Finite Group Databasebeta

← #29663809

Subgroups — #29663809

Order 15616.

690 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
1#21456401C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3#21455612C211yesmaximal, characteristic
4C2×#16839827C211yesmaximal, characteristic
5#21445273C212yesmaximal, complemented
6#21445269C212yesmaximal, complemented
7#21445217C212yesmaximal, complemented
8#21445183C212yesmaximal, complemented
9#21444577C211yesmaximal, characteristic, complemented
10#21444507C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C22×C8×D61—21no—
13#16840319C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840165C2211yescharacteristic, complemented
16#16840141C2211yescharacteristic, complemented
17#16840114C2211yescharacteristic, complemented
18#16840112C2211yescharacteristic, complemented
19#16840063—22no—
20#16840059—22no—
21#16840053C2212yes—
22#16840049C2212yes—
23#16840043C2212yes—
24#16840039C2212yes—
25#16840001—21no—
26#16840001—21no—
27C4×#11758202—21no—
28C4×#11758202—21no—
29#16839827—21no—
30#16839827—21no—
31#16839827C2212yes—
32#16839827C2212yes—
33#16839805—21no—
34#16839804—21no—
35#16839786C2211yescharacteristic
36#16839768—21no—
37#16839766—21no—
38C2×#14519689—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839514C2212yescomplemented
42#16839510C2212yescomplemented
43#16839509C2212yescomplemented
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yescomplemented
48C61×#332C2212yescomplemented
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758202—21no—
52C2×#11758202—21no—
53C2×#11758194—21no—
54C22×#92418D411yescharacteristic
55C22×#92418D411yescharacteristic
56C2×C8×D61—22no—
57C2×C8×D61—22no—
58C2×C8×D61—44no—
59#14519777C2311yescharacteristic
60#14519777C2311yescharacteristic
61#14519769—21no—
62#14519756—21no—
63#14519756—21no—
64C4×Dic122—21no—
65C4×Dic122—21no—
66C4×D244—21no—
67C4×D244—21no—
68C4×D244D412yes—
69C4×D244C2312yes—
70#14519736—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×#11758198—21no—
73#14519711—22no—
74#14519710—22no—
75#14519710—21no—
76#14519710—21no—
77#14519709—22no—
78#14519708—22no—
79C4⋊#92418—21no—
80C4⋊#92418C2312yes—
81#14519701—21no—
82#14519701—21no—
83#14519700—21no—
84#14519700—21no—
85#14519699—22no—
86#14519699—22no—
87#14519697—22no—
88#14519697—22no—
89#14519689—22no—
90#14519689—22no—
91C4×#92418D411yescharacteristic
92C4×#92418D411yescharacteristic
93C4×#92418C2311yescharacteristic
94C4×#92418C2311yescharacteristic
95C2×M16×C61—21no—
96C22×C8×C61—21no—
97C4⋊Q8×C61C2311yescharacteristic
98C4⋊Q8×C61C2311yescharacteristic
99#125×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61—21no—
104#117×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C4⋊C8×C61C2312yes—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×Dic61—21no—
117C2×D244—21no—
118C2×D244—22no—
119C2×D244Q1614yescomplemented
120C2×D244C2×D412yes—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127M16×C61—42no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C2×C8×C61—42no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C42×C61—41no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143C22⋊Dic61—41no—
144#11758202—42no—
145#11758202—42no—
146#11758198—22no—
147#11758198—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61C2×D412yes—
154#11758196—41no—
155#11758196—41no—
156C4×Dic61—41no—
157#11758194—42no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—41no—
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C8×D61—44no—
166C8×D61—44no—
167C23×C61#12111yescharacteristic
168C22×D61—22no—
169C22×D61—24no—
170C22×D61—42no—
171C22×D61—42no—
172Q8×C61—41no—
173Q8×C61—41no—
174Q8×C61—81no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61C2×Q1612yes—
184C2×C4×C61#13512yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×Dic61—22no—
189C2×Dic61—42no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192D244—22no—
193D244—24no—
194D244C2×Q1614yes—
195C4×D61—44no—
196C4×D61—42no—
197C4×D61—42no—
198Dic122—41no—
199Dic122—41no—
200Dic122—81no—
201C8×C61—42no—
202C8×C61—42no—
203#92418—21no—
204#92418—21no—
205#92418—21no—
206#92418—42no—
207#92418AGL(1,ℤ/8)12yes—
208#19256—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61#46712yes—
212C22×C61C2×#12111yescharacteristic
213C2×D61—24no—
214C2×D61—48no—
215C2×D61—42no—
216C2×D61—42no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—81no—
221C4×C61—81no—
222C4×C61#46712yes—
223C4×C61C22×Q1611yescharacteristic
224C4×C61C2×#13511yescharacteristic
225C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
226C4×C61#57511yescharacteristic
227Dic61—42no—
228Dic61—81no—
229Dic61—81no—
230#3170—611no—
231#3077—611no—
232#3077—611no—
233C2×C8×D4—611no—
234#1683—612no—
235#1683—612no—
236#1675—612no—
237#1669—612no—
238#1567—611no—
239#1550—611no—
240#1550D6111yescharacteristic, complemented
241C2×C61—22no—
242C2×C61#303611yescharacteristic
243C2×C61C2×#46711yescharacteristic
244C2×C61#306711yescharacteristic
245D61—44no—
246C23×C8—1221no—
247#533—611no—
248#533C2×D6111yescharacteristic
249C2×C4×D4—611no—
250#500—611no—
251#494—611no—
252#487—611no—
253#487—611no—
254#478—1222no—
255#478—1222no—
256#476—612no—
257#476—612no—
258#474—612no—
259#474—612no—
260#467—1221no—
261#467—1221no—
262C4×Q16—1221no—
263C4×Q16—1221no—
264C8×D4—612no—
265C8×D4—612no—
266C8×D4—1221no—
267C8×D4—1221no—
268#428—1221no—
269#428—1221no—
270#423—611no—
271#423C2×D6111yescharacteristic
272C2×C4⋊C8C2×D6111yescharacteristic
273#418—1221no—
274#418—1221no—
275C2×C22⋊C8—1221no—
276C2×C4×C8—611no—
277#335—612no—
278#335—612no—
279#333—612no—
280#333C2×D6112yescomplemented
281#332—612no—
282#332C2×D6112yescomplemented
283C61#1925611yescharacteristic, complemented, Sylow
284C22×D4—611no—
285C23×C4—1221no—
286C2×Q16—1221no—
287C2×Q16—1221no—
288C2×M16—21no—
289C2×M16—1221no—
290C22×C8—21no—
291C22×C8—611no—
292C22×C8—611no—
293C22×C8—1222no—
294C22×C8—1222no—
295C22×C8—2444no—
296C4⋊Q8—611no—
297C4⋊Q8—611no—
298C4⋊Q8C22×D6111yescharacteristic
299C4⋊Q8C22×D6111yescharacteristic
300#125—21no—
301#125—1221no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304C22⋊Q8—1221no—
305C22⋊Q8—1221no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—1221no—
309C4×Q8—1221no—
310C4×D4—612no—
311C4×D4—612no—
312C4×D4—1221no—
313C4×D4—1221no—
314#117—21no—
315#117—1221no—
316C2×C4⋊C4—611no—
317C2×C22⋊C4—1221no—
318C2×C42C22×D6111yescharacteristic
319#107—1222no—
320#107—1221no—
321#107—1221no—
322#107—1222no—
323#106—1222no—
324#106—1222no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—612no—
328C4⋊C8—1221no—
329C4⋊C8C22×D6112yes—
330C4⋊C8C22×D6112yes—
331#103—1221no—
332#103—1221no—
333#103—1221no—
334#103—1221no—
335#102—1222no—
336#102—1222no—
337#102—1222no—
338#102—1222no—
339C22⋊C8—1222no—
340C22⋊C8—1222no—
341C4×C8—611no—
342C4×C8—611no—
343C4×C8—611no—
344C4×C8—611no—
345C24—1221no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—1221no—
349C2×Q8—1221no—
350C2×D4—612no—
351C2×D4—614no—
352C2×D4—1221no—
353C2×D4—1222no—
354C22×C4—1221no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4#1175822211yescharacteristic
360C22×C4D4×D6111yescharacteristic
361C22×C4D4×D6111yescharacteristic
362Q16—2442no—
363Q16—2442no—
364M16—42no—
365M16—2442no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—42no—
369C2×C8—611no—
370C2×C8—611no—
371C2×C8—611no—
372C2×C8—611no—
373C2×C8—1222no—
374C2×C8—1222no—
375C2×C8—2444no—
376C2×C8—2444no—
377C2×C8—2441no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—612no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—2441no—
391C4⋊C4—2441no—
392C22⋊C4—41no—
393C22⋊C4—1222no—
394C22⋊C4—1222no—
395C22⋊C4—2441no—
396C42—41no—
397C42—2441no—
398C42#1175822211yescharacteristic
399C42D4×D6111yescharacteristic
400C42D4×D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—1222no—
403C23—1224no—
404C23—2442no—
405C23—2442no—
406C23#1451975011yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—81no—
410Q8—4881no—
411Q8—2441no—
412Q8—2441no—
413D4—614no—
414D4—1224no—
415D4—1222no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—1222no—
422C2×C4—2444no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2442no—
426C2×C4—2442no—
427C2×C4—2442no—
428C2×C4#1451975011yescharacteristic
429C2×C4#1451975011yescharacteristic
430C2×C4#1451975011yescharacteristic
431C2×C4Q16×D6112yes—
432C2×C4#1451979812yes—
433C2×C4C2×D4×D6111yescharacteristic
434C2×C4C2×D4×D6111yescharacteristic
435C2×C4C2×#1175822211yescharacteristic
436C8—42no—
437C8—42no—
438C8—612no—
439C8—1221no—
440C8—1221no—
441C8—1221no—
442C8—2442no—
443C22—21no—
444C22—21no—
445C22—1224no—
446C22—2448no—
447C22—2442no—
448C22—2442no—
449C22#1684000212yes—
450C22C2×#1451975011yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—81no—
455C4—81no—
456C4—4881no—
457C4—4881no—
458C4—2442no—
459C4#1684000212yes—
460C4C2×Q16×D6111yescharacteristic
461C4C2×#1451979811yescharacteristic
462C4AGL(1,ℤ/8)×D6111yescharacteristic
463C4#1684056111yescharacteristic
464C2—22no—
465C2—2444no—
466C2#2145528211yescharacteristic
467C2C2×#1684000211yescharacteristic
468C2#2145553811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1550D61
hypocenterC61#19256
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21456401C2semidirect
5. #21445273C2semidirect
6. #21445269C2semidirect
7. #21445217C2semidirect
8. #21445183C2semidirect
9. #21444577C2semidirect
10. #21444507C2semidirect
11. C61×#1550C2semidirect
15. #16840165C22semidirect
16. #16840141C22semidirect
17. #16840114C22semidirect
18. #16840112C22semidirect
40. #16839514C22semidirect
41. #16839514C22semidirect
42. #16839510C22semidirect
43. #16839509C22semidirect
47. C61×#333C22semidirect
48. C61×#332C22semidirect
119. C2×D244Q16semidirect
240. #1550D61semidirect
280. #333C2×D61semidirect
282. #332C2×D61semidirect
283. C61#19256semidirect

Maximal subgroups

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

subgroupindex
#214564012
#214556182
#214556122
C2×#168398272
#214452732
#214452692
#214452172
#214451832
#214445772
#214445072
C61×#15502
#1925661