Finite Group Databasebeta

← #30639902

Subgroups — #30639902

Order 16256.

244 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#22550433C211yesmaximal, characteristic, complemented
2#22549893C211yesmaximal, characteristic
3#22549850C211yesmaximal, characteristic
4#22549648C211yesmaximal, characteristic, complemented
5#22549636C211yesmaximal, characteristic, complemented
6#22549633C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9#16861209—21no—
10#16861201—21no—
11#16861188—22no—
12C4×Dic254—22no—
13#16861168C2211yescharacteristic
14#16861168C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861136—22no—
17#16861124—22no—
18C2×#41×C127C2211yescharacteristic
19C2×M16×C127C2211yescharacteristic
20C2×C42×C127C2211yescharacteristic
21#104×C127—21no—
22#104×C127—21no—
23C2×Dic254—22no—
24#14523232—21no—
25#14523232—21no—
26C2×C127⋊C8—21no—
27C4⋊Dic127—22no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31Dic127⋊C4—24no—
32C22⋊Dic127—21no—
33C22⋊Dic127—21no—
34C22⋊Dic127D412yes—
35C4×Dic127—21no—
36C4×Dic127—21no—
37C4×Dic127D412yes—
38#41×C127—21no—
39#41×C127—21no—
40#41×C127—21no—
41#41×C127—21no—
42C2×Q8×C127Q811yescharacteristic, complemented
43C2×D4×C127—21no—
44C2×D4×C127Q811yescharacteristic, complemented
45C22×C4×C127—21no—
46C22×C4×C127—21no—
47C22×C4×C127C2311yescharacteristic
48M16×C127—21no—
49M16×C127—21no—
50C2×C8×C127—21no—
51C42×C127—21no—
52C42×C127D411yescharacteristic
53C42×C127D411yescharacteristic
54Dic254—44no—
55C2×Dic127—22no—
56C2×Dic127—22no—
57C2×Dic127—41no—
58C127⋊C8—41no—
59C23×C127—21no—
60C23×C127C2×D411yescharacteristic
61Q8×C127—21no—
62Q8×C127—21no—
63D4×C127—21no—
64D4×C127—21no—
65D4×C127—41no—
66D4×C127—41no—
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—42no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×Q811yescharacteristic
76C2×C4×C127#4111yescharacteristic
77C2×C4×C127#4111yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C8×C127—41no—
80C22×C127—41no—
81C22×C127—41no—
82C22×C127—81no—
83C22×C127#12011yescharacteristic
84C22×C127C22⋊Q811yescharacteristic
85C22×C127#12411yescharacteristic
86C4×C127—22no—
87C4×C127—41no—
88C4×C127—41no—
89C4×C127C22⋊Q811yescharacteristic
90C4×C127C22⋊Q811yescharacteristic
91C4×C127#12411yescharacteristic
92C4×C127#12411yescharacteristic
93Dic127—42no—
94Dic127—42no—
95Dic127—81no—
96C2×C127—21no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127—81no—
100C2×C127#39311yescharacteristic
101#2031—1271nomaximal, Sylow
102C127#203111yescharacteristic, complemented, Sylow
103#533—1271no—
104#421—1271no—
105C2×#104D12711yescharacteristic
106#410—1271no—
107#344—1271no—
108#339—1271no—
109H(3,ℤ/4)—1271no—
110C2×#41C2×D12711yescharacteristic
111C2×M16—1271no—
112C2×M16C2×D12711yescharacteristic
113C4⋊Q8—2541no—
114#125—2541no—
115C22⋊Q8—2542no—
116C4×Q8—2542no—
117#117—1271no—
118#117—1271no—
119#117—1271no—
120C2×C42C2×D12711yescharacteristic
121#104—21no—
122#104—21no—
123#104—2542no—
124#99—2542no—
125#41—21no—
126#41—21no—
127#41—21no—
128#41—21no—
129C2×Q8—2542no—
130C2×Q8Dic25411yescharacteristic, complemented
131C2×D4—21no—
132C2×D4Dic25411yescharacteristic, complemented
133C22×C4—21no—
134C22×C4—21no—
135C22×C4C22×D12711yescharacteristic
136M16—21no—
137M16—21no—
138M16—2541no—
139M16—2541no—
140C2×C8—21no—
141C2×C8—2541no—
142C4⋊C4—2544no—
143C4⋊C4—2542no—
144C4⋊C4—2541no—
145C4⋊C4—2541no—
146C4⋊C4—2541no—
147C22⋊C4—1272no—
148C22⋊C4—2541no—
149C22⋊C4—2541no—
150C42—21no—
151C42—1272no—
152C42—2541no—
153C42—2541no—
154C42C22⋊D12711yescharacteristic
155C42C22⋊D12711yescharacteristic
156C23—21no—
157C23D4×D12711yescharacteristic
158Q8—21no—
159Q8—21no—
160Q8—5084no—
161D4—21no—
162D4—21no—
163D4—41no—
164D4—41no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—5081no—
172C2×C4—2542no—
173C2×C4—2542no—
174C2×C4D4×D12711yescharacteristic
175C2×C4C2×C22⋊D12711yescharacteristic
176C2×C4C2×Dic25411yescharacteristic
177C2×C4#1452324711yescharacteristic
178C2×C4#1452324711yescharacteristic
179C2×C4#1452324811yescharacteristic
180C8—41no—
181C8—5081no—
182C22—41no—
183C22—41no—
184C22—81no—
185C22#1686117711yescharacteristic
186C22#1686118611yescharacteristic
187C22#1686119511yescharacteristic
188C4—22no—
189C4—41no—
190C4—41no—
191C4—10161no—
192C4—5082no—
193C4—5082no—
194C4#1686118611yescharacteristic
195C4#1686118811yescharacteristic
196C4#1686119511yescharacteristic
197C4#1686119611yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#2254978811yescharacteristic
subgroupG/H
centerC4#16861195
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC2×C127#393
prime radicalC2×#41×C127C22
hypercenterC2×#104D127
hypocenterC127#2031
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. #22550433C2semidirect
4. #22549648C2semidirect
5. #22549636C2semidirect
6. #22549633C2semidirect
42. C2×Q8×C127Q8semidirect
44. C2×D4×C127Q8semidirect
102. C127#2031semidirect
130. C2×Q8Dic254semidirect
132. C2×D4Dic254semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225498932
#225498502
#225496482
#225496362
#225496332
C2×#104×C1272
#2031127