Finite Group Databasebeta

← #30639405

Subgroups — #30639405

Order 16256.

256 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16861132C211yesmaximal, characteristic
2C2×#16861132C211yesmaximal, characteristic
3C2×C4×C127⋊C8C211yesmaximal, characteristic
4#22549782C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549638C211yesmaximal, characteristic
7#389×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12#16861132—22no—
13#16861132C414yescomplemented
14C4×C127⋊C8—22no—
15C127⋊#95—21no—
16C127⋊#95—21no—
17C2×C4⋊C4×C127—21no—
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×C42×C127C2211yescharacteristic
21#95×C127—21no—
22C2×Dic254—21no—
23C2×Dic254—21no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254C2×C412yes—
28C22×Dic127—21no—
29C22×Dic127—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8C2×C412yes—
35C2×C127⋊C8D412yes—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C22×C4×C127—21no—
39C22×C4×C127—21no—
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C22×C4×C127C2311yescharacteristic
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C4⋊C4×C127—21no—
46C4⋊C4×C127—42no—
47C4⋊C4×C127C2×C412yes—
48C42×C127—21no—
49C42×C127—21no—
50Dic254—22no—
51Dic254—22no—
52Dic254—41no—
53Dic254—41no—
54C2×Dic127—22no—
55C2×Dic127—41no—
56C2×Dic127—41no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C127⋊C8—22no—
61C127⋊C8—22no—
62C23×C127C2×D411yescharacteristic
63C2×C4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—22no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127SD1612yes—
74C2×C4×C127Q1612yes—
75C2×C4×C127C22×C411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127#4111yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C22×C127C4×D411yescharacteristic
82C22×C127#12111yescharacteristic
83C22×C127#12411yescharacteristic
84C22×C127C2×SD1611yescharacteristic
85C22×C127C2×Q1611yescharacteristic
86C22×C127#13511yescharacteristic
87C22×C127#13511yescharacteristic
88C4×C127—22no—
89C4×C127—22no—
90C4×C127—42no—
91C4×C127—81no—
92C4×C127#11711yescharacteristic
93C4×C127C4×D411yescharacteristic
94C4×C127#12311yescharacteristic
95C4×C127#12611yescharacteristic
96Dic127—42no—
97Dic127—81no—
98C2×C127#38811yescharacteristic
99C2×C127C4×SD1611yescharacteristic
100C2×C127C4×Q1611yescharacteristic
101C2×C127#46511yescharacteristic
102C2×C127#46711yescharacteristic
103C2×C127#48711yescharacteristic
104C2×C127#48811yescharacteristic
105#1933—1271nomaximal, Sylow
106C127#193311yescharacteristic, complemented, Sylow
107C2×#103—1271no—
108C2×#103—1271no—
109C2×C4×C8—1271no—
110#391—1271no—
111#389D12711yescharacteristic
112#340—1271no—
113#340—1271no—
114C22×Q8—1271no—
115C22×C8—1271no—
116C22×C8—1271no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—1271no—
119C2×C4⋊C4C2×D12711yescharacteristic
120C2×C4⋊C4C2×D12711yescharacteristic
121C2×C42C2×D12711yescharacteristic
122#103—1274no—
123#103—2542no—
124C4×C8—2542no—
125#95—21no—
126#95—2541no—
127#95—2541no—
128C2×Q8—1272no—
129C2×Q8—2541no—
130C2×Q8—2541no—
131C2×Q8—2541no—
132C2×Q8—2541no—
133C2×Q8—2541no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—2541no—
137C22×C4—2541no—
138C22×C4C22⋊D12711yescharacteristic
139C22×C4C22⋊D12711yescharacteristic
140C22×C4C22×D12711yescharacteristic
141C2×C8—1272no—
142C2×C8—1272no—
143C2×C8—2541no—
144C2×C8—2541no—
145C2×C8—2541no—
146C2×C8—2541no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—42no—
151C4⋊C4—2541no—
152C4⋊C4—2541no—
153C4⋊C4C4×D12712yes—
154C42—21no—
155C42—21no—
156C23C2×C22⋊D12711yescharacteristic
157Q8—5081no—
158Q8—5081no—
159Q8—2542no—
160Q8—2542no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—5081no—
172C2×C4—5081no—
173C2×C4—5081no—
174C2×C4—5081no—
175C2×C4—5081no—
176C2×C4—2542no—
177C2×C4#1452323612yes—
178C2×C4#1452323812yes—
179C2×C4C2×C4×D12711yescharacteristic
180C2×C4D4×D12711yescharacteristic
181C2×C4#1452324711yescharacteristic
182C2×C4#1452324811yescharacteristic
183C2×C4#1452324911yescharacteristic
184C2×C4#1452324911yescharacteristic
185C8—2542no—
186C8—2542no—
187C22C4×C22⋊D12711yescharacteristic
188C22#1686118111yescharacteristic
189C22#1686119811yescharacteristic
190C22C2×#1452323611yescharacteristic
191C22C2×#1452323811yescharacteristic
192C22#1686122911yescharacteristic
193C22#1686122911yescharacteristic
194C4—22no—
195C4—22no—
196C4—42no—
197C4—81no—
198C4—10161no—
199C4—5082no—
200C4#1686116711yescharacteristic
201C4#1686117111yescharacteristic
202C4#1686119411yescharacteristic
203C4#1686120411yescharacteristic
204C2#2254976611yescharacteristic
205C2C4×#1452323611yescharacteristic
206C2C4×#1452323811yescharacteristic
207C2#2255010311yescharacteristic
208C2#2255011511yescharacteristic
209C2#2255022811yescharacteristic
210C2#2255023311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#389×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#389D127
hypocenterC127#1933
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
13. #16861132C4semidirect
106. C127#1933semidirect

Maximal subgroups

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

subgroupindex
C2×#168611322
C2×C4×C127⋊C82
#225497822
#225496382
#389×C1272
#1933127