Finite Group Databasebeta

← #29593799

Subgroups — #29593799

Order 15616.

366 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 69 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#21457578C211yesmaximal, characteristic
2#21448044C211yesmaximal, characteristic, complemented
3#21445205C211yesmaximal, characteristic, complemented
4#21443866C211yesmaximal, characteristic, complemented
5#21443783C211yesmaximal, characteristic, complemented
6#21443773C211yesmaximal, characteristic, complemented
7C61×#2225C211yesmaximal, characteristic, complemented
8C2×#14519800—21no—
9#16840434—21no—
10#16840139C2211yescharacteristic
11#16839999—21no—
12#16839964—21no—
13#16839962—21no—
14#16839910—22no—
15C8×D244C2211yescharacteristic
16#16839562—22no—
17C4×#11758185C2211yescharacteristic
18#16839515C2211yescharacteristic
19#16839508—21no—
20C61×#500C2211yescharacteristic
21C4×Q16×C61C2211yescharacteristic
22C61×#367—21no—
23C61×C4⋊C16C2211yescharacteristic
24C61×#358—21no—
25C2×#11758226—22no—
26#14519800—42no—
27#14519800—42no—
28C2×#11758201—22no—
29C2×#11758201—41no—
30C2×D488—21no—
31C2×D488D412yes—
32C2×C8×D61—21no—
33#14519774—21no—
34#14519774—41no—
35#14519752—22no—
36#14519752—42no—
37C4×D244—41no—
38C4×D244D411yescharacteristic
39C2×#11758185—21no—
40C4⋊#92418—21no—
41C4⋊#92418D411yescharacteristic
42#14519701—41no—
43#14519697—22no—
44#14519689—21no—
45C2×Q16×C61D411yescharacteristic
46C4⋊Q8×C61—21no—
47C4×Q8×C61—21no—
48C2×C16×C61—21no—
49#107×C61—21no—
50#107×C61—21no—
51#107×C61D411yescharacteristic
52#103×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C2×Q8×C61—21no—
55C2×Q8×C61—41no—
56#11758226—44no—
57#11758226—82no—
58C2×D244—22no—
59C2×D244—42no—
60C2×D244—41no—
61C2×D244—42no—
62C2×D244—41no—
63C2×D244D811yescharacteristic
64C2×C4×D61—21no—
65C2×C4×D61—42no—
66Q16×C61—21no—
67Q16×C61SD1612yescomplemented
68C2×C8×C61C2×D411yescharacteristic
69C2×C8×C61C2×D411yescharacteristic
70C4⋊C4×C61—21no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61—41no—
73C4⋊C4×C61—41no—
74C42×C61—41no—
75C42×C61C2×D411yescharacteristic
76#11758201—44no—
77#11758201—82no—
78#11758198—21no—
79#11758198—42no—
80C4⋊Dic61D811yescharacteristic
81C2×#92418—21no—
82C2×#92418—41no—
83D488—42no—
84D488—41no—
85D488—41no—
86C8×D61—41no—
87C8×D61—41no—
88C16×C61—41no—
89#11758185—41no—
90C22×D61—21no—
91C22×D61—42no—
92Q8×C61—22no—
93Q8×C61—41no—
94Q8×C61—81no—
95Q8×C61—81no—
96C2×C4×C61—41no—
97C2×C4×C61—41no—
98C2×C4×C61—81no—
99C2×C4×C61#12011yescharacteristic
100C2×C4×C61C2×D811yescharacteristic
101C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
102C2×Dic61—21no—
103D244—21no—
104D244—21no—
105D244—42no—
106D244—42no—
107D244—84no—
108D244—81no—
109D244—81no—
110D244—161no—
111D244—162no—
112C4×D61—41no—
113C4×D61—41no—
114C4×D61—84no—
115C8×C61—21no—
116C8×C61C2×SD1611yescharacteristic
117C8×C61AGL(1,ℤ/8)11yescharacteristic
118#92418—41no—
119#92418—81no—
120#12369—611nomaximal, Sylow
121C22×C61#44711yescharacteristic
122C2×D61—41no—
123C2×D61—41no—
124C2×D61—41no—
125C2×D61—82no—
126C2×D61—82no—
127C2×D61—82no—
128C4×C61—21no—
129C4×C61—42no—
130C4×C61—81no—
131C4×C61—161no—
132C4×C61#45011yescharacteristic
133C4×C61#45511yescharacteristic
134C4×C61#50811yescharacteristic
135C4×C61#50911yescharacteristic
136Dic61—41no—
137#3287—611no—
138#2240—611no—
139#2225D6111yescharacteristic, complemented
140#1673—611no—
141#1361—611no—
142#1332—611no—
143D4⋊C16—611no—
144C2×C61#163911yescharacteristic
145C2×C61#218611yescharacteristic
146C2×C61#219411yescharacteristic
147D61—81no—
148D61—162no—
149C2×#135—1221no—
150#550—1221no—
151#500C2×D6111yescharacteristic
152#493—611no—
153#466—1221no—
154#461—1221no—
155#460—1221no—
156#449—1222no—
157C4×Q16C2×D6111yescharacteristic
158C8×D4—611no—
159#367—21no—
160C4⋊C16C2×D6111yescharacteristic
161#358—21no—
162#357—1222no—
163C4×C16—611no—
164#335—611no—
165#332—1221no—
166C61#1236911yescharacteristic, complemented, Sylow
167C2×#41—1222no—
168#135—2442no—
169#135—2442no—
170C2×Q16C22⋊D6111yescharacteristic
171C2×SD16—1222no—
172C2×SD16—2441no—
173C2×D8—612no—
174C2×D8—1221no—
175C22×C8—1221no—
176C4⋊Q8—21no—
177#127—1221no—
178#127—2441no—
179#121—1222no—
180#121—2442no—
181C4×Q8—21no—
182C4×D4—611no—
183C4×D4—2441no—
184C2×C16—21no—
185C2×C16—1221no—
186#107—21no—
187#107—21no—
188#107C22⋊D6111yescharacteristic
189C4⋊C8—611no—
190C4⋊C8—1221no—
191#103—21no—
192#103—2441no—
193#102—1222no—
194C22⋊C8—1221no—
195C4×C8C22×D6111yescharacteristic
196#41—4882no—
197#41—2444no—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×D4—611no—
201C2×D4—1222no—
202C2×D4—2442no—
203C2×D4—2442no—
204C2×D4—2441no—
205C2×D4—2441no—
206C22×C4—1221no—
207C22×C4—2442no—
208Q16—21no—
209Q16#1175820112yescomplemented
210SD16—4882no—
211SD16—2444no—
212D8—2442no—
213D8—2441no—
214D8—2441no—
215C2×C8—1221no—
216C2×C8—2441no—
217C2×C8—2441no—
218C2×C8—2441no—
219C2×C8D4×D6111yescharacteristic
220C2×C8C2×C22⋊D6111yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—611no—
226C22⋊C4—1221no—
227C22⋊C4—2442no—
228C42—41no—
229C42D4×D6111yescharacteristic
230C16—41no—
231C16—2441no—
232C23—1221no—
233C23—2442no—
234Q8—22no—
235Q8—41no—
236Q8—81no—
237Q8—81no—
238D4—1221no—
239D4—1221no—
240D4—9761no—
241D4—9762no—
242D4—4884no—
243D4—4881no—
244D4—4881no—
245D4—2442no—
246D4—2442no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—81no—
250C2×C4—1221no—
251C2×C4—4884no—
252C2×C4—2441no—
253C2×C4—2441no—
254C2×C4#1451974511yescharacteristic
255C2×C4D8×D6111yescharacteristic
256C2×C4#1451980111yescharacteristic
257C8—21no—
258C8—4881no—
259C8—2441no—
260C8C2×#1175820111yescharacteristic
261C8#1451980211yescharacteristic
262C22—4882no—
263C22—4882no—
264C22—4882no—
265C22—2441no—
266C22—2441no—
267C22—2441no—
268C22#1683989411yescharacteristic
269C4—21no—
270C4—42no—
271C4—81no—
272C4—161no—
273C4—2441no—
274C4#1683991211yescharacteristic
275C4#1683993811yescharacteristic
276C4#1684019311yescharacteristic
277C4#1684019911yescharacteristic
278C2—9762no—
279C2—4881no—
280C2#2144499711yescharacteristic
281C2#2144774111yescharacteristic
282C2#2144778711yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2225C2
socleC22×C61#447
prime radical#21457578C2
hypercenter#2225D61
hypocenterC61#12369
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2225

Complements

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

NH ≅ G/Ntype
2. #21448044C2semidirect
3. #21445205C2semidirect
4. #21443866C2semidirect
5. #21443783C2semidirect
6. #21443773C2semidirect
7. C61×#2225C2semidirect
67. Q16×C61SD16semidirect
139. #2225D61semidirect
166. C61#12369semidirect
209. Q16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214575782
#214480442
#214452052
#214438662
#214437832
#214437732
C61×#22252
#1236961