Finite Group Databasebeta

← #30640016

Subgroups — #30640016

Order 16256.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550674C211yesmaximal, characteristic
2#22549900C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic, complemented
4#22549860C211yesmaximal, characteristic, complemented
5#22549858C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7C2×#104×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239—21no—
11#16861235—21no—
12#16861234—21no—
13#16861229—22no—
14C2×#14523232C2211yescharacteristic
15C2×#14523232C2211yescharacteristic
16C2×#14523232C2211yescharacteristic
17C4⋊(C127⋊C8)—22no—
18#16861137—21no—
19#16861137—21no—
20#16861128—21no—
21#16861127—21no—
22#16861126—21no—
23#16861125—21no—
24C2×#41×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27#104×C127—21no—
28#104×C127—21no—
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—22no—
38#14523236—22no—
39#14523235—22no—
40#14523233—22no—
41#14523232—21no—
42#14523232—21no—
43#14523232—21no—
44#14523232—21no—
45#14523232—21no—
46#14523232—21no—
47C2×C127⋊C8—21no—
48C2×C127⋊C8—21no—
49C2×C127⋊C8D412yes—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—21no—
53#41×C127—21no—
54C2×Q8×C127D411yescharacteristic
55C2×D4×C127—21no—
56C2×D4×C127D411yescharacteristic
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60M16×C127—21no—
61M16×C127—21no—
62C2×C8×C127—21no—
63C42×C127—21no—
64C42×C127D411yescharacteristic
65C42×C127D411yescharacteristic
66C22×D127—21no—
67Dic254—21no—
68Dic254—21no—
69D508—21no—
70D508—21no—
71C22⋊D127—41no—
72C22⋊D127—41no—
73C2×Dic127—21no—
74C4×D127—41no—
75C4×D127—41no—
76C127⋊C8—22no—
77C127⋊C8—41no—
78C127⋊C8—41no—
79C23×C127—21no—
80C23×C127#4111yescharacteristic
81Q8×C127—21no—
82Q8×C127—21no—
83D4×C127—21no—
84D4×C127—21no—
85D4×C127—41no—
86D4×C127—41no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—21no—
90C2×C4×C127—42no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C2×C4×C127#4111yescharacteristic
99C8×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—81no—
103C22×C127#12111yescharacteristic
104C22×C127#12311yescharacteristic
105C22×C127#12311yescharacteristic
106C2×D127—41no—
107C2×D127—81no—
108C4×C127—22no—
109C4×C127—41no—
110C4×C127—41no—
111C4×C127#12011yescharacteristic
112C4×C127#12111yescharacteristic
113C4×C127#12111yescharacteristic
114C4×C127#12411yescharacteristic
115Dic127—41no—
116C2×C127—21no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—81no—
120C2×C127#39411yescharacteristic
121D127—81no—
122#2050—1271nomaximal, Sylow
123C127#205011yescharacteristic, complemented, Sylow
124#575—1271no—
125#423—1271no—
126C2×#104—1271no—
127C2×#104D12711yescharacteristic, complemented
128#413—1271no—
129#413—1271no—
130#338—1271no—
131C2×#41—1271no—
132C2×#41C2×D12711yescharacteristic
133#137—2541no—
134#137—2541no—
135AGL(1,ℤ/8)—2541no—
136AGL(1,ℤ/8)—2541no—
137#135—2542no—
138C2×M16—1271no—
139C2×M16—1271no—
140C2×M16—1271no—
141C2×M16C2×D12711yescharacteristic
142C2×C42C2×D12711yescharacteristic
143C4⋊C8—2542no—
144#104—21no—
145#104—21no—
146#104—2541no—
147#104—2541no—
148#101—2541no—
149#101—2541no—
150#100—2541no—
151#100—2541no—
152#41—21no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—2541no—
157#41—2541no—
158#41—2541no—
159#41—2541no—
160C2×Q8—1271no—
161C2×Q8D50811yescharacteristic
162C2×D4—21no—
163C2×D4—1271no—
164C2×D4—2541no—
165C2×D4D50811yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—2541no—
169C22×C4C22×D12711yescharacteristic
170Q16—2542no—
171SD16—2542no—
172SD16—2542no—
173D8—2542no—
174M16—21no—
175M16—21no—
176M16—2541no—
177M16—2541no—
178M16—2541no—
179M16—2541no—
180M16—2541no—
181M16—2541no—
182C2×C8—21no—
183C2×C8—1272no—
184C2×C8—2541no—
185C2×C8—2541no—
186C42—21no—
187C42C22⋊D12711yescharacteristic
188C42C22⋊D12711yescharacteristic
189C23—21no—
190C23—2541no—
191C23#1452324711yescharacteristic
192Q8—21no—
193Q8—21no—
194Q8—2541no—
195Q8—2541no—
196D4—21no—
197D4—21no—
198D4—41no—
199D4—41no—
200D4—5081no—
201D4—5081no—
202D4—2541no—
203D4—2541no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—5081no—
211C2×C4—5081no—
212C2×C4—2541no—
213C2×C4D4×D12711yescharacteristic
214C2×C4D4×D12711yescharacteristic
215C2×C4C2×C22⋊D12711yescharacteristic
216C2×C4C2×D50811yescharacteristic
217C2×C4#1452324711yescharacteristic
218C2×C4#1452324811yescharacteristic
219C8—41no—
220C8—5081no—
221C8—5081no—
222C8—2542no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—10161no—
227C22—5081no—
228C22#1686118111yescharacteristic
229C22#1686119111yescharacteristic
230C22#1686119311yescharacteristic
231C4—22no—
232C4—41no—
233C4—41no—
234C4—5081no—
235C4#1686117911yescharacteristic
236C4#1686118011yescharacteristic
237C4#1686118311yescharacteristic
238C4#1686119511yescharacteristic
239C2—21no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—10161no—
244C2#2254979211yescharacteristic
subgroupG/H
centerC4#16861195
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#104×C127C2
socleC2×C127#394
prime radical#22550674C2
hypercenterC2×#104D127
hypocenterC127#2050
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #22549900C2semidirect
3. C2×#16861137C2semidirect
4. #22549860C2semidirect
5. #22549858C2semidirect
6. C127⋊#338C2semidirect
7. C2×#104×C127C2semidirect
123. C127#2050semidirect
127. C2×#104D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225499002
C2×#168611372
#225498602
#225498582
C127⋊#3382
C2×#104×C1272
#2050127