Finite Group Databasebeta

← #30639767

Subgroups — #30639767

Order 16256.

362 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 73 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
2#22550452C211yesmaximal, characteristic
3C2×#16861207C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#419×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10#16861240—22no—
11#16861240—22no—
12C2×#14523238—21no—
13C2×#14523235—21no—
14C2×#14523232C2211yescharacteristic
15#16861209—21no—
16#16861207—22no—
17#16861207—22no—
18#16861197—21no—
19#16861193—22no—
20#16861189—22no—
21C2×Dic127⋊C4—22no—
22C2×C4×Dic127C2211yescharacteristic
23#16861135—22no—
24#16861127—22no—
25C22×Q8×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27#117×C127C2211yescharacteristic
28#103×C127—22no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—21no—
38C2×Dic254—21no—
39C2×Dic254D411yescharacteristic
40C2×D508D411yescharacteristic
41C2×C22⋊D127—21no—
42C22×Dic127—21no—
43C22×Dic127D412yes—
44C2×C4×D127—21no—
45#14523238—24no—
46#14523235—24no—
47#14523232—21no—
48#14523232D412yes—
49C2×C127⋊C8—21no—
50C4⋊Dic127—24no—
51Dic127⋊C4—28no—
52(C2×D127)⋊C4—24no—
53C4×Dic127—21no—
54C4×Dic127D412yes—
55C2×Q8×C127—21no—
56C2×Q8×C127—21no—
57C2×Q8×C127—21no—
58C2×Q8×C127—21no—
59C2×Q8×C127—21no—
60C2×Q8×C127D412yes—
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63M16×C127—21no—
64M16×C127—21no—
65C2×C8×C127—21no—
66C4⋊C4×C127D412yes—
67C22⋊C4×C127—21no—
68C42×C127—21no—
69C22×D127—21no—
70Dic254—21no—
71Dic254—21no—
72Dic254—41no—
73Dic254—41no—
74Dic254—41no—
75Dic254—41no—
76D508—21no—
77D508—21no—
78C22⋊D127—41no—
79C22⋊D127—41no—
80C2×Dic127—24no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—21no—
85C2×Dic127—22no—
86C4×D127—41no—
87C4×D127—41no—
88C127⋊C8—22no—
89C23×C127C2×D411yescharacteristic
90Q8×C127—22no—
91Q8×C127—22no—
92Q8×C127—41no—
93Q8×C127—41no—
94C2×C4×C127—22no—
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×C127#4111yescharacteristic
104C8×C127—41no—
105C22×C127—22no—
106C22×C127#12011yescharacteristic
107C22×C127#12011yescharacteristic
108C22×C127#12111yescharacteristic
109C2×D127—41no—
110C2×D127—42no—
111C4×C127—42no—
112C4×C127—42no—
113C4×C127#12011yescharacteristic
114C4×C127#12111yescharacteristic
115C4×C127#12111yescharacteristic
116C4×C127#12711yescharacteristic
117Dic127—24no—
118Dic127—41no—
119Dic127—42no—
120C2×C127—21no—
121C2×C127—21no—
122C2×C127#39211yescharacteristic
123C2×C127#45612yes—
124D127—81no—
125#2005—1271nomaximal, Sylow
126C127#200511yescharacteristic, complemented, Sylow
127C2×#137—1271no—
128#536—1271no—
129C2×C4⋊Q8—1271no—
130C2×#104—1271no—
131#419D12711yescharacteristic, complemented
132C2×#101—1271no—
133#339—1271no—
134C2×#41—1271no—
135C22×Q8—1271no—
136C22×Q8C2×D12711yescharacteristic
137#137—2542no—
138#137—2542no—
139C2×Q16—2541no—
140C2×SD16—2541no—
141C2×M16—1271no—
142C2×M16C2×D12711yescharacteristic
143C4⋊Q8—2541no—
144C4⋊Q8—2542no—
145C4⋊Q8—2542no—
146#124—2541no—
147#123—2542no—
148C22⋊Q8—2542no—
149#117C2×D12711yescharacteristic
150C2×C4⋊C4—2542no—
151C2×C42—1271no—
152#104—2542no—
153#103—22no—
154#101—2542no—
155#41—2541no—
156#41—2541no—
157#41—2541no—
158#41—2541no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—1271no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8—2541no—
168C2×Q8—2541no—
169C2×Q8—2541no—
170C2×Q8—2541no—
171C2×Q8D50812yes—
172C2×D4—1271no—
173C2×D4—2541no—
174C22×C4—21no—
175C22×C4—1272no—
176C22×C4—2541no—
177C22×C4—2541no—
178C22×C4C22×D12711yescharacteristic
179Q16—2544no—
180SD16—2544no—
181M16—21no—
182M16—21no—
183M16—1272no—
184M16—2541no—
185C2×C8—21no—
186C2×C8—2541no—
187C4⋊C4—2548no—
188C4⋊C4—2544no—
189C4⋊C4C22⋊D12712yes—
190C22⋊C4—21no—
191C22⋊C4—2544no—
192C42—21no—
193C42—1272no—
194C42—2541no—
195C23—2541no—
196C23D4×D12711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—5081no—
202Q8—5081no—
203Q8—5081no—
204Q8—5081no—
205Q8—2541no—
206Q8—2541no—
207D4—5081no—
208D4—5081no—
209D4—2541no—
210D4—2541no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—5081no—
216C2×C4—5081no—
217C2×C4—2542no—
218C2×C4—2541no—
219C2×C4—2542no—
220C2×C4—2542no—
221C2×C4—2544no—
222C2×C4—2542no—
223C2×C4D4×D12711yescharacteristic
224C2×C4D4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4C2×C22⋊D12711yescharacteristic
227C2×C4C2×D50811yescharacteristic
228C2×C4#1452324811yescharacteristic
229C8—41no—
230C8—2542no—
231C22—22no—
232C22—5081no—
233C22—5082no—
234C22#1686117711yescharacteristic
235C22#1686117911yescharacteristic
236C22#1686118211yescharacteristic
237C4—42no—
238C4—42no—
239C4—5081no—
240C4—5082no—
241C4—2544no—
242C4#1686117911yescharacteristic
243C4#1686118211yescharacteristic
244C4#1686118311yescharacteristic
245C4#1686120511yescharacteristic
246C2—21no—
247C2—21no—
248C2—10161no—
249C2#2254978511yescharacteristic
250C2#2255005712yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#120
prime radicalC2×#14523248C22
hypercenter#419D127
hypocenterC127#2005
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. C2×#16861207C2semidirect
5. C2×#16861127C2semidirect
6. #22549635C2semidirect
7. #419×C127C2semidirect
126. C127#2005semidirect
131. #419D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225504522
C2×#168612072
C2×#168611352
C2×#168611272
#225496352
#419×C1272
#2005127