Finite Group Databasebeta

← #30640004

Subgroups — #30640004

Order 16256.

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

normal subgroups only

This group has 274 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
1C2×#16861234C211yesmaximal, characteristic
2#22549925C211yesmaximal, characteristic, complemented
3#22549880C211yesmaximal, characteristic, complemented
4#22549860C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9#16861234—22no—
10#16861234—22no—
11C2×#14523235—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C2×#14523232C2211yescharacteristic
16#16861145—22no—
17#16861129—22no—
18#16861128—21no—
19#16861126—21no—
20#16861125—22no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#104×C127—22no—
25C23×D127—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D412yes—
32#14523235—24no—
33#14523233—24no—
34#14523232—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39#14523232D412yes—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43#41×C127—21no—
44#41×C127—21no—
45#41×C127—21no—
46#41×C127—21no—
47C2×Q8×C127D411yescharacteristic
48C2×D4×C127—21no—
49C2×D4×C127D411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52M16×C127—21no—
53M16×C127D412yes—
54C2×C8×C127—21no—
55C4⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C42×C127—21no—
58C22×D127—22no—
59C22×D127—41no—
60C22×D127—41no—
61C22×D127—82no—
62D508—22no—
63D508—22no—
64D508—41no—
65D508—41no—
66C127⋊C8—22no—
67C127⋊C8—41no—
68C127⋊C8—41no—
69C23×C127—21no—
70C23×C127#4111yescharacteristic
71Q8×C127—21no—
72Q8×C127—21no—
73D4×C127—21no—
74D4×C127—21no—
75D4×C127—41no—
76D4×C127—41no—
77C2×C4×C127—21no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C8×C127—22no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127—81no—
91C22×C127#12111yescharacteristic
92C22×C127#12311yescharacteristic
93C22×C127#12411yescharacteristic
94C2×D127—42no—
95C2×D127—82no—
96C2×D127—82no—
97C2×D127—82no—
98C4×C127—41no—
99C4×C127—81no—
100C4×C127#12011yescharacteristic
101C4×C127#12111yescharacteristic
102C4×C127#12111yescharacteristic
103C4×C127#12311yescharacteristic
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—81no—
108C2×C127#39411yescharacteristic
109D127—82no—
110#2048—1271nomaximal, Sylow
111C127#204811yescharacteristic, complemented, Sylow
112C2×AGL(1,ℤ/8)—1271no—
113#430—1271no—
114#421D12711yescharacteristic, complemented
115#418—1271no—
116#413—1271no—
117C2×#100—1271no—
118#344—1271no—
119C2×#41C2×D12711yescharacteristic
120C22×D4—1271no—
121AGL(1,ℤ/8)—2542no—
122AGL(1,ℤ/8)—2542no—
123C2×SD16—2541no—
124C2×D8—2541no—
125C2×M16—1271no—
126C2×M16—1271no—
127C2×M16—1271no—
128C2×M16C2×D12711yescharacteristic
129#117C2×D12711yescharacteristic
130#108—2542no—
131#104—22no—
132#102—2542no—
133#101—2541no—
134#100—2541no—
135#100—2542no—
136C24—2541no—
137#41—21no—
138#41—21no—
139#41—21no—
140#41—21no—
141C2×Q8D50811yescharacteristic
142C2×D4—21no—
143C2×D4—1272no—
144C2×D4—2541no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4D50811yescharacteristic
150C22×C4—21no—
151C22×C4C22×D12711yescharacteristic
152SD16—2544no—
153D8—2544no—
154M16—21no—
155M16—1272no—
156M16—2541no—
157M16—2541no—
158M16—2541no—
159M16—2541no—
160M16—2541no—
161M16C22⋊D12712yes—
162C2×C8—21no—
163C2×C8—2541no—
164C2×C8—2541no—
165C2×C8—2541no—
166C4⋊C4—21no—
167C22⋊C4—21no—
168C42—21no—
169C23—21no—
170C23—10162no—
171C23—5081no—
172C23—5081no—
173C23—2542no—
174C23#1452324711yescharacteristic
175Q8—21no—
176Q8—21no—
177D4—21no—
178D4—21no—
179D4—41no—
180D4—41no—
181D4—5081no—
182D4—5081no—
183D4—2542no—
184D4—2542no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4D4×D12711yescharacteristic
190C2×C4D4×D12711yescharacteristic
191C2×C4C2×C22⋊D12711yescharacteristic
192C2×C4C2×D50811yescharacteristic
193C2×C4#1452324711yescharacteristic
194C2×C4#1452324811yescharacteristic
195C8—22no—
196C8—5081no—
197C8—5081no—
198C8—2542no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22—10162no—
203C22—10162no—
204C22—10162no—
205C22—5082no—
206C22#1686118011yescharacteristic
207C22#1686119311yescharacteristic
208C22#1686119511yescharacteristic
209C4—41no—
210C4—81no—
211C4#1686117911yescharacteristic
212C4#1686118111yescharacteristic
213C4#1686118311yescharacteristic
214C4#1686119111yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—81no—
219C2—10162no—
220C2#2254979211yescharacteristic
subgroupG/H
centerC2#22549792
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#394
prime radicalC2×#16861234C2
hypercenter#421D127
hypocenterC127#2048
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #22549925C2semidirect
3. #22549880C2semidirect
4. #22549860C2semidirect
5. C2×#16861125C2semidirect
6. #22549646C2semidirect
7. #421×C127C2semidirect
111. C127#2048semidirect
114. #421D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
#225499252
#225498802
#225498602
C2×#168611252
#225496462
#421×C1272
#2048127