Finite Group Databasebeta

← #30640045

Subgroups — #30640045

Order 16256.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C22×#14523235C211yesmaximal, characteristic
2C2×C4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic
5#22549778C211yesmaximal, characteristic
6#22549638C211yesmaximal, characteristic, complemented
7#391×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523235—22no—
10C2×#14523235—22no—
11C2×#14523235—22no—
12C22×C127⋊C8C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15C2×(C2×D127)⋊C4—21no—
16C2×(C2×D127)⋊C4—21no—
17C4⋊(C127⋊C8)—22no—
18#16861133—22no—
19#16861129—22no—
20C22×Q8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C42×C127C2211yescharacteristic
23#95×C127—21no—
24#95×C127—21no—
25C23×D127—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D412yes—
32C22×Dic127—21no—
33#14523235—28no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8D412yes—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42(C2×D127)⋊C4—42no—
43(C2×D127)⋊C4—42no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127D412yes—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127D411yescharacteristic
53C22×C4×C127D411yescharacteristic
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—21no—
56C4⋊C4×C127—21no—
57C42×C127—21no—
58C42×C127—21no—
59C22×D127—22no—
60C22×D127—41no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127—41no—
66D508—22no—
67D508—22no—
68D508—41no—
69D508—41no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C127⋊C8—22no—
74C127⋊C8—41no—
75C23×C127C2×D411yescharacteristic
76Q8×C127—22no—
77Q8×C127—22no—
78Q8×C127—41no—
79Q8×C127—41no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—22no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127SD1612yescomplemented
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C2×C4×C127#4111yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C22×C127#12111yescharacteristic
99C22×C127#12111yescharacteristic
100C22×C127#12111yescharacteristic
101C22×C127C2×SD1611yescharacteristic
102C22×C127#13511yescharacteristic
103C22×C127AGL(1,ℤ/8)11yescharacteristic
104C22×C127#13711yescharacteristic
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—81no—
109C2×D127—81no—
110C2×D127—81no—
111C2×D127—81no—
112C4×C127—22no—
113C4×C127—42no—
114C4×C127—41no—
115C4×C127—81no—
116C4×C127#12011yescharacteristic
117C4×C127#12311yescharacteristic
118C4×C127#12311yescharacteristic
119C4×C127#12411yescharacteristic
120Dic127—81no—
121C2×C127#39411yescharacteristic
122C2×C127#46011yescharacteristic
123C2×C127#46111yescharacteristic
124C2×C127#46311yescharacteristic
125C2×C127#46411yescharacteristic
126C2×C127#46511yescharacteristic
127C2×C127#46811yescharacteristic
128D127—82no—
129#2055—1271nomaximal, Sylow
130C127#205511yescharacteristic, complemented, Sylow
131C22×SD16—1271no—
132C2×C4⋊C8—1271no—
133C2×#103—1271no—
134C2×#102—1271no—
135#391D12711yescharacteristic, complemented
136#390—1271no—
137#340—1271no—
138C22×Q8C2×D12711yescharacteristic
139C22×D4—1271no—
140C2×SD16—2542no—
141C2×SD16—2542no—
142C2×SD16—2542no—
143C22×C8—1271no—
144C22×C8—1271no—
145C2×C4⋊C4—1271no—
146C2×C4⋊C4C2×D12711yescharacteristic
147C2×C22⋊C4—2541no—
148C2×C22⋊C4—2541no—
149C2×C42C2×D12711yescharacteristic
150C4⋊C8—2542no—
151#103—2542no—
152#102—2542no—
153#95—21no—
154#95—21no—
155C24—2541no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8D50812yes—
162C2×D4—1272no—
163C2×D4—2541no—
164C2×D4—2541no—
165C2×D4—2541no—
166C2×D4—2541no—
167C2×D4—2541no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2541no—
171C22×C4C22⋊D12711yescharacteristic
172C22×C4C22⋊D12711yescharacteristic
173C22×C4C22×D12711yescharacteristic
174SD16—2548no—
175C2×C8—1272no—
176C2×C8—2541no—
177C2×C8—2541no—
178C2×C8—2541no—
179C2×C8—2541no—
180C2×C8—2541no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—2541no—
184C4⋊C4—2541no—
185C22⋊C4—5082no—
186C22⋊C4—5082no—
187C42—21no—
188C42—21no—
189C23—5081no—
190C23—5081no—
191C23—5081no—
192C23—5081no—
193C23—5081no—
194C23—5081no—
195C23—2542no—
196C23C2×C22⋊D12711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201D4—5081no—
202D4—5081no—
203D4—2542no—
204D4—2542no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—5081no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4#1452323512yescomplemented
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×D50811yescharacteristic
223C2×C4#1452324711yescharacteristic
224C2×C4#1452324711yescharacteristic
225C2×C4#1452324811yescharacteristic
226C8—5081no—
227C8—2542no—
228C22—10161no—
229C22—10161no—
230C22—10161no—
231C22—10161no—
232C22—5082no—
233C22—5082no—
234C22—5082no—
235C22#1686118011yescharacteristic
236C22#1686118111yescharacteristic
237C22#1686118311yescharacteristic
238C22C2×#1452323511yescharacteristic
239C22#1686122911yescharacteristic
240C22#1686123411yescharacteristic
241C22#1686124011yescharacteristic
242C4—22no—
243C4—42no—
244C4—41no—
245C4—81no—
246C4—10161no—
247C4#1686117911yescharacteristic
248C4#1686119111yescharacteristic
249C4#1686119311yescharacteristic
250C4#1686119511yescharacteristic
251C2—10162no—
252C2#2254979211yescharacteristic
253C2#2255007611yescharacteristic
254C2#2255007811yescharacteristic
255C2#2255008811yescharacteristic
256C2#2255009811yescharacteristic
257C2#2255010711yescharacteristic
258C2#2255012411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenter#391D127
hypocenterC127#2055
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C127⋊C8)C2semidirect
3. C2×#16861133C2semidirect
6. #22549638C2semidirect
7. #391×C127C2semidirect
91. C2×C4×C127SD16semidirect
130. C127#2055semidirect
135. #391D127semidirect
219. C2×C4#14523235semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
C2×C4⋊(C127⋊C8)2
C2×#168611332
C2×#168611292
#225497782
#225496382
#391×C1272
#2055127