Finite Group Databasebeta

← #30639905

Subgroups — #30639905

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#22550429C211yesmaximal, characteristic, complemented
2#22549892C211yesmaximal, characteristic
3C2×#16861135C211yesmaximal, characteristic
4#22549849C211yesmaximal, characteristic
5#22549637C211yesmaximal, characteristic, complemented
6#22549633C211yesmaximal, characteristic, complemented
7#344×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861208—21no—
10#16861200—21no—
11#16861186—22no—
12C4×Dic254—22no—
13#16861168C2211yescharacteristic
14#16861168C2211yescharacteristic
15C2×C4×Dic127C2211yescharacteristic
16#16861135—21no—
17#16861135—21no—
18#16861135—22no—
19#16861123—22no—
20C2×M16×C127C2211yescharacteristic
21C2×M16×C127C2211yescharacteristic
22#117×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—
31C22×Dic127—21no—
32C2×C4×D127—21no—
33C4⋊Dic127—22no—
34C4⋊Dic127—21no—
35C4⋊Dic127—21no—
36Dic127⋊C4—22no—
37Dic127⋊C4—22no—
38C22⋊Dic127—21no—
39C22⋊Dic127—21no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C4×Dic127D411yescharacteristic
43C4×Dic127D411yescharacteristic
44C4×Dic127D412yes—
45C22×C4×C127C2311yescharacteristic
46M16×C127—21no—
47M16×C127—21no—
48M16×C127—21no—
49M16×C127—21no—
50C2×C8×C127—21no—
51C2×C8×C127—21no—
52C4⋊C4×C127—21no—
53C22⋊C4×C127D412yes—
54C42×C127—21no—
55C22×D127—21no—
56Dic254—21no—
57Dic254—21no—
58Dic254—44no—
59D508—21no—
60D508—21no—
61C22⋊D127—41no—
62C22⋊D127—41no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—22no—
66C2×Dic127—21no—
67C2×Dic127—42no—
68C2×Dic127—41no—
69C4×D127—41no—
70C4×D127—41no—
71C23×C127C2×D411yescharacteristic
72C2×C4×C127—22no—
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—
80C8×C127—41no—
81C22×C127—41no—
82C22×C127#12011yescharacteristic
83C22×C127C22⋊Q811yescharacteristic
84C22×C127#12411yescharacteristic
85C2×D127—41no—
86C2×D127—81no—
87C4×C127—42no—
88C4×C127C22⋊Q811yescharacteristic
89C4×C127C22⋊Q811yescharacteristic
90C4×C127#12411yescharacteristic
91C4×C127#12411yescharacteristic
92Dic127—22no—
93Dic127—41no—
94Dic127—42no—
95Dic127—41no—
96Dic127—81no—
97C2×C127—21no—
98C2×C127—21no—
99C2×C127—21no—
100C2×C127#39311yescharacteristic
101D127—81no—
102#2031—1271nomaximal, Sylow
103C127#203111yescharacteristic, complemented, Sylow
104#533—1271no—
105#421—1271no—
106C2×#104—1271no—
107#410—1271no—
108#344D12711yescharacteristic, complemented
109#339—1271no—
110H(3,ℤ/4)—1271no—
111C2×#41—1271no—
112C2×M16C2×D12711yescharacteristic
113C2×M16C2×D12711yescharacteristic
114C4⋊Q8—2541no—
115#125—2541no—
116C22⋊Q8—2542no—
117C4×Q8—2542no—
118#117—1271no—
119#117—1271no—
120#117C2×D12711yescharacteristic
121C2×C42—1271no—
122#104—2541no—
123#104—2541no—
124#104—2542no—
125#99—2542no—
126#41—2541no—
127#41—2541no—
128#41—2541no—
129#41—2541no—
130C2×Q8—1271no—
131C2×Q8—2542no—
132C2×D4—1271no—
133C2×D4—2541no—
134C22×C4—2541no—
135C22×C4—2541no—
136C22×C4C22×D12711yescharacteristic
137M16—21no—
138M16—21no—
139M16—21no—
140M16—21no—
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—2542no—
145C4⋊C4—2542no—
146C4⋊C4—2542no—
147C4⋊C4—2541no—
148C4⋊C4—2541no—
149C22⋊C4—2541no—
150C22⋊C4—2541no—
151C22⋊C4D50812yes—
152C42—21no—
153C42—1271no—
154C42—1271no—
155C42—1272no—
156C42—2541no—
157C42—2541no—
158C23—2541no—
159C23C2×D50811yescharacteristic
160Q8—5084no—
161Q8—2541no—
162Q8—2541no—
163D4—5081no—
164D4—5081no—
165D4—2541no—
166D4—2541no—
167C2×C4—22no—
168C2×C4—5081no—
169C2×C4—5082no—
170C2×C4—5081no—
171C2×C4—5081no—
172C2×C4—2541no—
173C2×C4—2541no—
174C2×C4—2542no—
175C2×C4—2541no—
176C2×C4D4×D12711yescharacteristic
177C2×C4D4×D12711yescharacteristic
178C2×C4Q8×D12711yescharacteristic
179C2×C4#1452324711yescharacteristic
180C2×C4#1452324811yescharacteristic
181C2×C4#1452324811yescharacteristic
182C8—41no—
183C8—41no—
184C22—41no—
185C22—10161no—
186C22—5081no—
187C22#1686117911yescharacteristic
188C22#1686118911yescharacteristic
189C22#1686119711yescharacteristic
190C4—42no—
191C4—10161no—
192C4—5081no—
193C4—5082no—
194C4—5081no—
195C4—2542no—
196C4#1686118711yescharacteristic
197C4#1686118711yescharacteristic
198C4#1686119511yescharacteristic
199C4#1686119511yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—10161no—
204C2#2254978911yescharacteristic
subgroupG/H
centerC4#16861195
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#344×C127C2
socleC2×C127#393
prime radicalC2×#14523248C22
hypercenter#344D127
hypocenterC127#2031
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #22550429C2semidirect
5. #22549637C2semidirect
6. #22549633C2semidirect
7. #344×C127C2semidirect
28. C2×Dic254Q8semidirect
29. C2×D508Q8semidirect
103. C127#2031semidirect
108. #344D127semidirect

Maximal subgroups

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

subgroupindex
#225504292
#225498922
C2×#168611352
#225498492
#225496372
#225496332
#344×C1272
#2031127