Finite Group Databasebeta

← #29665190

Subgroups — #29665190

Order 15616.

792 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456737C211yesmaximal, characteristic, complemented
2#21455285C211yesmaximal, characteristic, complemented
3C2×#16839954C211yesmaximal, characteristic, complemented
4#21454842C211yesmaximal, characteristic, complemented
5#21445255C214yesmaximal, complemented
6#21445173C214yesmaximal, complemented
7#21444595C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic, complemented
9C61×#1545C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11C22×#11758199—21no—
12#16840312C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840138C2212yescomplemented
15#16840108C2212yescomplemented
16#16839978—22no—
17#16839968C2212yescomplemented
18#16839958C2212yescomplemented
19#16839954—22no—
20#16839954—22no—
21#16839954C2214yes—
22#16839896—22no—
23C8×D244—22no—
24#16839827C2214yescomplemented
25#16839815C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27#16839764—21no—
28C2×#14519698—21no—
29C22⋊C8×D61—22no—
30#16839514C2214yescomplemented
31#16839507C2212yescomplemented
32#16839505C2212yescomplemented
33C2×#127×C61C2211yescharacteristic
34C61×#431C2211yescharacteristic
35C61×#423C2211yescharacteristic
36C61×#331C2214yescomplemented
37C2×#11758222—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40#14519805—44no—
41C2×#11758190—22no—
42C2×D488—22no—
43C2×#11758199—24no—
44C2×#11758199—24no—
45C2×#11758199—44no—
46C22×#92418—21no—
47C2×C8×D61—24no—
48#14519777C2311yescharacteristic
49#14519774C2311yescharacteristic
50#14519765—21no—
51#14519751—21no—
52#14519751—21no—
53C4×D244—21no—
54C4×D244—21no—
55C4×D244—21no—
56C4×D244—21no—
57C4×D244D412yes—
58C4×D244C2312yes—
59C2×C4⋊Dic61D411yescharacteristic
60C2×#11758198—21no—
61C4⋊#92418—21no—
62C4⋊#92418D412yes—
63C4⋊#92418C2312yes—
64#14519702—22no—
65#14519699—22no—
66#14519698—24no—
67#14519698—24no—
68C22⋊#92418D412yes—
69#14519689—24no—
70C4×#92418C2312yes—
71C22×D4×C61—21no—
72C22×D4×C61—21no—
73C22×D4×C61—21no—
74C2×M16×C61—21no—
75#127×C61—21no—
76#127×C61—21no—
77#127×C61—21no—
78#127×C61C2312yes—
79C2×C42×C61C2311yescharacteristic
80C4⋊C8×C61—21no—
81C4⋊C8×C61C2312yes—
82C4⋊C8×C61C2312yes—
83C22⋊C8×C61—21no—
84C4×C8×C61—21no—
85C24×C61—41no—
86C23×D61—21no—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—41no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61—42no—
94C2×D4×C61—42no—
95C2×D4×C61—82no—
96C2×D4×C61—82no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C2×C22⋊D61—41no—
101C22×Dic61—21no—
102#11758222—42no—
103#11758222—42no—
104C2×D244—21no—
105C2×D244—22no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244D814yescomplemented
110C2×D244C2×D412yes—
111C2×C4×D61—22no—
112C2×C4×D61—22no—
113C2×C4×D61—41no—
114C2×C4×D61—82no—
115C2×Dic122—21no—
116M16×C61—42no—
117C2×C8×C61—22no—
118C2×C8×C61—22no—
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122C42×C61C2411yescharacteristic
123#11758199—48no—
124#11758199—84no—
125#11758198—22no—
126#11758198—22no—
127#11758198—41no—
128#11758198—41no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61C2×D412yes—
133C2×#92418—22no—
134C2×#92418—42no—
135C2×#92418C2×D414yes—
136D488—44no—
137#11758190—44no—
138C8×D61—84no—
139C23×C61—42no—
140C23×C61—82no—
141C23×C61—82no—
142C23×C61—82no—
143C23×C61#12111yescharacteristic
144C22×D61—22no—
145C22×D61—24no—
146C22×D61—41no—
147C22×D61—81no—
148C22×D61—81no—
149D4×C61—42no—
150D4×C61—42no—
151D4×C61—44no—
152D4×C61—82no—
153D4×C61—82no—
154D4×C61—82no—
155D4×C61—82no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61C2×D812yes—
162C2×C4×C61#13512yes—
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C22⋊D61—82no—
167C2×Dic61—22no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170D244—22no—
171D244—24no—
172D244—41no—
173D244—41no—
174D244—42no—
175D244—81no—
176D244C2×D814yes—
177C4×D61—44no—
178C4×D61—82no—
179C4×D61—82no—
180Dic122—41no—
181Dic122—41no—
182C8×C61—42no—
183C8×C61—42no—
184#92418—24no—
185#92418—42no—
186#19386—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—82no—
190C22×C61—82no—
191C22×C61—82no—
192C22×C61—164no—
193C22×C61#46612yes—
194C22×C61C2×#12111yescharacteristic
195C2×D61—24no—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61—84no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61#46612yes—
206C4×C61C22×D811yescharacteristic
207C4×C61C2×#13511yescharacteristic
208C4×C61#57612yes—
209Dic61—42no—
210Dic61—81no—
211#3205—611no—
212#3036—611no—
213C2×#459—611no—
214#2972—611no—
215#1681—614no—
216#1667—614no—
217#1571—611no—
218#1545D6111yescharacteristic, complemented
219#1540—611no—
220C2×C61—22no—
221C2×C61—162no—
222C2×C61#304111yescharacteristic
223C2×C61C2×#46611yescharacteristic
224C2×C61#306811yescharacteristic
225D61—44no—
226D61—161no—
227C2×AGL(1,ℤ/8)—1221no—
228C22×D8—1221no—
229#532—611no—
230C2×#127C2×D6111yescharacteristic
231C2×C4×D4—611no—
232#493—612no—
233#486—612no—
234#463—1222no—
235#461—612no—
236#459—614no—
237#459—612no—
238#459—1222no—
239#459—1222no—
240#447—1222no—
241C8×D4—614no—
242C8×D4—1222no—
243#431—611no—
244#431C2×D6111yescharacteristic
245#423C2×D6111yescharacteristic
246C2×C4⋊C8—611no—
247#417—1221no—
248C2×#102—1221no—
249C2×C22⋊C8—1222no—
250#335—614no—
251#332—612no—
252#331—612no—
253#331C2×D6114yescomplemented
254C61#1938611yescharacteristic, complemented, Sylow
255C2×#41—1221no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—611no—
260C23×C4—1221no—
261AGL(1,ℤ/8)—2444no—
262C2×SD16—1222no—
263C2×D8—1222no—
264C2×D8—1224no—
265C2×D8—1224no—
266C2×D8—2444no—
267C2×M16—21no—
268C22×C8—1224no—
269C22×C8—1221no—
270C4⋊Q8—611no—
271#127—21no—
272#127—21no—
273#127—21no—
274#127—611no—
275#127C22×D6112yes—
276#124—1221no—
277#121—1221no—
278#121—1221no—
279C4×D4—612no—
280C4×D4—612no—
281C4×D4—1221no—
282C4×D4—1221no—
283C4×D4—1221no—
284C4×D4—1221no—
285C2×C4⋊C4—611no—
286C2×C22⋊C4—1221no—
287C2×C42C22×D6111yescharacteristic
288C4⋊C8—21no—
289C4⋊C8—612no—
290C4⋊C8—612no—
291C4⋊C8—1221no—
292C4⋊C8C22×D6112yes—
293C4⋊C8C22×D6112yes—
294#103—1222no—
295#102—1222no—
296#102—1224no—
297#102—1224no—
298C22⋊C8—21no—
299C22⋊C8—612no—
300C22⋊C8—1224no—
301C4×C8—21no—
302C4×C8—612no—
303C24—41no—
304C24—1221no—
305#41—2442no—
306#41—2442no—
307C2×Q8—1221no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—82no—
317C2×D4—82no—
318C2×D4—612no—
319C2×D4—614no—
320C2×D4—1221no—
321C2×D4—1221no—
322C2×D4—1221no—
323C2×D4—1221no—
324C2×D4—1222no—
325C2×D4—2441no—
326C22×C4—1221no—
327C22×C4—1222no—
328C22×C4—1222no—
329C22×C4—4882no—
330C22×C4—2441no—
331C22×C4#1175822211yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4D4×D6111yescharacteristic
334SD16—2444no—
335D8—4884no—
336D8—2444no—
337D8—2448no—
338M16—42no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—614no—
342C2×C8—1222no—
343C2×C8—4884no—
344C2×C8—2442no—
345C4⋊C4—612no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C22⋊C4—1222no—
350C22⋊C4—1222no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C42#1175822211yescharacteristic
354C42D4×D6111yescharacteristic
355C42D4×D6111yescharacteristic
356C42C23×D6111yescharacteristic
357C23—42no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—1224no—
362C23—1222no—
363C23—4881no—
364C23—4881no—
365C23—2441no—
366C23#1451975011yescharacteristic
367Q8—2441no—
368Q8—2441no—
369D4—42no—
370D4—42no—
371D4—44no—
372D4—82no—
373D4—82no—
374D4—82no—
375D4—82no—
376D4—614no—
377D4—1224no—
378D4—1222no—
379D4—4881no—
380D4—4882no—
381D4—2441no—
382D4—2441no—
383D4—2442no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—1222no—
387C2×C4—4882no—
388C2×C4—4882no—
389C2×C4—4881no—
390C2×C4—2441no—
391C2×C4—2444no—
392C2×C4#1451975011yescharacteristic
393C2×C4#1451975011yescharacteristic
394C2×C4#1451975011yescharacteristic
395C2×C4D8×D6112yes—
396C2×C4#1451979812yes—
397C2×C4C2×D4×D6111yescharacteristic
398C2×C4C2×D4×D6111yescharacteristic
399C2×C4C2×#1175822211yescharacteristic
400C8—42no—
401C8—42no—
402C8—1224no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—164no—
410C22—1224no—
411C22—4884no—
412C22—4881no—
413C22—4881no—
414C22—4881no—
415C22—2442no—
416C22—2442no—
417C22#1683999712yes—
418C22C2×#1451975011yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—4881no—
423C4—2442no—
424C4#1683999712yes—
425C4C2×D8×D6111yescharacteristic
426C4C2×#1451979811yescharacteristic
427C4#1684056612yes—
428C2—22no—
429C2—162no—
430C2—9761no—
431C2—2444no—
432C2#2145532111yescharacteristic
433C2C2×#1683999711yescharacteristic
434C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C2×#121
hypercenter#1545D61
hypocenterC61#19386
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
1. #21456737C2semidirect
2. #21455285C2semidirect
3. C2×#16839954C2semidirect
4. #21454842C2semidirect
5. #21445255C2semidirect
6. #21445173C2semidirect
7. #21444595C2semidirect
8. #21444466C2semidirect
9. C61×#1545C2semidirect
14. #16840138C22semidirect
15. #16840108C22semidirect
17. #16839968C22semidirect
18. #16839958C22semidirect
24. #16839827C22semidirect
30. #16839514C22semidirect
31. #16839507C22semidirect
32. #16839505C22semidirect
36. C61×#331C22semidirect
109. C2×D244D8semidirect
218. #1545D61semidirect
253. #331C2×D61semidirect
254. C61#19386semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214552852
C2×#168399542
#214548422
#214452552
#214451732
#214445952
#214444662
C61×#15452
#1938661