Finite Group Databasebeta

← #30640000

Subgroups — #30640000

Order 16256.

274 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 67 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#22549924C211yesmaximal, characteristic, complemented
2#22549893C211yesmaximal, characteristic, complemented
3#22549881C211yesmaximal, characteristic, complemented
4#22549860C211yesmaximal, characteristic, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6#22549648C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861168C2211yescharacteristic
12#16861144—22no—
13#16861136—22no—
14#16861130—22no—
15#16861128—21no—
16#16861126—22no—
17#16861126—21no—
18C2×#41×C127C2211yescharacteristic
19C22×D4×C127C2211yescharacteristic
20AGL(1,ℤ/8)×C127—22no—
21AGL(1,ℤ/8)×C127—22no—
22C2×SD16×C127—21no—
23C2×D8×C127—21no—
24C2×M16×C127C2211yescharacteristic
25#14523232—21no—
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29#14523232—21no—
30#14523232D412yes—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C4⋊Dic127—21no—
35C22⋊Dic127—21no—
36C4×Dic127—21no—
37C24×C127—21no—
38#41×C127—21no—
39#41×C127—21no—
40#41×C127—21no—
41#41×C127—21no—
42C2×Q8×C127D411yescharacteristic
43C2×D4×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127D411yescharacteristic
50C2×D4×C127D412yes—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53SD16×C127—24no—
54D8×C127—24no—
55M16×C127—21no—
56M16×C127D412yes—
57C2×C8×C127—21no—
58C2×Dic127—41no—
59C127⋊C8—22no—
60C127⋊C8—41no—
61C127⋊C8—41no—
62C23×C127—22no—
63C23×C127—21no—
64C23×C127—41no—
65C23×C127—41no—
66C23×C127—82no—
67C23×C127#4111yescharacteristic
68Q8×C127—21no—
69Q8×C127—21no—
70D4×C127—21no—
71D4×C127—22no—
72D4×C127—21no—
73D4×C127—22no—
74D4×C127—41no—
75D4×C127—41no—
76D4×C127—41no—
77D4×C127—41no—
78C2×C4×C127—21no—
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—42no—
91C22×C127—81no—
92C22×C127—82no—
93C22×C127—82no—
94C22×C127—82no—
95C22×C127#12111yescharacteristic
96C22×C127#12311yescharacteristic
97C22×C127#12411yescharacteristic
98C4×C127—41no—
99C4×C127#12011yescharacteristic
100C4×C127#12111yescharacteristic
101C4×C127#12111yescharacteristic
102C4×C127#12311yescharacteristic
103Dic127—81no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—81no—
108C2×C127—82no—
109C2×C127#39411yescharacteristic
110#2048—1271nomaximal, Sylow
111C127#204811yescharacteristic, complemented, Sylow
112C2×AGL(1,ℤ/8)D12711yescharacteristic
113#430—1271no—
114#421—1271no—
115#418—1271no—
116#413—1271no—
117C2×#100—1271no—
118#344—1271no—
119C2×#41C2×D12711yescharacteristic
120C22×D4C2×D12711yescharacteristic
121AGL(1,ℤ/8)—22no—
122AGL(1,ℤ/8)—22no—
123C2×SD16—21no—
124C2×D8—21no—
125C2×M16—1271no—
126C2×M16—1271no—
127C2×M16—1271no—
128C2×M16C2×D12711yescharacteristic
129#117—1271no—
130#108—2542no—
131#104—2542no—
132#102—2542no—
133#101—2541no—
134#100—2542no—
135#100—2541no—
136C24—21no—
137#41—21no—
138#41—21no—
139#41—21no—
140#41—21no—
141C2×Q8C22⋊D12711yescharacteristic
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C22⋊D12711yescharacteristic
149C2×D4C22⋊D12712yes—
150C22×C4—21no—
151C22×C4C22×D12711yescharacteristic
152SD16—24no—
153D8—24no—
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—2541no—
167C22⋊C4—2541no—
168C42—2541no—
169C23—21no—
170C23—22no—
171C23—41no—
172C23—41no—
173C23—82no—
174C23#1452324711yescharacteristic
175Q8—21no—
176Q8—21no—
177D4—22no—
178D4—21no—
179D4—22no—
180D4—21no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—41no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—5081no—
189C2×C4D4×D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4C2×C22⋊D12711yescharacteristic
192C2×C4C2×C22⋊D12711yescharacteristic
193C2×C4#1452324711yescharacteristic
194C2×C4#1452324711yescharacteristic
195C8—22no—
196C8—5081no—
197C8—5081no—
198C8—2542no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—82no—
203C22—81no—
204C22—82no—
205C22—82no—
206C22#1686118111yescharacteristic
207C22#1686119011yescharacteristic
208C22#1686119611yescharacteristic
209C4—41no—
210C4—10161no—
211C4#1686117811yescharacteristic
212C4#1686118011yescharacteristic
213C4#1686118111yescharacteristic
214C4#1686119011yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—82no—
219C2—81no—
220C2#2254979011yescharacteristic
subgroupG/H
centerC2#22549790
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC2×C127#394
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#2048
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22549924C2semidirect
2. #22549893C2semidirect
3. #22549881C2semidirect
4. #22549860C2semidirect
5. C2×#16861126C2semidirect
6. #22549648C2semidirect
111. C127#2048semidirect

Maximal subgroups

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

subgroupindex
#225499242
#225498932
#225498812
#225498602
C2×#168611262
#225496482
C2×AGL(1,ℤ/8)×C1272
#2048127