Finite Group Databasebeta

← #30639590

Subgroups — #30639590

Order 16256.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#14523238C211yesmaximal, characteristic
2C2×#16861142C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic
5C2×C4×C127⋊C8C211yesmaximal, characteristic
6#22549782C211yesmaximal, characteristic
7#391×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523238—22no—
10C2×#14523238—22no—
11C2×#14523238C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15#16861142—22no—
16#16861133—22no—
17#16861132—22no—
18C4×C127⋊C8—22no—
19C127⋊#95—22no—
20C22×Q8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C42×C127C2211yescharacteristic
23#95×C127—22no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C22×Dic127—21no—
31C22×Dic127—21no—
32#14523238—28no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8C2×C412yes—
36C2×C127⋊C8D412yes—
37C2×C127⋊C8D412yes—
38C2×C127⋊C8D412yes—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127C2×C412yes—
47C22×C4×C127—21no—
48C22×C4×C127—21no—
49C22×C4×C127D412yes—
50C22×C4×C127C2311yescharacteristic
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C42×C127—21no—
54C42×C127—21no—
55Dic254—22no—
56Dic254—22no—
57Dic254—41no—
58Dic254—41no—
59C2×Dic127—22no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C2×Dic127—41no—
63C2×Dic127—42no—
64C127⋊C8—22no—
65C127⋊C8C22⋊C414yes—
66C23×C127C2×D411yescharacteristic
67Q8×C127—22no—
68Q8×C127—22no—
69Q8×C127—41no—
70Q8×C127—41no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—42no—
78C2×C4×C127Q1614yescomplemented
79C2×C4×C127C22×C411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C22×C127C4×D411yescharacteristic
86C22×C127#12111yescharacteristic
87C22×C127#12711yescharacteristic
88C22×C127C2×Q1612yes—
89C22×C127#13512yes—
90C4×C127—24no—
91C4×C127—42no—
92C4×C127—81no—
93C4×C127C2×C22⋊C411yescharacteristic
94C4×C127C4×D411yescharacteristic
95C4×C127#12111yescharacteristic
96C4×C127#12411yescharacteristic
97Dic127—42no—
98Dic127—81no—
99C2×C127#39011yescharacteristic
100C2×C127C4×Q1612yes—
101C2×C127#46712yes—
102C2×C127#49411yescharacteristic
103C2×C127#49511yescharacteristic
104#1969—1271nomaximal, Sylow
105C127#196911yescharacteristic, complemented, Sylow
106C22×Q16—1271no—
107C2×#107—1271no—
108C2×#103—1271no—
109C2×#103—1271no—
110C2×C4×C8—1271no—
111#391—1271no—
112#391D12711yescharacteristic
113C22×Q8—1271no—
114C22×Q8C2×D12711yescharacteristic
115C2×Q16—1274no—
116C2×Q16—2542no—
117C2×Q16—2542no—
118C22×C8—1271no—
119C22×C8—1271no—
120C2×C4⋊C4—1271no—
121C2×C4⋊C4C2×D12711yescharacteristic
122C2×C42C2×D12711yescharacteristic
123#107—2542no—
124#103—2542no—
125#103—2542no—
126C4×C8—2542no—
127#95—22no—
128#95—2542no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—1272no—
135C2×Q8—2541no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×Q8C4×D12712yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—2541no—
144C22×C4—2541no—
145C22×C4C22⋊D12712yes—
146C22×C4C22×D12711yescharacteristic
147Q16—2548no—
148C2×C8—1272no—
149C2×C8—1272no—
150C2×C8—1272no—
151C2×C8—1272no—
152C2×C8—2541no—
153C2×C8—2541no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2541no—
157C4⋊C4—2541no—
158C42—21no—
159C42—21no—
160C23C2×C22⋊D12711yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—5081no—
166Q8—5081no—
167Q8—2542no—
168Q8—2542no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—5081no—
177C2×C4—5082no—
178C2×C4—5081no—
179C2×C4—5081no—
180C2×C4—2542no—
181C2×C4#1452323814yescomplemented
182C2×C4C2×C4×D12711yescharacteristic
183C2×C4D4×D12711yescharacteristic
184C2×C4D4×D12711yescharacteristic
185C2×C4D4×D12711yescharacteristic
186C2×C4#1452324711yescharacteristic
187C2×C4#1452324811yescharacteristic
188C8—1274no—
189C8—2542no—
190C22C4×C22⋊D12711yescharacteristic
191C22#1686118111yescharacteristic
192C22#1686120511yescharacteristic
193C22C2×#1452323812yes—
194C22#1686122912yes—
195C4—24no—
196C4—42no—
197C4—81no—
198C4—10161no—
199C4—5082no—
200C4C22⋊C4×D12711yescharacteristic
201C4(C22⋊D127)⋊C411yescharacteristic
202C4#1686118211yescharacteristic
203C4#1686119511yescharacteristic
204C2#2254977611yescharacteristic
205C2C4×#1452323812yes—
206C2#2255011512yes—
207C2#2255025511yescharacteristic
208C2#2255025811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#391D127
hypocenterC127#1969
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
11. C2×#14523238C4semidirect
78. C2×C4×C127Q16semidirect
105. C127#1969semidirect
181. C2×C4#14523238semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
C2×#168611422
C2×#168611332
C2×#168611322
C2×C4×C127⋊C82
#225497822
#391×C1272
#1969127