Finite Group Databasebeta

← #30639212

Subgroups — #30639212

Order 16256.

324 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 93 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#22550643C211yesmaximal, characteristic, complemented
2#22549893C211yesmaximal, characteristic, complemented
3#22549881C211yesmaximal, characteristic, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5#22549647C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
8#16861216—22no—
9#16861216—22no—
10C2×#14523232—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14#16861168C2211yescharacteristic
15#16861136—22no—
16#16861130—22no—
17#16861126—22no—
18C2×#41×C127C2211yescharacteristic
19C22×D4×C127C2211yescharacteristic
20AGL(1,ℤ/8)×C127—22no—
21AGL(1,ℤ/8)×C127—22no—
22C2×SD16×C127C412yescomplemented
23C2×D8×C127C412yescomplemented
24C2×M16×C127C2211yescharacteristic
25#14523232—24no—
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29#14523232—21no—
30#14523232—21no—
31#14523232D412yes—
32C2×C127⋊C8—24no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8D412yes—
36C4⋊Dic127—21no—
37C22⋊Dic127—21no—
38C4×Dic127—21no—
39C24×C127—21no—
40#41×C127—21no—
41#41×C127—21no—
42#41×C127D412yes—
43#41×C127D412yes—
44C2×Q8×C127C2×C411yescharacteristic
45C2×D4×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127C2×C412yes—
52C2×D4×C127C2×C411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55SD16×C127—24no—
56D8×C127—24no—
57M16×C127—21no—
58M16×C127—21no—
59C2×C8×C127C2×C412yes—
60C2×Dic127—41no—
61C127⋊C8—22no—
62C127⋊C8—22no—
63C127⋊C8—41no—
64C23×C127—21no—
65C23×C127—22no—
66C23×C127—41no—
67C23×C127—41no—
68C23×C127—82no—
69C23×C127#4111yescharacteristic
70Q8×C127—21no—
71Q8×C127C22⋊C412yes—
72D4×C127—22no—
73D4×C127—22no—
74D4×C127—22no—
75D4×C127—21no—
76D4×C127—22no—
77D4×C127—41no—
78D4×C127—41no—
79D4×C127C22⋊C412yes—
80C2×C4×C127—21no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127C22×C411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C8×C127—22no—
90C22×C127—22no—
91C22×C127—41no—
92C22×C127—42no—
93C22×C127—81no—
94C22×C127—82no—
95C22×C127—82no—
96C22×C127—82no—
97C22×C127C4×D411yescharacteristic
98C22×C127#12111yescharacteristic
99C22×C127#12311yescharacteristic
100C4×C127—22no—
101C4×C127C2×C22⋊C411yescharacteristic
102C4×C127C4×D411yescharacteristic
103C4×C127#12011yescharacteristic
104C4×C127#12111yescharacteristic
105Dic127—81no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127—42no—
110C2×C127—82no—
111C2×C127#38611yescharacteristic
112#1903—1271nomaximal, Sylow
113C127#190311yescharacteristic, complemented, Sylow
114C2×AGL(1,ℤ/8)D12711yescharacteristic
115#568—1271no—
116#421—1271no—
117#418—1271no—
118C2×#100—1271no—
119#344—1271no—
120#338—1271no—
121C2×#41C2×D12711yescharacteristic
122C22×D4C2×D12711yescharacteristic
123AGL(1,ℤ/8)—22no—
124AGL(1,ℤ/8)—22no—
125C2×SD16Dic12712yescomplemented
126C2×D8Dic12712yescomplemented
127#131—2542no—
128#131—2542no—
129C2×M16—1271no—
130C2×M16—1271no—
131C2×M16—1271no—
132C2×M16—2542no—
133C2×M16C2×D12711yescharacteristic
134#117—1271no—
135#104—2542no—
136#102—2542no—
137#100—2542no—
138C24—21no—
139#41—21no—
140#41—21no—
141#41C22⋊D12712yes—
142#41C22⋊D12712yes—
143C2×Q8C2×Dic12711yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C2×Dic12712yes—
151C2×D4C2×Dic12711yescharacteristic
152C22×C4—21no—
153C22×C4C22×D12711yescharacteristic
154SD16—24no—
155D8—24no—
156M16—21no—
157M16—21no—
158M16—1272no—
159M16—2541no—
160M16—2541no—
161M16—2544no—
162M16—2541no—
163M16—2541no—
164M16—2541no—
165C2×C8—1272no—
166C2×C8—2541no—
167C2×C8—2544no—
168C2×C8—2541no—
169C2×C8C2×Dic12712yes—
170C4⋊C4—2541no—
171C22⋊C4—2541no—
172C42—2541no—
173C23—21no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—82no—
178C23#1452324711yescharacteristic
179Q8—21no—
180Q8C22⋊Dic12712yes—
181D4—22no—
182D4—22no—
183D4—22no—
184D4—21no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4C22⋊Dic12712yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—5081no—
193C2×C4C22×Dic12711yescharacteristic
194C2×C4D4×D12711yescharacteristic
195C2×C4D4×D12711yescharacteristic
196C2×C4C2×C22⋊D12711yescharacteristic
197C2×C4C2×C22⋊D12711yescharacteristic
198C2×C4#1452324711yescharacteristic
199C8—22no—
200C8—5081no—
201C8—2542no—
202C8—2542no—
203C22—22no—
204C22—41no—
205C22—42no—
206C22—81no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22D4×Dic12711yescharacteristic
211C22#1686118111yescharacteristic
212C22#1686119011yescharacteristic
213C4—22no—
214C4—10161no—
215C4C2×C22⋊Dic12711yescharacteristic
216C4D4×Dic12711yescharacteristic
217C4#1686117711yescharacteristic
218C4#1686118111yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—82no—
224C2#2254975611yescharacteristic
subgroupG/H
centerC2#22549756
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC2×C127#386
cosocleC22×C4×C127C23
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#1903
LCSG ▷ C2×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. #22550643C2semidirect
2. #22549893C2semidirect
3. #22549881C2semidirect
4. C2×#16861126C2semidirect
5. #22549647C2semidirect
6. C127⋊#338C2semidirect
22. C2×SD16×C127C4semidirect
23. C2×D8×C127C4semidirect
113. C127#1903semidirect
125. C2×SD16Dic127semidirect
126. C2×D8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225498932
#225498812
C2×#168611262
#225496472
C127⋊#3382
C2×AGL(1,ℤ/8)×C1272
#1903127