Finite Group Databasebeta

← #29663581

Subgroups — #29663581

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457131C211yesmaximal, characteristic, complemented
2#21456269C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21445301C212yesmaximal, complemented
5#21445289C212yesmaximal, complemented
6#21445177C212yesmaximal, complemented
7#21445165C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9#21444466C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic, complemented
11C61×#3208C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519803—21no—
14#16840312C2211yescharacteristic
15#16840149—22no—
16#16840138C2212yes—
17#16840134C2212yes—
18#16840049—21no—
19#16840047C2212yescomplemented
20#16839993—22no—
21#16839964—21no—
22#16839962C2212yescomplemented
23#16839893—22no—
24#16839893C2212yescomplemented
25Q8×#92418C2212yescomplemented
26#16839789—21no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839718—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yescomplemented
32#16839512C2212yescomplemented
33#16839507C2211yescharacteristic, complemented
34#16839507C2211yescharacteristic, complemented
35#16839506C2212yes—
36#16839506C2212yes—
37#16839505C2211yescharacteristic, complemented
38#16839505C2211yescharacteristic, complemented
39C61×#533C2211yescharacteristic
40C2×#127×C61C2211yescharacteristic
41C61×#492—21no—
42C61×#486—21no—
43C61×#460C2212yes—
44C61×#460C2212yes—
45C61×#450—22no—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48#14519803—44no—
49#14519803—44no—
50C2×#11758200—22no—
51C2×#11758200—22no—
52C2×#11758201—22no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C2×#11758194D411yescharacteristic
56C2×#11758194D411yescharacteristic
57C22×#92418—21no—
58C22×#92418—21no—
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519708—22no—
67C4⋊#92418—21no—
68C4⋊#92418—21no—
69C4⋊#92418—22no—
70C4⋊#92418—22no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74#14519701—22no—
75#14519697—22no—
76#14519688—22no—
77#14519688D412yes—
78C4×#92418—22no—
79C4×#92418C2312yes—
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C2×SD16×C61—22no—
84C2×SD16×C61—22no—
85C4⋊Q8×C61D411yescharacteristic
86C4⋊Q8×C61D411yescharacteristic
87#127×C61—21no—
88#127×C61—21no—
89#127×C61—21no—
90#127×C61C2312yes—
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61D412yes—
94C4×Q8×C61—21no—
95C4×Q8×C61C2312yes—
96#117×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2311yescharacteristic
100#102×C61—22no—
101#102×C61—22no—
102C22⋊C8×C61—21no—
103C4×C8×C61—21no—
104C24×C61—41no—
105C2×Q8×C61—21no—
106C2×Q8×C61C2×D412yes—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61—81no—
116C2×D4×C61—81no—
117C2×D4×C61—82no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C2×C22⋊D61—41no—
122#11758222—42no—
123#11758222—42no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129SD16×C61—82no—
130SD16×C61—82no—
131C2×C8×C61—21no—
132C2×C8×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61C2×D412yes—
139C22⋊C4×C61—22no—
140C42×C61—22no—
141C42×C61C2×D411yescharacteristic
142C42×C61C2×D411yescharacteristic
143C42×C61#4111yescharacteristic
144C42×C61C2411yescharacteristic
145#11758201—82no—
146#11758200—44no—
147#11758200—44no—
148#11758199—44no—
149#11758199—44no—
150#11758198—41no—
151#11758198—41no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154#11758194—22no—
155#11758194SD1614yescomplemented
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—22no—
159C2×#92418—42no—
160C2×#92418—81no—
161C2×#92418C2×D412yes—
162C2×#92418C2×D412yes—
163C23×C61—42no—
164C23×C61—82no—
165C23×C61—82no—
166C23×C61—82no—
167C23×C61#12111yescharacteristic
168C22×D61—41no—
169Q8×C61—22no—
170Q8×C61—22no—
171Q8×C61—81no—
172D4×C61—42no—
173D4×C61—42no—
174D4×C61—42no—
175D4×C61—42no—
176D4×C61—82no—
177D4×C61—82no—
178D4×C61—82no—
179D4×C61—82no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61C2×SD1612yes—
189C2×C4×C61AGL(1,ℤ/8)12yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C22⋊D61—82no—
194C2×Dic61—41no—
195D244—41no—
196D244—41no—
197D244—41no—
198D244—41no—
199D244—81no—
200C4×D61—82no—
201Dic122—41no—
202Dic122—41no—
203C8×C61—41no—
204C8×C61—41no—
205#92418—22no—
206#92418—41no—
207#92418—41no—
208#92418—42no—
209#92418C2×SD1614yes—
210#19238—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—82no—
214C22×C61—82no—
215C22×C61—82no—
216C22×C61—164no—
217C22×C61#46012yes—
218C22×C61C2×#12111yescharacteristic
219C2×D61—81no—
220C2×D61—81no—
221C2×D61—81no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—42no—
227C4×C61#46012yes—
228C4×C61C22×SD1611yescharacteristic
229C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C61#57611yescharacteristic
231C4×C61#57711yescharacteristic
232Dic61—81no—
233#3246—611no—
234#3208D6111yescharacteristic, complemented
235#3151—611no—
236#2962—611no—
237#1688—612no—
238#1686—612no—
239#1668—612no—
240#1666—612no—
241#1565—611no—
242#1540—611no—
243C2×#331—611no—
244C2×C61—22no—
245C2×C61—162no—
246C2×C61C2×#46011yescharacteristic
247C2×C61#306211yescharacteristic
248C2×C61#306811yescharacteristic
249D61—161no—
250C2×AGL(1,ℤ/8)—1221no—
251C2×AGL(1,ℤ/8)—1221no—
252#533C2×D6111yescharacteristic
253#532—611no—
254C2×#127C2×D6111yescharacteristic
255#496—1222no—
256#493—612no—
257#492—21no—
258#492—612no—
259#486—21no—
260#476—1221no—
261#475—612no—
262#465—1222no—
263#461—1221no—
264#460—612no—
265#460C2×D6112yes—
266#460C2×D6112yes—
267#450—22no—
268#446—612no—
269#446—1222no—
270C8×Q8—612no—
271#431C2×D6111yescharacteristic
272#424—1221no—
273C2×C4⋊C8—611no—
274C2×C4⋊C8—611no—
275#405—1221no—
276C4×M16—611no—
277#335—612no—
278#334—612no—
279#332—611no—
280#332—611no—
281#331—612no—
282#331—612no—
283#331—611no—
284#331—611no—
285C61#1923811yescharacteristic, complemented, Sylow
286C2×#41—1221no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—21no—
290AGL(1,ℤ/8)—2444no—
291AGL(1,ℤ/8)—2444no—
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16—1222no—
295C2×SD16—1222no—
296C2×SD16—1222no—
297C2×D8—1222no—
298C2×D8—1222no—
299C2×M16—611no—
300C2×M16—611no—
301C22×C8—1221no—
302C22×C8—1221no—
303C4⋊Q8—611no—
304C4⋊Q8C22⋊D6111yescharacteristic
305C4⋊Q8C22⋊D6111yescharacteristic
306#127—21no—
307#127—21no—
308#127—21no—
309#127—611no—
310#127C22×D6112yes—
311#125—21no—
312#124—1221no—
313C22⋊Q8—21no—
314C22⋊Q8C22⋊D6112yes—
315#121—1221no—
316#121—1221no—
317C4×Q8—21no—
318C4×Q8C22×D6112yes—
319C4×D4—1221no—
320C4×D4—1221no—
321#117—21no—
322C2×C42C22×D6111yescharacteristic
323#106—1222no—
324C4⋊C8—611no—
325C4⋊C8—611no—
326C4⋊C8—612no—
327C4⋊C8—1222no—
328C4⋊C8—1222no—
329C4⋊C8—1221no—
330C4⋊C8—1221no—
331C4⋊C8C22×D6111yescharacteristic
332C4⋊C8C22×D6111yescharacteristic
333#103—1222no—
334#102—22no—
335#102—22no—
336#102—1222no—
337C22⋊C8—21no—
338#97—612no—
339#97—1222no—
340C4×C8—21no—
341C4×C8—612no—
342C4×C8—1222no—
343C24—41no—
344#41—2442no—
345#41—2442no—
346C2×Q8—21no—
347C2×Q8—1221no—
348C2×Q8C2×C22⋊D6112yes—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—82no—
360C2×D4—1221no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—2441no—
364C22×C4—2441no—
365C22×C4D4×D6111yescharacteristic
366C22×C4#1175822411yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368SD16—82no—
369SD16—82no—
370SD16—4882no—
371SD16—2444no—
372SD16—2444no—
373D8—2444no—
374D8—2444no—
375M16—614no—
376M16—1222no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—612no—
380C2×C8—612no—
381C2×C8—1221no—
382C2×C8—1221no—
383C2×C8—1222no—
384C2×C8—4881no—
385C2×C8—2442no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4C2×C22⋊D6112yes—
394C22⋊C4—22no—
395C22⋊C4—2441no—
396C22⋊C4—2441no—
397C42—22no—
398C42D4×D6111yescharacteristic
399C42#1175822411yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—42no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—2441no—
407C23#1451974911yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—81no—
411Q8—2441no—
412Q8—2441no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—82no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—4881no—
422D4—4882no—
423D4—2441no—
424D4—2441no—
425D4—2441no—
426D4—2441no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—4882no—
433C2×C4—2441no—
434C2×C4#1451974911yescharacteristic
435C2×C4#1451974911yescharacteristic
436C2×C4#1451974911yescharacteristic
437C2×C4SD16×D6112yes—
438C2×C4#1451980412yes—
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C22×C22⋊D6111yescharacteristic
441C2×C4C2×#1175822411yescharacteristic
442C8—41no—
443C8—41no—
444C8—614no—
445C8—1222no—
446C8—2441no—
447C8—2441no—
448C8—2442no—
449C22—21no—
450C22—21no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—164no—
455C22—4881no—
456C22—4881no—
457C22—4881no—
458C22#1683996012yes—
459C22C2×#1451974911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—4881no—
466C4#1683996012yes—
467C4C2×SD16×D6111yescharacteristic
468C4C2×#1451980411yescharacteristic
469C4#1684056511yescharacteristic
470C4#1684057311yescharacteristic
471C2—22no—
472C2—162no—
473C2—9761no—
474C2C2×#1683996011yescharacteristic
475C2#2145549511yescharacteristic
476C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3208C2
socleC22×C61C2×#121
prime radical#21456269C2
hypercenter#3208D61
hypocenterC61#19238
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #21457131C2semidirect
3. #21454768C2semidirect
4. #21445301C2semidirect
5. #21445289C2semidirect
6. #21445177C2semidirect
7. #21445165C2semidirect
8. #21444572C2semidirect
9. #21444466C2semidirect
10. C2×#16839506C2semidirect
11. C61×#3208C2semidirect
19. #16840047C22semidirect
22. #16839962C22semidirect
24. #16839893C22semidirect
25. Q8×#92418C22semidirect
31. #16839514C22semidirect
32. #16839512C22semidirect
33. #16839507C22semidirect
34. #16839507C22semidirect
37. #16839505C22semidirect
38. #16839505C22semidirect
155. #11758194SD16semidirect
234. #3208D61semidirect
285. C61#19238semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214562692
#214547682
#214453012
#214452892
#214451772
#214451652
#214445722
#214444662
C2×#168395062
C61×#32082
#1923861