Finite Group Databasebeta

← #29665188

Subgroups — #29665188

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#21455283C211yesmaximal, characteristic, complemented
3#21454843C211yesmaximal, characteristic, complemented
4#21445254C214yesmaximal, complemented
5#21445172C214yesmaximal, complemented
6#21444593C211yesmaximal, characteristic, complemented
7#21444487C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic, complemented
9C2×C61×#459C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12#16840138C2212yescomplemented
13#16840108C2212yescomplemented
14#16839977—22no—
15#16839967C2212yescomplemented
16#16839957C2212yescomplemented
17#16839954—22no—
18#16839896—22no—
19D4×#92418—22no—
20D4×#92418C2214yescomplemented
21#16839815C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839786C2211yescharacteristic
24#16839761—21no—
25C2×C22⋊#92418—22no—
26#16839514C2214yescomplemented
27#16839509C2212yescomplemented
28#16839506C2214yescomplemented
29#16839505C2212yescomplemented
30C22×D8×C61—21no—
31C2×#127×C61C2211yescharacteristic
32C2×C4×D4×C61C2211yescharacteristic
33C61×#459—22no—
34C61×#459C2214yes—
35C2×C4⋊C8×C61C2211yescharacteristic
36C2×#102×C61—21no—
37C2×#11758222—21no—
38#14519803—44no—
39C2×#11758200—22no—
40C2×#11758199—22no—
41C2×#11758199—24no—
42C2×#11758194—21no—
43C22×#92418—24no—
44#14519777C2311yescharacteristic
45#14519774C2311yescharacteristic
46#14519765—21no—
47#14519751—21no—
48#14519751—21no—
49C4×D244—21no—
50C4×D244—21no—
51C4⋊#92418—21no—
52C4⋊#92418D412yes—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519700—22no—
56#14519698—24no—
57#14519697—22no—
58C22⋊#92418—21no—
59C22⋊#92418—24no—
60C22⋊#92418D412yes—
61C4×#92418—21no—
62C4×#92418C2312yes—
63C22×D4×C61—21no—
64C22×D4×C61—21no—
65C22×D4×C61—21no—
66C22×D4×C61D411yescharacteristic
67C23×C4×C61—21no—
68C2×D8×C61—24no—
69C2×D8×C61—44no—
70C22×C8×C61—21no—
71#127×C61—21no—
72#127×C61—21no—
73#127×C61—21no—
74#127×C61C2312yes—
75C4×D4×C61—21no—
76C4×D4×C61—21no—
77C4×D4×C61D412yes—
78C4×D4×C61C2312yes—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C22⋊C4×C61—21no—
81C2×C42×C61C2311yescharacteristic
82C4⋊C8×C61—21no—
83C4⋊C8×C61C2312yes—
84#102×C61—24no—
85C24×C61—21no—
86C24×C61—41no—
87C2×D4×C61—21no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—22no—
91C2×D4×C61—22no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×D4×C61—41no—
95C2×D4×C61—42no—
96C2×D4×C61—42no—
97C2×D4×C61—82no—
98C2×D4×C61—82no—
99C2×D4×C61D814yescomplemented
100C2×D4×C61C2×D412yes—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—82no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C2×C22⋊D61—41no—
109#11758222—42no—
110#11758222—42no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×C4×D61—41no—
115C2×Dic122—21no—
116D8×C61—48no—
117C2×C8×C61—22no—
118C2×C8×C61—42no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61C2×D412yes—
121C22⋊C4×C61—22no—
122C22⋊C4×C61—22no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127#11758200—44no—
128#11758199—44no—
129#11758199—84no—
130#11758198—41no—
131#11758198—41no—
132C4⋊Dic61—21no—
133C4⋊Dic61—21no—
134#11758194—42no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418—84no—
138C2×#92418C2×D414yes—
139C23×C61—22no—
140C23×C61—24no—
141C23×C61—42no—
142C23×C61—81no—
143C23×C61—81no—
144C23×C61—82no—
145C23×C61—82no—
146C23×C61—82no—
147C23×C61#12111yescharacteristic
148C22×D61—41no—
149D4×C61—22no—
150D4×C61—24no—
151D4×C61—42no—
152D4×C61—42no—
153D4×C61—44no—
154D4×C61—82no—
155D4×C61—82no—
156D4×C61—82no—
157D4×C61—82no—
158D4×C61C2×D814yes—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—41no—
162C2×C4×C61—44no—
163C2×C4×C61—82no—
164C2×C4×C61—81no—
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61C2×D812yes—
169C2×C4×C61#13512yes—
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C22⋊D61—82no—
174C2×Dic61—41no—
175D244—41no—
176D244—41no—
177D244—42no—
178D244—81no—
179C4×D61—82no—
180Dic122—41no—
181Dic122—41no—
182C8×C61—42no—
183#92418—24no—
184#92418—42no—
185#92418—42no—
186#19386—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—24no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—84no—
193C22×C61—82no—
194C22×C61—82no—
195C22×C61—82no—
196C22×C61—164no—
197C22×C61#46612yes—
198C22×C61C2×#12111yescharacteristic
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—42no—
206C4×C61#46612yes—
207C4×C61C22×D811yescharacteristic
208C4×C61C2×#13511yescharacteristic
209C4×C61#57612yes—
210Dic61—81no—
211#3205—611no—
212#3036—611no—
213C2×#459D6111yescharacteristic, complemented
214#2972—611no—
215#1681—614no—
216#1667—614no—
217#1571—611no—
218#1545—611no—
219#1540—611no—
220C2×C61—22no—
221C2×C61—44no—
222C2×C61—162no—
223C2×C61#304111yescharacteristic
224C2×C61C2×#46611yescharacteristic
225C2×C61#306811yescharacteristic
226D61—161no—
227C2×AGL(1,ℤ/8)—1221no—
228C22×D8—21no—
229#532—611no—
230C2×#127C2×D6111yescharacteristic
231C2×C4×D4C2×D6111yescharacteristic
232#493—612no—
233#486—612no—
234#463—1222no—
235#461—612no—
236#459—22no—
237#459—612no—
238#459—1222no—
239#459C2×D6114yes—
240#447—1222no—
241C8×D4—614no—
242C8×D4—1222no—
243#431—611no—
244#431—611no—
245#423—611no—
246C2×C4⋊C8C2×D6111yescharacteristic
247#417—1221no—
248C2×#102—21no—
249C2×C22⋊C8—1222no—
250#335—614no—
251#332—612no—
252#331—614no—
253#331—612no—
254C61#1938611yescharacteristic, complemented, Sylow
255C2×#41—1221no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4C22⋊D6111yescharacteristic
260C23×C4—21no—
261AGL(1,ℤ/8)—2444no—
262C2×SD16—1222no—
263C2×D8—24no—
264C2×D8—44no—
265C2×D8—1224no—
266C2×D8—1222no—
267C2×M16—1221no—
268C22×C8—21no—
269C22×C8—1224no—
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—21no—
280C4×D4—21no—
281C4×D4—1221no—
282C4×D4—1221no—
283C4×D4C22⋊D6112yes—
284C4×D4C22×D6112yes—
285C2×C4⋊C4C22⋊D6111yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42C22×D6111yescharacteristic
288C4⋊C8—21no—
289C4⋊C8—612no—
290C4⋊C8—612no—
291C4⋊C8—612no—
292C4⋊C8—1221no—
293C4⋊C8C22×D6112yes—
294#103—1222no—
295#102—24no—
296#102—1224no—
297#102—1222no—
298C22⋊C8—612no—
299C22⋊C8—1224no—
300C22⋊C8—1221no—
301C4×C8—612no—
302C4×C8—1221no—
303C24—21no—
304C24—41no—
305#41—2442no—
306#41—2442no—
307C2×Q8—1221no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—1221no—
321C2×D4—1221no—
322C2×D4—1221no—
323C2×D4—2441no—
324C2×D4#1175819914yescomplemented
325C2×D4C2×C22⋊D6112yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—82no—
330C22×C4—2441no—
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—2444no—
335D8—48no—
336D8—4884no—
337D8—2444no—
338M16—2442no—
339C2×C8—22no—
340C2×C8—42no—
341C2×C8—614no—
342C2×C8—1222no—
343C2×C8—1222no—
344C2×C8—4884no—
345C4⋊C4—21no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4C2×C22⋊D6112yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C42D4×D6111yescharacteristic
354C42#1175822411yescharacteristic
355C42C2×C22⋊D6111yescharacteristic
356C42C23×D6111yescharacteristic
357C23—22no—
358C23—24no—
359C23—42no—
360C23—81no—
361C23—81no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—2441no—
366C23#1451974911yescharacteristic
367Q8—2441no—
368Q8—2441no—
369D4—24no—
370D4—22no—
371D4—42no—
372D4—42no—
373D4—44no—
374D4—82no—
375D4—82no—
376D4—82no—
377D4—82no—
378D4—4881no—
379D4—4882no—
380D4—2441no—
381D4—2441no—
382D4—2442no—
383D4C2×#1175819914yes—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—4882no—
391C2×C4—2441no—
392C2×C4#1451974911yescharacteristic
393C2×C4#1451974911yescharacteristic
394C2×C4#1451974911yescharacteristic
395C2×C4C2×#1175819912yes—
396C2×C4#1451979712yes—
397C2×C4C2×D4×D6111yescharacteristic
398C2×C4C22×C22⋊D6111yescharacteristic
399C2×C4C2×#1175822411yescharacteristic
400C8—42no—
401C8—1224no—
402C8—2442no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—24no—
407C22—42no—
408C22—42no—
409C22—84no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—164no—
414C22—4881no—
415C22—4881no—
416C22—4881no—
417C22#1683999612yes—
418C22C2×#1451974911yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—4881no—
424C4#1683999612yes—
425C4C22×#1175819911yescharacteristic
426C4C2×#1451979711yescharacteristic
427C4#1684056312yes—
428C2—22no—
429C2—44no—
430C2—162no—
431C2—9761no—
432C2#2145531911yescharacteristic
433C2C2×#1683999611yescharacteristic
434C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#459C2
socleC22×C61C2×#121
hypercenterC2×#459D61
hypocenterC61#19386
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #21456737C2semidirect
2. #21455283C2semidirect
3. #21454843C2semidirect
4. #21445254C2semidirect
5. #21445172C2semidirect
6. #21444593C2semidirect
7. #21444487C2semidirect
8. #21444469C2semidirect
9. C2×C61×#459C2semidirect
12. #16840138C22semidirect
13. #16840108C22semidirect
15. #16839967C22semidirect
16. #16839957C22semidirect
20. D4×#92418C22semidirect
26. #16839514C22semidirect
27. #16839509C22semidirect
28. #16839506C22semidirect
29. #16839505C22semidirect
99. C2×D4×C61D8semidirect
213. C2×#459D61semidirect
254. C61#19386semidirect
324. C2×D4#11758199semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214552832
#214548432
#214452542
#214451722
#214445932
#214444872
#214444692
C2×C61×#4592
#1938661