Finite Group Databasebeta

← #30648217

Subgroups — #30648217

Order 16256.

480 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16861239C212yesmaximal
2C2×#16861209C211yesmaximal, characteristic, complemented
3#22550086C214yesmaximal, complemented
4#22550081C214yesmaximal, complemented
5#22549900C211yesmaximal, characteristic, complemented
6#22549884C212yesmaximal, complemented
7C4×#41×C127C211yesmaximal, characteristic
8C22×Dic254C2212yes—
9#16861239—28no—
10C2×#14523238C2214yes—
11C2×#14523236C2214yes—
12C2×#14523232C2212yes—
13#16861209—21no—
14#16861209—21no—
15#16861209C2214yes—
16C2×C4⋊Dic127—22no—
17C4⋊(C127⋊C8)C2214yes—
18#16861132C2218yes—
19C2×#41×C127C2211yescharacteristic
20C4×Q8×C127C2212yes—
21C4×D4×C127—22no—
22C4×D4×C127C2212yes—
23#117×C127—21no—
24#117×C127C2211yescharacteristic
25C2×C42×C127—21no—
26C2×C42×C127C2211yescharacteristic
27C2×Dic254—22no—
28C2×Dic254—24no—
29C2×Dic254—24no—
30C2×Dic254C2314yes—
31C22×Dic127—22no—
32#14523238—28no—
33#14523236—28no—
34#14523232—24no—
35C2×C127⋊C8C2314yes—
36C4⋊Dic127—24no—
37C4⋊Dic127—24no—
38#41×C127—22no—
39#41×C127D414yes—
40C2×Q8×C127C2311yescharacteristic
41C2×D4×C127—21no—
42C2×D4×C127C2311yescharacteristic
43C22×C4×C127—21no—
44C22×C4×C127—22no—
45C22×C4×C127D412yes—
46C22×C4×C127C2311yescharacteristic
47C4⋊C4×C127—22no—
48C4⋊C4×C127C2312yes—
49C22⋊C4×C127—21no—
50C22⋊C4×C127—22no—
51C42×C127—21no—
52C42×C127—22no—
53C42×C127D411yescharacteristic
54C42×C127D411yescharacteristic
55C42×C127C2312yes—
56Dic254—24no—
57Dic254—24no—
58Dic254—44no—
59C2×Dic127—24no—
60C2×Dic127—44no—
61C127⋊C8—24no—
62C23×C127—21no—
63C23×C127C2×D411yescharacteristic
64Q8×C127—21no—
65Q8×C127C2×D412yes—
66D4×C127—24no—
67D4×C127—21no—
68D4×C127C2×D412yes—
69C2×C4×C127—21no—
70C2×C4×C127—24no—
71C2×C4×C127—24no—
72C2×C4×C127—22no—
73C2×C4×C127—44no—
74C2×C4×C127C2×D412yes—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D412yes—
77C2×C4×C127C2×D412yes—
78C2×C4×C127#4112yes—
79C2×C4×C127C2411yescharacteristic
80C22×C127—22no—
81C22×C127—22no—
82C22×C127—42no—
83C22×C127#12112yes—
84C22×C127C22×D411yescharacteristic
85C4×C127—22no—
86C4×C127—22no—
87C4×C127—42no—
88C4×C127#12112yes—
89C4×C127#13714yescomplemented
90C4×C127C22×D411yescharacteristic
91C4×C127C2×#4111yescharacteristic
92Dic127—44no—
93C2×C127—22no—
94C2×C127—42no—
95C2×C127C2×#12111yescharacteristic
96C2×C127C2×#13712yes—
97#3039—1271nomaximal, Sylow
98C127#303911yescharacteristic, complemented, Sylow
99C2×#137—1272no—
100C2×C4⋊Q8—1271no—
101C4×#41D12711yescharacteristic
102#462—1274no—
103#461—1274no—
104#423—1271no—
105#419—1272no—
106C2×#41C2×D12711yescharacteristic
107C22×Q8—1272no—
108#137—2548no—
109C2×Q16—1274no—
110C2×SD16—1274no—
111C2×M16—1272no—
112C4⋊Q8—1274no—
113C4⋊Q8—2541no—
114C4⋊Q8—2541no—
115C4×Q8C2×D12712yes—
116C4×D4—22no—
117C4×D4C2×D12712yes—
118#117—21no—
119#117C2×D12711yescharacteristic
120C2×C4⋊C4—2542no—
121C2×C42—21no—
122C2×C42C2×D12711yescharacteristic
123C4⋊C8—1274no—
124#103—1278no—
125#41—22no—
126#41D50814yes—
127C2×Q8—1274no—
128C2×Q8—2544no—
129C2×Q8—2544no—
130C2×Q8—2542no—
131C2×Q8C22×D12711yescharacteristic
132C2×D4—21no—
133C2×D4C22×D12711yescharacteristic
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—2542no—
137C22×C4C22⋊D12712yes—
138C22×C4C22×D12711yescharacteristic
139Q16—2548no—
140SD16—2548no—
141M16—2544no—
142C2×C8—1274no—
143C4⋊C4—22no—
144C4⋊C4—2544no—
145C4⋊C4—2544no—
146C4⋊C4C22×D12712yes—
147C22⋊C4—21no—
148C22⋊C4—22no—
149C42—21no—
150C42—22no—
151C42C22⋊D12711yescharacteristic
152C42C22⋊D12711yescharacteristic
153C42C22×D12712yes—
154C23—21no—
155C23C2×C22⋊D12711yescharacteristic
156Q8—21no—
157Q8—5084no—
158Q8—2544no—
159Q8—2544no—
160Q8C2×D50812yes—
161D4—24no—
162D4—21no—
163D4C2×D50812yes—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—44no—
169C2×C4—5084no—
170C2×C4—2544no—
171C2×C4C2×C22⋊D12712yes—
172C2×C4C2×C22⋊D12712yes—
173C2×C4C2×C22⋊D12711yescharacteristic
174C2×C4C2×D50812yes—
175C2×C4#1452324812yes—
176C2×C4C23×D12711yescharacteristic
177C8—2544no—
178C22—22no—
179C22—22no—
180C22—42no—
181C22#1686118312yes—
182C22C22×C22⋊D12711yescharacteristic
183C4—22no—
184C4—22no—
185C4—42no—
186C4—5084no—
187C4#1686118312yes—
188C4#1686123914yescomplemented
189C4C22×D50811yescharacteristic
190C4C2×#1452324811yescharacteristic
191C2—22no—
192C2—42no—
193C2C2×#1686118311yescharacteristic
194C2C2×#1686123912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×#41×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×C127C22
hypercenterC4×#41D127
hypocenterC127#3039
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. C2×#16861209C2semidirect
3. #22550086C2semidirect
4. #22550081C2semidirect
5. #22549900C2semidirect
6. #22549884C2semidirect
89. C4×C127#137semidirect
98. C127#3039semidirect
188. C4#16861239semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612092
#225500862
#225500812
#225499002
#225498842
C4×#41×C1272
#3039127