Finite Group Databasebeta

← #29598774

Subgroups — #29598774

Order 15616.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16840034C212yesmaximal
2C2×#16839578C212yesmaximal, complemented
3#21447432C211yesmaximal, characteristic
4#21443913C211yesmaximal, characteristic, complemented
5C22×Q32×C61C211yesmaximal, characteristic, complemented
6C2×#14519808—22no—
7C2×#14519783C2211yescharacteristic
8C2×#14519717C2211yescharacteristic
9#16840034—24no—
10#16840034—24no—
11C2×#14519713C2212yes—
12C2×#14519696—22no—
13#16839578—24no—
14#16839541—22no—
15C22×Q16×C61C2212yes—
16C2×Q32×C61—24no—
17C2×Q32×C61—22no—
18C22×C16×C61C2211yescharacteristic
19C2×#11758222D411yescharacteristic
20#14519808—28no—
21#14519808—44no—
22C2×#11758202—42no—
23C2×#11758201—42no—
24#14519783—22no—
25#14519783—22no—
26C2×#11758194—22no—
27C2×#11758194D411yescharacteristic
28C2×#11758189—21no—
29C2×C8×D61—21no—
30#14519717—22no—
31C2×#11758185—21no—
32#14519713—22no—
33#14519713D414yes—
34#14519696—28no—
35C22×Q8×C61—22no—
36C2×Q16×C61—22no—
37C2×Q16×C61—22no—
38C2×Q16×C61—22no—
39C2×Q16×C61—22no—
40C2×Q16×C61—22no—
41C2×Q16×C61D414yes—
42C22×C8×C61C2311yescharacteristic
43Q32×C61—28no—
44C2×C16×C61—22no—
45C2×C16×C61D412yes—
46C2×Q8×C61—24no—
47C2×Q8×C61—42no—
48C2×Q8×C61—42no—
49C2×Q8×C61—42no—
50C2×Q8×C61—42no—
51C2×Q8×C61—42no—
52C22×C4×C61—42no—
53C22×C4×C61C2×D411yescharacteristic
54C2×C22⋊D61—21no—
55#11758222—22no—
56#11758222—22no—
57C2×D244D811yescharacteristic
58C2×C4×D61—21no—
59C2×Dic122D811yescharacteristic
60Q16×C61—24no—
61Q16×C61—24no—
62Q16×C61—42no—
63Q16×C61—42no—
64C2×C8×C61C2×D412yes—
65C2×C8×C61C2×D411yescharacteristic
66C2×C8×C61C2×D412yes—
67C2×C8×C61#4111yescharacteristic
68#11758202—48no—
69#11758201—48no—
70#11758194—24no—
71#11758194—22no—
72#11758194—42no—
73C2×#92418—21no—
74C2×#92418—42no—
75#11758189—42no—
76C8×D61—41no—
77C8×D61—41no—
78C16×C61—22no—
79#11758185—41no—
80C23×C61#13511yescharacteristic
81C22×D61—21no—
82Q8×C61—44no—
83Q8×C61—44no—
84Q8×C61—82no—
85Q8×C61—82no—
86C2×C4×C61—44no—
87C2×C4×C61—82no—
88C2×C4×C61—82no—
89C2×C4×C61#12012yes—
90C2×C4×C61#12111yescharacteristic
91C2×C4×C61C2×D811yescharacteristic
92C2×C4×C61AGL(1,ℤ/8)12yes—
93C22⋊D61—42no—
94C2×Dic61—21no—
95D244—22no—
96C4×D61—41no—
97C4×D61—41no—
98Dic122—22no—
99C8×C61#12011yescharacteristic
100C8×C61#12112yes—
101C8×C61#12711yescharacteristic
102#92418—41no—
103#92418—44no—
104#13315—611nomaximal, Sylow
105C22×C61—22no—
106C22×C61#44912yes—
107C22×C61#46611yescharacteristic
108C2×D61—42no—
109C2×D61—41no—
110C4×C61—84no—
111C4×C61#39211yescharacteristic
112C4×C61#44712yes—
113C4×C61#46911yescharacteristic
114Dic61—41no—
115C22×Q32D6111yescharacteristic, complemented
116C2×#473—612no—
117C2×#362—612no—
118#2113—611no—
119#1381—611no—
120C2×C61—22no—
121C2×C61#199911yescharacteristic
122C2×C61#222212yes—
123D61—81no—
124C2×#137—1222no—
125C22×Q16C2×D6112yes—
126C2×#131—611no—
127C2×Q32—24no—
128C2×Q32—22no—
129C2×M32—611no—
130C22×C16C2×D6111yescharacteristic
131#473—1224no—
132#473—1224no—
133C2×#108—612no—
134C2×#101—1222no—
135#362—1224no—
136C22⋊C16—1222no—
137C61#1331511yescharacteristic, complemented, Sylow
138C2×#41—611no—
139C22×Q8—22no—
140#137—1228no—
141#137—2444no—
142C2×Q16—22no—
143C2×Q16—22no—
144C2×Q16—22no—
145C2×Q16—22no—
146C2×Q16—22no—
147C2×Q16—2442no—
148C2×Q16C22⋊D6114yes—
149C2×SD16—2442no—
150#131—1222no—
151#131—1222no—
152C2×M16—611no—
153C2×M16—1221no—
154C2×M16—1222no—
155C22×C8—1221no—
156C22×C8C22×D6111yescharacteristic
157Q32—28no—
158M32—1222no—
159C2×C16—22no—
160C2×C16—1221no—
161C2×C16C22⋊D6112yes—
162#108—614no—
163#108—1222no—
164#101—1228no—
165#41—1222no—
166#41—1222no—
167C2×Q8—24no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—611no—
174C2×D4—611no—
175C2×D4—1221no—
176C22×C4—42no—
177C22×C4—1221no—
178C22×C4D4×D6111yescharacteristic
179Q16—24no—
180Q16—24no—
181Q16—42no—
182Q16—42no—
183Q16—2448no—
184SD16—2448no—
185M16—1222no—
186M16—1224no—
187M16—2442no—
188M16—2442no—
189C2×C8—1221no—
190C2×C8—2442no—
191C2×C8—2441no—
192C2×C8—2441no—
193C2×C8D4×D6112yes—
194C2×C8#1175822411yescharacteristic
195C2×C8C2×C22⋊D6112yes—
196C2×C8C2×C22⋊D6111yescharacteristic
197C16—22no—
198C16—2441no—
199C23—1221no—
200C23#1451979611yescharacteristic
201Q8—44no—
202Q8—44no—
203Q8—82no—
204Q8—82no—
205Q8—1222no—
206D4—1222no—
207D4—2442no—
208C2×C4—44no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—1221no—
212C2×C4—2441no—
213C2×C4—2441no—
214C2×C4#1451974512yes—
215C2×C4#1451974811yescharacteristic
216C2×C4D8×D6111yescharacteristic
217C2×C4#1451980112yes—
218C8—2444no—
219C8—2441no—
220C8#1451974611yescharacteristic
221C8#1451974912yes—
222C8#1451977311yescharacteristic
223C22—22no—
224C22—2442no—
225C22—2441no—
226C22#1683990412yes—
227C22#1683999511yescharacteristic
228C4—84no—
229C4—2441no—
230C4#1683967011yescharacteristic
231C4#1683989412yes—
232C4#1684001211yescharacteristic
233C2—22no—
234C2—4881no—
235C2#2144681411yescharacteristic
236C2#2144794112yes—
subgroupG/H
centerC22#16839995
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC22×Q32×C61C2
socleC22×C61#466
prime radicalC2×#11758222D4
hypercenterC22×Q32D61
hypocenterC61#13315
LCSG ▷ C22×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×Q32

Complements

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

NH ≅ G/Ntype
2. C2×#16839578C2semidirect
4. #21443913C2semidirect
5. C22×Q32×C61C2semidirect
115. C22×Q32D61semidirect
137. C61#13315semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
C2×#168395782
#214474322
#214439132
C22×Q32×C612
#1331561