Finite Group Databasebeta

← #30639114

Subgroups — #30639114

Order 16256.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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#22550643C211yesmaximal, characteristic, complemented
2#22549893C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic, complemented
4#22549636C212yesmaximal, complemented
5#532×C127C211yesmaximal, characteristic
6#16861216—24no—
7C2×#14523232—22no—
8C2×#14523232C2212yes—
9#16861168C2212yes—
10#16861136—24no—
11#16861124—22no—
12C2×#41×C127C2211yescharacteristic
13C2×#41×C127C2211yescharacteristic
14C4⋊Q8×C127C411yescharacteristic, complemented
15#127×C127C411yescharacteristic, complemented
16#124×C127C411yescharacteristic, complemented
17#124×C127C411yescharacteristic, complemented
18#121×C127—22no—
19C4×D4×C127—22no—
20C2×C42×C127C2211yescharacteristic
21#14523232—24no—
22#14523232—22no—
23#14523232D414yescomplemented
24C2×C127⋊C8—24no—
25C2×C127⋊C8—22no—
26C4⋊Dic127—22no—
27C22⋊Dic127—22no—
28C4×Dic127—22no—
29#41×C127—22no—
30#41×C127—22no—
31#41×C127—22no—
32#41×C127—22no—
33C2×Q8×C127C2×C411yescharacteristic
34C2×Q8×C127C2×C411yescharacteristic
35C2×D4×C127—22no—
36C2×D4×C127—21no—
37C2×D4×C127C2×C411yescharacteristic
38C2×D4×C127C2×C411yescharacteristic
39C2×D4×C127D412yes—
40C22×C4×C127—21no—
41C22×C4×C127—21no—
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C22⋊C4×C127—22no—
46C22⋊C4×C127—22no—
47C42×C127—21no—
48C42×C127C2×C411yescharacteristic
49C42×C127C2×C411yescharacteristic
50C2×Dic127—42no—
51C127⋊C8—24no—
52C23×C127—21no—
53C23×C127C22⋊C412yes—
54C23×C127C2×D411yescharacteristic
55Q8×C127—22no—
56Q8×C127—22no—
57D4×C127—24no—
58D4×C127—22no—
59D4×C127—22no—
60D4×C127—44no—
61D4×C127—42no—
62C2×C4×C127—21no—
63C2×C4×C127—22no—
64C2×C4×C127—22no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—42no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127C22⋊C412yes—
71C2×C4×C127C22×C411yescharacteristic
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127#4112yes—
75C22×C127—22no—
76C22×C127—41no—
77C22×C127—42no—
78C22×C127—42no—
79C22×C127—41no—
80C22×C127C2×C22⋊C411yescharacteristic
81C22×C127#12112yes—
82C4×C127—22no—
83C4×C127—22no—
84C4×C127—41no—
85C4×C127—41no—
86C4×C127C4×D412yes—
87C4×C127#12011yescharacteristic
88C4×C127#12311yescharacteristic
89Dic127—82no—
90C2×C127—22no—
91C2×C127—21no—
92C2×C127—42no—
93C2×C127—81no—
94C2×C127#38611yescharacteristic
95#1884—1271nomaximal, Sylow
96C127#188411yescharacteristic, complemented, Sylow
97#568—1271no—
98#532D12711yescharacteristic
99#421—1272no—
100#410—1271no—
101#339—1272no—
102C2×#41C2×D12711yescharacteristic
103C2×#41C2×D12711yescharacteristic
104#131—2544no—
105C2×M16—1272no—
106C2×M16—2542no—
107C4⋊Q8Dic12711yescharacteristic, complemented
108#127Dic12711yescharacteristic, complemented
109#124Dic12711yescharacteristic, complemented
110#124Dic12711yescharacteristic, complemented
111#121—22no—
112C4×D4—22no—
113#117—1272no—
114C2×C42C2×D12711yescharacteristic
115#104—2544no—
116#99—2542no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121C2×Q8C2×Dic12711yescharacteristic
122C2×Q8C2×Dic12711yescharacteristic
123C2×D4—22no—
124C2×D4—21no—
125C2×D4C2×Dic12711yescharacteristic
126C2×D4C2×Dic12711yescharacteristic
127C2×D4C22⋊D12712yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4C22⋊D12712yes—
131C22×C4C22×D12711yescharacteristic
132M16—1274no—
133M16—2544no—
134M16—2542no—
135C2×C8—2544no—
136C2×C8—2542no—
137C4⋊C4—22no—
138C4⋊C4—2542no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—2542no—
142C42—21no—
143C42—2542no—
144C42C2×Dic12711yescharacteristic
145C42C2×Dic12711yescharacteristic
146C23—21no—
147C23C22⋊Dic12712yes—
148C23C2×C22⋊D12711yescharacteristic
149Q8—22no—
150Q8—22no—
151D4—24no—
152D4—22no—
153D4—22no—
154D4—44no—
155D4—42no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—5082no—
165C2×C4C22⋊Dic12712yes—
166C2×C4C22×Dic12711yescharacteristic
167C2×C4D4×D12712yes—
168C2×C4C2×C22⋊D12711yescharacteristic
169C2×C4#1452324712yes—
170C8—2544no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic12711yescharacteristic
177C22#1686118112yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—41no—
182C4—10162no—
183C4D4×Dic12712yes—
184C4#1686117711yescharacteristic
185C4#1686119011yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—81no—
190C2#2254975611yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#532×C127C2
socleC2×C127#386
cosocleC22×C4×C127C23
hypercenter#532D127
hypocenterC127#1884
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #22550643C2semidirect
2. #22549893C2semidirect
3. #22549850C2semidirect
4. #22549636C2semidirect
14. C4⋊Q8×C127C4semidirect
15. #127×C127C4semidirect
16. #124×C127C4semidirect
17. #124×C127C4semidirect
23. #14523232D4semidirect
96. C127#1884semidirect
107. C4⋊Q8Dic127semidirect
108. #127Dic127semidirect
109. #124Dic127semidirect
110. #124Dic127semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225498932
#225498502
#225496362
#532×C1272
#1884127