Finite Group Databasebeta

← #30639749

Subgroups — #30639749

Order 16256.

382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 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#22550447C211yesmaximal, characteristic, complemented
2C2×#16861196C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5C2×#16861127C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861207—21no—
11#16861196—22no—
12#16861196—22no—
13#16861196—21no—
14#16861190—22no—
15#16861188—22no—
16#16861168C2211yescharacteristic
17C2×C22⋊Dic127—22no—
18C2×C4×Dic127C2211yescharacteristic
19#16861136—22no—
20#16861130—22no—
21#16861127—22no—
22C2×#41×C127C2211yescharacteristic
23C22×D4×C127C2211yescharacteristic
24AGL(1,ℤ/8)×C127—22no—
25AGL(1,ℤ/8)×C127—22no—
26C2×SD16×C127—21no—
27C2×D8×C127—21no—
28C2×M16×C127C2211yescharacteristic
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C22×Dic127—21no—
36C22×Dic127D412yes—
37#14523232—21no—
38#14523232—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127D412yes—
41Dic127⋊C4—24no—
42C22⋊Dic127—24no—
43C22⋊Dic127—28no—
44C22⋊Dic127—21no—
45C4×Dic127—21no—
46C4×Dic127—21no—
47C4×Dic127D412yes—
48C24×C127—21no—
49#41×C127—21no—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—21no—
53C2×Q8×C127D411yescharacteristic
54C2×D4×C127—21no—
55C2×D4×C127—21no—
56C2×D4×C127—21no—
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127D411yescharacteristic
61C2×D4×C127D412yes—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64SD16×C127—24no—
65D8×C127—24no—
66M16×C127—21no—
67M16×C127D412yes—
68C2×C8×C127—21no—
69Dic254—41no—
70Dic254—41no—
71Dic254—41no—
72Dic254—41no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—42no—
79C127⋊C8—41no—
80C23×C127—21no—
81C23×C127—22no—
82C23×C127—44no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127C2×D411yescharacteristic
86Q8×C127—21no—
87Q8×C127—21no—
88D4×C127—21no—
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—21no—
92D4×C127—41no—
93D4×C127—41no—
94D4×C127—41no—
95D4×C127—41no—
96C2×C4×C127—21no—
97C2×C4×C127—41no—
98C2×C4×C127—41no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127#4111yescharacteristic
105C8×C127—22no—
106C22×C127—22no—
107C22×C127—44no—
108C22×C127—41no—
109C22×C127—42no—
110C22×C127—42no—
111C22×C127—82no—
112C22×C127—82no—
113C22×C127#12011yescharacteristic
114C22×C127#12011yescharacteristic
115C22×C127#12111yescharacteristic
116C4×C127—41no—
117C4×C127#12011yescharacteristic
118C4×C127#12111yescharacteristic
119C4×C127#12111yescharacteristic
120C4×C127#12711yescharacteristic
121Dic127—42no—
122Dic127—42no—
123Dic127—42no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—81no—
127C2×C127—82no—
128C2×C127#39211yescharacteristic
129C2×C127#45412yes—
130#2003—1271nomaximal, Sylow
131C127#200311yescharacteristic, complemented, Sylow
132C2×AGL(1,ℤ/8)D12711yescharacteristic
133#536—1271no—
134C2×#124—1271no—
135C2×#104—1271no—
136#418—1271no—
137C2×#101—1271no—
138#339—1271no—
139C2×#41C2×D12711yescharacteristic
140C22×Q8—1271no—
141C22×D4C2×D12711yescharacteristic
142AGL(1,ℤ/8)—22no—
143AGL(1,ℤ/8)—22no—
144C2×SD16—21no—
145C2×D8—21no—
146C2×M16—1271no—
147C2×M16C2×D12711yescharacteristic
148C4⋊Q8—2541no—
149#124—2542no—
150#124—2542no—
151#124—2541no—
152#123—2542no—
153C22⋊Q8—2542no—
154#117—1271no—
155C2×C22⋊C4—2542no—
156C2×C42—1271no—
157#104—2542no—
158#102—2542no—
159#101—2542no—
160C24—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8—2541no—
168C2×Q8—2541no—
169C2×Q8—2541no—
170C2×Q8—2541no—
171C2×Q8C22⋊D12711yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4C22⋊D12711yescharacteristic
179C2×D4C22⋊D12712yes—
180C22×C4—21no—
181C22×C4—1272no—
182C22×C4—2541no—
183C22×C4C22×D12711yescharacteristic
184SD16—24no—
185D8—24no—
186M16—21no—
187M16—2541no—
188M16—2541no—
189M16C22⋊D12712yes—
190C2×C8—21no—
191C2×C8—2541no—
192C4⋊C4—1272no—
193C4⋊C4—2544no—
194C22⋊C4—2548no—
195C22⋊C4—2544no—
196C22⋊C4—2541no—
197C42—1272no—
198C42—2541no—
199C42—2541no—
200C23—22no—
201C23—21no—
202C23—44no—
203C23—41no—
204C23—41no—
205C23D4×D12711yescharacteristic
206Q8—21no—
207Q8—21no—
208Q8—5081no—
209Q8—5081no—
210Q8—5081no—
211Q8—5081no—
212D4—21no—
213D4—22no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—5082no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4—2542no—
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4C2×C22⋊D12711yescharacteristic
232C2×C4C2×C22⋊D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4#1452324711yescharacteristic
235C8—22no—
236C8—5081no—
237C22—22no—
238C22—44no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22—82no—
244C22#1686117711yescharacteristic
245C22#1686117711yescharacteristic
246C22#1686118011yescharacteristic
247C4—41no—
248C4—5082no—
249C4—5082no—
250C4—5082no—
251C4#1686117811yescharacteristic
252C4#1686118111yescharacteristic
253C4#1686118111yescharacteristic
254C4#1686120511yescharacteristic
255C2—21no—
256C2—21no—
257C2—82no—
258C2—81no—
259C2#2254978411yescharacteristic
260C2#2255004712yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#120
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#2003
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. #22550447C2semidirect
2. C2×#16861196C2semidirect
3. C2×#16861136C2semidirect
4. #22549881C2semidirect
5. C2×#16861127C2semidirect
6. #22549637C2semidirect
131. C127#2003semidirect

Maximal subgroups

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

subgroupindex
#225504472
C2×#168611962
C2×#168611362
#225498812
C2×#168611272
#225496372
C2×AGL(1,ℤ/8)×C1272
#2003127