacteristic
425C2×C4D8×D6111yescharacteristic
426C2×C4C2×D4×D6111yescharacteristic
427C8—21no—
428C8—612no—
429C8—1221no—
430C8C2×D4×D6111yescharacteristic
431C8#41×D6111yescharacteristic
432C22—42no—
433C22—81no—
434C22—81no—
435C22—612no—
436C22—1221no—
437C22—1221no—
438C22—4888no—
439C22—4882no—
440C22—4882no—
441C22—4882no—
442C22—4881no—
443C22—4881no—
444C22—2442no—
445C22C2×D8×D6111yescharacteristic
446C4—21no—
447C4—82no—
448C4—81no—
449C4—612no—
450C4—1221no—
451C4—1221no—
452C4—4881no—
453C4#1684019912yescomplemented
454C4#121×D6111yescharacteristic
455C4AGL(1,ℤ/8)×D6111yescharacteristic
456C2—82no—
457C2—1222no—
458C2—9762no—
459C2—4882no—
460C2#459×D6111yescharacteristic
461C2C2×#1684019911yescharacteristic
462C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2208C2
socleC22×C61C22×D8
hypercenter#2208D61
hypocenterC61#33580
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2208

Complements

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

NH ≅ G/Ntype
1. C2×#16840199C2semidirect
2. #21456248C2semidirect
3. C4×D8×D61C2semidirect
4. #21447877C2semidirect
5. #21447876C2semidirect
6. #21447861C2semidirect
7. #21447855C2semidirect
8. #21447854C2semidirect
9. #21447573C2semidirect
10. #21447530C2semidirect
11. C61×#2208C2semidirect
23. #16840147C22semidirect
24. #16840139C22semidirect
34. C4⋊#11758185C22semidirect
35. #16839580C22semidirect
36. #16839566C22semidirect
37. #16839564C22semidirect
38. #16839562C22semidirect
40. C61×#494C22semidirect
42. C61×C4⋊C16C22semidirect
43. C61×#358C22semidirect
57. D8×D61D4semidirect
218. C4×C61#509semidirect
231. #2208D61semidirect
256. #494C2×D61semidirect
270. C4⋊C16C2×D61semidirect
272. #358C2×D61semidirect
275. C61#33580semidirect
453. C4#16840199semidirect

Maximal subgroups

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

subgroupindex
C2×#168401992
#214562482
C4×D8×D612
#214478772
#214478762
#214478612
#214478552
#214478542
#214475732
#214475302
C61×#22082
#3358061