Finite Group Databasebeta

← #30639895

Subgroups — #30639895

Order 16256.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#22550502C211yesmaximal, characteristic, complemented
2#22549731C212yesmaximal
3#22549731C211yesmaximal, characteristic
4#22549642C212yesmaximal, complemented
5#342×C127C211yesmaximal, characteristic, complemented
6C24×D127C2211yescharacteristic
7#16861188—22no—
8#16861186—24no—
9C2×C22⋊Dic127C2211yescharacteristic
10C2×C22⋊Dic127C2212yes—
11C2×(C2×D127)⋊C4—24no—
12C2×(C2×D127)⋊C4—22no—
13C2×C22⋊C4×C127C2212yes—
14C2×C22⋊C4×C127C2211yescharacteristic
15C23×D127—22no—
16C23×D127—21no—
17C23×D127—21no—
18C23×D127—22no—
19C23×D127—44no—
20C23×D127Q812yescomplemented
21C2×Dic254—22no—
22C22×Dic127—21no—
23C22×Dic127D414yes—
24C4⋊Dic127—22no—
25Dic127⋊C4—24no—
26C22⋊Dic127—22no—
27C22⋊Dic127—22no—
28C22⋊Dic127—22no—
29(C2×D127)⋊C4—44no—
30(C2×D127)⋊C4—48no—
31C24×C127C2311yescharacteristic
32C22×C4×C127—22no—
33C22×C4×C127D412yes—
34C22⋊C4×C127—22no—
35C22⋊C4×C127—22no—
36C22⋊C4×C127—22no—
37C22×D127—28no—
38C22×D127—24no—
39C22×D127—24no—
40C22×D127—22no—
41C22×D127—44no—
42C22×D127—44no—
43C22×D127—48no—
44C22×D127—44no—
45C22×D127—42no—
46C22×D127—41no—
47C22×D127—42no—
48C22×D127—41no—
49Dic254—82no—
50C2×Dic127—24no—
51C2×Dic127—44no—
52C2×Dic127—42no—
53C2×Dic127—41no—
54C23×C127—42no—
55C23×C127C2×D412yes—
56C23×C127C2×D411yescharacteristic
57C23×C127C2×Q811yescharacteristic
58C23×C127#4111yescharacteristic
59C23×C127#4112yes—
60C2×C4×C127—22no—
61C2×C4×C127—42no—
62C2×C4×C127—44no—
63C2×C4×C127—42no—
64C22×C127—24no—
65C22×C127—22no—
66C22×C127—41no—
67C22×C127—41no—
68C22×C127—42no—
69C22×C127#12011yescharacteristic
70C22×C127C22⋊Q812yes—
71C22×C127C22⋊Q811yescharacteristic
72C22×C127#12412yes—
73C22×C127#12411yescharacteristic
74C2×D127—28no—
75C2×D127—24no—
76C2×D127—44no—
77C2×D127—48no—
78C2×D127—44no—
79C2×D127—42no—
80C2×D127—42no—
81C2×D127—44no—
82C4×C127—42no—
83C4×C127—82no—
84Dic127—44no—
85Dic127—81no—
86C2×C127—21no—
87C2×C127—22no—
88C2×C127—22no—
89C2×C127—21no—
90C2×C127#39311yescharacteristic
91C2×C127B(4,𝔽2)12yes—
92D127—44no—
93#2029—1271nomaximal, Sylow
94C127#202911yescharacteristic, complemented, Sylow
95#543—1271no—
96#379—1272no—
97#379—1271no—
98#342—1272no—
99#342D12711yescharacteristic, complemented
100C25—1271no—
101C22⋊Q8—2544no—
102C22⋊Q8—2542no—
103C2×C22⋊C4—1272no—
104C2×C22⋊C4—1271no—
105C2×C22⋊C4—2544no—
106C2×C22⋊C4—2542no—
107C2×C22⋊C4C2×D12712yes—
108C2×C22⋊C4C2×D12711yescharacteristic
109C24—1272no—
110C24—5084no—
111C24—2541no—
112C24—2542no—
113C24—2542no—
114C24—2541no—
115C24C22×D12711yescharacteristic
116C2×Q8—2542no—
117C22×C4—22no—
118C22×C4—1274no—
119C22×C4—2541no—
120C22×C4D50812yes—
121C4⋊C4—2544no—
122C4⋊C4—2542no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—5084no—
127C22⋊C4—5088no—
128C22⋊C4—2542no—
129C22⋊C4—2542no—
130C22⋊C4—2542no—
131C23—42no—
132C23—5084no—
133C23—5082no—
134C23—5084no—
135C23—5088no—
136C23—5084no—
137C23—5082no—
138C23—5081no—
139C23—5081no—
140C23—2548no—
141C23—2544no—
142C23—2544no—
143C23—2542no—
144C23D4×D12712yes—
145C23C2×D50811yescharacteristic
146C23Q8×D12711yescharacteristic
147C23#1452324711yescharacteristic
148C23#1452324812yes—
149Q8—10162no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—5082no—
155C2×C4—5081no—
156C2×C4—5084no—
157C2×C4—2544no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—5088no—
164C22—5084no—
165C22—5084no—
166C22—5082no—
167C22—5082no—
168C22—5084no—
169C22—2548no—
170C22—2544no—
171C22#1686117911yescharacteristic
172C22#1686118712yes—
173C22#1686118911yescharacteristic
174C22#1686119512yes—
175C22#1686119711yescharacteristic
176C4—42no—
177C4—82no—
178C4—10161no—
179C4—5084no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—5084no—
185C2#2254978911yescharacteristic
186C2#2255006412yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC24×C127C23
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127#120
prime radicalC24×D127C22
hypercenter#342D127
hypocenterC127#2029
LCSG ▷ C24×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #22550502C2semidirect
4. #22549642C2semidirect
5. #342×C127C2semidirect
20. C23×D127Q8semidirect
94. C127#2029semidirect
99. #342D127semidirect

Maximal subgroups

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

subgroupindex
#225505022
#225497312
#225496422
#342×C1272
#2029127