Finite Group Databasebeta

← #30640142

Subgroups — #30640142

Order 16256.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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×#16861134C211yesmaximal, characteristic
2C2×#16861132C211yesmaximal, characteristic
3#22549781C211yesmaximal, characteristic
4#22549745C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—21no—
12C2×Dic127⋊C4—21no—
13C2×C4×Dic127C2211yescharacteristic
14#16861134—22no—
15#16861132—22no—
16C127⋊#95—21no—
17C127⋊#95—21no—
18C22×C8×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2211yescharacteristic
21C2×Dic254—21no—
22C2×Dic254—21no—
23C2×Dic254—21no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254D412yes—
27C22×Dic127—21no—
28C22×Dic127—21no—
29C22×Dic127D411yescharacteristic
30C22×Dic127D411yescharacteristic
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36Dic127⋊C4—42no—
37Dic127⋊C4—42no—
38C4×Dic127—21no—
39C4×Dic127—21no—
40C22×C4×C127—21no—
41C22×C4×C127—21no—
42C22×C4×C127C2311yescharacteristic
43C2×C8×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50Dic254—22no—
51Dic254—22no—
52Dic254—41no—
53Dic254—41no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—22no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127SD1612yes—
65C2×Dic127Q1612yes—
66C127⋊C8—41no—
67C23×C127C2×D411yescharacteristic
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127#4111yescharacteristic
76C2×C4×C127#4111yescharacteristic
77C2×C4×C127#4111yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C8×C127—41no—
81C22×C127#12111yescharacteristic
82C22×C127#12311yescharacteristic
83C22×C127#12411yescharacteristic
84C22×C127C2×SD1611yescharacteristic
85C22×C127C2×Q1611yescharacteristic
86C22×C127AGL(1,ℤ/8)11yescharacteristic
87C22×C127#13711yescharacteristic
88C4×C127—81no—
89C4×C127—81no—
90C4×C127#12311yescharacteristic
91C4×C127#12311yescharacteristic
92C4×C127#12611yescharacteristic
93C4×C127#12611yescharacteristic
94Dic127—22no—
95Dic127—22no—
96Dic127—42no—
97Dic127—81no—
98C2×C127#39711yescharacteristic
99C2×C127#46011yescharacteristic
100C2×C127#46211yescharacteristic
101C2×C127#48311yescharacteristic
102C2×C127#48411yescharacteristic
103C2×C127#48711yescharacteristic
104C2×C127#48911yescharacteristic
105#2072—1271nomaximal, Sylow
106C127#207211yescharacteristic, complemented, Sylow
107C2×#103—1271no—
108C2×#103—1271no—
109#391—1271no—
110#384—1271no—
111#340—1271no—
112#340D12711yescharacteristic
113#336—1271no—
114C22×Q8—1271no—
115C22×C8—1271no—
116C22×C8C2×D12711yescharacteristic
117C2×C4⋊C4—1271no—
118C2×C4⋊C4—2541no—
119C2×C4⋊C4—2541no—
120C2×C4⋊C4C2×D12711yescharacteristic
121C2×C4⋊C4C2×D12711yescharacteristic
122C2×C42—1271no—
123#103—2542no—
124#103—2542no—
125#95—2541no—
126#95—2541no—
127C2×Q8—1272no—
128C2×Q8—2541no—
129C2×Q8—2541no—
130C2×Q8—2541no—
131C2×Q8—2541no—
132C2×Q8—2541no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—1271no—
136C22×C4—1271no—
137C22×C4—2541no—
138C22×C4—2541no—
139C22×C4C22×D12711yescharacteristic
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—2541no—
144C2×C8—2541no—
145C2×C8—2541no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—5082no—
151C4⋊C4—5082no—
152C4⋊C4—2541no—
153C4⋊C4—2541no—
154C42—2541no—
155C42—2541no—
156C23D4×D12711yescharacteristic
157Q8—5081no—
158Q8—5081no—
159Q8—2542no—
160Q8—2542no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—1272no—
168C2×C4—1272no—
169C2×C4—5081no—
170C2×C4—5081no—
171C2×C4—5081no—
172C2×C4—5081no—
173C2×C4—5081no—
174C2×C4—2542no—
175C2×C4—2541no—
176C2×C4—2541no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4D4×D12711yescharacteristic
180C2×C4#1452324711yescharacteristic
181C2×C4#1452324811yescharacteristic
182C2×C4#1452324811yescharacteristic
183C2×C4#1452324811yescharacteristic
184C2×C4#1452324911yescharacteristic
185C8—41no—
186C8—5081no—
187C22#1686118211yescharacteristic
188C22#1686119411yescharacteristic
189C22#1686119511yescharacteristic
190C22SD16×D12711yescharacteristic
191C22Q16×D12711yescharacteristic
192C22#1686123311yescharacteristic
193C22#1686123811yescharacteristic
194C4—81no—
195C4—81no—
196C4—10161no—
197C4—5082no—
198C4—2542no—
199C4—2542no—
200C4#1686119111yescharacteristic
201C4#1686119411yescharacteristic
202C4#1686120311yescharacteristic
203C4#1686120411yescharacteristic
204C2#2254980511yescharacteristic
205C2#2255007511yescharacteristic
206C2#2255008511yescharacteristic
207C2#2255020811yescharacteristic
208C2#2255021311yescharacteristic
209C2#2255022511yescharacteristic
210C2#2255023411yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socle, prime radicalC23×C127C2×D4
hypercenter#340D127
hypocenterC127#2072
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
106. C127#2072semidirect

Maximal subgroups

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

subgroupindex
C2×#168611342
C2×#168611322
#225497812
#225497452
#225496382
#225496302
#340×C1272
#2072127