Finite Group Databasebeta

← #30639903

Subgroups — #30639903

Order 16256.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 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#22549892C211yesmaximal, characteristic
3C2×#16861137C211yesmaximal, characteristic
4#22549849C211yesmaximal, characteristic
5#22549647C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#339×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861209—21no—
11#16861201—21no—
12#16861188—22no—
13C4×Dic254—22no—
14#16861168C2211yescharacteristic
15#16861168C2211yescharacteristic
16#16861137—21no—
17#16861137—21no—
18#16861135—22no—
19#16861123—22no—
20C2×M16×C127C2211yescharacteristic
21#117×C127C2211yescharacteristic
22C2×C42×C127C2211yescharacteristic
23#14523248—21no—
24#14523248—21no—
25#14523248—21no—
26#14523248—21no—
27C2×Dic254—22no—
28C2×Dic254Q811yescharacteristic, complemented
29C2×D508Q811yescharacteristic, complemented
30C2×C22⋊D127—21no—
31C2×C4×D127—21no—
32#14523232—21no—
33#14523232—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—22no—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38Dic127⋊C4—24no—
39C22⋊Dic127—21no—
40C22⋊Dic127D412yes—
41C4×Dic127—21no—
42C4×Dic127D412yes—
43C22×C4×C127—21no—
44C22×C4×C127C2311yescharacteristic
45M16×C127—21no—
46M16×C127—21no—
47C2×C8×C127—21no—
48C4⋊C4×C127—21no—
49C22⋊C4×C127—21no—
50C42×C127—21no—
51C42×C127—21no—
52C42×C127D411yescharacteristic
53C42×C127D411yescharacteristic
54C22×D127—21no—
55Dic254—21no—
56Dic254—21no—
57Dic254—44no—
58D508—21no—
59D508—21no—
60C22⋊D127—41no—
61C22⋊D127—41no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—21no—
65C4×D127—41no—
66C4×D127—41no—
67C127⋊C8—41no—
68C23×C127C2×D411yescharacteristic
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—42no—
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#12011yescharacteristic
82C22×C127C22⋊Q811yescharacteristic
83C22×C127#12411yescharacteristic
84C2×D127—41no—
85C2×D127—81no—
86C4×C127—22no—
87C4×C127—41no—
88C4×C127—81no—
89C4×C127C22⋊Q811yescharacteristic
90C4×C127C22⋊Q811yescharacteristic
91C4×C127#12411yescharacteristic
92C4×C127#12411yescharacteristic
93Dic127—42no—
94Dic127—42no—
95Dic127—41no—
96C2×C127—21no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127#39311yescharacteristic
100D127—81no—
101#2031—1271nomaximal, Sylow
102C127#203111yescharacteristic, complemented, Sylow
103#533—1271no—
104#421—1271no—
105C2×#104—1271no—
106#410—1271no—
107#344—1271no—
108#339D12711yescharacteristic, complemented
109H(3,ℤ/4)—1271no—
110C2×#41—1271no—
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#117C2×D12711yescharacteristic
120C2×C42C2×D12711yescharacteristic
121#104—2542no—
122#104—2541no—
123#104—2541no—
124#99—2542no—
125#41—2541no—
126#41—2541no—
127#41—2541no—
128#41—2541no—
129C2×Q8—1271no—
130C2×Q8—2542no—
131C2×D4—1271no—
132C2×D4—2541no—
133C22×C4—21no—
134C22×C4—2541no—
135C22×C4C22×D12711yescharacteristic
136M16—21no—
137M16—21no—
138M16—2541no—
139M16—2541no—
140C2×C8—21no—
141C2×C8—2541no—
142C4⋊C4—21no—
143C4⋊C4—2544no—
144C4⋊C4—2542no—
145C4⋊C4—2541no—
146C4⋊C4—2541no—
147C22⋊C4—21no—
148C22⋊C4—1272no—
149C22⋊C4—2541no—
150C42—21no—
151C42—21no—
152C42—1272no—
153C42—2541no—
154C42D50811yescharacteristic
155C42D50811yescharacteristic
156C23—2541no—
157C23D4×D12711yescharacteristic
158Q8—5084no—
159Q8—2541no—
160Q8—2541no—
161D4—5081no—
162D4—5081no—
163D4—2541no—
164D4—2541no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—5081no—
170C2×C4—5081no—
171C2×C4—2541no—
172C2×C4—2542no—
173C2×C4—2542no—
174C2×C4D4×D12711yescharacteristic
175C2×C4C2×D50811yescharacteristic
176C2×C4Q8×D12711yescharacteristic
177C2×C4#1452324711yescharacteristic
178C2×C4#1452324811yescharacteristic
179C2×C4#1452324811yescharacteristic
180C8—41no—
181C8—5081no—
182C22—41no—
183C22—10161no—
184C22—5081no—
185C22#1686117911yescharacteristic
186C22#1686118711yescharacteristic
187C22#1686119511yescharacteristic
188C4—22no—
189C4—41no—
190C4—81no—
191C4—5082no—
192C4—5082no—
193C4—5081no—
194C4#1686118711yescharacteristic
195C4#1686118911yescharacteristic
196C4#1686119511yescharacteristic
197C4#1686119711yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—10161no—
202C2#2254978911yescharacteristic
subgroupG/H
centerC4#16861195
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#339×C127C2
socleC2×C127#393
prime radicalC2×#14523248C22
hypercenter#339D127
hypocenterC127#2031
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #22550433C2semidirect
5. #22549647C2semidirect
6. #22549632C2semidirect
7. #339×C127C2semidirect
28. C2×Dic254Q8semidirect
29. C2×D508Q8semidirect
102. C127#2031semidirect
108. #339D127semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225498922
C2×#168611372
#225498492
#225496472
#225496322
#339×C1272
#2031127