Finite Group Databasebeta

← #30639113

Subgroups — #30639113

Order 16256.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550422C211yesmaximal, characteristic
2#22549892C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic, complemented
4#22549637C212yesmaximal, complemented
5#568×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2211yescharacteristic
7#16861207C411yescharacteristic, complemented
8#16861205C411yescharacteristic, complemented
9#16861198C411yescharacteristic, complemented
10#16861196C411yescharacteristic, complemented
11#16861181—22no—
12C4×C22⋊D127—22no—
13#16861168C2212yes—
14C2×C4×Dic127C2211yescharacteristic
15#16861135—24no—
16#16861124—22no—
17C2×#41×C127C2211yescharacteristic
18#131×C127—24no—
19C2×M16×C127—22no—
20C2×M16×C127C2212yes—
21#14523248—22no—
22#14523248—22no—
23C2×Dic254C2×C411yescharacteristic
24C2×D508C2×C411yescharacteristic
25C2×C22⋊D127—22no—
26C2×C22⋊D127—21no—
27C22×Dic127—21no—
28C2×C4×D127—21no—
29C4⋊Dic127—22no—
30Dic127⋊C4—22no—
31C22⋊Dic127—22no—
32C22⋊Dic127—22no—
33(C2×D127)⋊C4—22no—
34C4×Dic127—21no—
35C4×Dic127—22no—
36C4×Dic127C2×C411yescharacteristic
37C4×Dic127C2×C411yescharacteristic
38#41×C127—22no—
39#41×C127—22no—
40C2×Q8×C127C2×C411yescharacteristic
41C2×D4×C127C2×C411yescharacteristic
42C2×D4×C127D412yes—
43C22×C4×C127D412yes—
44C22×C4×C127C2311yescharacteristic
45M16×C127—24no—
46M16×C127—22no—
47M16×C127D414yescomplemented
48C2×C8×C127—24no—
49C2×C8×C127—22no—
50C22×D127—21no—
51Dic254—22no—
52D508—22no—
53C22⋊D127—44no—
54C22⋊D127—42no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—42no—
59C2×Dic127—42no—
60C4×D127—41no—
61C4×D127—41no—
62C23×C127C22⋊C412yes—
63C23×C127C2×D411yescharacteristic
64Q8×C127—22no—
65D4×C127—24no—
66D4×C127—22no—
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127C22⋊C412yes—
70C2×C4×C127C22×C411yescharacteristic
71C2×C4×C127C2×D412yes—
72C2×C4×C127C2×D411yescharacteristic
73C2×C4×C127#4112yes—
74C8×C127—24no—
75C22×C127—22no—
76C22×C127—41no—
77C22×C127—42no—
78C22×C127C2×C22⋊C411yescharacteristic
79C22×C127#12112yes—
80C2×D127—42no—
81C2×D127—41no—
82C4×C127—22no—
83C4×C127C4×D412yes—
84C4×C127#12011yescharacteristic
85C4×C127#12311yescharacteristic
86Dic127—22no—
87Dic127—41no—
88Dic127—41no—
89Dic127—82no—
90C2×C127—21no—
91C2×C127—22no—
92C2×C127—42no—
93C2×C127#38611yescharacteristic
94D127—81no—
95#1884—1271nomaximal, Sylow
96C127#188411yescharacteristic, complemented, Sylow
97#568D12711yescharacteristic, complemented
98#532—1271no—
99#421—1272no—
100#410—1271no—
101#339—1272no—
102C2×#41—1271no—
103C2×#41C2×D12711yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D12712yes—
107C4⋊Q8—1271no—
108#127—1271no—
109#124—1271no—
110#124—1271no—
111#121—2542no—
112C4×D4—2542no—
113#117—1272no—
114C2×C42—1271no—
115#104—2544no—
116#99—2542no—
117#41—22no—
118#41—22no—
119#41—2542no—
120#41—2542no—
121C2×Q8—1271no—
122C2×Q8C4×D12711yescharacteristic
123C2×D4—1271no—
124C2×D4—2542no—
125C2×D4—2541no—
126C2×D4C4×D12711yescharacteristic
127C2×D4D50812yes—
128C22×C4—2541no—
129C22×C4—2541no—
130C22×C4C22⋊D12712yes—
131C22×C4C22×D12711yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D12714yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—2542no—
138C4⋊C4—2542no—
139C22⋊C4—2542no—
140C22⋊C4—2542no—
141C22⋊C4—2542no—
142C42—1271no—
143C42—1271no—
144C42—2542no—
145C42—2541no—
146C23—2541no—
147C23(C2×D127)⋊C412yes—
148C23C2×D50811yescharacteristic
149Q8—22no—
150Q8—2542no—
151D4—24no—
152D4—22no—
153D4—5082no—
154D4—5084no—
155D4—2542no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—5082no—
159C2×C4—5082no—
160C2×C4—5081no—
161C2×C4—5081no—
162C2×C4—2541no—
163C2×C4—2541no—
164C2×C4—2541no—
165C2×C4(C2×D127)⋊C412yes—
166C2×C4C2×C4×D12711yescharacteristic
167C2×C4C2×C22⋊D12711yescharacteristic
168C2×C4C2×C22⋊D12712yes—
169C2×C4#1452324812yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—5082no—
175C22—5081no—
176C22C2×(C2×D127)⋊C411yescharacteristic
177C22#1686118312yes—
178C4—22no—
179C4—10162no—
180C4—5081no—
181C4—5081no—
182C4—2542no—
183C4C4×C22⋊D12712yes—
184C4#1686117811yescharacteristic
185C4#1686119211yescharacteristic
186C2—22no—
187C2—21no—
188C2—42no—
189C2—10161no—
190C2#2254975511yescharacteristic
subgroupG/H
centerC4#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#386
prime radical#22550422C2
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1884
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22549892C2semidirect
3. #22549850C2semidirect
4. #22549637C2semidirect
5. #568×C127C2semidirect
7. #16861207C4semidirect
8. #16861205C4semidirect
9. #16861198C4semidirect
10. #16861196C4semidirect
47. M16×C127D4semidirect
96. C127#1884semidirect
97. #568D127semidirect
134. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504222
#225498922
#225498502
#225496372
#568×C1272
#1884127