Finite Group Databasebeta

← #30607920

Subgroups — #30607920

Order 15616.

274 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 86 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×#16840689C211yesmaximal, characteristic
2#21462027C211yesmaximal, characteristic
3#21461937C212yesmaximal
4#21460895C212yesmaximal
5#368×D61C211yesmaximal, characteristic
6D61⋊M32C2211yescharacteristic
7#16840773—22no—
8C2×#14519846C2212yes—
9#16840725C2211yescharacteristic
10#16840689—21no—
11#16840689—21no—
12#16840649—21no—
13C16×#6923—21no—
14#16840641—22no—
15C2×C16×D61C2211yescharacteristic
16#108×D61C2212yes—
17#16839592C411yescharacteristic, complemented
18#16839591C412yescomplemented
19C61×#368C411yescharacteristic, complemented
20D61⋊M16D411yescharacteristic
21#14519856D411yescharacteristic
22C2×C4⋊#6923—22no—
23#14519852Q812yes—
24#14519851—21no—
25D61⋊C16—21no—
26#14519846—22no—
27#14519846—22no—
28#14519846—42no—
29#14519845—42no—
30#14519831—21no—
31C8×#6923—21no—
32M16×D61—22no—
33C2×C8×D61C2311yescharacteristic
34C2×#11758185C2×C411yescharacteristic
35C16×D61D411yescharacteristic
36C16×D61D411yescharacteristic
37C16×D61Q811yescharacteristic
38C16×D61Q811yescharacteristic
39#14519713C2×C412yes—
40#14519712—22no—
41C2×C16×C61C2×C411yescharacteristic
42#108×C61C2×C412yes—
43C22×#6923—42no—
44C2×C4×D61C2×D411yescharacteristic
45C22⋊#6923—21no—
46#11758217Q1612yes—
47C4⋊#6923—42no—
48C4⋊#6923—42no—
49C4⋊#6923D812yes—
50C4×#6923—21no—
51#11758213—21no—
52D61⋊C8—21no—
53M16×C61—22no—
54C2×C8×C61C22×C411yescharacteristic
55#11758194—22no—
56C2×#92418#4111yescharacteristic
57#11758189—42no—
58C8×D61—42no—
59C8×D61C2×D411yescharacteristic
60C8×D61C2×Q811yescharacteristic
61C8×D61C2×Q811yescharacteristic
62C8×D61#4111yescharacteristic
63#11758187—22no—
64C16×C61C4⋊C411yescharacteristic
65C16×C61C4⋊C411yescharacteristic
66#11758185C4⋊C411yescharacteristic
67#11758185C4⋊C411yescharacteristic
68C22×D61#13511yescharacteristic
69C2×#6923—22no—
70C2×#6923—82no—
71C2×#6923—162no—
72C2×C4×C61C4×D411yescharacteristic
73C2×Dic61C2×Q1611yescharacteristic
74C4×D61#12111yescharacteristic
75C4×D61C4⋊Q811yescharacteristic
76C4×D61C2×D811yescharacteristic
77C4×D61#13511yescharacteristic
78#92420—22no—
79C8×C61—42no—
80C8×C61C2×C4⋊C411yescharacteristic
81C8×C61C4×Q811yescharacteristic
82#92418—42no—
83#92418C22⋊Q811yescharacteristic
84#92418#12511yescharacteristic
85#13172—611nomaximal, Sylow
86C22×C61C4×Q1611yescharacteristic
87C2×D61—22no—
88C2×D61#46611yescharacteristic
89C2×D61#49911yescharacteristic
90#6923—42no—
91#6923—162no—
92C4×C61#38911yescharacteristic
93C4×C61C4×D811yescharacteristic
94Dic61#46711yescharacteristic
95Dic61#50011yescharacteristic
96C2×#368—611no—
97C2×#367—611no—
98#2108—611no—
99#1952—612no—
100#1387—612no—
101C2×C61—21no—
102C2×C61#194211yescharacteristic
103D61—21no—
104D61#222312yes—
105C2×M32—611no—
106C22×C16—611no—
107C2×#108—612no—
108#428—1222no—
109C2×#107—612no—
110#405—611no—
111#368—611no—
112#368—612no—
113#368#692311yescharacteristic, complemented
114#367—1221no—
115#367—1221no—
116#346—1221no—
117C4×C16—1221no—
118#343—1222no—
119C61#1317211yescharacteristic, complemented, Sylow
120C2×M16—611no—
121C2×M16—1222no—
122C22×C8—611no—
123#117—611no—
124C2×C4⋊C4—1222no—
125M32—612no—
126M32—1221no—
127C2×C16—611no—
128C2×C16—611no—
129C2×C16—611no—
130C2×C16—611no—
131C2×C16—611no—
132C2×C16—1221no—
133C2×C16C2×#692311yescharacteristic
134#108—612no—
135#108—1222no—
136#108C2×#692312yes—
137#107—1222no—
138#107—1222no—
139#107—2442no—
140#106—2442no—
141#97—1221no—
142C4×C8—1221no—
143C22×C4—611no—
144C22×C4—2442no—
145M16—22no—
146M16—612no—
147M16—1221no—
148M16—1222no—
149M16—2442no—
150C2×C8—611no—
151C2×C8—611no—
152C2×C8—611no—
153C2×C8—611no—
154C2×C8—611no—
155C2×C8—1221no—
156C2×C8—2442no—
157C2×C8C22×#692311yescharacteristic
158C4⋊C4—612no—
159C4⋊C4—2442no—
160C4⋊C4—2442no—
161C22⋊C4—1221no—
162C42—1221no—
163C16—611no—
164C16—611no—
165C16—1222no—
166C16C4⋊#692311yescharacteristic
167C16C4⋊#692311yescharacteristic
168C23—611no—
169C2×C4—611no—
170C2×C4—611no—
171C2×C4—611no—
172C2×C4—611no—
173C2×C4—611no—
174C2×C4—1222no—
175C2×C4—9762no—
176C2×C4—4882no—
177C2×C4D4×#692311yescharacteristic
178C8—42no—
179C8—611no—
180C8—611no—
181C8—1222no—
182C8—2442no—
183C8C2×C4⋊#692311yescharacteristic
184C8Q8×#692311yescharacteristic
185C22—611no—
186C22—611no—
187C22—1222no—
188C22Q16×#692311yescharacteristic
189C4—611no—
190C4—611no—
191C4—9762no—
192C4—2442no—
193C4#1684071311yescharacteristic
194C4D8×#692311yescharacteristic
195C2—21no—
196C2—612no—
197C2—1221no—
198C2#2146192311yescharacteristic
subgroupG/H
centerC2#21461923
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#368C4
socleC2×C61#1942
prime radicalC22×D61#135
cosocleC2×C8×D61C23
hypercenter#368#6923
hypocenterC61#13172
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #368

Complements

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

NH ≅ G/Ntype
17. #16839592C4semidirect
18. #16839591C4semidirect
19. C61×#368C4semidirect
113. #368#6923semidirect
119. C61#13172semidirect

Maximal subgroups

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

subgroupindex
C2×#168406892
#214620272
#214619372
#214608952
#368×D612
#1317261