Finite Group Databasebeta

← #29835240

Subgroups — #29835240

Order 15616.

564 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456552C211yesmaximal, characteristic, complemented
2#21454729C211yesmaximal, characteristic, complemented
3#21448034C211yesmaximal, characteristic, complemented
4#21448031C211yesmaximal, characteristic, complemented
5#21447988C212yesmaximal, complemented
6#21447893C212yesmaximal, complemented
7#21447765C211yesmaximal, characteristic, complemented
8#21447718C212yesmaximal, complemented
9#21447714C212yesmaximal, complemented
10#21447628C211yesmaximal, characteristic, complemented
11C61×#2190C211yesmaximal, characteristic, complemented
12#16840529—21no—
13#16840307—21no—
14#16840197C2212yes—
15#16840159C2211yescharacteristic, complemented
16#16840137C2211yescharacteristic, complemented
17#16840107—21no—
18#16840015C2211yescharacteristic
19#16840013—21no—
20#16839884—21no—
21#16839884C2212yes—
22#16839879—21no—
23#16839879C2212yes—
24#16839874C2211yescharacteristic, complemented
25#16839873C2211yescharacteristic, complemented
26#16839871—21no—
27#16839811C2211yescharacteristic
28C2×#14519704—21no—
29M16×Dic61C2211yescharacteristic
30#16839585C2212yescomplemented
31#16839584C2212yescomplemented
32#16839582C2212yescomplemented
33#16839576C2212yescomplemented
34#16839575C2212yescomplemented
35#16839572C2212yes—
36#16839544C2211yescharacteristic
37#16839539C2212yescomplemented
38C61×#575C2211yescharacteristic
39C61×#509C2212yes—
40C61×#469C2211yescharacteristic
41C61×#360C2212yes—
42C61×#349C2211yescharacteristic
43C2×#11758224—41no—
44C2×D488C2311yescharacteristic
45C2×#11758199—21no—
46#14519784—22no—
47#14519784—22no—
48#14519784—42no—
49C2×#11758194—21no—
50C2×#11758194—42no—
51C2×#11758194D411yescharacteristic
52#14519776—21no—
53#14519773—21no—
54#14519763—41no—
55#14519751—21no—
56#14519749—41no—
57D4×Dic61—21no—
58#14519739—21no—
59#14519737—41no—
60C2×C4×Dic61D411yescharacteristic
61#14519720—22no—
62#14519718—22no—
63#14519717C2312yes—
64#14519713—21no—
65#14519713C2312yes—
66#14519709C2311yescharacteristic
67#14519708—21no—
68#14519704—22no—
69#14519704—22no—
70#14519699—21no—
71#14519698—21no—
72#14519697—21no—
73#14519687D411yescharacteristic
74#14519687C2311yescharacteristic
75C8×Dic61D411yescharacteristic
76C8×Dic61C2311yescharacteristic
77C2×#41×C61—21no—
78#137×C61—22no—
79AGL(1,ℤ/8)×C61—22no—
80#135×C61—21no—
81#135×C61C2312yes—
82C2×D8×C61C2311yescharacteristic
83C2×M16×C61C2311yescharacteristic
84#121×C61—21no—
85SD32×C61—22no—
86D16×C61—22no—
87M32×C61C2312yes—
88#106×C61C2311yescharacteristic
89#102×C61—21no—
90#41×C61—22no—
91#41×C61—22no—
92#41×C61—41no—
93#41×C61—41no—
94C2×Q8×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—41no—
102C2×C22⋊D61—41no—
103C2×C22⋊D61—41no—
104C22×Dic61—21no—
105C22×Dic61—41no—
106#11758224—84no—
107C2×D244—21no—
108C2×C4×D61—41no—
109C2×Dic122—41no—
110Q16×C61—21no—
111Q16×C61#4112yescomplemented
112SD16×C61—22no—
113SD16×C61—22no—
114D8×C61—21no—
115D8×C61—22no—
116D8×C61#4112yescomplemented
117M16×C61—22no—
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2411yescharacteristic
120C4⋊C4×C61—21no—
121C22⋊C4×C61—41no—
122C22⋊Dic61—41no—
123#11758199—42no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127#11758196—41no—
128C4×Dic61—21no—
129C4×Dic61C2×D411yescharacteristic
130C4×Dic61C2×D411yescharacteristic
131#11758194—22no—
132#11758194—22no—
133#11758194—42no—
134#11758194—41no—
135#11758194—42no—
136C2×#92418—42no—
137C2×#92418—41no—
138C2×#92418—42no—
139C2×#92418C2×D411yescharacteristic
140C2×#92418C2×D411yescharacteristic
141D488—22no—
142C16×C61—22no—
143#11758185—22no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61AGL(1,ℤ/8)11yescharacteristic
147C22×D61—41no—
148Q8×C61—22no—
149Q8×C61—41no—
150D4×C61—22no—
151D4×C61—41no—
152D4×C61—42no—
153D4×C61—42no—
154D4×C61—41no—
155D4×C61—41no—
156D4×C61—82no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—82no—
166C22⋊D61—82no—
167C22⋊D61—82no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172C2×Dic61—82no—
173D244—41no—
174D244—41no—
175C4×D61—82no—
176Dic122—82no—
177C8×C61—21no—
178C8×C61C2×#4112yes—
179#92418—21no—
180#92418—42no—
181#92418AGL(1,ℤ/8)12yes—
182#33688—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—41no—
185C22×C61—42no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—161no—
190C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C4×C61—21no—
195C4×C61—42no—
196C4×C61—81no—
197C4×C61#52511yescharacteristic
198C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
199Dic61—22no—
200Dic61—41no—
201Dic61—81no—
202#3184—611no—
203#2956—611no—
204#2238—611no—
205#2238—611no—
206#2230—612no—
207#2213—612no—
208#2190—611no—
209#2190D6111yescharacteristic, complemented
210AGL(1,ℤ/16)—612no—
211AGL(1,ℤ/16)—612no—
212#2165—611no—
213C2×C61—21no—
214C2×C61—41no—
215C2×C61—82no—
216C2×C61—161no—
217C2×C61#310911yescharacteristic
218D61—161no—
219#575C2×D6111yescharacteristic
220#568—1221no—
221#532—1221no—
222#509—612no—
223#509C2×D6112yes—
224#498—611no—
225#493—611no—
226#486—1221no—
227#469—611no—
228#469—1221no—
229#469C2×D6111yescharacteristic
230#444—612no—
231#444—1221no—
232#443—612no—
233#443—1221no—
234#442—611no—
235#442—611no—
236#442—1221no—
237#430—611no—
238C2×#104—1221no—
239C4×M16—611no—
240#365—612no—
241#365—612no—
242#364—612no—
243#361—612no—
244#361—612no—
245#360—612no—
246#360C2×D6112yes—
247#349—611no—
248#349C2×D6111yescharacteristic
249#347—612no—
250C61#3368811yescharacteristic, complemented, Sylow
251C2×#41—21no—
252C2×#41—2441no—
253#137—22no—
254AGL(1,ℤ/8)—22no—
255#135—21no—
256#135C22×D6112yes—
257C2×D8—611no—
258C2×D8—1221no—
259C2×D8C22×D6111yescharacteristic
260#131—1222no—
261#131—1222no—
262#131—2442no—
263C2×M16—611no—
264C2×M16—1221no—
265C2×M16—2442no—
266C2×M16C22×D6111yescharacteristic
267C4⋊Q8—1221no—
268#127—1221no—
269#124—2441no—
270#121—21no—
271#121—1221no—
272#121—2441no—
273C4×D4—1221no—
274C4×D4—1221no—
275C4×D4—2441no—
276C2×C42—611no—
277SD32—22no—
278SD32—1222no—
279D16—22no—
280D16—1222no—
281M32—612no—
282M32C22×D6112yes—
283#108—612no—
284#108—1221no—
285#106—611no—
286#106—1221no—
287#106C22×D6111yescharacteristic
288#104—1222no—
289#104—1222no—
290#102—21no—
291#102—1221no—
292#102—1221no—
293#102—1221no—
294#97—611no—
295#97—611no—
296C4×C8—611no—
297C4×C8—611no—
298#41—22no—
299#41—22no—
300#41—41no—
301#41—41no—
302#41—4884no—
303C2×Q8—21no—
304C2×Q8—2441no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—1221no—
310C2×D4—2441no—
311C2×D4—2441no—
312C2×D4—2441no—
313C22×C4—41no—
314C22×C4—1221no—
315C22×C4—2441no—
316C22×C4—2441no—
317C22×C4D4×D6111yescharacteristic
318Q16—21no—
319Q16#1175822412yescomplemented
320SD16—22no—
321SD16—22no—
322D8—21no—
323D8—22no—
324D8—1222no—
325D8—2442no—
326D8#1175822412yescomplemented
327M16—22no—
328M16—1222no—
329M16—1222no—
330M16—2442no—
331M16—2442no—
332M16—2441no—
333C2×C8—611no—
334C2×C8—611no—
335C2×C8—2442no—
336C2×C8—2442no—
337C2×C8—2441no—
338C2×C8D4×D6111yescharacteristic
339C2×C8C23×D6111yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—1221no—
342C4⋊C4—2441no—
343C22⋊C4—41no—
344C22⋊C4—2441no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C42—611no—
348C42—611no—
349C42—1221no—
350C16—22no—
351C16—1222no—
352C23—41no—
353C23—41no—
354C23—2441no—
355C23#1451980111yescharacteristic
356Q8—22no—
357Q8—41no—
358Q8—4882no—
359D4—22no—
360D4—41no—
361D4—42no—
362D4—42no—
363D4—41no—
364D4—41no—
365D4—82no—
366D4—4882no—
367D4—4882no—
368D4—4882no—
369D4—2441no—
370D4—2441no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—1221no—
378C2×C4—1221no—
379C2×C4—4882no—
380C2×C4—4882no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4#1451980111yescharacteristic
384C2×C4C2×D4×D6111yescharacteristic
385C8—21no—
386C8—612no—
387C8—1221no—
388C8—2442no—
389C8C2×#1175822412yes—
390C22—21no—
391C22—41no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—161no—
397C22—4881no—
398C22—4881no—
399C22—4881no—
400C22C2×#1451980111yescharacteristic
401C4—21no—
402C4—42no—
403C4—81no—
404C4—1222no—
405C4—4881no—
406C4—2441no—
407C4#1684026811yescharacteristic
408C4AGL(1,ℤ/8)×D6111yescharacteristic
409C2—21no—
410C2—41no—
411C2—82no—
412C2—161no—
413C2—9761no—
414C2#2145590511yescharacteristic
subgroupG/H
centerC2#21455905
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2190C2
socleC2×C61#3109
hypercenter#2190D61
hypocenterC61#33688
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2190

Complements

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

NH ≅ G/Ntype
1. #21456552C2semidirect
2. #21454729C2semidirect
3. #21448034C2semidirect
4. #21448031C2semidirect
5. #21447988C2semidirect
6. #21447893C2semidirect
7. #21447765C2semidirect
8. #21447718C2semidirect
9. #21447714C2semidirect
10. #21447628C2semidirect
11. C61×#2190C2semidirect
15. #16840159C22semidirect
16. #16840137C22semidirect
24. #16839874C22semidirect
25. #16839873C22semidirect
30. #16839585C22semidirect
31. #16839584C22semidirect
32. #16839582C22semidirect
33. #16839576C22semidirect
34. #16839575C22semidirect
37. #16839539C22semidirect
111. Q16×C61#41semidirect
116. D8×C61#41semidirect
209. #2190D61semidirect
250. C61#33688semidirect
319. Q16#11758224semidirect
326. D8#11758224semidirect

Maximal subgroups

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

subgroupindex
#214565522
#214547292
#214480342
#214480312
#214479882
#214478932
#214477652
#214477182
#214477142
#214476282
C61×#21902
#3368861