Finite Group Databasebeta

← #29835248

Subgroups — #29835248

Order 15616.

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

normal subgroups only

This group has 520 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#21456559C211yesmaximal, characteristic, complemented
2#21454729C211yesmaximal, characteristic
3#21448039C211yesmaximal, characteristic, complemented
4#21448036C211yesmaximal, characteristic, complemented
5#21447988C212yesmaximal, complemented
6#21447898C212yesmaximal, complemented
7#21447770C211yesmaximal, characteristic
8#21447711C212yesmaximal, complemented
9#21447707C212yesmaximal, complemented
10#21447628C211yesmaximal, characteristic, complemented
11C61×#2191C211yesmaximal, characteristic
12#16840529—21no—
13#16840314—21no—
14#16840202C2212yes—
15#16840159C2211yescharacteristic, complemented
16#16840140C2211yescharacteristic, complemented
17#16840111—21no—
18#16840020C2211yescharacteristic
19#16840018—21no—
20#16839884—21no—
21#16839884C2212yes—
22#16839879—21no—
23#16839879C2212yes—
24#16839869C2211yescharacteristic, complemented
25#16839868C2211yescharacteristic, complemented
26#16839866—21no—
27#16839811C2211yescharacteristic
28C2×#14519704—21no—
29M16×Dic61C2211yescharacteristic
30#16839585C2212yescomplemented
31#16839584C2212yescomplemented
32#16839582C2212yescomplemented
33#16839579C2212yescomplemented
34#16839578C2212yescomplemented
35#16839572C2212yes—
36#16839544C2211yescharacteristic
37#16839539C2212yescomplemented
38C61×#575C2211yescharacteristic
39C61×#510C2212yes—
40C61×#470C2211yescharacteristic
41C61×#360C2212yes—
42C61×#349C2211yescharacteristic
43C2×Dic244C2311yescharacteristic
44C2×#11758202—21no—
45#14519784—22no—
46#14519784—22no—
47#14519784—42no—
48C2×#11758194—21no—
49C2×#11758194—42no—
50C2×#11758194D411yescharacteristic
51#14519776—21no—
52#14519775—21no—
53#14519767—41no—
54#14519756—21no—
55#14519754—41no—
56Q8×Dic61—21no—
57#14519742—21no—
58#14519742—41no—
59#14519733—41no—
60C2×C4×Dic61D411yescharacteristic
61#14519723—22no—
62#14519721—22no—
63#14519717C2312yes—
64#14519713—21no—
65#14519713C2312yes—
66#14519709C2311yescharacteristic
67#14519708—21no—
68#14519704—22no—
69#14519704—22no—
70#14519702—21no—
71#14519701—21no—
72#14519700—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×Q16×C61C2311yescharacteristic
83C2×M16×C61C2311yescharacteristic
84C22⋊Q8×C61—21no—
85Q32×C61—22no—
86SD32×C61—22no—
87M32×C61C2312yes—
88#106×C61C2311yescharacteristic
89#103×C61—21no—
90#41×C61—22no—
91#41×C61—22no—
92#41×C61—41no—
93#41×C61—41no—
94C2×Q8×C61—21no—
95C2×Q8×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C22×Dic61—21no—
101C2×Dic122—21no—
102C2×Dic122—41no—
103Q16×C61—21no—
104Q16×C61—22no—
105Q16×C61#4112yes—
106SD16×C61—22no—
107SD16×C61—22no—
108D8×C61—21no—
109D8×C61#4112yes—
110M16×C61—22no—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2411yescharacteristic
113C4⋊C4×C61—21no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊Dic61—41no—
117#11758202—42no—
118C4⋊Dic61—21no—
119C4⋊Dic61—41no—
120#11758196—41no—
121#11758196—41no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—21no—
126C4×Dic61—41no—
127C4×Dic61—41no—
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130#11758194—22no—
131#11758194—22no—
132#11758194—42no—
133#11758194—41no—
134#11758194—42no—
135C2×#92418—42no—
136C2×#92418—41no—
137C2×#92418—42no—
138C2×#92418C2×D411yescharacteristic
139C2×#92418C2×D411yescharacteristic
140Dic244—22no—
141C16×C61—22no—
142#11758185—22no—
143C23×C61—41no—
144C23×C61AGL(1,ℤ/8)11yescharacteristic
145Q8×C61—22no—
146Q8×C61—41no—
147Q8×C61—41no—
148Q8×C61—41no—
149D4×C61—22no—
150D4×C61—41no—
151D4×C61—42no—
152D4×C61—42no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C2×Dic61—21no—
163C2×Dic61—21no—
164C2×Dic61—42no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167Dic122—41no—
168Dic122—41no—
169Dic122—82no—
170C8×C61—21no—
171C8×C61C2×#4112yes—
172#92418—21no—
173#92418—42no—
174#92418AGL(1,ℤ/8)12yes—
175#33689—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—41no—
178C22×C61—42no—
179C22×C61—161no—
180C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
181C4×C61—21no—
182C4×C61—42no—
183C4×C61—81no—
184C4×C61—81no—
185C4×C61#52511yescharacteristic
186C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
187Dic61—22no—
188Dic61—41no—
189Dic61—81no—
190Dic61—81no—
191#3185—611no—
192#2956—611no—
193#2239—611no—
194#2239—611no—
195#2230—612no—
196#2214—612no—
197#2191—611no—
198#2191D6111yescharacteristic
199#2180—612no—
200#2180—612no—
201#2165—611no—
202C2×C61—21no—
203C2×C61—41no—
204C2×C61—82no—
205C2×C61#310911yescharacteristic
206#575C2×D6111yescharacteristic
207#568—1221no—
208#533—1221no—
209#510—612no—
210#510C2×D6112yes—
211#498—611no—
212#494—611no—
213#487—1221no—
214#470—611no—
215#470—1221no—
216#470C2×D6111yescharacteristic
217#444—612no—
218#444—1221no—
219#443—612no—
220#443—1221no—
221#441—611no—
222#441—611no—
223#441—1221no—
224#430—611no—
225C2×#104—1221no—
226C4×M16—611no—
227#365—612no—
228#365—612no—
229#364—612no—
230#362—612no—
231#362—612no—
232#360—612no—
233#360C2×D6112yes—
234#349—611no—
235#349C2×D6111yescharacteristic
236#347—612no—
237C61#3368911yescharacteristic, complemented, Sylow
238C2×#41—21no—
239#137—22no—
240AGL(1,ℤ/8)—22no—
241#135—21no—
242#135C22×D6112yes—
243C2×Q16—611no—
244C2×Q16—1221no—
245C2×Q16C22×D6111yescharacteristic
246#131—1222no—
247#131—1222no—
248#131—2442no—
249C2×M16—611no—
250C2×M16—1221no—
251C2×M16—2442no—
252C2×M16C22×D6111yescharacteristic
253C4⋊Q8—1221no—
254C4⋊Q8—1221no—
255#125—2441no—
256C22⋊Q8—21no—
257C22⋊Q8—1221no—
258C22⋊Q8—2441no—
259C4×Q8—1221no—
260C4×Q8—1221no—
261C4×Q8—2441no—
262#117—2441no—
263C2×C42—611no—
264Q32—22no—
265Q32—1222no—
266SD32—22no—
267SD32—1222no—
268M32—612no—
269M32C22×D6112yes—
270#108—612no—
271#108—1221no—
272#106—611no—
273#106—1221no—
274#106C22×D6111yescharacteristic
275#104—1222no—
276#104—1222no—
277#103—21no—
278#103—1221no—
279#103—1221no—
280#103—1221no—
281#97—611no—
282#97—611no—
283C4×C8—611no—
284C4×C8—611no—
285#41—22no—
286#41—22no—
287#41—41no—
288#41—41no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—1221no—
292C2×Q8—2441no—
293C2×D4—21no—
294C2×D4—41no—
295C22×C4—41no—
296C22×C4—1221no—
297C22×C4D4×D6111yescharacteristic
298Q16—21no—
299Q16—22no—
300Q16—1222no—
301Q16—2442no—
302Q16#1175822412yes—
303SD16—22no—
304SD16—22no—
305D8—21no—
306D8#1175822412yes—
307M16—22no—
308M16—1222no—
309M16—1222no—
310M16—2442no—
311M16—2442no—
312M16—2441no—
313C2×C8—611no—
314C2×C8—611no—
315C2×C8—2442no—
316C2×C8—2442no—
317C2×C8—2441no—
318C2×C8D4×D6111yescharacteristic
319C2×C8C23×D6111yescharacteristic
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—1221no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C22⋊C4—41no—
330C22⋊C4—2441no—
331C42—611no—
332C42—611no—
333C42—1221no—
334C42—2441no—
335C42—2441no—
336C16—22no—
337C16—1222no—
338C23—41no—
339C23#1451980111yescharacteristic
340Q8—22no—
341Q8—41no—
342Q8—41no—
343Q8—41no—
344Q8—4882no—
345Q8—2441no—
346Q8—2441no—
347D4—22no—
348D4—41no—
349D4—42no—
350D4—42no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—1221no—
359C2×C4—1221no—
360C2×C4—2442no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4#1451980111yescharacteristic
364C2×C4C2×D4×D6111yescharacteristic
365C8—21no—
366C8—612no—
367C8—1221no—
368C8—2442no—
369C8C2×#1175822412yes—
370C22—21no—
371C22—41no—
372C22—42no—
373C22—161no—
374C22C2×#1451980111yescharacteristic
375C4—21no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—1222no—
380C4—4881no—
381C4—4881no—
382C4—2441no—
383C4#1684026811yescharacteristic
384C4AGL(1,ℤ/8)×D6111yescharacteristic
385C2—21no—
386C2—41no—
387C2—82no—
388C2#2145590511yescharacteristic
subgroupG/H
centerC2#21455905
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2191C2
socleC2×C61#3109
prime radicalC61×#575C22
hypercenter#2191D61
hypocenterC61#33689
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2191

Complements

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

NH ≅ G/Ntype
1. #21456559C2semidirect
3. #21448039C2semidirect
4. #21448036C2semidirect
5. #21447988C2semidirect
6. #21447898C2semidirect
8. #21447711C2semidirect
9. #21447707C2semidirect
10. #21447628C2semidirect
15. #16840159C22semidirect
16. #16840140C22semidirect
24. #16839869C22semidirect
25. #16839868C22semidirect
30. #16839585C22semidirect
31. #16839584C22semidirect
32. #16839582C22semidirect
33. #16839579C22semidirect
34. #16839578C22semidirect
37. #16839539C22semidirect
237. C61#33689semidirect

Maximal subgroups

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

subgroupindex
#214565592
#214547292
#214480392
#214480362
#214479882
#214478982
#214477702
#214477112
#214477072
#214476282
C61×#21912
#3368961