Finite Group Databasebeta

← #29593561

Subgroups — #29593561

Order 15616.

306 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 71 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#21458825C211yesmaximal, characteristic
2#21448071C211yesmaximal, characteristic
3#21447866C211yesmaximal, characteristic
4#21443867C211yesmaximal, characteristic
5Q8⋊#11758185C211yesmaximal, characteristic
6#21443782C211yesmaximal, characteristic
7C61×#1670C211yesmaximal, characteristic
8#16840491—21no—
9C2×#14519723—22no—
10C2×#14519723—21no—
11#16840142—21no—
12#16840142C2211yescharacteristic
13#16839973—22no—
14#16839973—21no—
15C4×Dic244—21no—
16C4×Dic244C2211yescharacteristic
17C4⋊#11758185C2211yescharacteristic
18#16839566—21no—
19C4×#11758185C2211yescharacteristic
20C61×#494C2211yescharacteristic
21C61×#475—21no—
22C8×Q8×C61C2211yescharacteristic
23C61×#335C2211yescharacteristic
24C61×#332—21no—
25C2×Dic244—21no—
26C2×Dic244—22no—
27C2×Dic244D411yescharacteristic
28C2×Dic244D412yes—
29#14519777—22no—
30#14519777—21no—
31#14519768—42no—
32#14519768—41no—
33Q8×Dic61—41no—
34C4×Dic122—22no—
35C4×Dic122—21no—
36#14519723—44no—
37#14519723—42no—
38C2×#11758185—21no—
39C2×#11758185—21no—
40#14519711D411yescharacteristic
41#14519702—22no—
42#14519702—21no—
43C2×Q16×C61—21no—
44C2×Q16×C61—21no—
45C4⋊Q8×C61—21no—
46C4×Q8×C61D411yescharacteristic
47#106×C61—41no—
48C4⋊C8×C61—21no—
49C4⋊C8×C61—21no—
50C4⋊C8×C61D411yescharacteristic
51#103×C61—41no—
52C4×C8×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C2×Q8×C61—41no—
55C2×Q8×C61D811yescharacteristic
56C2×Dic122—21no—
57C2×Dic122—22no—
58Q16×C61—42no—
59Q16×C61—41no—
60Q16×C61—41no—
61C2×C8×C61—21no—
62C2×C8×C61—41no—
63C2×C8×C61C2×D411yescharacteristic
64C2×C8×C61C2×D411yescharacteristic
65C4⋊C4×C61—21no—
66C4⋊C4×C61—41no—
67C4⋊C4×C61—41no—
68C4⋊C4×C61D811yescharacteristic
69C42×C61—21no—
70C42×C61C2×D411yescharacteristic
71C4⋊Dic61—21no—
72C4⋊Dic61—42no—
73C4⋊Dic61—42no—
74C4⋊Dic61—41no—
75C4⋊Dic61—41no—
76#11758196—42no—
77#11758196—41no—
78C4×Dic61—42no—
79C4×Dic61—41no—
80Dic244—21no—
81Dic244—21no—
82Dic244—42no—
83Dic244—41no—
84Dic244—41no—
85Dic244—82no—
86#11758185—22no—
87#11758185—41no—
88Q8×C61—21no—
89Q8×C61—81no—
90Q8×C61—81no—
91Q8×C61Q3212yescomplemented
92C2×C4×C61—21no—
93C2×C4×C61—21no—
94C2×C4×C61—81no—
95C2×C4×C61#12011yescharacteristic
96C2×C4×C61C2×D811yescharacteristic
97C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
98C2×Dic61—41no—
99C2×Dic61—42no—
100C2×Dic61—41no—
101Dic122—41no—
102Dic122—42no—
103Dic122—42no—
104Dic122—41no—
105C8×C61—21no—
106C8×C61—41no—
107C8×C61—81no—
108C8×C61#13511yescharacteristic
109C8×C61AGL(1,ℤ/8)11yescharacteristic
110#12334—611nomaximal, Sylow
111C22×C61#44711yescharacteristic
112C4×C61—22no—
113C4×C61—21no—
114C4×C61—41no—
115C4×C61—161no—
116C4×C61#44911yescharacteristic
117C4×C61#45311yescharacteristic
118C4×C61C2×Q3211yescharacteristic
119C4×C61#51011yescharacteristic
120Dic61—81no—
121Dic61—82no—
122Dic61—81no—
123#3395—611no—
124#2247—611no—
125#2209—611no—
126#1670D6111yescharacteristic
127#1362—611no—
128Q8⋊C16—611no—
129#1332—611no—
130C2×C61#162111yescharacteristic
131C2×C61#218811yescharacteristic
132C2×C61#219511yescharacteristic
133#557—1221no—
134C2×Q32—1221no—
135C2×Q32—1222no—
136#494—611no—
137#494—1221no—
138#494C2×D6111yescharacteristic
139#475—21no—
140#462—1222no—
141#462—1221no—
142C8×Q8C2×D6111yescharacteristic
143C4×Q16—611no—
144C4×Q16—1221no—
145C4⋊C16—611no—
146#358—1221no—
147C4×C16—611no—
148#335C2×D6111yescharacteristic
149#332—21no—
150C61#1233411yescharacteristic, complemented, Sylow
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—611no—
154C2×Q16—612no—
155C2×Q16—1222no—
156C2×Q16—1221no—
157C4⋊Q8—21no—
158C4⋊Q8—1222no—
159C4⋊Q8—1221no—
160#125—2442no—
161#125—2441no—
162C4×Q8—1222no—
163C4×Q8—1221no—
164C4×Q8—2441no—
165C4×Q8C22⋊D6111yescharacteristic
166Q32—2444no—
167Q32—2442no—
168C2×C16—1221no—
169C2×C16—1221no—
170#107—611no—
171#106—41no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8C22⋊D6111yescharacteristic
175#103—41no—
176#103—1222no—
177#103—1221no—
178C4×C8—21no—
179C4×C8C22×D6111yescharacteristic
180C2×Q8—41no—
181C2×Q8—1221no—
182C2×Q8—1222no—
183C2×Q8#1175819911yescharacteristic
184Q16—42no—
185Q16—41no—
186Q16—41no—
187Q16—1221no—
188Q16—1221no—
189Q16—4882no—
190Q16—2442no—
191Q16—2441no—
192Q16—2441no—
193C2×C8—21no—
194C2×C8—41no—
195C2×C8D4×D6111yescharacteristic
196C2×C8D4×D6111yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—1221no—
201C4⋊C4—2442no—
202C4⋊C4—2442no—
203C4⋊C4—2442no—
204C4⋊C4—2441no—
205C4⋊C4—2441no—
206C4⋊C4—2441no—
207C4⋊C4#1175819911yescharacteristic
208C42—21no—
209C42—2442no—
210C42—2441no—
211C42C2×C22⋊D6111yescharacteristic
212C16—1222no—
213C16—2441no—
214Q8—21no—
215Q8—81no—
216Q8—81no—
217Q8—2441no—
218Q8—2442no—
219Q8—2441no—
220Q8—2442no—
221Q8#1451972312yescomplemented
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—81no—
225C2×C4—2441no—
226C2×C4—2442no—
227C2×C4—2441no—
228C2×C4#1451974511yescharacteristic
229C2×C4C2×#1175819911yescharacteristic
230C2×C4#1451980211yescharacteristic
231C8—21no—
232C8—41no—
233C8—81no—
234C8#1451979811yescharacteristic
235C8#1451980411yescharacteristic
236C22#1683989511yescharacteristic
237C4—22no—
238C4—21no—
239C4—41no—
240C4—161no—
241C4—4881no—
242C4—4882no—
243C4—4881no—
244C4#1683990611yescharacteristic
245C4#1683992811yescharacteristic
246C4C2×#1451972311yescharacteristic
247C4#1684020211yescharacteristic
248C2#2144488211yescharacteristic
249C2#2144775211yescharacteristic
250C2#2144779511yescharacteristic
subgroupG/H
centerC22#16839895
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1670C2
socle, prime radicalC22×C61#447
hypercenter#1670D61
hypocenterC61#12334
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
91. Q8×C61Q32semidirect
150. C61#12334semidirect
221. Q8#14519723semidirect

Maximal subgroups

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

subgroupindex
#214588252
#214480712
#214478662
#214438672
Q8⋊#117581852
#214437822
C61×#16702
#1233461