Finite Group Databasebeta

← #30639090

Subgroups — #30639090

Order 16256.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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#22550643C211yesmaximal, characteristic, complemented
3#22549894C211yesmaximal, characteristic, complemented
4#22549842C211yesmaximal, characteristic, complemented
5#22549646C211yesmaximal, characteristic, complemented
6#22549641C211yesmaximal, characteristic, complemented
7#416×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239C412yescomplemented
11#16861235—21no—
12#16861234C412yescomplemented
13#16861229—22no—
14#16861216—22no—
15#16861216—22no—
16C2×#14523232—22no—
17C2×#14523232C2211yescharacteristic
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20#16861137—22no—
21(C2×D127)⋊C8—22no—
22C2×#41×C127C2211yescharacteristic
23C22×C8×C127C2211yescharacteristic
24#117×C127C2211yescharacteristic
25#103×C127—21no—
26#102×C127—21no—
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254C2×C411yescharacteristic
32C2×D508C2×C411yescharacteristic
33C2×C22⋊D127—21no—
34C2×C4×D127—21no—
35#14523238—22no—
36#14523236—22no—
37#14523235—22no—
38#14523233—22no—
39#14523232—24no—
40#14523232—21no—
41#14523232—21no—
42#14523232—21no—
43#14523232—21no—
44#14523232C2×C412yes—
45#14523232D412yes—
46C2×C127⋊C8—24no—
47C2×C127⋊C8—21no—
48C2×C127⋊C8—21no—
49C2×C127⋊C8D412yes—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127C2×C412yes—
53#41×C127D412yes—
54C2×Q8×C127D411yescharacteristic
55C2×D4×C127—21no—
56C2×D4×C127D411yescharacteristic
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C2×C8×C127—21no—
62C4⋊C4×C127—21no—
63C22⋊C4×C127—21no—
64C42×C127—21no—
65C22×D127—21no—
66Dic254—21no—
67Dic254—21no—
68D508—21no—
69D508—21no—
70C22⋊D127—41no—
71C22⋊D127—41no—
72C2×Dic127—21no—
73C4×D127—41no—
74C4×D127—41no—
75C127⋊C8—22no—
76C127⋊C8—22no—
77C127⋊C8—22no—
78C23×C127—21no—
79C23×C127#4111yescharacteristic
80Q8×C127—21no—
81Q8×C127C22⋊C412yes—
82D4×C127—22no—
83D4×C127—22no—
84D4×C127—21no—
85D4×C127C22⋊C412yes—
86C2×C4×C127—21no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—41no—
90C2×C4×C127C22×C411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127#4111yescharacteristic
96C8×C127—41no—
97C22×C127—22no—
98C22×C127—41no—
99C22×C127—81no—
100C22×C127C4×D411yescharacteristic
101C22×C127#12111yescharacteristic
102C22×C127#12311yescharacteristic
103C2×D127—42no—
104C2×D127—41no—
105C4×C127—22no—
106C4×C127—81no—
107C4×C127C2×C22⋊C411yescharacteristic
108C4×C127C4×D411yescharacteristic
109C4×C127#12011yescharacteristic
110C4×C127#12111yescharacteristic
111Dic127—41no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—42no—
116C2×C127#38611yescharacteristic
117D127—81no—
118#1881—1271nomaximal, Sylow
119C127#188111yescharacteristic, complemented, Sylow
120#575—1271no—
121#568—1271no—
122#421—1271no—
123#416D12711yescharacteristic, complemented
124#408—1271no—
125#344—1271no—
126#341—1271no—
127C2×#41—1271no—
128C2×#41C2×D12711yescharacteristic
129#137—1272no—
130#137—2541no—
131AGL(1,ℤ/8)—1272no—
132AGL(1,ℤ/8)—2541no—
133#135—2542no—
134#131—2542no—
135#131—2542no—
136C2×M16—1271no—
137C2×M16—1271no—
138C2×M16—1271no—
139C2×M16—2542no—
140C22×C8C2×D12711yescharacteristic
141#117C2×D12711yescharacteristic
142#104—2542no—
143#103—21no—
144#102—21no—
145C22⋊C8—2542no—
146#41—21no—
147#41—21no—
148#41—2541no—
149#41—2541no—
150#41—2541no—
151#41—2541no—
152#41C4×D12712yes—
153#41D50812yes—
154C2×Q8—1271no—
155C2×Q8C22⋊D12711yescharacteristic
156C2×D4—21no—
157C2×D4—1271no—
158C2×D4—2541no—
159C2×D4C22⋊D12711yescharacteristic
160C22×C4—21no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163Q16—2542no—
164SD16—2542no—
165SD16—2542no—
166D8—2542no—
167M16—1272no—
168M16—1272no—
169M16—2544no—
170M16—2541no—
171M16—2541no—
172M16—2541no—
173M16—2541no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—1272no—
178C2×C8—2544no—
179C2×C8—2541no—
180C2×C8—2541no—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C42—21no—
184C23—21no—
185C23—2541no—
186C23#1452324711yescharacteristic
187Q8—21no—
188Q8—2541no—
189Q8—2541no—
190Q8(C2×D127)⋊C412yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—5081no—
195D4—5081no—
196D4—2541no—
197D4—2541no—
198D4(C2×D127)⋊C412yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—2541no—
206C2×C4C2×C4×D12711yescharacteristic
207C2×C4D4×D12711yescharacteristic
208C2×C4D4×D12711yescharacteristic
209C2×C4C2×C22⋊D12711yescharacteristic
210C2×C4C2×D50811yescharacteristic
211C2×C4#1452324711yescharacteristic
212C8—41no—
213C8—2542no—
214C8—2542no—
215C8—2542no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—5082no—
220C22—5081no—
221C22(C22⋊D127)⋊C411yescharacteristic
222C22#1686118111yescharacteristic
223C22#1686119311yescharacteristic
224C4—22no—
225C4—81no—
226C4—5081no—
227C4C2×(C2×D127)⋊C411yescharacteristic
228C4(C22⋊D127)⋊C411yescharacteristic
229C4#1686117911yescharacteristic
230C4#1686118111yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—10161no—
236C2#2254975311yescharacteristic
subgroupG/H
centerC4#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#416×C127C2
socleC2×C127#386
prime radical#22550674C2
cosocleC22×C4×C127C23
hypercenter#416D127
hypocenterC127#1881
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #22550643C2semidirect
3. #22549894C2semidirect
4. #22549842C2semidirect
5. #22549646C2semidirect
6. #22549641C2semidirect
7. #416×C127C2semidirect
10. #16861239C4semidirect
12. #16861234C4semidirect
119. C127#1881semidirect
123. #416D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225506432
#225498942
#225498422
#225496462
#225496412
#416×C1272
#1881127