Finite Group Databasebeta

← #30639967

Subgroups — #30639967

Order 16256.

318 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861198C211yesmaximal, characteristic
2C2×#16861133C211yesmaximal, characteristic, complemented
3C2×#16861131C211yesmaximal, characteristic
4C2×#16861129C211yesmaximal, characteristic
5#22549772C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#103×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10#16861198—22no—
11#16861198—22no—
12C2×C4⋊Dic127C2211yescharacteristic
13C2×Dic127⋊C4—21no—
14C2×(C2×D127)⋊C4—21no—
15C2×(C2×D127)⋊C4—21no—
16C2×C4×Dic127C2211yescharacteristic
17#16861133—22no—
18#16861131—22no—
19#16861129—22no—
20C127⋊#95—21no—
21C22×Q8×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#103×C127—22no—
25C23×D127—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D412yes—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C22×Dic127D411yescharacteristic
35C2×C127⋊C8—21no—
36C2×C127⋊C8—21no—
37C2×C127⋊C8—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127D412yes—
40Dic127⋊C4—42no—
41(C2×D127)⋊C4—24no—
42(C2×D127)⋊C4—24no—
43C4×Dic127—21no—
44C4×Dic127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C2×Q8×C127D412yes—
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C4⋊C4×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—21no—
71C2×Dic127—21no—
72C2×Dic127—22no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—41no—
78C2×Dic127—41no—
79C127⋊C8—41no—
80C23×C127C2×D411yescharacteristic
81Q8×C127—22no—
82Q8×C127—22no—
83Q8×C127—41no—
84Q8×C127—41no—
85C2×C4×C127—22no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C2×C4×C127#4111yescharacteristic
97C8×C127—41no—
98C22×C127#12011yescharacteristic
99C22×C127#12111yescharacteristic
100C22×C127#12411yescharacteristic
101C22×C127#13511yescharacteristic
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—42no—
113C4×C127—81no—
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12311yescharacteristic
117C4×C127#12311yescharacteristic
118Dic127—41no—
119Dic127—41no—
120Dic127—42no—
121C2×C127#39411yescharacteristic
122C2×C127#44911yescharacteristic
123C2×C127#45211yescharacteristic
124C2×C127#46311yescharacteristic
125C2×C127#46411yescharacteristic
126C2×C127#48811yescharacteristic
127C2×C127#48911yescharacteristic
128D127—82no—
129#2042—1271nomaximal, Sylow
130C127#204211yescharacteristic, complemented, Sylow
131C2×#124—1271no—
132C2×#103—1271no—
133C2×#103D12711yescharacteristic, complemented
134C2×#102—1271no—
135C2×#102—1271no—
136#389—1271no—
137#336—1271no—
138C22×Q8C2×D12711yescharacteristic
139C22×D4—1271no—
140C22×C8—1271no—
141C22×C8C2×D12711yescharacteristic
142#124—2542no—
143#124—2542no—
144C2×C4⋊C4—1271no—
145C2×C4⋊C4—2541no—
146C2×C4⋊C4C2×D12711yescharacteristic
147C2×C22⋊C4—2541no—
148C2×C22⋊C4—2541no—
149C2×C42—1271no—
150#103—22no—
151#103—2542no—
152#102—2542no—
153#102—2542no—
154#95—2541no—
155C24—2541no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8C22⋊D12712yes—
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—1271no—
171C22×C4—1271no—
172C22×C4—2541no—
173C22×C4C22×D12711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—2541no—
178C2×C8—2541no—
179C2×C8—2541no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—1272no—
183C4⋊C4—5082no—
184C4⋊C4—2541no—
185C22⋊C4—2544no—
186C22⋊C4—2544no—
187C42—2541no—
188C42—2541no—
189C23—5081no—
190C23—5081no—
191C23—5081no—
192C23—5081no—
193C23—5081no—
194C23—5081no—
195C23—2542no—
196C23D4×D12711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201D4—5081no—
202D4—5081no—
203D4—2542no—
204D4—2542no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—2541no—
214C2×C4—2541no—
215C2×C4—2541no—
216C2×C4—2541no—
217C2×C4—2541no—
218C2×C4—2541no—
219C2×C4—2542no—
220C2×C4D4×D12711yescharacteristic
221C2×C4D4×D12711yescharacteristic
222C2×C4C2×C22⋊D12711yescharacteristic
223C2×C4#1452324711yescharacteristic
224C2×C4#1452324811yescharacteristic
225C2×C4#1452324811yescharacteristic
226C8—41no—
227C8—5081no—
228C22—10161no—
229C22—10161no—
230C22—10161no—
231C22—10161no—
232C22—5082no—
233C22—5082no—
234C22—5082no—
235C22#1686117711yescharacteristic
236C22#1686118211yescharacteristic
237C22#1686119511yescharacteristic
238C22#1686123011yescharacteristic
239C22#1686123211yescharacteristic
240C22#1686123611yescharacteristic
241C22#1686124111yescharacteristic
242C4—42no—
243C4—81no—
244C4—5081no—
245C4—5081no—
246C4—5082no—
247C4#1686118111yescharacteristic
248C4#1686118211yescharacteristic
249C4#1686119111yescharacteristic
250C4#1686119211yescharacteristic
251C2—10162no—
252C2#2254979111yescharacteristic
253C2#2255002011yescharacteristic
254C2#2255003711yescharacteristic
255C2#2255009411yescharacteristic
256C2#2255010111yescharacteristic
257C2#2255023211yescharacteristic
258C2#2255023711yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenterC2×#103D127
hypocenterC127#2042
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#16861133C2semidirect
5. #22549772C2semidirect
6. #22549630C2semidirect
7. C2×#103×C127C2semidirect
130. C127#2042semidirect
133. C2×#103D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611982
C2×#168611332
C2×#168611312
C2×#168611292
#225497722
#225496302
C2×#103×C1272
#2042127