Finite Group Databasebeta

← #30638784

Subgroups — #30638784

Order 16256.

256 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16861145C211yesmaximal, characteristic, complemented
2#22549914C211yesmaximal, characteristic, complemented
3#22549893C212yesmaximal
4#22549648C212yesmaximal, complemented
5C2×#131×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7#16861168C2212yes—
8#16861145—21no—
9#16861145—21no—
10#16861143—21no—
11#16861141—21no—
12#16861136—24no—
13C2×#41×C127C2211yescharacteristic
14#131×C127—21no—
15#131×C127—21no—
16#131×C127C412yescomplemented
17#131×C127C412yescomplemented
18C2×M16×C127—21no—
19C2×M16×C127C2211yescharacteristic
20C22×C8×C127—21no—
21C22×C8×C127C2211yescharacteristic
22#14523232—22no—
23#14523232—22no—
24C2×C127⋊C8—22no—
25C4⋊Dic127—22no—
26C22⋊Dic127—22no—
27C4×Dic127—22no—
28#41×C127—21no—
29#41×C127—21no—
30#41×C127C2×C412yes—
31#41×C127D412yes—
32C2×Q8×C127Q811yescharacteristic
33C2×D4×C127—21no—
34C2×D4×C127Q811yescharacteristic
35C22×C4×C127—21no—
36C22×C4×C127C2311yescharacteristic
37M16×C127—21no—
38M16×C127—22no—
39M16×C127—22no—
40M16×C127C2×C412yes—
41C2×C8×C127—21no—
42C2×C8×C127—22no—
43C2×C8×C127—22no—
44C2×C8×C127C2×C411yescharacteristic
45C2×C8×C127C2×C411yescharacteristic
46C2×C8×C127D411yescharacteristic
47C2×C8×C127D411yescharacteristic
48C2×C8×C127D411yescharacteristic
49C2×C8×C127D411yescharacteristic
50C2×Dic127—42no—
51C127⋊C8—42no—
52C23×C127—21no—
53C23×C127#4111yescharacteristic
54Q8×C127—21no—
55Q8×C127C4⋊C412yes—
56D4×C127—22no—
57D4×C127—22no—
58D4×C127—21no—
59D4×C127C4⋊C412yes—
60C2×C4×C127—21no—
61C2×C4×C127—22no—
62C2×C4×C127—22no—
63C2×C4×C127C22×C411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×Q811yescharacteristic
68C2×C4×C127#4111yescharacteristic
69C8×C127—22no—
70C8×C127C22⋊C411yescharacteristic
71C8×C127C22⋊C411yescharacteristic
72C8×C127C22⋊C411yescharacteristic
73C8×C127C22⋊C411yescharacteristic
74C22×C127—22no—
75C22×C127—41no—
76C22×C127—42no—
77C22×C127#11711yescharacteristic
78C22×C127#12111yescharacteristic
79C22×C127C22⋊Q811yescharacteristic
80C4×C127—22no—
81C4×C127C2×C22⋊C411yescharacteristic
82C4×C127C2×C4⋊C411yescharacteristic
83C4×C127#12111yescharacteristic
84C4×C127C22⋊Q811yescharacteristic
85Dic127—82no—
86C2×C127—21no—
87C2×C127—21no—
88C2×C127—21no—
89C2×C127—42no—
90C2×C127#38011yescharacteristic
91#1814—1271nomaximal, Sylow
92C127#181411yescharacteristic, complemented, Sylow
93C2×#131D12711yescharacteristic
94C2×#108—1271no—
95#427—1271no—
96#421—1272no—
97#344—1272no—
98C2×#41C2×D12711yescharacteristic
99#131—21no—
100#131—21no—
101#131Dic12712yescomplemented
102#131Dic12712yescomplemented
103C2×M16—21no—
104C2×M16—1272no—
105C2×M16C2×D12711yescharacteristic
106C22×C8—21no—
107C22×C8C2×D12711yescharacteristic
108#117—1272no—
109#108—2541no—
110#108—2541no—
111#107—2541no—
112#106—2541no—
113#104—2544no—
114#41—21no—
115#41—21no—
116#41C2×Dic12712yes—
117#41D50812yes—
118C2×Q8Dic25411yescharacteristic
119C2×D4—21no—
120C2×D4Dic25411yescharacteristic
121C22×C4—21no—
122C22×C4C22×D12711yescharacteristic
123M16—21no—
124M16—22no—
125M16—22no—
126M16—2542no—
127M16—2542no—
128M16C2×Dic12712yes—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—2542no—
133C2×C8C2×Dic12711yescharacteristic
134C2×C8C2×Dic12711yescharacteristic
135C2×C8C22⋊D12711yescharacteristic
136C2×C8C22⋊D12711yescharacteristic
137C2×C8C22⋊D12711yescharacteristic
138C2×C8C22⋊D12711yescharacteristic
139C4⋊C4—2542no—
140C22⋊C4—2542no—
141C42—2542no—
142C23—21no—
143C23#1452324811yescharacteristic
144Q8—21no—
145Q8C4⋊Dic12712yes—
146D4—22no—
147D4—22no—
148D4—21no—
149D4C4⋊Dic12712yes—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—5082no—
154C2×C4C22×Dic12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C2×C4C2×C22⋊D12711yescharacteristic
157C2×C4C2×D50811yescharacteristic
158C2×C4C2×Dic25411yescharacteristic
159C2×C4#1452324811yescharacteristic
160C8—22no—
161C8—5082no—
162C8C22⋊Dic12711yescharacteristic
163C8C22⋊Dic12711yescharacteristic
164C8C22⋊Dic12711yescharacteristic
165C8C22⋊Dic12711yescharacteristic
166C22—22no—
167C22—41no—
168C22—42no—
169C22#1686116811yescharacteristic
170C22#1686118311yescharacteristic
171C22#1686118811yescharacteristic
172C4—22no—
173C4—10162no—
174C4C2×C22⋊Dic12711yescharacteristic
175C4C2×C4⋊Dic12711yescharacteristic
176C4#1686118311yescharacteristic
177C4#1686118811yescharacteristic
178C2—21no—
179C2—21no—
180C2—21no—
181C2—42no—
182C2#2254973611yescharacteristic
subgroupG/H
centerC4#16861183
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#131×C127C2
socleC2×C127#380
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#131D127
hypocenterC127#1814
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#16861145C2semidirect
2. #22549914C2semidirect
4. #22549648C2semidirect
16. #131×C127C4semidirect
17. #131×C127C4semidirect
92. C127#1814semidirect
101. #131Dic127semidirect
102. #131Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611452
#225499142
#225498932
#225496482
C2×#131×C1272
#1814127