Finite Group Databasebeta

← #30640019

Subgroups — #30640019

Order 16256.

282 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 71 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×#16861144C211yesmaximal, characteristic, complemented
2#22549877C211yesmaximal, characteristic, complemented
3C2×#16861130C211yesmaximal, characteristic, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5#22549843C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12#16861144—22no—
13#16861133—21no—
14#16861130—22no—
15#16861130—21no—
16#16861126—22no—
17C22⋊(C127⋊C8)—22no—
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
25C22×Dic127—21no—
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29#14523232—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8D412yes—
35C4⋊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—
59C2×Dic127—81no—
60C127⋊C8—22no—
61C127⋊C8—41no—
62C127⋊C8—41no—
63C23×C127—21no—
64C23×C127—22no—
65C23×C127—41no—
66C23×C127—41no—
67C23×C127—81no—
68C23×C127—81no—
69C23×C127#4111yescharacteristic
70Q8×C127—21no—
71Q8×C127—21no—
72D4×C127—21no—
73D4×C127—22no—
74D4×C127—22no—
75D4×C127—21no—
76D4×C127—41no—
77D4×C127—41no—
78D4×C127—41no—
79D4×C127—41no—
80C2×C4×C127—21no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C8×C127—22no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—41no—
93C22×C127—42no—
94C22×C127—41no—
95C22×C127—41no—
96C22×C127—82no—
97C22×C127—82no—
98C22×C127—81no—
99C22×C127—81no—
100C22×C127#12111yescharacteristic
101C22×C127#12311yescharacteristic
102C22×C127#12411yescharacteristic
103C4×C127—41no—
104C4×C127#12011yescharacteristic
105C4×C127#12111yescharacteristic
106C4×C127#12111yescharacteristic
107C4×C127#12311yescharacteristic
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—81no—
112C2×C127—82no—
113C2×C127#39411yescharacteristic
114C2×C127#47111yescharacteristic
115C2×C127#47211yescharacteristic
116#2051—1271nomaximal, Sylow
117C127#205111yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)D12711yescharacteristic
119C2×#108—1271no—
120#417—1271no—
121C2×#102—1271no—
122C2×#100—1271no—
123#408—1271no—
124#343—1271no—
125C2×#41C2×D12711yescharacteristic
126C22×D4C2×D12711yescharacteristic
127AGL(1,ℤ/8)—22no—
128AGL(1,ℤ/8)—22no—
129C2×SD16—21no—
130C2×D8—21no—
131C2×M16—1271no—
132C2×M16—1271no—
133C2×M16C2×D12711yescharacteristic
134C22×C8—1271no—
135C2×C4⋊C4—1271no—
136#108—2542no—
137#103—2541no—
138#102—2542no—
139#102—2541no—
140#100—2542no—
141C22⋊C8—2542no—
142C24—21no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C22⋊D12711yescharacteristic
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4C22⋊D12712yes—
155C2×D4C22⋊D12711yescharacteristic
156C22×C4—21no—
157C22×C4—2541no—
158C22×C4C22×D12711yescharacteristic
159SD16—24no—
160D8—24no—
161M16—21no—
162M16—2541no—
163M16—2541no—
164M16—2541no—
165M16—2541no—
166M16C22⋊D12712yes—
167C2×C8—21no—
168C2×C8—1272no—
169C2×C8—2541no—
170C2×C8—2541no—
171C2×C8—2541no—
172C2×C8—2541no—
173C4⋊C4—2541no—
174C4⋊C4—2541no—
175C23—21no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—81no—
180C23—81no—
181C23#1452324711yescharacteristic
182Q8—21no—
183Q8—21no—
184D4—21no—
185D4—21no—
186D4—22no—
187D4—22no—
188D4—41no—
189D4—41no—
190D4—41no—
191D4—41no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—10161no—
196C2×C4—5081no—
197C2×C4D4×D12711yescharacteristic
198C2×C4C2×C22⋊D12711yescharacteristic
199C2×C4C2×C22⋊D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4#1452324711yescharacteristic
202C2×C4#1452324711yescharacteristic
203C8—22no—
204C8—5081no—
205C8—5081no—
206C8—2542no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—41no—
213C22—82no—
214C22—82no—
215C22—81no—
216C22—81no—
217C22#1686118111yescharacteristic
218C22#1686119011yescharacteristic
219C22#1686119611yescharacteristic
220C4—41no—
221C4—10161no—
222C4#1686117811yescharacteristic
223C4#1686118011yescharacteristic
224C4#1686118111yescharacteristic
225C4#1686119011yescharacteristic
226C2—21no—
227C2—21no—
228C2—81no—
229C2—82no—
230C2#2254979011yescharacteristic
231C2#2255013711yescharacteristic
232C2#2255014411yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#121
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#2051
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16861144C2semidirect
2. #22549877C2semidirect
3. C2×#16861130C2semidirect
4. C2×#16861126C2semidirect
5. #22549843C2semidirect
6. #22549644C2semidirect
117. C127#2051semidirect

Maximal subgroups

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

subgroupindex
C2×#168611442
#225498772
C2×#168611302
C2×#168611262
#225498432
#225496442
C2×AGL(1,ℤ/8)×C1272
#2051127