Finite Group Databasebeta

← #29593735

Subgroups — #29593735

Order 15616.

296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 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#21458688C211yesmaximal, characteristic
2#21447861C211yesmaximal, characteristic
3#21445186C211yesmaximal, characteristic, complemented
4#21443877C211yesmaximal, characteristic, complemented
5#21443799C211yesmaximal, characteristic, complemented
6#21443781C211yesmaximal, characteristic
7C61×#2241C211yesmaximal, characteristic, complemented
8#16840473—21no—
9C2×#14519720—21no—
10#16840163—21no—
11#16840065—22no—
12#16840049—21no—
13#16840044—21no—
14C8×Dic122C2211yescharacteristic
15C4×D488C2211yescharacteristic
16#16839844—21no—
17C4⋊#11758185C2211yescharacteristic
18#16839566—21no—
19#16839515C2211yescharacteristic
20#16839508—21no—
21C61×#500C2211yescharacteristic
22C61×#494C2211yescharacteristic
23C61×#358—22no—
24C4×C16×C61C2211yescharacteristic
25C2×D488D411yescharacteristic
26#14519768—22no—
27#14519757—41no—
28#14519753—42no—
29C4×Dic122D411yescharacteristic
30C4×D244—21no—
31#14519739—22no—
32C4⋊C4×D61—41no—
33#14519720—42no—
34C2×#11758185—21no—
35#14519711D411yescharacteristic
36#14519708—22no—
37#14519708—41no—
38C4⋊#92418—21no—
39C4⋊#92418D411yescharacteristic
40#14519706—21no—
41#14519700—41no—
42#14519699—21no—
43#14519697—22no—
44C8×Dic61—21no—
45C2×Q16×C61—21no—
46C2×Q16×C61—21no—
47C4⋊Q8×C61—21no—
48C4⋊Q8×C61—21no—
49C2×C16×C61—21no—
50#107×C61—21no—
51#107×C61D412yes—
52C4×C8×C61C2311yescharacteristic
53C2×Q8×C61—41no—
54C2×Q8×C61—41no—
55C2×D244—21no—
56C2×C4×D61—41no—
57C2×C4×D61—42no—
58C2×Dic122D811yescharacteristic
59Q16×C61—42no—
60Q16×C61—41no—
61Q16×C61—41no—
62C2×C8×C61C2×D411yescharacteristic
63C2×C8×C61C2×D411yescharacteristic
64C4⋊C4×C61—22no—
65C4⋊C4×C61—41no—
66C4⋊C4×C61—41no—
67C4⋊C4×C61—41no—
68C42×C61C2×D411yescharacteristic
69#11758198—42no—
70#11758198—41no—
71C4⋊Dic61—21no—
72C4⋊Dic61—42no—
73C4⋊Dic61D811yescharacteristic
74#11758196—21no—
75#11758196—42no—
76C4×Dic61—21no—
77C2×#92418—21no—
78C2×#92418—41no—
79D488—21no—
80D488SD1612yes—
81C16×C61—41no—
82#11758185—41no—
83C22×D61—41no—
84Q8×C61—81no—
85Q8×C61—81no—
86Q8×C61—161no—
87C2×C4×C61—42no—
88C2×C4×C61—81no—
89C2×C4×C61#12011yescharacteristic
90C2×C4×C61C2×D811yescharacteristic
91C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
92C2×Dic61—21no—
93C2×Dic61—21no—
94C2×Dic61—41no—
95D244—22no—
96D244—41no—
97C4×D61—81no—
98C4×D61—81no—
99C4×D61—162no—
100Dic122—21no—
101Dic122—21no—
102C8×C61—21no—
103C8×C61C2×SD1611yescharacteristic
104C8×C61#13711yescharacteristic
105#92418—41no—
106#92418—81no—
107#12359—611nomaximal, Sylow
108C22×C61#44711yescharacteristic
109C2×D61—42no—
110C2×D61—81no—
111C2×D61—81no—
112C4×C61—21no—
113C4×C61—82no—
114C4×C61—161no—
115C4×C61#44811yescharacteristic
116C4×C61#45411yescharacteristic
117C4×C61#50811yescharacteristic
118C4×C61#51011yescharacteristic
119Dic61—41no—
120Dic61—41no—
121Dic61—81no—
122#3380—611no—
123#2241D6111yescharacteristic, complemented
124#2208—611no—
125#1670—611no—
126#1365—611no—
127Q8⋊C16—611no—
128#1331—611no—
129C2×C61#162711yescharacteristic
130C2×C61#218911yescharacteristic
131C2×C61#219411yescharacteristic
132D61—82no—
133#554—1221no—
134C2×SD32—1221no—
135#500C2×D6111yescharacteristic
136#499—1221no—
137#494C2×D6111yescharacteristic
138#478—1222no—
139#476—1221no—
140#475—1221no—
141C8×Q8—611no—
142C4×D8—611no—
143C4×D8—1221no—
144C4⋊C16—611no—
145#358—22no—
146#358—1221no—
147C4×C16C2×D6111yescharacteristic
148#335—611no—
149#332—1221no—
150C61#1235911yescharacteristic, complemented, Sylow
151C2×Q16—21no—
152C2×Q16—21no—
153C2×D8—611no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156#125—1222no—
157C22⋊Q8—2441no—
158C22⋊Q8—2442no—
159C4×Q8—611no—
160C4×D4—1222no—
161C4×D4—1221no—
162C2×C4⋊C4—2441no—
163SD32—2442no—
164C2×C16—21no—
165C2×C16—1221no—
166#107—21no—
167#107—611no—
168#107C22⋊D6112yes—
169#106—1222no—
170#106—2441no—
171C4⋊C8—611no—
172C4⋊C8—1221no—
173C4⋊C8—1221no—
174#103—2441no—
175#102—1222no—
176#102—1221no—
177C4×C8—1221no—
178C4×C8C22×D6111yescharacteristic
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—611no—
182C2×D4—1221no—
183C22×C4—2442no—
184C22×C4—2441no—
185Q16—42no—
186Q16—41no—
187Q16—41no—
188D8—612no—
189D8—1221no—
190C2×C8—1221no—
191C2×C8—2441no—
192C2×C8D4×D6111yescharacteristic
193C2×C8C2×C22⋊D6111yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—611no—
199C4⋊C4—1221no—
200C4⋊C4—1221no—
201C4⋊C4—2442no—
202C4⋊C4—2442no—
203C22⋊C4—2442no—
204C22⋊C4—2441no—
205C42—1221no—
206C42D4×D6111yescharacteristic
207C16—41no—
208C16—2441no—
209C23—2441no—
210Q8—81no—
211Q8—81no—
212Q8—161no—
213Q8—1221no—
214Q8—1221no—
215D4—1222no—
216D4—2441no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—1221no—
220C2×C4—1221no—
221C2×C4—9762no—
222C2×C4—4881no—
223C2×C4—4881no—
224C2×C4—2441no—
225C2×C4#1451974511yescharacteristic
226C2×C4D8×D6111yescharacteristic
227C2×C4#1451980111yescharacteristic
228C8—21no—
229C8—4881no—
230C8—2441no—
231C8SD16×D6111yescharacteristic
232C8#1451980611yescharacteristic
233C22—4881no—
234C22—4881no—
235C22—2442no—
236C22#1683989411yescharacteristic
237C4—21no—
238C4—82no—
239C4—161no—
240C4—4881no—
241C4—2441no—
242C4—2441no—
243C4#1683989911yescharacteristic
244C4#1683993411yescharacteristic
245C4#1684019511yescharacteristic
246C4#1684020411yescharacteristic
247C2—4882no—
248C2#2144491911yescharacteristic
249C2#2144775811yescharacteristic
250C2#2144778911yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2241C2
socleC22×C61#447
prime radicalC2×D488D4
hypercenter#2241D61
hypocenterC61#12359
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2241

Complements

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

NH ≅ G/Ntype
3. #21445186C2semidirect
4. #21443877C2semidirect
5. #21443799C2semidirect
7. C61×#2241C2semidirect
123. #2241D61semidirect
150. C61#12359semidirect

Maximal subgroups

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

subgroupindex
#214586882
#214478612
#214451862
#214438772
#214437992
#214437812
C61×#22412
#1235961