Finite Group Databasebeta

← #29593363

Subgroups — #29593363

Order 15616.

322 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 53 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#21456730C211yesmaximal, characteristic
2#21448038C211yesmaximal, characteristic, complemented
3#21447371C211yesmaximal, characteristic, complemented
4#21443808C211yesmaximal, characteristic, complemented
5#21443804C211yesmaximal, characteristic, complemented
6#21443756C211yesmaximal, characteristic, complemented
7C61×#2238C211yesmaximal, characteristic, complemented
8#16840341—21no—
9#16840094—21no—
10#16840079—21no—
11#16840020C2211yescharacteristic
12#16839989—21no—
13#16839925—21no—
14#16839904—21no—
15#16839790—21no—
16#16839778—21no—
17#16839730C2211yescharacteristic
18#16839586—21no—
19#16839577—21no—
20#16839544C2211yescharacteristic
21#16839519C2211yescharacteristic
22C61×#469C2211yescharacteristic
23C61×#430C2211yescharacteristic
24C61×#365—21no—
25C61×#361—21no—
26C61×#349C2211yescharacteristic
27C2×#11758222D411yescharacteristic
28C2×#11758224—21no—
29C2×Dic244D411yescharacteristic
30C2×#11758200—21no—
31C22×#92418D411yescharacteristic
32#14519757—21no—
33#14519756—21no—
34#14519754—41no—
35#14519750—41no—
36#14519749—41no—
37#14519748—21no—
38#14519748—41no—
39#14519736—21no—
40C4⋊C4×D61—21no—
41#14519717—21no—
42#14519709D411yescharacteristic
43#14519708—21no—
44#14519706—42no—
45#14519703—42no—
46#14519702—21no—
47#14519700—21no—
48#14519698—21no—
49#14519689—21no—
50#14519689—21no—
51C2×D8×C61D411yescharacteristic
52C2×M16×C61—21no—
53C2×M16×C61C2311yescharacteristic
54#121×C61—21no—
55M32×C61—21no—
56#108×C61—21no—
57#108×C61—21no—
58#106×C61D411yescharacteristic
59#102×C61—21no—
60C2×D4×C61—21no—
61C2×D4×C61—41no—
62C22×C4×C61C2×D411yescharacteristic
63C2×C22⋊D61—21no—
64C2×C22⋊D61—41no—
65C2×C22⋊D61—41no—
66C22×Dic61—41no—
67#11758224—81no—
68#11758224—81no—
69#11758222—41no—
70#11758222—41no—
71C2×D244SD1611yescharacteristic
72C2×C4×D61—21no—
73C2×C4×D61—41no—
74C2×Dic122—21no—
75C2×Dic122SD1611yescharacteristic
76D8×C61—41no—
77M16×C61—41no—
78M16×C61—41no—
79M16×C61—41no—
80C2×C8×C61—41no—
81C2×C8×C61C2×D411yescharacteristic
82C2×C8×C61C2×D411yescharacteristic
83C4⋊C4×C61—21no—
84C22⋊C4×C61—41no—
85C22⋊Dic61—41no—
86C22⋊Dic61—41no—
87C22⋊Dic61—41no—
88#11758200—81no—
89#11758198—41no—
90C4⋊Dic61—21no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93#11758196—41no—
94#11758196—41no—
95C4×Dic61—22no—
96C2×#92418—21no—
97C2×#92418—41no—
98Dic244—41no—
99C16×C61—41no—
100#11758185—41no—
101C23×C61—41no—
102C23×C61AGL(1,ℤ/8)11yescharacteristic
103C22×D61—21no—
104D4×C61—41no—
105D4×C61—41no—
106D4×C61—161no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61#12011yescharacteristic
111C2×C4×C61C2×SD1611yescharacteristic
112C22⋊D61—81no—
113C22⋊D61—82no—
114C22⋊D61—161no—
115C2×Dic61—21no—
116C2×Dic61—42no—
117C2×Dic61—41no—
118C2×Dic61—41no—
119C2×Dic61—82no—
120D244—41no—
121C4×D61—41no—
122C4×D61—41no—
123C4×D61—161no—
124Dic122—41no—
125Dic122—41no—
126Dic122—41no—
127C8×C61—21no—
128C8×C61—21no—
129C8×C61—81no—
130#92418—41no—
131#12303—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—41no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61—81no—
137C22×C61#45011yescharacteristic
138C2×D61—41no—
139C2×D61—41no—
140C2×D61—41no—
141C4×C61—21no—
142C4×C61—81no—
143C4×C61#45311yescharacteristic
144C4×C61#45811yescharacteristic
145Dic61—41no—
146Dic61—81no—
147Dic61—82no—
148Dic61—81no—
149#3204—611no—
150#2239—611no—
151#2238D6111yescharacteristic, complemented
152#2098—611no—
153#1340—611no—
154#1339—611no—
155#1321—611no—
156C2×C61—21no—
157C2×C61—41no—
158C2×C61—161no—
159C2×C61#161311yescharacteristic
160D61—81no—
161#537—1221no—
162#483—1221no—
163#481—1221no—
164#470—611no—
165#469C2×D6111yescharacteristic
166#465—1221no—
167#452—1221no—
168#449—1221no—
169#430C2×D6111yescharacteristic
170#424—1221no—
171#421—1221no—
172#408—611no—
173#365—21no—
174#365—1221no—
175#362—1221no—
176#361—21no—
177#349—611no—
178#349C2×D6111yescharacteristic
179H(3,ℤ/4)—611no—
180C61#1230311yescharacteristic, complemented, Sylow
181C2×#41—611no—
182C2×#41—1221no—
183C2×Q16—611no—
184C2×SD16—1221no—
185C2×D8C22⋊D6111yescharacteristic
186C2×M16—21no—
187C2×M16C22×D6111yescharacteristic
188C22×C8—611no—
189C22⋊Q8—1221no—
190C22⋊Q8—1221no—
191C22⋊Q8—2441no—
192#121—21no—
193#121—1221no—
194#121—2441no—
195#121—2441no—
196#121—2441no—
197#117—1221no—
198C2×C4⋊C4—1221no—
199M32—21no—
200M32—1221no—
201#108—21no—
202#108—21no—
203#106—611no—
204#106—1221no—
205#106C22⋊D6111yescharacteristic
206C4⋊C8—2442no—
207#104—2442no—
208#103—1221no—
209#103—1221no—
210#102—21no—
211#102—1221no—
212C22⋊C8—1221no—
213C22⋊C8—1221no—
214#41—4881no—
215#41—4881no—
216#41—2441no—
217#41—2441no—
218C2×Q8—611no—
219C2×Q8—1221no—
220C2×D4—21no—
221C2×D4—41no—
222C2×D4—611no—
223C2×D4—1221no—
224C2×D4—2441no—
225C2×D4—2441no—
226C22×C4—1221no—
227C22×C4—2441no—
228C22×C4—2441no—
229C22×C4D4×D6111yescharacteristic
230Q16—2441no—
231SD16—4881no—
232D8—41no—
233M16—41no—
234M16—41no—
235M16—41no—
236C2×C8—41no—
237C2×C8—1221no—
238C2×C8—2441no—
239C2×C8D4×D6111yescharacteristic
240C2×C8C2×C22⋊D6111yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—1221no—
243C4⋊C4—2441no—
244C4⋊C4—2441no—
245C4⋊C4—2441no—
246C4⋊C4—2441no—
247C22⋊C4—41no—
248C22⋊C4—2441no—
249C22⋊C4—2441no—
250C22⋊C4—2441no—
251C22⋊C4—2441no—
252C42—1222no—
253C16—41no—
254C16—2441no—
255C23—41no—
256C23—1221no—
257C23#1451980411yescharacteristic
258Q8—2441no—
259Q8—2441no—
260Q8—2441no—
261D4—41no—
262D4—41no—
263D4—161no—
264D4—9761no—
265D4—4882no—
266D4—4881no—
267D4—2441no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—1221no—
272C2×C4—9761no—
273C2×C4—4882no—
274C2×C4—2442no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4—2441no—
278C2×C4—2441no—
279C2×C4#1451974511yescharacteristic
280C2×C4SD16×D6111yescharacteristic
281C8—21no—
282C8—21no—
283C8—81no—
284C8—2441no—
285C22—21no—
286C22—41no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—2441no—
291C22—2441no—
292C22—2441no—
293C22#1683991111yescharacteristic
294C4—21no—
295C4—81no—
296C4—4881no—
297C4—4882no—
298C4—4881no—
299C4—2441no—
300C4#1683992811yescharacteristic
301C4#1683995111yescharacteristic
302C2—21no—
303C2—41no—
304C2—161no—
305C2—4881no—
306C2#2144483711yescharacteristic
subgroupG/H
centerC2#21444837
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#2238C2
socleC2×C61#1613
prime radical#21456730C2
hypercenter#2238D61
hypocenterC61#12303
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2238

Complements

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

NH ≅ G/Ntype
2. #21448038C2semidirect
3. #21447371C2semidirect
4. #21443808C2semidirect
5. #21443804C2semidirect
6. #21443756C2semidirect
7. C61×#2238C2semidirect
151. #2238D61semidirect
180. C61#12303semidirect

Maximal subgroups

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

subgroupindex
#214567302
#214480382
#214473712
#214438082
#214438042
#214437562
C61×#22382
#1230361