Finite Group Databasebeta

← #30639842

Subgroups — #30639842

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×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic
3C2×#16861128C211yesmaximal, characteristic, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5#22549842C211yesmaximal, characteristic, complemented
6#22549641C211yesmaximal, characteristic, complemented
7C22×SD16×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240—22no—
11#16861235—22no—
12#16861235—22no—
13C2×#14523238—21no—
14C2×#14523236—21no—
15C2×#14523235—21no—
16C2×#14523233—21no—
17C2×#14523232C2211yescharacteristic
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20#16861128—22no—
21#16861126—22no—
22(C2×D127)⋊C8—22no—
23C22×Q8×C127C2211yescharacteristic
24C22×D4×C127C2211yescharacteristic
25C2×SD16×C127—22no—
26C2×SD16×C127—22no—
27C2×SD16×C127—22no—
28C22×C8×C127C2211yescharacteristic
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33C2×Dic254D411yescharacteristic
34C2×D508D411yescharacteristic
35C2×C22⋊D127—21no—
36C2×C4×D127—21no—
37#14523238—24no—
38#14523236—24no—
39#14523235—24no—
40#14523233—24no—
41#14523232—21no—
42#14523232—21no—
43#14523232—21no—
44#14523232—21no—
45#14523232D412yes—
46#14523232D412yes—
47C2×C127⋊C8—21no—
48C2×C127⋊C8—21no—
49C2×C127⋊C8—21no—
50C24×C127—21no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—21no—
56C2×Q8×C127D412yes—
57C2×D4×C127—21no—
58C2×D4×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127D412yes—
63C22×C4×C127—21no—
64C22×C4×C127C2311yescharacteristic
65SD16×C127—28no—
66C2×C8×C127—21no—
67C2×C8×C127—21no—
68C2×C8×C127D412yes—
69C22×D127—21no—
70Dic254—21no—
71Dic254—21no—
72D508—21no—
73D508—21no—
74C22⋊D127—41no—
75C22⋊D127—41no—
76C2×Dic127—21no—
77C4×D127—41no—
78C4×D127—41no—
79C127⋊C8—22no—
80C127⋊C8—22no—
81C127⋊C8—41no—
82C23×C127—22no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—82no—
86C23×C127#4111yescharacteristic
87Q8×C127—22no—
88Q8×C127—22no—
89Q8×C127—41no—
90Q8×C127—41no—
91D4×C127—22no—
92D4×C127—22no—
93D4×C127—41no—
94D4×C127—41no—
95C2×C4×C127—22no—
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—42no—
107C22×C127—82no—
108C22×C127—82no—
109C22×C127—82no—
110C22×C127#12111yescharacteristic
111C22×C127#12111yescharacteristic
112C22×C127#12111yescharacteristic
113C2×D127—41no—
114C2×D127—42no—
115C4×C127—42no—
116C4×C127#12011yescharacteristic
117C4×C127#12011yescharacteristic
118C4×C127#12011yescharacteristic
119C4×C127#12711yescharacteristic
120Dic127—41no—
121C2×C127—21no—
122C2×C127—21no—
123C2×C127—82no—
124C2×C127#39211yescharacteristic
125C2×C127#47112yes—
126D127—81no—
127#2019—1271nomaximal, Sylow
128C127#201911yescharacteristic, complemented, Sylow
129C2×#137—1271no—
130C2×AGL(1,ℤ/8)—1271no—
131C22×SD16D12711yescharacteristic, complemented
132C2×#101—1271no—
133C2×#100—1271no—
134#408—1271no—
135#341—1271no—
136C2×#41—1271no—
137C22×Q8C2×D12711yescharacteristic
138C22×D4C2×D12711yescharacteristic
139#137—2542no—
140#137—2542no—
141AGL(1,ℤ/8)—2542no—
142AGL(1,ℤ/8)—2542no—
143C2×Q16—2541no—
144C2×SD16—22no—
145C2×SD16—22no—
146C2×SD16—22no—
147C2×SD16—2541no—
148C2×SD16—2541no—
149C2×D8—2541no—
150C2×M16—1271no—
151C2×M16—1271no—
152C2×M16—1271no—
153C22×C8C2×D12711yescharacteristic
154#101—2542no—
155#100—2542no—
156C22⋊C8—2542no—
157C24—21no—
158#41—2541no—
159#41—2541no—
160#41—2541no—
161#41—2541no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—1271no—
168C2×Q8C22⋊D12712yes—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—1271no—
175C2×D4—2541no—
176C2×D4C22⋊D12712yes—
177C22×C4—21no—
178C22×C4—2541no—
179C22×C4C22×D12711yescharacteristic
180Q16—2544no—
181SD16—28no—
182SD16—2544no—
183SD16—2544no—
184D8—2544no—
185M16—1272no—
186M16—1272no—
187M16—2541no—
188M16—2541no—
189M16—2541no—
190M16—2541no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—2541no—
194C2×C8—2541no—
195C2×C8—2541no—
196C2×C8C22⋊D12712yes—
197C23—22no—
198C23—41no—
199C23—41no—
200C23—82no—
201C23—2541no—
202C23#1452324711yescharacteristic
203Q8—22no—
204Q8—22no—
205Q8—41no—
206Q8—41no—
207Q8—2541no—
208Q8—2541no—
209D4—22no—
210D4—22no—
211D4—41no—
212D4—41no—
213D4—5081no—
214D4—5081no—
215D4—2541no—
216D4—2541no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—5081no—
221C2×C4—5081no—
222C2×C4—2541no—
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—42no—
235C22—82no—
236C22—82no—
237C22—82no—
238C22—5081no—
239C22—5082no—
240C22#1686118011yescharacteristic
241C22#1686118111yescharacteristic
242C22#1686118111yescharacteristic
243C4—42no—
244C4—5081no—
245C4#1686117711yescharacteristic
246C4#1686117711yescharacteristic
247C4#1686117811yescharacteristic
248C4#1686120511yescharacteristic
249C2—21no—
250C2—21no—
251C2—82no—
252C2—10161no—
253C2#2254978411yescharacteristic
254C2#2255013612yes—
subgroupG/H
centerC22#16861180
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC22×SD16×C127C2
socleC22×C127#121
prime radicalC2×#16861235C2
hypercenterC22×SD16D127
hypocenterC127#2019
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. C2×#16861128C2semidirect
4. C2×#16861126C2semidirect
5. #22549842C2semidirect
6. #22549641C2semidirect
7. C22×SD16×C127C2semidirect
128. C127#2019semidirect
131. C22×SD16D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612352
C2×#168611282
C2×#168611262
#225498422
#225496412
C22×SD16×C1272
#2019127