Finite Group Databasebeta

← #30639866

Subgroups — #30639866

Order 16256.

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

normal subgroups only

This group has 326 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
1C2×#16861209C211yesmaximal, characteristic, complemented
2C2×#16861130C212yesmaximal
3#22549638C212yesmaximal, complemented
4#22549631C211yesmaximal, characteristic, complemented
5#390×C127C211yesmaximal, characteristic
6C22×Dic254—21no—
7C22×C127⋊C8C2212yes—
8#16861209—24no—
9C2×C4⋊Dic127—21no—
10C2×C4⋊Dic127C2212yes—
11#16861130—24no—
12C22×D4×C127C2211yescharacteristic
13C2×C4⋊C4×C127C2211yescharacteristic
14C2×C22⋊C4×C127—21no—
15C2×C22⋊C4×C127—21no—
16C2×C42×C127C2211yescharacteristic
17C2×Dic254—24no—
18C2×Dic254—42no—
19C2×Dic254—42no—
20C22×Dic127—22no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23C2×C127⋊C8—22no—
24C4⋊Dic127—24no—
25C4⋊Dic127—22no—
26C4⋊Dic127D414yes—
27C24×C127—21no—
28C2×D4×C127—22no—
29C2×D4×C127—22no—
30C2×D4×C127—21no—
31C2×D4×C127Q812yescomplemented
32C22×C4×C127—21no—
33C22×C4×C127D411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—22no—
37C22⋊C4×C127—42no—
38C22⋊C4×C127—42no—
39C42×C127—22no—
40Dic254—48no—
41C2×Dic127—24no—
42C2×Dic127—42no—
43C2×Dic127—42no—
44C127⋊C8—42no—
45C23×C127—22no—
46C23×C127—41no—
47C23×C127—41no—
48C23×C127—41no—
49C23×C127—41no—
50C23×C127—41no—
51C23×C127—41no—
52C23×C127C2×D411yescharacteristic
53D4×C127—24no—
54D4×C127—42no—
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—41no—
60C2×C4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127D812yes—
63C2×C4×C127SD1612yes—
64C2×C4×C127C2×D412yes—
65C2×C4×C127C2×Q811yescharacteristic
66C2×C4×C127#4112yes—
67C2×C4×C127#4111yescharacteristic
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—42no—
71C22×C127—81no—
72C22×C127—81no—
73C22×C127—81no—
74C22×C127—81no—
75C22×C127#12011yescharacteristic
76C22×C127C22⋊Q811yescharacteristic
77C22×C127#12411yescharacteristic
78C22×C127C2×D811yescharacteristic
79C22×C127C2×SD1611yescharacteristic
80C22×C127AGL(1,ℤ/8)11yescharacteristic
81C22×C127#13711yescharacteristic
82C4×C127—22no—
83C4×C127—22no—
84C4×C127—81no—
85C4×C127C22⋊Q812yes—
86C4×C127#12412yes—
87Dic127—44no—
88C2×C127—82no—
89C2×C127#39311yescharacteristic
90C2×C127#44711yescharacteristic
91C2×C127#44811yescharacteristic
92C2×C127#47411yescharacteristic
93C2×C127#47611yescharacteristic
94C2×C127#48611yescharacteristic
95C2×C127#48911yescharacteristic
96#2023—1271nomaximal, Sylow
97C127#202311yescharacteristic, complemented, Sylow
98C2×C4⋊Q8—1271no—
99C2×#102—1272no—
100#390D12711yescharacteristic
101#340—1272no—
102#336—1271no—
103C22×Q8—2541no—
104C22×D4C2×D12711yescharacteristic
105C22×C8—1272no—
106C4⋊Q8—2544no—
107C2×C4⋊C4—1272no—
108C2×C4⋊C4—2541no—
109C2×C4⋊C4C2×D12711yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C42C2×D12711yescharacteristic
113#102—2544no—
114C24—21no—
115C2×Q8—5082no—
116C2×Q8—5082no—
117C2×Q8—2544no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—21no—
121C2×D4Dic25412yescomplemented
122C22×C4—21no—
123C22×C4—2542no—
124C22×C4C22⋊D12711yescharacteristic
125C22×C4C22⋊D12711yescharacteristic
126C22×C4C22×D12711yescharacteristic
127C2×C8—2542no—
128C2×C8—2542no—
129C2×C8—2542no—
130C4⋊C4—22no—
131C4⋊C4—1274no—
132C4⋊C4—2544no—
133C4⋊C4—2542no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C42—22no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23C2×C22⋊D12711yescharacteristic
145Q8—5088no—
146D4—24no—
147D4—42no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—5082no—
156C2×C4—5082no—
157C2×C4—2544no—
158C2×C4#1452323312yes—
159C2×C4#1452323612yes—
160C2×C4D4×D12712yes—
161C2×C4C2×Dic25411yescharacteristic
162C2×C4#1452324712yes—
163C2×C4#1452324811yescharacteristic
164C8—5082no—
165C22—42no—
166C22—42no—
167C22—42no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22#1686117711yescharacteristic
173C22#1686118811yescharacteristic
174C22#1686119611yescharacteristic
175C22C2×#1452323311yescharacteristic
176C22C2×#1452323611yescharacteristic
177C22#1686123411yescharacteristic
178C22#1686123911yescharacteristic
179C4—22no—
180C4—22no—
181C4—81no—
182C4—5084no—
183C4#1686118612yes—
184C4#1686119512yes—
185C2—82no—
186C2#2254978811yescharacteristic
187C2#2255000911yescharacteristic
188C2#2255001411yescharacteristic
189C2#2255015611yescharacteristic
190C2#2255016611yescharacteristic
191C2#2255022211yescharacteristic
192C2#2255023511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenter#390D127
hypocenterC127#2023
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16861209C2semidirect
3. #22549638C2semidirect
4. #22549631C2semidirect
31. C2×D4×C127Q8semidirect
97. C127#2023semidirect
121. C2×D4Dic254semidirect

Maximal subgroups

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

subgroupindex
C2×#168612092
C2×#168611302
#225496382
#225496312
#390×C1272
#2023127