Finite Group Databasebeta

← #30639879

Subgroups — #30639879

Order 16256.

258 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16861200C211yesmaximal, characteristic
2C2×#16861133C211yesmaximal, characteristic
3#22549783C211yesmaximal, characteristic
4#22549639C211yesmaximal, characteristic
5#22549638C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7C2×#103×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9#16861200—22no—
10#16861200—22no—
11C2×C4⋊Dic127—21no—
12C2×C4⋊Dic127C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—21no—
15C2×C4×Dic127C2211yescharacteristic
16#16861133—22no—
17C127⋊#95—21no—
18C127⋊#95—21no—
19C22×Q8×C127C2211yescharacteristic
20C22×C8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22#103×C127—22no—
23C22×Dic127—21no—
24C22×Dic127—21no—
25C22×Dic127D411yescharacteristic
26C22×Dic127D411yescharacteristic
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C4⋊Dic127—24no—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34C4⋊Dic127D412yes—
35Dic127⋊C4—24no—
36C4×Dic127—21no—
37C4×Dic127—21no—
38C2×Q8×C127—21no—
39C2×Q8×C127—21no—
40C2×Q8×C127—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127Q812yes—
44C22×C4×C127—21no—
45C22×C4×C127—21no—
46C22×C4×C127C2311yescharacteristic
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127D412yes—
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—22no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C127⋊C8—41no—
65C23×C127C2×D411yescharacteristic
66Q8×C127—22no—
67Q8×C127—22no—
68Q8×C127—41no—
69Q8×C127—41no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×Q811yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C8×C127—41no—
83C22×C127#12011yescharacteristic
84C22×C127C22⋊Q811yescharacteristic
85C22×C127#12411yescharacteristic
86C22×C127#13511yescharacteristic
87C22×C127#13511yescharacteristic
88C22×C127#13711yescharacteristic
89C22×C127#13711yescharacteristic
90C4×C127—42no—
91C4×C127—42no—
92C4×C127C22⋊Q811yescharacteristic
93C4×C127C22⋊Q811yescharacteristic
94C4×C127#12411yescharacteristic
95C4×C127#12411yescharacteristic
96Dic127—41no—
97Dic127—41no—
98Dic127—42no—
99Dic127—81no—
100C2×C127#39311yescharacteristic
101C2×C127#45211yescharacteristic
102C2×C127#45211yescharacteristic
103C2×C127#47911yescharacteristic
104C2×C127#47911yescharacteristic
105C2×C127#48811yescharacteristic
106C2×C127#49111yescharacteristic
107#2026—1271nomaximal, Sylow
108C127#202611yescharacteristic, complemented, Sylow
109C2×#125—1271no—
110C2×#103—1271no—
111C2×#103D12711yescharacteristic
112#391—1271no—
113#340—1271no—
114#340—1271no—
115#336—1271no—
116C22×Q8C2×D12711yescharacteristic
117C22×C8—1271no—
118C22×C8C2×D12711yescharacteristic
119#125—2542no—
120#125—2542no—
121C2×C4⋊C4—1271no—
122C2×C4⋊C4—1271no—
123C2×C4⋊C4—2541no—
124C2×C4⋊C4—2541no—
125C2×C4⋊C4C2×D12711yescharacteristic
126C2×C42—1271no—
127#103—22no—
128#103—2542no—
129#95—2541no—
130#95—2541no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8Dic25412yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—1271no—
140C22×C4—1271no—
141C22×C4—2541no—
142C22×C4—2541no—
143C22×C4C22×D12711yescharacteristic
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—2541no—
148C2×C8—2541no—
149C2×C8—2541no—
150C4⋊C4—21no—
151C4⋊C4—1272no—
152C4⋊C4—2544no—
153C4⋊C4—2544no—
154C4⋊C4—2541no—
155C4⋊C4—2541no—
156C4⋊C4—2541no—
157C4⋊C4C22⋊D12712yes—
158C42—2541no—
159C42—2541no—
160C23D4×D12711yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165C2×C4—22no—
166C2×C4—22no—
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—2541no—
177C2×C4—2541no—
178C2×C4—2541no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4—2541no—
182C2×C4—2542no—
183C2×C4D4×D12711yescharacteristic
184C2×C4C2×C22⋊D12711yescharacteristic
185C2×C4C2×Dic25411yescharacteristic
186C2×C4#1452324711yescharacteristic
187C2×C4#1452324711yescharacteristic
188C2×C4#1452324811yescharacteristic
189C8—41no—
190C8—5081no—
191C22#1686117711yescharacteristic
192C22#1686118611yescharacteristic
193C22#1686119511yescharacteristic
194C22#1686123011yescharacteristic
195C22#1686123211yescharacteristic
196C22#1686123811yescharacteristic
197C22#1686124111yescharacteristic
198C4—42no—
199C4—42no—
200C4—10161no—
201C4—5081no—
202C4—5082no—
203C4—5081no—
204C4#1686118611yescharacteristic
205C4#1686118811yescharacteristic
206C4#1686119511yescharacteristic
207C4#1686119611yescharacteristic
208C2#2254978811yescharacteristic
209C2#2255003511yescharacteristic
210C2#2255003711yescharacteristic
211C2#2255018111yescharacteristic
212C2#2255018511yescharacteristic
213C2#2255023211yescharacteristic
214C2#2255024311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC2×#103D127
hypocenterC127#2026
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
108. C127#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#168612002
C2×#168611332
#225497832
#225496392
#225496382
#225496302
C2×#103×C1272
#2026127