Finite Group Databasebeta

← #30639094

Subgroups — #30639094

Order 16256.

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

normal subgroups only

This group has 310 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#22549872C211yesmaximal, characteristic, complemented
4#22549843C211yesmaximal, characteristic, complemented
5#22549648C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7#575×C127C211yesmaximal, characteristic
8#16861216—22no—
9#16861216—22no—
10C2×#14523232—22no—
11C2×#14523232C2211yescharacteristic
12C2×#14523232C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861168C2211yescharacteristic
15#16861136—22no—
16#16861133—21no—
17#16861130—21no—
18C22⋊(C127⋊C8)—22no—
19C2×#41×C127C2211yescharacteristic
20C2×#41×C127C2211yescharacteristic
21#137×C127—21no—
22#137×C127C412yescomplemented
23AGL(1,ℤ/8)×C127—21no—
24AGL(1,ℤ/8)×C127C412yescomplemented
25#135×C127—22no—
26C2×M16×C127C2211yescharacteristic
27#14523232—24no—
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⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8D412yes—
38C4⋊Dic127—21no—
39C22⋊Dic127—21no—
40C4×Dic127—21no—
41#41×C127—21no—
42#41×C127—21no—
43#41×C127—21no—
44#41×C127—21no—
45#41×C127—21no—
46#41×C127—21no—
47#41×C127C2×C412yes—
48#41×C127D412yes—
49C2×Q8×C127C2×C411yescharacteristic
50C2×Q8×C127D411yescharacteristic
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C2×D4×C127C2×C411yescharacteristic
54C2×D4×C127D411yescharacteristic
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127C2311yescharacteristic
58Q16×C127—22no—
59SD16×C127—22no—
60SD16×C127—22no—
61D8×C127—22no—
62M16×C127—21no—
63M16×C127C2×C412yes—
64C2×C8×C127—21no—
65C2×Dic127—41no—
66C127⋊C8—22no—
67C127⋊C8—22no—
68C127⋊C8—41no—
69C23×C127—21no—
70C23×C127—21no—
71C23×C127#4111yescharacteristic
72Q8×C127—21no—
73Q8×C127—21no—
74Q8×C127—21no—
75Q8×C127C22⋊C412yes—
76D4×C127—22no—
77D4×C127—22no—
78D4×C127—21no—
79D4×C127—21no—
80D4×C127—21no—
81D4×C127—41no—
82D4×C127—41no—
83D4×C127C22⋊C412yes—
84C2×C4×C127—21no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—21no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127C22×C411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C8×C127—22no—
97C22×C127—22no—
98C22×C127—41no—
99C22×C127—42no—
100C22×C127—41no—
101C22×C127—81no—
102C22×C127C4×D411yescharacteristic
103C22×C127#12111yescharacteristic
104C22×C127#12311yescharacteristic
105C4×C127—22no—
106C4×C127—41no—
107C4×C127C2×C22⋊C411yescharacteristic
108C4×C127C4×D411yescharacteristic
109C4×C127#12011yescharacteristic
110C4×C127#12111yescharacteristic
111Dic127—81no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—42no—
116C2×C127—81no—
117C2×C127#38611yescharacteristic
118#1881—1271nomaximal, Sylow
119C127#188111yescharacteristic, complemented, Sylow
120#575D12711yescharacteristic
121#568—1271no—
122#421—1271no—
123#416—1271no—
124#408—1271no—
125#344—1271no—
126#341—1271no—
127C2×#41C2×D12711yescharacteristic
128C2×#41C2×D12711yescharacteristic
129#137—21no—
130#137Dic12712yescomplemented
131AGL(1,ℤ/8)—21no—
132AGL(1,ℤ/8)Dic12712yescomplemented
133#135—22no—
134#131—2542no—
135#131—2542no—
136C2×M16—1271no—
137C2×M16—1271no—
138C2×M16—2542no—
139C2×M16C2×D12711yescharacteristic
140C22×C8—1271no—
141#117—1271no—
142#104—2542no—
143#103—2541no—
144#102—2541no—
145C22⋊C8—2542no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152#41C2×Dic12712yes—
153#41C22⋊D12712yes—
154C2×Q8C2×Dic12711yescharacteristic
155C2×Q8C22⋊D12711yescharacteristic
156C2×D4—21no—
157C2×D4—21no—
158C2×D4C2×Dic12711yescharacteristic
159C2×D4C22⋊D12711yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4C22×D12711yescharacteristic
163Q16—22no—
164SD16—22no—
165SD16—22no—
166D8—22no—
167M16—21no—
168M16—1272no—
169M16—2544no—
170M16—2541no—
171M16—2541no—
172M16—2541no—
173M16C2×Dic12712yes—
174C2×C8—21no—
175C2×C8—1272no—
176C2×C8—2541no—
177C2×C8—2541no—
178C2×C8—2541no—
179C2×C8—2544no—
180C2×C8—2541no—
181C4⋊C4—2541no—
182C22⋊C4—2541no—
183C42—2541no—
184C23—21no—
185C23—21no—
186C23#1452324711yescharacteristic
187Q8—21no—
188Q8—21no—
189Q8—21no—
190Q8C22⋊Dic12712yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—21no—
195D4—21no—
196D4—41no—
197D4—41no—
198D4C22⋊Dic12712yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—5081no—
206C2×C4C22×Dic12711yescharacteristic
207C2×C4D4×D12711yescharacteristic
208C2×C4D4×D12711yescharacteristic
209C2×C4C2×C22⋊D12711yescharacteristic
210C2×C4C2×C22⋊D12711yescharacteristic
211C2×C4#1452324711yescharacteristic
212C8—22no—
213C8—5081no—
214C8—2542no—
215C8—2542no—
216C22—22no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22—81no—
221C22D4×Dic12711yescharacteristic
222C22#1686118111yescharacteristic
223C22#1686119011yescharacteristic
224C4—22no—
225C4—41no—
226C4—10161no—
227C4C2×C22⋊Dic12711yescharacteristic
228C4D4×Dic12711yescharacteristic
229C4#1686117711yescharacteristic
230C4#1686118111yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—81no—
236C2#2254975611yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#575×C127C2
socleC2×C127#386
cosocleC22×C4×C127C23
hypercenter#575D127
hypocenterC127#1881
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22550643C2semidirect
2. #22549893C2semidirect
3. #22549872C2semidirect
4. #22549843C2semidirect
5. #22549648C2semidirect
6. #22549640C2semidirect
22. #137×C127C4semidirect
24. AGL(1,ℤ/8)×C127C4semidirect
119. C127#1881semidirect
130. #137Dic127semidirect
132. AGL(1,ℤ/8)Dic127semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225498932
#225498722
#225498432
#225496482
#225496402
#575×C1272
#1881127