Finite Group Databasebeta

← #30639754

Subgroups — #30639754

Order 16256.

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

normal subgroups only

This group has 382 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×#16861237C211yesmaximal, characteristic
2#22550450C211yesmaximal, characteristic, complemented
3C2×#16861198C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549882C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7C2×#101×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×D508C2211yescharacteristic
10#16861237—22no—
11#16861237—22no—
12C2×#14523237—21no—
13C2×D1016—21no—
14#16861207—21no—
15#16861198—22no—
16#16861198—21no—
17#16861198—22no—
18#16861192—22no—
19#16861185—22no—
20#16861168C2211yescharacteristic
21C2×(C2×D127)⋊C4—22no—
22C2×C4×Dic127C2211yescharacteristic
23#16861135—22no—
24#16861131—22no—
25C22×Q8×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27C2×M16×C127C2211yescharacteristic
28#101×C127—22no—
29C23×D127—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34C2×Dic254D411yescharacteristic
35C2×D508—21no—
36C2×D508—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508D411yescharacteristic
41C2×D508D412yes—
42C2×C22⋊D127—21no—
43C22×Dic127—21no—
44C2×C4×D127—21no—
45#14523237—24no—
46D1016—24no—
47C4⋊Dic127D412yes—
48Dic127⋊C4—24no—
49C22⋊Dic127—21no—
50(C2×D127)⋊C4—24no—
51(C2×D127)⋊C4—28no—
52C4×Dic127—21no—
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×C127—21no—
61C22×C4×C127D412yes—
62C22×C4×C127C2311yescharacteristic
63M16×C127—21no—
64M16×C127—21no—
65M16×C127—21no—
66M16×C127D412yes—
67C2×C8×C127—21no—
68C2×C8×C127—21no—
69C22×D127—21no—
70C22×D127—22no—
71C22×D127—44no—
72C22×D127—41no—
73C22×D127—41no—
74Dic254—21no—
75Dic254—21no—
76D508—21no—
77D508—22no—
78D508—22no—
79D508—21no—
80D508—41no—
81D508—41no—
82C22⋊D127—41no—
83C22⋊D127—41no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—21no—
87C2×Dic127—42no—
88C4×D127—41no—
89C4×D127—41no—
90C23×C127C2×D411yescharacteristic
91Q8×C127—41no—
92Q8×C127—41no—
93Q8×C127—41no—
94Q8×C127—41no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
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—22no—
105C8×C127—41no—
106C22×C127—22no—
107C22×C127#12011yescharacteristic
108C22×C127#12011yescharacteristic
109C22×C127#12111yescharacteristic
110C2×D127—44no—
111C2×D127—41no—
112C2×D127—42no—
113C2×D127—42no—
114C2×D127—82no—
115C2×D127—82no—
116C4×C127—42no—
117C4×C127#12011yescharacteristic
118C4×C127#12111yescharacteristic
119C4×C127#12111yescharacteristic
120C4×C127#12711yescharacteristic
121Dic127—42no—
122Dic127—41no—
123Dic127—42no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127#39211yescharacteristic
127C2×C127#45412yes—
128D127—81no—
129D127—82no—
130#2003—1271nomaximal, Sylow
131C127#200311yescharacteristic, complemented, Sylow
132C2×AGL(1,ℤ/8)—1271no—
133#536—1271no—
134C2×#124—1271no—
135C2×#104—1271no—
136#418—1271no—
137C2×#101D12711yescharacteristic, complemented
138#339—1271no—
139C2×#41—1271no—
140C22×Q8C2×D12711yescharacteristic
141C22×D4—1271no—
142AGL(1,ℤ/8)—2542no—
143AGL(1,ℤ/8)—2542no—
144C2×SD16—2541no—
145C2×D8—2541no—
146C2×M16C2×D12711yescharacteristic
147C2×M16C2×D12711yescharacteristic
148C4⋊Q8—2541no—
149#124—2542no—
150#124—2542no—
151#124—2541no—
152#123—2542no—
153C22⋊Q8—2542no—
154#117—1271no—
155C2×C22⋊C4—2542no—
156C2×C42—1271no—
157#104—2542no—
158#102—2542no—
159#101—22no—
160C24—2541no—
161#41—2541no—
162#41—2541no—
163#41—2541no—
164#41—2541no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—1271no—
172C2×D4—1271no—
173C2×D4—1272no—
174C2×D4—2541no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4—2541no—
179C2×D4—2541no—
180C22×C4—2541no—
181C22×C4—2541no—
182C22×C4D50812yes—
183C22×C4C22×D12711yescharacteristic
184SD16—2544no—
185D8—2544no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16C22⋊D12712yes—
190C2×C8—21no—
191C2×C8—21no—
192C4⋊C4—1272no—
193C4⋊C4—2544no—
194C22⋊C4—2548no—
195C22⋊C4—2544no—
196C22⋊C4—2541no—
197C42—1272no—
198C42—2541no—
199C42—2541no—
200C23—5084no—
201C23—5081no—
202C23—5081no—
203C23—2542no—
204C23—2541no—
205C23C2×D50811yescharacteristic
206Q8—41no—
207Q8—41no—
208Q8—41no—
209Q8—41no—
210Q8—2541no—
211Q8—2541no—
212D4—5081no—
213D4—5081no—
214D4—5081no—
215D4—5081no—
216D4—2541no—
217D4—2542no—
218D4—2542no—
219D4—2541no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—5082no—
224C2×C4—5081no—
225C2×C4—5081no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4—2541no—
229C2×C4D4×D12711yescharacteristic
230C2×C4D4×D12711yescharacteristic
231C2×C4D4×D12711yescharacteristic
232C2×C4D4×D12711yescharacteristic
233C2×C4C2×C22⋊D12711yescharacteristic
234C2×C4#1452324811yescharacteristic
235C8—22no—
236C8—41no—
237C22—22no—
238C22—10162no—
239C22—10162no—
240C22—5084no—
241C22—5081no—
242C22—5082no—
243C22—5082no—
244C22#1686117911yescharacteristic
245C22#1686117911yescharacteristic
246C22#1686118311yescharacteristic
247C4—42no—
248C4—5082no—
249C4—5081no—
250C4—5082no—
251C4#1686117711yescharacteristic
252C4#1686118211yescharacteristic
253C4#1686118211yescharacteristic
254C4#1686120511yescharacteristic
255C2—21no—
256C2—21no—
257C2—10162no—
258C2—10161no—
259C2#2254978511yescharacteristic
260C2#2255005112yes—
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC22×C127#120
prime radicalC2×#16861237C2
hypercenterC2×#101D127
hypocenterC127#2003
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #22550450C2semidirect
3. C2×#16861198C2semidirect
4. C2×#16861135C2semidirect
5. #22549882C2semidirect
6. #22549637C2semidirect
7. C2×#101×C127C2semidirect
131. C127#2003semidirect
137. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
#225504502
C2×#168611982
C2×#168611352
#225498822
#225496372
C2×#101×C1272
#2003127