Finite Group Databasebeta

← #29663807

Subgroups — #29663807

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#21455616C211yesmaximal, characteristic
3C2×D4×#92418C211yesmaximal, characteristic
4#21445272C212yesmaximal, complemented
5#21445268C212yesmaximal, complemented
6#21445216C212yesmaximal, complemented
7#21445182C212yesmaximal, complemented
8#21444577C211yesmaximal, characteristic, complemented
9#21444510C211yesmaximal, characteristic, complemented
10#21444509C211yesmaximal, characteristic, complemented
11C61×#3077C211yesmaximal, characteristic
12C23×#92418—21no—
13#16840319C2211yescharacteristic
14#16840165C2211yescharacteristic, complemented
15#16840141C2211yescharacteristic, complemented
16#16840114C2211yescharacteristic, complemented
17#16840112C2211yescharacteristic, complemented
18#16840062—22no—
19#16840052C2212yes—
20#16840042C2212yes—
21#16840001—21no—
22#16840001—21no—
23C4×#11758202—21no—
24C4×#11758202—21no—
25D4×#92418—21no—
26D4×#92418—21no—
27D4×#92418C2212yes—
28D4×#92418C2212yes—
29#16839802—21no—
30#16839786C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839767—21no—
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yescomplemented
36#16839514C2212yescomplemented
37#16839511C2212yescomplemented
38#16839510C2212yescomplemented
39#16839508C2212yescomplemented
40#16839507C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#478—22no—
44C61×#476C2212yes—
45C61×#474C2212yes—
46C61×#428—21no—
47C61×#423C2211yescharacteristic
48C61×#418—21no—
49C2×#11758202—21no—
50C2×#11758202—21no—
51C2×#11758194—21no—
52C22×#92418—21no—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418—44no—
56C22×#92418D411yescharacteristic
57C22×#92418D411yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66#14519710—22no—
67#14519710—21no—
68#14519710—21no—
69#14519708—22no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519701—21no—
75#14519701—21no—
76#14519700—21no—
77#14519700—21no—
78#14519698—22no—
79#14519698—22no—
80C22⋊#92418—22no—
81C22⋊#92418—22no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×M16×C61—21no—
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100#117×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#107×C61—22no—
105#106×C61—22no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#102×C61—22no—
109#102×C61—22no—
110C24×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61Q1614yescomplemented
116C2×D4×C61C2×D412yes—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61#4111yescharacteristic
125C2×Dic122—21no—
126C2×Dic122—21no—
127M16×C61—42no—
128C2×C8×C61—22no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61C2×D412yes—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—41no—
140C42×C61—41no—
141C42×C61C2×D411yescharacteristic
142C42×C61C2×D411yescharacteristic
143C42×C61#4111yescharacteristic
144C42×C61C2411yescharacteristic
145C22⋊Dic61—41no—
146#11758202—42no—
147#11758202—42no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152#11758196—41no—
153#11758196—41no—
154C4×Dic61—41no—
155#11758194—42no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—41no—
160C2×#92418—44no—
161C2×#92418—42no—
162C2×#92418—44no—
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C23×C61—22no—
168C23×C61—24no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61#12111yescharacteristic
172Q8×C61—41no—
173Q8×C61—41no—
174Q8×C61—81no—
175D4×C61—22no—
176D4×C61—24no—
177D4×C61C2×Q1614yes—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—41no—
182C2×C4×C61—44no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61C2×Q1612yes—
192C2×C4×C61#13512yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×Dic61—41no—
197C2×Dic61—41no—
198Dic122—41no—
199Dic122—41no—
200Dic122—81no—
201C8×C61—42no—
202#92418—21no—
203#92418—21no—
204#92418—21no—
205#92418—42no—
206#92418—42no—
207#92418AGL(1,ℤ/8)12yes—
208#19256—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—24no—
212C22×C61—48no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61#46712yes—
216C22×C61C2×#12111yescharacteristic
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—42no—
221C4×C61—81no—
222C4×C61—81no—
223C4×C61#46712yes—
224C4×C61C22×Q1611yescharacteristic
225C4×C61C2×#13511yescharacteristic
226C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C4×C61#57511yescharacteristic
228Dic61—81no—
229Dic61—81no—
230#3170—611no—
231#3077—611no—
232#3077D6111yescharacteristic
233C2×C8×D4—611no—
234#1683—612no—
235#1683—612no—
236#1675—612no—
237#1669—612no—
238#1567—611no—
239#1550—611no—
240#1550—611no—
241C2×C61—22no—
242C2×C61—44no—
243C2×C61#303611yescharacteristic
244C2×C61C2×#46711yescharacteristic
245C2×C61#306711yescharacteristic
246C23×C8—1221no—
247#533—611no—
248#533C2×D6111yescharacteristic
249C2×C4×D4C2×D6111yescharacteristic
250#500—611no—
251#494—611no—
252#487—611no—
253#487—611no—
254#478—22no—
255#478—1222no—
256#476—612no—
257#476C2×D6112yes—
258#474—612no—
259#474C2×D6112yes—
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—21no—
269#428—1221no—
270#423—611no—
271#423C2×D6111yescharacteristic
272C2×C4⋊C8—611no—
273#418—21no—
274#418—1221no—
275C2×C22⋊C8—1221no—
276C2×C4×C8—611no—
277#335—612no—
278#335—612no—
279#333—612no—
280#333—612no—
281#332—612no—
282#332—612no—
283C61#1925611yescharacteristic, complemented, Sylow
284C22×D4C22⋊D6111yescharacteristic
285C23×C4—21no—
286C2×Q16—1221no—
287C2×Q16—1221no—
288C2×M16—21no—
289C2×M16—1221no—
290C22×C8—611no—
291C22×C8—611no—
292C22×C8—1222no—
293C22×C8—1222no—
294C22×C8—1221no—
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—21no—
311C4×D4—21no—
312C4×D4C22⋊D6112yes—
313C4×D4C22×D6112yes—
314#117—21no—
315#117—1221no—
316C2×C4⋊C4C22⋊D6111yescharacteristic
317C2×C22⋊C4—21no—
318C2×C42C22×D6111yescharacteristic
319#107—22no—
320#107—1221no—
321#107—1221no—
322#107—1222no—
323#106—22no—
324#106—1222no—
325C4⋊C8—21no—
326C4⋊C8—612no—
327C4⋊C8—612no—
328C4⋊C8—1221no—
329C4⋊C8—1221no—
330C4⋊C8C22×D6112yes—
331#103—1221no—
332#103—1221no—
333#103—1221no—
334#103—1221no—
335#102—22no—
336#102—22no—
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—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—1221no—
349C2×Q8—1221no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4#1175820214yescomplemented
353C2×D4C2×C22⋊D6112yes—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4D4×D6111yescharacteristic
360C22×C4#1175822411yescharacteristic
361C22×C4C2×C22⋊D6111yescharacteristic
362Q16—2442no—
363Q16—2442no—
364M16—42no—
365M16—2442no—
366C2×C8—22no—
367C2×C8—611no—
368C2×C8—611no—
369C2×C8—611no—
370C2×C8—611no—
371C2×C8—1222no—
372C2×C8—1222no—
373C2×C8—1222no—
374C2×C8—2444no—
375C2×C8—2441no—
376C2×C8—2444no—
377C2×C8—2442no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—2441no—
390C4⋊C4—2441no—
391C4⋊C4C2×C22⋊D6112yes—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—41no—
395C22⋊C4—2441no—
396C42—41no—
397C42—2441no—
398C42D4×D6111yescharacteristic
399C42#1175822411yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—22no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23#1451974911yescharacteristic
407Q8—41no—
408Q8—41no—
409Q8—81no—
410Q8—4881no—
411Q8—2441no—
412Q8—2441no—
413D4—24no—
414D4—22no—
415D4C2×#1175820214yes—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4#1451974911yescharacteristic
429C2×C4#1451974911yescharacteristic
430C2×C4#1451974911yescharacteristic
431C2×C4C2×#1175820212yes—
432C2×C4#1451979712yes—
433C2×C4C2×D4×D6111yescharacteristic
434C2×C4C22×C22⋊D6111yescharacteristic
435C2×C4C2×#1175822411yescharacteristic
436C8—42no—
437C8—612no—
438C8—1221no—
439C8—1221no—
440C8—1221no—
441C8—2442no—
442C8—2442no—
443C22—21no—
444C22—21no—
445C22—24no—
446C22—48no—
447C22—42no—
448C22—42no—
449C22#1684000112yes—
450C22C2×#1451974911yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—4881no—
458C4—4881no—
459C4#1684000112yes—
460C4C22×#1175820211yescharacteristic
461C4C2×#1451979711yescharacteristic
462C4AGL(1,ℤ/8)×D6111yescharacteristic
463C4#1684055811yescharacteristic
464C2—22no—
465C2—44no—
466C2#2145528011yescharacteristic
467C2C2×#1684000111yescharacteristic
468C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3077C2
socleC22×C61C2×#121
prime radicalC22×D4×C61D4
hypercenter#3077D61
hypocenterC61#19256
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #21456401C2semidirect
4. #21445272C2semidirect
5. #21445268C2semidirect
6. #21445216C2semidirect
7. #21445182C2semidirect
8. #21444577C2semidirect
9. #21444510C2semidirect
10. #21444509C2semidirect
14. #16840165C22semidirect
15. #16840141C22semidirect
16. #16840114C22semidirect
17. #16840112C22semidirect
35. #16839514C22semidirect
36. #16839514C22semidirect
37. #16839511C22semidirect
38. #16839510C22semidirect
39. #16839508C22semidirect
40. #16839507C22semidirect
115. C2×D4×C61Q16semidirect
283. C61#19256semidirect
352. C2×D4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214564012
#214556162
C2×D4×#924182
#214452722
#214452682
#214452162
#214451822
#214445772
#214445102
#214445092
C61×#30772
#1925661