Finite Group Databasebeta

← #29663587

Subgroups — #29663587

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#21456757C211yesmaximal, characteristic, complemented
2#21456272C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21445301C212yesmaximal, complemented
5#21445293C212yesmaximal, complemented
6#21445177C212yesmaximal, complemented
7#21445169C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444468C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#3246C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C2×#14519802—21no—
14C2×#14519774C2211yescharacteristic
15#16840152—22no—
16#16840138C2212yes—
17#16840136—21no—
18#16840136C2212yes—
19#16840110—21no—
20#16839962C2212yes—
21#16839962C2212yes—
22#16839912—22no—
23#16839893—22no—
24#16839893C2212yescomplemented
25Q8×#92418C2212yescomplemented
26#16839815C2211yescharacteristic
27#16839789—21no—
28C2×C4⋊#92418C2211yescharacteristic
29#16839718—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yescomplemented
32#16839512C2212yescomplemented
33#16839508C2211yescharacteristic, complemented
34#16839508C2211yescharacteristic, complemented
35#16839506C2211yescharacteristic, complemented
36#16839506C2211yescharacteristic, complemented
37#16839505C2212yes—
38#16839505C2212yes—
39C61×#533C2211yescharacteristic
40C61×#532C2211yescharacteristic
41C61×#476—21no—
42C61×#475C2212yescomplemented
43C61×#465—22no—
44C61×#461—21no—
45C61×#460C2212yescomplemented
46C2×C4⋊C8×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50#14519802—44no—
51#14519802—44no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758199—22no—
57C2×#11758199—22no—
58C2×#11758194D411yescharacteristic
59C2×#11758194D411yescharacteristic
60C22×#92418—21no—
61#14519774—21no—
62#14519774—21no—
63#14519774—21no—
64#14519774C2312yes—
65C4⋊#92418—21no—
66C4⋊#92418—22no—
67C4⋊#92418—22no—
68C4⋊#92418C2311yescharacteristic
69C4⋊#92418C2311yescharacteristic
70C4⋊#92418C2311yescharacteristic
71C4⋊#92418C2311yescharacteristic
72#14519697—22no—
73#14519697—22no—
74C22⋊#92418—21no—
75#14519688—22no—
76#14519688D412yes—
77C4×#92418—21no—
78C4×#92418—22no—
79C4×#92418C2312yes—
80C2×#41×C61—21no—
81C2×SD16×C61—22no—
82C22×C8×C61—21no—
83C4⋊Q8×C61D411yescharacteristic
84C4⋊Q8×C61D411yescharacteristic
85C4⋊Q8×C61C2311yescharacteristic
86#127×C61C2311yescharacteristic
87#125×C61—21no—
88#124×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61D412yes—
91#121×C61—21no—
92#121×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61C2312yes—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97#117×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#106×C61—22no—
100C4⋊C8×C61—21no—
101C4⋊C8×C61C2312yes—
102#103×C61—22no—
103#102×C61—22no—
104C23×D61—41no—
105#41×C61—42no—
106#41×C61—42no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×Q8×C61C2×D412yes—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—42no—
122C2×D244—42no—
123C2×D244—41no—
124C2×D244—41no—
125C2×D244—41no—
126C2×D244—81no—
127C2×D244—81no—
128C2×D244—82no—
129SD16×C61—82no—
130C2×C8×C61—22no—
131C2×C8×C61—81no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61C2×D412yes—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C42×C61—22no—
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61#4111yescharacteristic
147C42×C61C2411yescharacteristic
148#11758201—44no—
149#11758201—44no—
150#11758201—82no—
151#11758201—82no—
152#11758199—44no—
153#11758199—44no—
154#11758194—22no—
155#11758194SD1614yescomplemented
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—42no—
161C2×#92418C2×D412yes—
162C2×#92418C2×D412yes—
163C23×C61—41no—
164C23×C61#12111yescharacteristic
165C22×D61—42no—
166C22×D61—82no—
167C22×D61—82no—
168C22×D61—82no—
169Q8×C61—22no—
170Q8×C61—22no—
171Q8×C61—41no—
172Q8×C61—41no—
173Q8×C61—81no—
174D4×C61—41no—
175D4×C61—41no—
176D4×C61—41no—
177D4×C61—41no—
178D4×C61—81no—
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—41no—
186C2×C4×C61—82no—
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61C2×SD1612yes—
191C2×C4×C61AGL(1,ℤ/8)12yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195D244—42no—
196D244—42no—
197D244—42no—
198D244—42no—
199D244—82no—
200D244—82no—
201D244—82no—
202D244—82no—
203C8×C61—42no—
204#92418—22no—
205#92418—41no—
206#92418—41no—
207#92418—41no—
208#92418—41no—
209#92418C2×SD1614yes—
210#19238—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—81no—
214C22×C61—81no—
215C22×C61—81no—
216C22×C61#46012yes—
217C22×C61C2×#12111yescharacteristic
218C2×D61—82no—
219C2×D61—82no—
220C2×D61—82no—
221C2×D61—164no—
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—42no—
227C4×C61—81no—
228C4×C61#46012yes—
229C4×C61C22×SD1611yescharacteristic
230C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C61#57611yescharacteristic
232C4×C61#57711yescharacteristic
233#3246D6111yescharacteristic, complemented
234#3208—611no—
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—161no—
246C2×C61C2×#46011yescharacteristic
247C2×C61#306211yescharacteristic
248C2×C61#306811yescharacteristic
249D61—162no—
250C2×AGL(1,ℤ/8)—1221no—
251C2×AGL(1,ℤ/8)—1221no—
252#533C2×D6111yescharacteristic
253#532C2×D6111yescharacteristic
254C2×#127—611no—
255#496—1222no—
256#493—612no—
257#492—612no—
258#492—1221no—
259#486—1221no—
260#476—21no—
261#475C2×D6112yescomplemented
262#465—22no—
263#461—21no—
264#460—612no—
265#460—612no—
266#460C2×D6112yescomplemented
267#450—1222no—
268#446—612no—
269#446—1222no—
270C8×Q8—612no—
271#431—611no—
272#424—1221no—
273C2×C4⋊C8—611no—
274C2×C4⋊C8C2×D6111yescharacteristic
275#405—1221no—
276C4×M16—611no—
277#335—612no—
278#334—612no—
279#332—611no—
280#332—611no—
281#331—611no—
282#331—611no—
283#331—612no—
284#331—612no—
285C61#1923811yescharacteristic, complemented, Sylow
286C2×#41—21no—
287C22×D4—1221no—
288C22×D4—1221no—
289C22×D4—1221no—
290AGL(1,ℤ/8)—2444no—
291AGL(1,ℤ/8)—2444no—
292C2×SD16—22no—
293C2×SD16—1222no—
294C2×SD16—1222no—
295C2×SD16—1222no—
296C2×SD16—1222no—
297C2×D8—1222no—
298C2×D8—1222no—
299C2×M16—611no—
300C2×M16—611no—
301C22×C8—21no—
302C22×C8—1221no—
303C4⋊Q8C22⋊D6111yescharacteristic
304C4⋊Q8C22⋊D6111yescharacteristic
305C4⋊Q8C22×D6111yescharacteristic
306#127—612no—
307#127—1221no—
308#127—1221no—
309#127—1221no—
310#127C22×D6111yescharacteristic
311#125—21no—
312#124—21no—
313C22⋊Q8—21no—
314C22⋊Q8C22⋊D6112yes—
315#121—21no—
316#121—21no—
317C4×Q8—21no—
318C4×Q8C22×D6112yes—
319C4×D4—21no—
320C4×D4—21no—
321#117—21no—
322C2×C42C22×D6111yescharacteristic
323#106—22no—
324C4⋊C8—21no—
325C4⋊C8—611no—
326C4⋊C8—611no—
327C4⋊C8—611no—
328C4⋊C8—611no—
329C4⋊C8—1222no—
330C4⋊C8—1222no—
331C4⋊C8—1221no—
332C4⋊C8C22×D6112yes—
333#103—22no—
334#102—22no—
335#102—1222no—
336#102—1222no—
337C22⋊C8—1221no—
338#97—612no—
339#97—1222no—
340C4×C8—612no—
341C4×C8—1222no—
342C4×C8—1221no—
343C24—2441no—
344#41—42no—
345#41—42no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8C2×C22⋊D6112yes—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—1222no—
354C2×D4—1222no—
355C2×D4—1222no—
356C2×D4—4881no—
357C2×D4—4881no—
358C2×D4—4882no—
359C2×D4—2441no—
360C2×D4—2441no—
361C2×D4—2441no—
362C2×D4—2442no—
363C2×D4—2442no—
364C22×C4—41no—
365C22×C4D4×D6111yescharacteristic
366C22×C4#1175822411yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368SD16—82no—
369SD16—4882no—
370SD16—4882no—
371SD16—2444no—
372SD16—2444no—
373D8—2444no—
374D8—2444no—
375M16—614no—
376M16—1222no—
377C2×C8—22no—
378C2×C8—81no—
379C2×C8—612no—
380C2×C8—612no—
381C2×C8—1221no—
382C2×C8—1221no—
383C2×C8—1221no—
384C2×C8—1221no—
385C2×C8—2442no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4C2×C22⋊D6112yes—
394C22⋊C4—22no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C42—22no—
398C42D4×D6111yescharacteristic
399C42#1175822411yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—41no—
403C23—4882no—
404C23—4882no—
405C23—4882no—
406C23—2442no—
407C23#1451974911yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—41no—
411Q8—41no—
412Q8—81no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—41no—
417D4—81no—
418D4—82no—
419D4—4882no—
420D4—4882no—
421D4—4882no—
422D4—4882no—
423D4—2442no—
424D4—2442no—
425D4—2442no—
426D4—2442no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—82no—
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—42no—
443C8—614no—
444C8—1222no—
445C8—2441no—
446C8—2441no—
447C8—2441no—
448C8—2441no—
449C22—21no—
450C22—21no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—9764no—
455C22—4882no—
456C22—4882no—
457C22—4882no—
458C22#1683996012yes—
459C22C2×#1451974911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—81no—
466C4#1683996012yes—
467C4C2×SD16×D6111yescharacteristic
468C4C2×#1451980411yescharacteristic
469C4#1684056511yescharacteristic
470C4#1684057011yescharacteristic
471C2—22no—
472C2—161no—
473C2—9762no—
474C2C2×#1683996011yescharacteristic
475C2#2145550111yescharacteristic
476C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3246C2
socleC22×C61C2×#121
prime radical#21456272C2
hypercenter#3246D61
hypocenterC61#19238
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3246

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
3. #21454768C2semidirect
4. #21445301C2semidirect
5. #21445293C2semidirect
6. #21445177C2semidirect
7. #21445169C2semidirect
8. #21444569C2semidirect
9. #21444468C2semidirect
10. C2×#16839505C2semidirect
11. C61×#3246C2semidirect
24. #16839893C22semidirect
25. Q8×#92418C22semidirect
31. #16839514C22semidirect
32. #16839512C22semidirect
33. #16839508C22semidirect
34. #16839508C22semidirect
35. #16839506C22semidirect
36. #16839506C22semidirect
42. C61×#475C22semidirect
45. C61×#460C22semidirect
155. #11758194SD16semidirect
233. #3246D61semidirect
261. #475C2×D61semidirect
266. #460C2×D61semidirect
285. C61#19238semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214562722
#214547682
#214453012
#214452932
#214451772
#214451692
#214445692
#214444682
C2×#168395052
C61×#32462
#1923861