Finite Group Databasebeta

← #30639974

Subgroups — #30639974

Order 16256.

298 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16861207C211yesmaximal, characteristic
2C2×#16861134C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic
5#22549772C211yesmaximal, characteristic
6#22549630C211yesmaximal, characteristic
7C2×#103×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861207—22no—
11#16861207—22no—
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4—21no—
14C2×Dic127⋊C4—21no—
15C2×Dic127⋊C4—21no—
16C2×C4×Dic127C2211yescharacteristic
17#16861134—22no—
18#16861133—22no—
19#16861132—22no—
20C127⋊#95—21no—
21C22×Q8×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#103×C127—22no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254D412yes—
31C22×Dic127—21no—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C22×Dic127D411yescharacteristic
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127D412yes—
40Dic127⋊C4—24no—
41Dic127⋊C4—24no—
42Dic127⋊C4—42no—
43C4×Dic127—21no—
44C4×Dic127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127D412yes—
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C4⋊C4×C127—21no—
59Dic254—22no—
60Dic254—22no—
61Dic254—41no—
62Dic254—41no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127SD1612yes—
74C2×Dic127Q1612yes—
75C127⋊C8—41no—
76C23×C127C2×D411yescharacteristic
77Q8×C127—22no—
78Q8×C127—22no—
79Q8×C127—41no—
80Q8×C127—41no—
81C2×C4×C127—22no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127#4111yescharacteristic
91C2×C4×C127#4111yescharacteristic
92C2×C4×C127#4111yescharacteristic
93C8×C127—41no—
94C22×C127#12011yescharacteristic
95C22×C127#12111yescharacteristic
96C22×C127#12411yescharacteristic
97C22×C127C2×SD1611yescharacteristic
98C22×C127C2×Q1611yescharacteristic
99C22×C127#13711yescharacteristic
100C22×C127#13711yescharacteristic
101C4×C127—42no—
102C4×C127—81no—
103C4×C127#12111yescharacteristic
104C4×C127#12111yescharacteristic
105C4×C127#12311yescharacteristic
106C4×C127#12311yescharacteristic
107Dic127—22no—
108Dic127—22no—
109Dic127—42no—
110Dic127—42no—
111C2×C127#39411yescharacteristic
112C2×C127#44811yescharacteristic
113C2×C127#45111yescharacteristic
114C2×C127#46111yescharacteristic
115C2×C127#46211yescharacteristic
116C2×C127#48711yescharacteristic
117C2×C127#49111yescharacteristic
118#2043—1271nomaximal, Sylow
119C127#204311yescharacteristic, complemented, Sylow
120C2×C4⋊Q8—1271no—
121C2×#103—1271no—
122C2×#103—1271no—
123C2×#103—1271no—
124C2×#103D12711yescharacteristic
125#389—1271no—
126#336—1271no—
127C22×Q8—1271no—
128C22×Q8C2×D12711yescharacteristic
129C22×C8—1271no—
130C22×C8C2×D12711yescharacteristic
131C4⋊Q8—2542no—
132C4⋊Q8—2542no—
133C2×C4⋊C4—1271no—
134C2×C4⋊C4—2541no—
135C2×C4⋊C4—2541no—
136C2×C4⋊C4—2541no—
137C2×C4⋊C4C2×D12711yescharacteristic
138C2×C42—1271no—
139#103—22no—
140#103—2542no—
141#103—2542no—
142#103—2542no—
143#95—2541no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1272no—
150C2×Q8—2541no—
151C2×Q8—2541no—
152C2×Q8—2541no—
153C2×Q8—2541no—
154C2×Q8—2541no—
155C2×Q8C22⋊D12712yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—1271no—
159C22×C4—1271no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—2541no—
167C2×C8—2541no—
168C2×C8—2541no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—1272no—
172C4⋊C4—5082no—
173C4⋊C4—2544no—
174C4⋊C4—2544no—
175C4⋊C4—2541no—
176C42—2541no—
177C42—2541no—
178C23D4×D12711yescharacteristic
179Q8—22no—
180Q8—22no—
181Q8—41no—
182Q8—41no—
183Q8—5081no—
184Q8—5081no—
185Q8—2542no—
186Q8—2542no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—1272no—
194C2×C4—1272no—
195C2×C4—5081no—
196C2×C4—5081no—
197C2×C4—5081no—
198C2×C4—5081no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4—2541no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4D4×D12711yescharacteristic
206C2×C4D4×D12711yescharacteristic
207C2×C4C2×C22⋊D12711yescharacteristic
208C2×C4#1452324711yescharacteristic
209C2×C4#1452324811yescharacteristic
210C2×C4#1452324811yescharacteristic
211C8—41no—
212C8—5081no—
213C22#1686117711yescharacteristic
214C22#1686118211yescharacteristic
215C22#1686119511yescharacteristic
216C22SD16×D12711yescharacteristic
217C22Q16×D12711yescharacteristic
218C22#1686123811yescharacteristic
219C22#1686124111yescharacteristic
220C4—42no—
221C4—81no—
222C4—5082no—
223C4—5082no—
224C4—2542no—
225C4—2542no—
226C4#1686118111yescharacteristic
227C4#1686118211yescharacteristic
228C4#1686119111yescharacteristic
229C4#1686119211yescharacteristic
230C2#2254979111yescharacteristic
231C2#2255001311yescharacteristic
232C2#2255003011yescharacteristic
233C2#2255008011yescharacteristic
234C2#2255008511yescharacteristic
235C2#2255022511yescharacteristic
236C2#2255024311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC2×#103D127
hypocenterC127#2043
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
119. C127#2043semidirect

Maximal subgroups

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

subgroupindex
C2×#168612072
C2×#168611342
C2×#168611332
C2×#168611322
#225497722
#225496302
C2×#103×C1272
#2043127