Finite Group Databasebeta

← #29835278

Subgroups — #29835278

Order 15616.

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

normal subgroups only

This group has 536 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#21456464C211yesmaximal, characteristic, complemented
2#21454722C211yesmaximal, characteristic, complemented
3#21448034C211yesmaximal, characteristic, complemented
4#21448031C211yesmaximal, characteristic, complemented
5#21447993C212yesmaximal, complemented
6#21447893C212yesmaximal, complemented
7#21447765C211yesmaximal, characteristic, complemented
8#21447718C212yesmaximal, complemented
9#21447714C212yesmaximal, complemented
10#21447632C211yesmaximal, characteristic, complemented
11C61×#2190C211yesmaximal, characteristic, complemented
12C2×#14519784—21no—
13#16840365—21no—
14#16840197C2212yes—
15#16840162C2211yescharacteristic, complemented
16#16840137C2211yescharacteristic, complemented
17#16840125—21no—
18#16840015C2211yescharacteristic
19#16839996—21no—
20#16839884—21no—
21#16839884C2212yes—
22#16839879—21no—
23#16839879C2212yes—
24#16839874C2211yescharacteristic, complemented
25#16839873C2211yescharacteristic, complemented
26#16839842—21no—
27#16839811C2211yescharacteristic
28#16839779—21no—
29#16839720C2211yescharacteristic
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×D488C2311yescharacteristic
44C2×#11758199—21no—
45#14519784—22no—
46#14519784—22no—
47#14519784—41no—
48#14519784—41no—
49C2×#11758194—21no—
50C2×#11758194—41no—
51C22×#92418—41no—
52C22×#92418D411yescharacteristic
53#14519773—21no—
54#14519768—21no—
55#14519761—41no—
56#14519759—41no—
57#14519751—21no—
58#14519749—41no—
59#14519748—41no—
60D4×Dic61—21no—
61#14519739—21no—
62#14519736D411yescharacteristic
63#14519720—22no—
64#14519718—22no—
65#14519717C2312yes—
66#14519713—21no—
67#14519713C2312yes—
68#14519710—21no—
69#14519709C2311yescharacteristic
70#14519704—22no—
71#14519704—22no—
72#14519699—21no—
73#14519698—21no—
74#14519697—21no—
75#14519687D411yescharacteristic
76#14519687C2311yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C2×#41×C61—21no—
80#137×C61—22no—
81AGL(1,ℤ/8)×C61—22no—
82#135×C61—21no—
83#135×C61C2312yes—
84C2×D8×C61C2311yescharacteristic
85C2×M16×C61C2311yescharacteristic
86#121×C61—21no—
87SD32×C61—22no—
88D16×C61—22no—
89M32×C61C2312yes—
90#106×C61C2311yescharacteristic
91#102×C61—21no—
92#41×C61—22no—
93#41×C61—22no—
94#41×C61—41no—
95#41×C61—41no—
96C2×Q8×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—41no—
104C2×C22⋊D61—41no—
105C2×C22⋊D61—41no—
106C22×Dic61—41no—
107C2×D244—21no—
108C2×C4×D61—41no—
109Q16×C61—21no—
110Q16×C61#4112yes—
111SD16×C61—22no—
112SD16×C61—22no—
113D8×C61—21no—
114D8×C61—22no—
115D8×C61#4112yes—
116M16×C61—22no—
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2411yescharacteristic
119C4⋊C4×C61—21no—
120C22⋊C4×C61—41no—
121C22⋊Dic61—21no—
122C22⋊Dic61—41no—
123#11758199—41no—
124#11758199—41no—
125#11758198—41no—
126#11758198—41no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—41no—
130#11758196—41no—
131C4×Dic61C2×D411yescharacteristic
132C4×Dic61C2×D411yescharacteristic
133#11758194—22no—
134#11758194—42no—
135#11758194—41no—
136#11758194—42no—
137C2×#92418—21no—
138C2×#92418—21no—
139C2×#92418—42no—
140C2×#92418—41no—
141C2×#92418—42no—
142C2×#92418C2×D411yescharacteristic
143C2×#92418C2×D411yescharacteristic
144D488—22no—
145C16×C61—22no—
146#11758185—22no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61AGL(1,ℤ/8)11yescharacteristic
150C22×D61—41no—
151Q8×C61—22no—
152Q8×C61—41no—
153D4×C61—22no—
154D4×C61—41no—
155D4×C61—42no—
156D4×C61—42no—
157D4×C61—41no—
158D4×C61—41no—
159D4×C61—161no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—42no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—41no—
166C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C22⋊D61—161no—
169C22⋊D61—161no—
170C22⋊D61—161no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—41no—
174C2×Dic61—161no—
175D244—41no—
176D244—41no—
177C4×D61—161no—
178C8×C61—21no—
179C8×C61C2×#4112yes—
180#92418—21no—
181#92418—42no—
182#92418AGL(1,ℤ/8)11yescharacteristic
183#92418AGL(1,ℤ/8)11yescharacteristic
184#33692—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—41no—
187C22×C61—42no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61—82no—
191C22×C61—81no—
192C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—81no—
196C4×C61—21no—
197C4×C61—42no—
198C4×C61—81no—
199C4×C61#52511yescharacteristic
200C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
201Dic61—41no—
202Dic61—41no—
203Dic61—81no—
204#3176—611no—
205#2955—611no—
206#2238—611no—
207#2238—611no—
208#2231—612no—
209#2213—612no—
210#2190—611no—
211#2190D6111yescharacteristic, complemented
212AGL(1,ℤ/16)—612no—
213AGL(1,ℤ/16)—612no—
214#2166—611no—
215C2×C61—21no—
216C2×C61—41no—
217C2×C61—82no—
218C2×C61—161no—
219C2×C61#310911yescharacteristic
220D61—161no—
221#575C2×D6111yescharacteristic
222C2×#131—1221no—
223#540—1221no—
224#509—612no—
225#509C2×D6112yes—
226#499—611no—
227#493—611no—
228#490—1221no—
229#469—611no—
230#469C2×D6111yescharacteristic
231#466—1221no—
232#444—612no—
233#444—1221no—
234#443—612no—
235#443—1221no—
236#442—611no—
237#442—611no—
238C4×D8—1221no—
239#430—611no—
240#421—1221no—
241#405—611no—
242#365—612no—
243#365—612no—
244#364—612no—
245#361—612no—
246#361—612no—
247#360—612no—
248#360C2×D6112yes—
249#349—611no—
250#349C2×D6111yescharacteristic
251#347—612no—
252C61#3369211yescharacteristic, complemented, Sylow
253C2×#41—21no—
254#137—22no—
255AGL(1,ℤ/8)—22no—
256#135—21no—
257#135C22×D6112yes—
258C2×D8—611no—
259C2×D8—1221no—
260C2×D8C22×D6111yescharacteristic
261#131—1222no—
262#131—1222no—
263#131—2441no—
264#131—2441no—
265C2×M16—1221no—
266C2×M16—2441no—
267C2×M16C22×D6111yescharacteristic
268C22×C8—611no—
269C22×C8—2441no—
270#127—1221no—
271#125—1221no—
272#123—2441no—
273#123—2441no—
274#121—21no—
275#121—1221no—
276#121—2441no—
277#121—2441no—
278C4×D4—1221no—
279C4×D4—1221no—
280#117—611no—
281SD32—22no—
282SD32—1222no—
283D16—22no—
284D16—1222no—
285M32—612no—
286M32C22×D6112yes—
287#108—612no—
288#108—1221no—
289#107—1221no—
290#106—611no—
291#106C22×D6111yescharacteristic
292#104—1222no—
293#104—1222no—
294#102—21no—
295#102—1221no—
296#102—1221no—
297#102—1221no—
298#97—611no—
299#97—611no—
300C4×C8—611no—
301C4×C8—611no—
302#41—22no—
303#41—22no—
304#41—41no—
305#41—41no—
306C2×Q8—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—1221no—
312C2×D4—2441no—
313C2×D4—2441no—
314C2×D4—2441no—
315C22×C4—41no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4D4×D6111yescharacteristic
319Q16—21no—
320Q16#1175822412yes—
321SD16—22no—
322SD16—22no—
323D8—21no—
324D8—22no—
325D8—1222no—
326D8—2441no—
327D8—2441no—
328D8#1175822412yes—
329M16—22no—
330M16—1222no—
331M16—2442no—
332M16—2442no—
333M16—2441no—
334C2×C8—611no—
335C2×C8—611no—
336C2×C8—1221no—
337C2×C8—1221no—
338C2×C8—2442no—
339C2×C8—2442no—
340C2×C8—2441no—
341C2×C8D4×D6111yescharacteristic
342C2×C8C23×D6111yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C4⋊C4—2441no—
347C4⋊C4—2441no—
348C22⋊C4—41no—
349C22⋊C4—1221no—
350C22⋊C4—2441no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C42—611no—
354C42—611no—
355C16—22no—
356C16—1222no—
357C23—41no—
358C23—41no—
359C23—2441no—
360C23#1451980111yescharacteristic
361Q8—22no—
362Q8—41no—
363D4—22no—
364D4—41no—
365D4—42no—
366D4—42no—
367D4—41no—
368D4—41no—
369D4—161no—
370D4—9761no—
371D4—9761no—
372D4—9761no—
373D4—2441no—
374D4—2441no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—9761no—
384C2×C4—9761no—
385C2×C4—2441no—
386C2×C4#1451980111yescharacteristic
387C2×C4C2×D4×D6111yescharacteristic
388C8—21no—
389C8—611no—
390C8—611no—
391C8—1221no—
392C8—2442no—
393C8C2×#1175822412yes—
394C22—21no—
395C22—41no—
396C22—42no—
397C22—81no—
398C22—81no—
399C22—82no—
400C22—81no—
401C22—4881no—
402C22—4881no—
403C22—4881no—
404C22C2×#1451980111yescharacteristic
405C4—21no—
406C4—42no—
407C4—81no—
408C4—4881no—
409C4—2441no—
410C4—2441no—
411C4#1684026811yescharacteristic
412C4AGL(1,ℤ/8)×D6111yescharacteristic
413C2—21no—
414C2—41no—
415C2—82no—
416C2—161no—
417C2—9761no—
418C2#2145590511yescharacteristic
subgroupG/H
centerC2#21455905
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2190C2
socleC2×C61#3109
hypercenter#2190D61
hypocenterC61#33692
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. #21456464C2semidirect
2. #21454722C2semidirect
3. #21448034C2semidirect
4. #21448031C2semidirect
5. #21447993C2semidirect
6. #21447893C2semidirect
7. #21447765C2semidirect
8. #21447718C2semidirect
9. #21447714C2semidirect
10. #21447632C2semidirect
11. C61×#2190C2semidirect
15. #16840162C22semidirect
16. #16840137C22semidirect
24. #16839874C22semidirect
25. #16839873C22semidirect
30. #16839585C22semidirect
31. #16839584C22semidirect
32. #16839582C22semidirect
33. #16839576C22semidirect
34. #16839575C22semidirect
37. #16839539C22semidirect
211. #2190D61semidirect
252. C61#33692semidirect

Maximal subgroups

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

subgroupindex
#214564642
#214547222
#214480342
#214480312
#214479932
#214478932
#214477652
#214477182
#214477142
#214476322
C61×#21902
#3369261