Finite Group Databasebeta

← #30639014

Subgroups — #30639014

Order 16256.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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#22550642C211yesmaximal, characteristic
2#22549902C211yesmaximal, characteristic, complemented
3#22549849C211yesmaximal, characteristic
4#22549842C211yesmaximal, characteristic
5#22549647C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#424×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861215—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13M16×D127—22no—
14#16861168C2211yescharacteristic
15Dic127⋊C8—22no—
16#16861123C414yes—
17(C2×D127)⋊C8—22no—
18C2×M16×C127C2211yescharacteristic
19C22×C8×C127C2211yescharacteristic
20#117×C127C2211yescharacteristic
21C4⋊C8×C127—22no—
22#14523248—21no—
23#14523248—21no—
24#14523248—21no—
25#14523248—21no—
26C2×Dic254Q811yescharacteristic
27C2×D508Q811yescharacteristic
28C2×C22⋊D127C2×C412yes—
29C2×C4×D127D412yes—
30#14523232—21no—
31#14523232—21no—
32#14523232—21no—
33#14523232—21no—
34#14523231—24no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8D412yes—
37C8×D127—24no—
38C4⋊Dic127—21no—
39C22⋊Dic127C2×C412yes—
40C4×Dic127—21no—
41C22×C4×C127C2311yescharacteristic
42M16×C127—21no—
43M16×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C2×C8×C127D412yes—
48C4⋊C4×C127—21no—
49C22⋊C4×C127C2×C412yes—
50C42×C127—21no—
51C22×D127C4⋊C412yes—
52Dic254—21no—
53Dic254—21no—
54D508—21no—
55D508—21no—
56C22⋊D127—22no—
57C22⋊D127—22no—
58C2×Dic127—22no—
59C2×Dic127C4⋊C412yes—
60C4×D127—22no—
61C4×D127—22no—
62C127⋊C8—22no—
63C127⋊C8—41no—
64C23×C127C22×C411yescharacteristic
65C2×C4×C127—22no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×Q811yescharacteristic
70C2×C4×C127#4111yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C8×C127—22no—
73C8×C127—41no—
74C22×C127—41no—
75C22×C127C2×C4⋊C411yescharacteristic
76C22×C127C4×D411yescharacteristic
77C22×C127C4×D411yescharacteristic
78C2×D127—22no—
79C2×D127—42no—
80C4×C127—42no—
81C4×C127#12011yescharacteristic
82C4×C127C22⋊Q811yescharacteristic
83C4×C127C22⋊Q811yescharacteristic
84C4×C127#12311yescharacteristic
85Dic127—22no—
86Dic127—42no—
87C2×C127—21no—
88C2×C127—21no—
89C2×C127—21no—
90C2×C127#38511yescharacteristic
91D127—42no—
92#1868—1271nomaximal, Sylow
93C127#186811yescharacteristic, complemented, Sylow
94#568—1271no—
95#424—1271no—
96#424D12711yescharacteristic, complemented
97#410—1271no—
98#408—1271no—
99#344—1271no—
100#344—1271no—
101C2×#41—1271no—
102#131—2542no—
103#131—2542no—
104C2×M16—1271no—
105C2×M16—1271no—
106C2×M16—2542no—
107C2×M16C2×D12711yescharacteristic
108C22×C8C2×D12711yescharacteristic
109#117—1271no—
110#117C2×D12711yescharacteristic
111C4⋊C8—22no—
112C4⋊C8—2542no—
113#99—1274no—
114C22⋊C8—2542no—
115#41—2541no—
116#41—2541no—
117#41—2541no—
118#41—2541no—
119C2×Q8—1271no—
120C2×D4—1272no—
121C2×D4—1271no—
122C22×C4—1272no—
123C22×C4C22×D12711yescharacteristic
124M16—21no—
125M16—21no—
126M16—2544no—
127M16—2541no—
128M16—2541no—
129M16—2541no—
130M16—2541no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—1272no—
135C2×C8—2544no—
136C2×C8—2541no—
137C2×C8C22⋊D12712yes—
138C4⋊C4—21no—
139C4⋊C4—2541no—
140C22⋊C4—1272no—
141C22⋊C4C4×D12712yes—
142C42—21no—
143C42—2541no—
144C23—1272no—
145C23C2×C4×D12711yescharacteristic
146Q8—2541no—
147Q8—2541no—
148D4—2542no—
149D4—2542no—
150D4—2541no—
151D4—2541no—
152C2×C4—22no—
153C2×C4—1272no—
154C2×C4—2542no—
155C2×C4—2542no—
156C2×C4—2542no—
157C2×C4D4×D12711yescharacteristic
158C2×C4D4×D12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C2×C4Q8×D12711yescharacteristic
161C2×C4#1452324811yescharacteristic
162C2×C4#1452324911yescharacteristic
163C8—22no—
164C8—41no—
165C8—5081no—
166C8—2542no—
167C22—41no—
168C22—5082no—
169C22—2542no—
170C22C4⋊C4×D12711yescharacteristic
171C22C4×C22⋊D12711yescharacteristic
172C22#1686117111yescharacteristic
173C4—42no—
174C4—5082no—
175C4—2542no—
176C4#1686117711yescharacteristic
177C4#1686118511yescharacteristic
178C4#1686118711yescharacteristic
179C4#1686119411yescharacteristic
180C2—21no—
181C2—21no—
182C2—21no—
183C2—5082no—
184C2#2254974711yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#424×C127C2
socleC2×C127#385
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#424D127
hypocenterC127#1868
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #22549902C2semidirect
5. #22549647C2semidirect
6. #22549646C2semidirect
7. #424×C127C2semidirect
93. C127#1868semidirect
96. #424D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225499022
#225498492
#225498422
#225496472
#225496462
#424×C1272
#1868127