Finite Group Databasebeta

← #30638943

Subgroups — #30638943

Order 16256.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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#22550353C211yesmaximal, characteristic
2#22549916C211yesmaximal, characteristic
3C2×#16861127C211yesmaximal, characteristic
4#22549781C211yesmaximal, characteristic
5#22549637C211yesmaximal, characteristic
6#22549635C211yesmaximal, characteristic
7#428×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C4×Dic254—21no—
11C4×Dic254—21no—
12#16861174—21no—
13#16861174—21no—
14#16861168C2211yescharacteristic
15#16861165—22no—
16C2×C4×Dic127C2211yescharacteristic
17#16861142—21no—
18#16861140—21no—
19#16861127C414yescomplemented
20C127⋊#95—22no—
21C2×M16×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#107×C127—21no—
25#106×C127—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254C2×C412yes—
31C2×Dic254Q812yes—
32C22×Dic127—21no—
33C22×Dic127D412yes—
34#14523232—21no—
35#14523232C2×C412yes—
36C2×C127⋊C8—21no—
37C4⋊Dic127D411yescharacteristic
38C4⋊Dic127D411yescharacteristic
39Dic127⋊C4—22no—
40Dic127⋊C4—22no—
41C22⋊Dic127—21no—
42C4×Dic127—22no—
43C4×Dic127—22no—
44C4×Dic127—21no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C22×C4×C127—21no—
48C22×C4×C127C2311yescharacteristic
49M16×C127—21no—
50M16×C127C2×C412yes—
51C2×C8×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127D411yescharacteristic
55C4⋊C4×C127D411yescharacteristic
56C22⋊C4×C127—21no—
57C42×C127—21no—
58Dic254—22no—
59Dic254—22no—
60Dic254—22no—
61Dic254—22no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—22no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127C4⋊C414yes—
70C127⋊C8—22no—
71C23×C127C2×D411yescharacteristic
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—42no—
75C2×C4×C127—41no—
76C2×C4×C127C22×C411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2×Q811yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C8×C127—22no—
83C22×C127—21no—
84C22×C127—21no—
85C22×C127C2×C4⋊C411yescharacteristic
86C22×C127#12011yescharacteristic
87C22×C127#12311yescharacteristic
88C4×C127—41no—
89C4×C127—41no—
90C4×C127—81no—
91C4×C127C4×D411yescharacteristic
92C4×C127C4×D411yescharacteristic
93C4×C127C22⋊Q811yescharacteristic
94C4×C127C22⋊Q811yescharacteristic
95Dic127—24no—
96Dic127—42no—
97Dic127—41no—
98Dic127—41no—
99C2×C127—21no—
100C2×C127—21no—
101C2×C127#38511yescharacteristic
102C2×C127#45411yescharacteristic
103C2×C127#45611yescharacteristic
104#1857—1271nomaximal, Sylow
105C127#185711yescharacteristic, complemented, Sylow
106#519—1271no—
107#428—1271no—
108#428D12711yescharacteristic
109C2×#101—1271no—
110#391—1271no—
111#339—1271no—
112#339—1271no—
113C22×Q8—1271no—
114C2×M16—1271no—
115C2×M16C2×D12711yescharacteristic
116C4×Q8—2541no—
117C4×Q8—2541no—
118C4×Q8—2541no—
119C4×Q8—2541no—
120#117—1271no—
121#117—2542no—
122#117C2×D12711yescharacteristic
123C2×C4⋊C4C2×D12711yescharacteristic
124C2×C42—1271no—
125#107—21no—
126#107—2541no—
127#106—21no—
128#106—2541no—
129#101—1274no—
130#95—2542no—
131C2×Q8—1272no—
132C2×Q8—1272no—
133C2×Q8—2541no—
134C2×Q8—2541no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C22×C4—21no—
138C22×C4—1272no—
139C22×C4—2541no—
140C22×C4C22×D12711yescharacteristic
141M16—21no—
142M16—1272no—
143M16—2541no—
144M16C4×D12712yes—
145C2×C8—21no—
146C2×C8—2541no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—1271no—
150C4⋊C4—1271no—
151C4⋊C4—2542no—
152C4⋊C4—2542no—
153C4⋊C4C22⋊D12711yescharacteristic
154C4⋊C4C22⋊D12711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—2541no—
157C42—21no—
158C42—2542no—
159C42—2542no—
160C42—2541no—
161C42—2541no—
162C42—2541no—
163C23D4×D12711yescharacteristic
164Q8—2542no—
165Q8—2542no—
166Q8—2542no—
167Q8—2542no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—1274no—
173C2×C4—5081no—
174C2×C4—5081no—
175C2×C4—2542no—
176C2×C4—2541no—
177C2×C4—2541no—
178C2×C4—2542no—
179C2×C4—2542no—
180C2×C4C2×C4×D12711yescharacteristic
181C2×C4D4×D12711yescharacteristic
182C2×C4C2×C22⋊D12711yescharacteristic
183C2×C4Q8×D12711yescharacteristic
184C2×C4#1452324811yescharacteristic
185C2×C4#1452324911yescharacteristic
186C8—22no—
187C8—2542no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D12711yescharacteristic
191C22#1686117711yescharacteristic
192C22#1686119411yescharacteristic
193C4—41no—
194C4—41no—
195C4—81no—
196C4—5082no—
197C4—5081no—
198C4—5081no—
199C4—2544no—
200C4C4×C22⋊D12711yescharacteristic
201C4#1686117111yescharacteristic
202C4#1686118511yescharacteristic
203C4#1686118711yescharacteristic
204C2—21no—
205C2—21no—
206C2#2254974711yescharacteristic
207C2#2255004711yescharacteristic
208C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#428×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#428D127
hypocenterC127#1857
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
19. #16861127C4semidirect
105. C127#1857semidirect

Maximal subgroups

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

subgroupindex
#225503532
#225499162
C2×#168611272
#225497812
#225496372
#225496352
#428×C1272
#1857127