Finite Group Databasebeta

← #30638985

Subgroups — #30638985

Order 16256.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861174C211yesmaximal, characteristic
2C2×#16861142C211yesmaximal, characteristic
3C2×#16861134C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic
5#22549775C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861174—22no—
11#16861174—22no—
12C2×C4⋊Dic127—21no—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—21no—
15C2×C4×Dic127—21no—
16C2×C4×Dic127C2211yescharacteristic
17#16861142—22no—
18#16861134—22no—
19#16861132C414yescomplemented
20C127⋊#95—21no—
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254C2×C412yes—
28C2×Dic254D412yes—
29C2×Dic254Q812yes—
30C22×Dic127—21no—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C22×Dic127D411yescharacteristic
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8C2×C412yes—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127—42no—
40Dic127⋊C4—24no—
41C4×Dic127—24no—
42C4×Dic127—21no—
43C4×Dic127—21no—
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×C127—21no—
52C4⋊C4×C127C2×C412yes—
53C4⋊C4×C127D412yes—
54Dic254—22no—
55Dic254—22no—
56Dic254—22no—
57Dic254C4⋊C414yescomplemented
58C2×Dic127—22no—
59C2×Dic127—21no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—41no—
67C2×Dic127—41no—
68C2×Dic127—41no—
69C2×Dic127Q1612yes—
70C127⋊C8—22no—
71C23×C127C2×D411yescharacteristic
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127C22×C411yescharacteristic
79C2×C4×C127C2×D411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×Q811yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C8×C127—41no—
85C22×C127C4×D411yescharacteristic
86C22×C127#12011yescharacteristic
87C22×C127C22⋊Q811yescharacteristic
88C22×C127C2×Q1611yescharacteristic
89C22×C127#13511yescharacteristic
90C22×C127AGL(1,ℤ/8)11yescharacteristic
91C22×C127#13711yescharacteristic
92C4×C127—42no—
93C4×C127—42no—
94C4×C127C2×C4⋊C411yescharacteristic
95C4×C127C4×D411yescharacteristic
96C4×C127C22⋊Q811yescharacteristic
97C4×C127#12311yescharacteristic
98Dic127—24no—
99Dic127—22no—
100Dic127—41no—
101Dic127—81no—
102C2×C127#38511yescharacteristic
103C2×C127C4×Q1611yescharacteristic
104C2×C127#44011yescharacteristic
105C2×C127#44911yescharacteristic
106C2×C127#45111yescharacteristic
107C2×C127#47711yescharacteristic
108C2×C127#47911yescharacteristic
109#1863—1271nomaximal, Sylow
110C127#186311yescharacteristic, complemented, Sylow
111C2×C4×Q8—1271no—
112C2×#107—1271no—
113C2×#103—1271no—
114C2×#103—1271no—
115#389—1271no—
116#340D12711yescharacteristic
117#336—1271no—
118C22×Q8—1271no—
119C22×C8—1271no—
120C22×C8C2×D12711yescharacteristic
121C4×Q8—2542no—
122C4×Q8—2542no—
123C2×C4⋊C4—1271no—
124C2×C4⋊C4—2541no—
125C2×C4⋊C4—2541no—
126C2×C4⋊C4C2×D12711yescharacteristic
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C42—1271no—
129C2×C42—2541no—
130#107—2542no—
131#103—1274no—
132#103—2542no—
133#95—2541no—
134C2×Q8—1272no—
135C2×Q8—1272no—
136C2×Q8—1272no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—1271no—
143C22×C4—1271no—
144C22×C4—2541no—
145C22×C4—2541no—
146C22×C4C22×D12711yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—1272no—
151C2×C8—2541no—
152C2×C8—2541no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—5082no—
156C4⋊C4—2544no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4C4×D12712yes—
160C4⋊C4D50812yes—
161C42—2544no—
162C42—2541no—
163C42—2541no—
164C23D4×D12711yescharacteristic
165Q8—1274no—
166Q8—2542no—
167Q8—2542no—
168Q8—2542no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—1272no—
176C2×C4—5081no—
177C2×C4—5081no—
178C2×C4—5081no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4—2541no—
182C2×C4—2542no—
183C2×C4—2541no—
184C2×C4—2542no—
185C2×C4—2541no—
186C2×C4—2542no—
187C2×C4C2×C4×D12711yescharacteristic
188C2×C4D4×D12711yescharacteristic
189C2×C4C2×D50811yescharacteristic
190C2×C4Q8×D12711yescharacteristic
191C2×C4#1452324711yescharacteristic
192C2×C4#1452324811yescharacteristic
193C8—41no—
194C8—2542no—
195C22(C22⋊D127)⋊C411yescharacteristic
196C22#1686117911yescharacteristic
197C22#1686118711yescharacteristic
198C22Q16×D12711yescharacteristic
199C22#1686123011yescharacteristic
200C22#1686123311yescharacteristic
201C22#1686123811yescharacteristic
202C4—42no—
203C4—42no—
204C4—10161no—
205C4—5081no—
206C4—2542no—
207C4—2544no—
208C4C4⋊C4×D12711yescharacteristic
209C4C4×D50811yescharacteristic
210C4#1686118911yescharacteristic
211C4#1686119111yescharacteristic
212C2#2254975011yescharacteristic
213C2#2254996811yescharacteristic
214C2#2254997311yescharacteristic
215C2#2255002211yescharacteristic
216C2#2255003311yescharacteristic
217C2#2255017011yescharacteristic
218C2#2255018611yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socle, prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#340D127
hypocenterC127#1863
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
19. #16861132C4semidirect
57. Dic254C4⋊C4semidirect
110. C127#1863semidirect

Maximal subgroups

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

subgroupindex
C2×#168611742
C2×#168611422
C2×#168611342
C2×#168611322
#225497752
#225496302
#340×C1272
#1863127