Finite Group Databasebeta

← #29665194

Subgroups — #29665194

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#21456734C211yesmaximal, characteristic, complemented
2#21455279C211yesmaximal, characteristic, complemented
3C2×#16839958C211yesmaximal, characteristic, complemented
4#21454842C211yesmaximal, characteristic, complemented
5#21445259C214yesmaximal, complemented
6#21445173C214yesmaximal, complemented
7#21444593C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#1540C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×D488—21no—
12C2×#14519774C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840138C2212yescomplemented
15#16840110C2212yescomplemented
16#16839974—22no—
17#16839964C2212yescomplemented
18#16839958—22no—
19#16839958C2214yes—
20#16839957—22no—
21#16839954C2212yescomplemented
22#16839895—22no—
23C8×D244—22no—
24#16839827C2214yescomplemented
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839763—21no—
28C2×#14519699—21no—
29C22⋊C8×D61—22no—
30#16839514C2214yescomplemented
31#16839505C2214yescomplemented
32C61×#532C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#332C2212yescomplemented
36C61×#331C2212yescomplemented
37C22×D244—21no—
38C22×D244—21no—
39C22×D244—21no—
40C22×D244D411yescharacteristic
41C22×C4×D61—21no—
42#14519802—44no—
43C2×#11758201—22no—
44C2×D488—24no—
45C2×D488—44no—
46C2×#11758199—22no—
47C2×#11758199—24no—
48C2×#11758194—21no—
49C2×C8×D61—24no—
50#14519774—21no—
51#14519774—21no—
52#14519774—21no—
53#14519774C2312yes—
54C4×D244—21no—
55C4×D244—21no—
56C4×D244D412yes—
57C4×D244C2312yes—
58C2×C4⋊Dic61D411yescharacteristic
59C2×#11758198—21no—
60C4⋊#92418—21no—
61C4⋊#92418D412yes—
62C4⋊#92418C2312yes—
63#14519701—22no—
64#14519699—24no—
65#14519698—22no—
66#14519697—24no—
67C22⋊#92418D412yes—
68#14519689—24no—
69C4×#92418C2312yes—
70C2×#41×C61—21no—
71C22×C8×C61—21no—
72C4⋊Q8×C61C2311yescharacteristic
73#127×C61C2311yescharacteristic
74#124×C61—21no—
75#121×C61—21no—
76#121×C61—21no—
77C4×D4×C61—21no—
78C4×D4×C61—21no—
79C2×C42×C61C2311yescharacteristic
80C4⋊C8×C61—21no—
81C4⋊C8×C61C2312yes—
82C4⋊C8×C61C2312yes—
83C22⋊C8×C61—21no—
84C4×C8×C61—21no—
85C23×D61—21no—
86C23×D61—41no—
87#41×C61—42no—
88#41×C61—42no—
89C2×Q8×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C22×Dic61—21no—
99C2×D244—21no—
100C2×D244—22no—
101C2×D244—22no—
102C2×D244—22no—
103C2×D244—22no—
104C2×D244—42no—
105C2×D244—42no—
106C2×D244—41no—
107C2×D244—41no—
108C2×D244—41no—
109C2×D244—82no—
110C2×D244—82no—
111C2×D244D814yescomplemented
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61—82no—
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C2×C8×C61—42no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127#11758201—44no—
128#11758199—44no—
129#11758199—84no—
130#11758198—22no—
131#11758198—22no—
132C4⋊Dic61—21no—
133C4⋊Dic61C2×D412yes—
134#11758194—42no—
135C2×#92418—22no—
136C2×#92418C2×D414yes—
137D488—48no—
138C8×D61—84no—
139C23×C61—41no—
140C23×C61#12111yescharacteristic
141C22×D61—22no—
142C22×D61—24no—
143C22×D61—42no—
144C22×D61—81no—
145C22×D61—81no—
146C22×D61—82no—
147C22×D61—82no—
148C22×D61—82no—
149Q8×C61—41no—
150Q8×C61—41no—
151D4×C61—41no—
152D4×C61—41no—
153D4×C61—42no—
154D4×C61—81no—
155D4×C61—82no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—82no—
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61C2×D812yes—
164C2×C4×C61#13512yes—
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C2×#4111yescharacteristic
168C2×Dic61—22no—
169C2×Dic61—81no—
170D244—22no—
171D244—24no—
172D244—42no—
173D244—42no—
174D244—44no—
175D244—82no—
176D244—82no—
177D244—82no—
178D244—82no—
179D244C2×D814yes—
180C4×D61—44no—
181C4×D61—82no—
182C8×C61—42no—
183C8×C61—42no—
184#92418—24no—
185#92418—42no—
186#19386—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—81no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#46612yes—
193C22×C61C2×#12111yescharacteristic
194C2×D61—24no—
195C2×D61—42no—
196C2×D61—42no—
197C2×D61—84no—
198C2×D61—82no—
199C2×D61—82no—
200C2×D61—82no—
201C2×D61—164no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—81no—
206C4×C61#46612yes—
207C4×C61C22×D811yescharacteristic
208C4×C61C2×#13511yescharacteristic
209C4×C61#57612yes—
210Dic61—42no—
211#3205—611no—
212#3036—611no—
213C2×#459—611no—
214#2972—611no—
215#1681—614no—
216#1667—614no—
217#1571—611no—
218#1545—611no—
219#1540D6111yescharacteristic, complemented
220C2×C61—22no—
221C2×C61—161no—
222C2×C61#304111yescharacteristic
223C2×C61C2×#46611yescharacteristic
224C2×C61#306811yescharacteristic
225D61—44no—
226D61—162no—
227C2×AGL(1,ℤ/8)—1221no—
228C22×D8—1221no—
229#532C2×D6111yescharacteristic
230C2×#127—611no—
231C2×C4×D4—611no—
232#493—612no—
233#486—612no—
234#463—1222no—
235#461—612no—
236#459—612no—
237#459—614no—
238#459—1222no—
239#459—1222no—
240#447—1222no—
241C8×D4—614no—
242C8×D4—1222no—
243#431—611no—
244#431C2×D6111yescharacteristic
245#423—611no—
246C2×C4⋊C8C2×D6111yescharacteristic
247#417—1221no—
248C2×#102—1221no—
249C2×C22⋊C8—1222no—
250#335—614no—
251#332C2×D6112yescomplemented
252#331—614no—
253#331C2×D6112yescomplemented
254C61#1938611yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C22×D4—611no—
257C22×D4—1221no—
258C22×D4—1221no—
259C22×D4—1221no—
260C23×C4—1221no—
261AGL(1,ℤ/8)—2444no—
262C2×SD16—1222no—
263C2×D8—1224no—
264C2×D8—1224no—
265C2×D8—1222no—
266C2×D8—2444no—
267C2×M16—1221no—
268C22×C8—21no—
269C22×C8—1224no—
270C4⋊Q8C22×D6111yescharacteristic
271#127—612no—
272#127—1221no—
273#127—1221no—
274#127—1221no—
275#127C22×D6111yescharacteristic
276#124—21no—
277#121—21no—
278#121—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—612no—
282C4×D4—612no—
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—1224no—
296#102—1224no—
297#102—1222no—
298C22⋊C8—21no—
299C22⋊C8—612no—
300C22⋊C8—1224no—
301C4×C8—21no—
302C4×C8—612no—
303C24—1221no—
304C24—2441no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—41no—
312C2×D4—612no—
313C2×D4—614no—
314C2×D4—1222no—
315C2×D4—1222no—
316C2×D4—1222no—
317C2×D4—1221no—
318C2×D4—1222no—
319C2×D4—4882no—
320C2×D4—4882no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2441no—
324C2×D4—2442no—
325C2×D4—2442no—
326C22×C4—41no—
327C22×C4—1221no—
328C22×C4—1222no—
329C22×C4—1222no—
330C22×C4—4882no—
331C22×C4#1175822211yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4D4×D6111yescharacteristic
334SD16—2444no—
335D8—4884no—
336D8—2448no—
337D8—2444no—
338M16—2442no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—42no—
342C2×C8—614no—
343C2×C8—1222no—
344C2×C8—4884no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—612no—
348C4⋊C4—1221no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—1222no—
352C22⋊C4—1222no—
353C42#1175822211yescharacteristic
354C42D4×D6111yescharacteristic
355C42D4×D6111yescharacteristic
356C42C23×D6111yescharacteristic
357C23—41no—
358C23—1224no—
359C23—1222no—
360C23—4882no—
361C23—4882no—
362C23—4881no—
363C23—4881no—
364C23—4882no—
365C23—2442no—
366C23#1451975011yescharacteristic
367Q8—41no—
368Q8—41no—
369D4—41no—
370D4—41no—
371D4—42no—
372D4—81no—
373D4—82no—
374D4—614no—
375D4—1224no—
376D4—1222no—
377D4—4882no—
378D4—4882no—
379D4—4882no—
380D4—4882no—
381D4—2442no—
382D4—2442no—
383D4—2444no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—82no—
388C2×C4—1222no—
389C2×C4—4882no—
390C2×C4—4881no—
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—81no—
407C22—81no—
408C22—81no—
409C22—1224no—
410C22—9764no—
411C22—4884no—
412C22—4882no—
413C22—4882no—
414C22—4882no—
415C22—2442no—
416C22—2442no—
417C22#1683999712yes—
418C22C2×#1451975011yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—81no—
423C4—2442no—
424C4#1683999712yes—
425C4C2×D8×D6111yescharacteristic
426C4C2×#1451979811yescharacteristic
427C4#1684056412yes—
428C2—22no—
429C2—161no—
430C2—9762no—
431C2—2444no—
432C2#2145532511yescharacteristic
433C2C2×#1683999711yescharacteristic
434C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1540C2
socleC22×C61C2×#121
hypercenter#1540D61
hypocenterC61#19386
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. #21455279C2semidirect
3. C2×#16839958C2semidirect
4. #21454842C2semidirect
5. #21445259C2semidirect
6. #21445173C2semidirect
7. #21444593C2semidirect
8. #21444486C2semidirect
9. C61×#1540C2semidirect
14. #16840138C22semidirect
15. #16840110C22semidirect
17. #16839964C22semidirect
21. #16839954C22semidirect
24. #16839827C22semidirect
30. #16839514C22semidirect
31. #16839505C22semidirect
35. C61×#332C22semidirect
36. C61×#331C22semidirect
111. C2×D244D8semidirect
219. #1540D61semidirect
251. #332C2×D61semidirect
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
#214567342
#214552792
C2×#168399582
#214548422
#214452592
#214451732
#214445932
#214444862
C61×#15402
#1938661