Finite Group Databasebeta

← #30639844

Subgroups — #30639844

Order 16256.

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

normal subgroups only

This group has 362 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
1C2×#16861239C211yesmaximal, characteristic, complemented
2C22×#14523236C211yesmaximal, characteristic, complemented
3C2×#16861127C211yesmaximal, characteristic, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5#22549843C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9#16861239—22no—
10#16861239—22no—
11C2×#14523238—21no—
12C2×#14523236—22no—
13C2×#14523236—22no—
14C2×#14523236—22no—
15C2×#14523236—21no—
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18C22×C127⋊C8C2211yescharacteristic
19#16861127—22no—
20#16861126—22no—
21C22⋊(C127⋊C8)—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×Dic254D412yes—
35C22×Dic127—21no—
36#14523238—24no—
37#14523236—28no—
38#14523236—24no—
39#14523232—21no—
40#14523232—21no—
41#14523232—21no—
42#14523232D412yes—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8D412yes—
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—22no—
70Dic254—22no—
71Dic254—41no—
72Dic254—41no—
73C2×Dic127—22no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C127⋊C8—22no—
77C127⋊C8—22no—
78C127⋊C8—41no—
79C23×C127—21no—
80C23×C127—22no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127—82no—
84C23×C127#4111yescharacteristic
85Q8×C127—21no—
86Q8×C127—21no—
87D4×C127—21no—
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—21no—
91D4×C127—41no—
92D4×C127—41no—
93D4×C127—41no—
94D4×C127—41no—
95C2×C4×C127—21no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D411yescharacteristic
104C8×C127—22no—
105C22×C127—22no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—82no—
110C22×C127—82no—
111C22×C127—82no—
112C22×C127#12111yescharacteristic
113C22×C127#12111yescharacteristic
114C22×C127#12111yescharacteristic
115C4×C127—41no—
116C4×C127#12011yescharacteristic
117C4×C127#12011yescharacteristic
118C4×C127#12011yescharacteristic
119C4×C127#12711yescharacteristic
120Dic127—42no—
121C2×C127—21no—
122C2×C127—21no—
123C2×C127—81no—
124C2×C127—82no—
125C2×C127#39211yescharacteristic
126C2×C127#47112yes—
127#2019—1271nomaximal, Sylow
128C127#201911yescharacteristic, complemented, Sylow
129C2×#137—1271no—
130C2×AGL(1,ℤ/8)D12711yescharacteristic
131C22×SD16—1271no—
132C2×#101—1271no—
133C2×#100—1271no—
134#408—1271no—
135#341—1271no—
136C2×#41C2×D12711yescharacteristic
137C22×Q8—1271no—
138C22×D4C2×D12711yescharacteristic
139#137—2542no—
140#137—2542no—
141AGL(1,ℤ/8)—22no—
142AGL(1,ℤ/8)—22no—
143C2×Q16—2541no—
144C2×SD16—21no—
145C2×SD16—2542no—
146C2×SD16—2542no—
147C2×SD16—2542no—
148C2×SD16—2541no—
149C2×D8—21no—
150C2×M16—1271no—
151C2×M16—1271no—
152C2×M16C2×D12711yescharacteristic
153C22×C8—1271no—
154#101—2542no—
155#100—2542no—
156C22⋊C8—2542no—
157C24—21no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—21no—
162C2×Q8—1272no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8—2541no—
168C2×Q8C22⋊D12711yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4C22⋊D12711yescharacteristic
176C2×D4C22⋊D12712yes—
177C22×C4—21no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180Q16—2544no—
181SD16—24no—
182SD16—2548no—
183SD16—2544no—
184D8—24no—
185M16—21no—
186M16—1272no—
187M16—2541no—
188M16—2541no—
189M16—2541no—
190M16C22⋊D12712yes—
191C2×C8—21no—
192C2×C8—1272no—
193C2×C8—2541no—
194C2×C8—2541no—
195C2×C8—2541no—
196C2×C8—2541no—
197C23—21no—
198C23—22no—
199C23—41no—
200C23—41no—
201C23—82no—
202C23#1452324711yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—5081no—
206Q8—5081no—
207Q8—2542no—
208Q8—2542no—
209D4—22no—
210D4—21no—
211D4—21no—
212D4—22no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—5081no—
221C2×C4—5081no—
222C2×C4—2542no—
223C2×C4D4×D12711yescharacteristic
224C2×C4D4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4C2×C22⋊D12711yescharacteristic
228C2×C4C2×C22⋊D12711yescharacteristic
229C8—22no—
230C8—5081no—
231C8—2542no—
232C8—2542no—
233C22—22no—
234C22—41no—
235C22—42no—
236C22—42no—
237C22—82no—
238C22—82no—
239C22—82no—
240C22#1686118011yescharacteristic
241C22#1686118111yescharacteristic
242C22#1686118111yescharacteristic
243C4—41no—
244C4—5082no—
245C4#1686117711yescharacteristic
246C4#1686117711yescharacteristic
247C4#1686117811yescharacteristic
248C4#1686120511yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—82no—
253C2#2254978411yescharacteristic
254C2#2255013712yes—
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×AGL(1,ℤ/8)×C127C2
socleC22×C127#121
hypercenterC2×AGL(1,ℤ/8)D127
hypocenterC127#2019
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. C2×#16861239C2semidirect
2. C22×#14523236C2semidirect
3. C2×#16861127C2semidirect
4. C2×#16861126C2semidirect
5. #22549843C2semidirect
6. #22549640C2semidirect
128. C127#2019semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232362
C2×#168611272
C2×#168611262
#225498432
#225496402
C2×AGL(1,ℤ/8)×C1272
#2019127