Finite Group Databasebeta

← #30648552

Subgroups — #30648552

Order 16256.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861201C211yesmaximal, characteristic, complemented
2#22550183C214yesmaximal, complemented
3#22550176C214yesmaximal, complemented
4#22549919C212yesmaximal, complemented
5#22549900C211yesmaximal, characteristic, complemented
6#22549877C212yesmaximal
7C4×#41×C127C211yesmaximal, characteristic
8C2×#14523232C2212yes—
9#16861201—21no—
10#16861201—21no—
11#16861201C2214yes—
12C2×C4⋊Dic127C2212yes—
13C2×Dic127⋊C4—22no—
14#16861142C2214yes—
15#16861140C2214yes—
16C4⋊(C127⋊C8)C2214yes—
17#16861133C2214yes—
18#16861130C2214yes—
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
27C22×Dic127—22no—
28#14523232—24no—
29C2×C127⋊C8C2314yes—
30C4⋊Dic127—22no—
31C4⋊Dic127C2314yes—
32Dic127⋊C4—24no—
33Dic127⋊C4—24no—
34#41×C127—22no—
35#41×C127Q814yes—
36C2×Q8×C127C2311yescharacteristic
37C2×D4×C127—21no—
38C2×D4×C127C2311yescharacteristic
39C22×C4×C127—21no—
40C22×C4×C127—22no—
41C22×C4×C127D412yes—
42C22×C4×C127C2311yescharacteristic
43C4⋊C4×C127—22no—
44C4⋊C4×C127C2312yes—
45C22⋊C4×C127—21no—
46C22⋊C4×C127—22no—
47C42×C127—21no—
48C42×C127—22no—
49C42×C127D411yescharacteristic
50C42×C127D411yescharacteristic
51C42×C127C2312yes—
52C2×Dic127—24no—
53C2×Dic127—44no—
54C127⋊C8—24no—
55C23×C127—21no—
56C23×C127C2×D411yescharacteristic
57Q8×C127—21no—
58Q8×C127C2×Q812yes—
59D4×C127—24no—
60D4×C127—21no—
61D4×C127C2×Q812yes—
62C2×C4×C127—21no—
63C2×C4×C127—24no—
64C2×C4×C127—22no—
65C2×C4×C127—42no—
66C2×C4×C127—44no—
67C2×C4×C127C2×D412yes—
68C2×C4×C127C2×D412yes—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×Q812yes—
71C2×C4×C127#4112yes—
72C2×C4×C127C2411yescharacteristic
73C22×C127—22no—
74C22×C127—22no—
75C22×C127—42no—
76C22×C127C22⋊Q812yes—
77C22×C127C22×D411yescharacteristic
78C4×C127—22no—
79C4×C127—22no—
80C4×C127—22no—
81C4×C127—42no—
82C4×C127C22⋊Q812yes—
83C4×C127C22×Q811yescharacteristic
84C4×C127C2×#4111yescharacteristic
85Dic127—44no—
86C2×C127—22no—
87C2×C127—42no—
88C2×C127C2×C22⋊Q811yescharacteristic
89C2×C127#57512yes—
90#3080—1271nomaximal, Sylow
91C127#308011yescharacteristic, complemented, Sylow
92C2×#125—1271no—
93C4×#41D12711yescharacteristic
94#479—1274no—
95#478—1274no—
96#428—1272no—
97#423—1271no—
98#417—1272no—
99C2×#41C2×D12711yescharacteristic
100C2×M16—1272no—
101#125—1274no—
102#125—2541no—
103#125—2541no—
104C4×Q8C2×D12712yes—
105C4×D4—22no—
106C4×D4C2×D12712yes—
107#117—21no—
108#117C2×D12711yescharacteristic
109C2×C4⋊C4—1272no—
110C2×C4⋊C4—2542no—
111C2×C42—21no—
112C2×C42C2×D12711yescharacteristic
113#107—1274no—
114#106—1274no—
115C4⋊C8—1274no—
116#103—1274no—
117#102—1274no—
118#41—22no—
119#41Dic25414yes—
120C2×Q8C22×D12711yescharacteristic
121C2×D4—21no—
122C2×D4C22×D12711yescharacteristic
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—2542no—
126C22×C4C22⋊D12712yes—
127C22×C4C22×D12711yescharacteristic
128M16—2544no—
129C2×C8—1274no—
130C4⋊C4—22no—
131C4⋊C4—1274no—
132C4⋊C4—2544no—
133C4⋊C4—2544no—
134C4⋊C4—2542no—
135C4⋊C4C22×D12712yes—
136C22⋊C4—21no—
137C22⋊C4—22no—
138C42—21no—
139C42—22no—
140C42C22⋊D12711yescharacteristic
141C42C22⋊D12711yescharacteristic
142C42C22×D12712yes—
143C23—21no—
144C23C2×C22⋊D12711yescharacteristic
145Q8—21no—
146Q8C2×Dic25412yes—
147D4—24no—
148D4—21no—
149D4C2×Dic25412yes—
150C2×C4—21no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—44no—
155C2×C4—5084no—
156C2×C4—2544no—
157C2×C4C2×C22⋊D12712yes—
158C2×C4C2×C22⋊D12712yes—
159C2×C4C2×C22⋊D12711yescharacteristic
160C2×C4C2×Dic25412yes—
161C2×C4#1452324812yes—
162C2×C4C23×D12711yescharacteristic
163C8—2544no—
164C22—22no—
165C22—22no—
166C22—42no—
167C22#1686118812yes—
168C22C22×C22⋊D12711yescharacteristic
169C4—22no—
170C4—22no—
171C4—22no—
172C4—42no—
173C4—5084no—
174C4#1686118812yes—
175C4C22×Dic25411yescharacteristic
176C4C2×#1452324811yescharacteristic
177C2—22no—
178C2—42no—
179C2C2×#1686118811yescharacteristic
180C2#2255067412yes—
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#3080
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
1. C2×#16861201C2semidirect
2. #22550183C2semidirect
3. #22550176C2semidirect
4. #22549919C2semidirect
5. #22549900C2semidirect
91. C127#3080semidirect

Maximal subgroups

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

subgroupindex
C2×#168612012
#225501832
#225501762
#225499192
#225499002
#225498772
C4×#41×C1272
#3080127