Finite Group Databasebeta

← #30639030

Subgroups — #30639030

Order 16256.

376 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#14523232C211yesmaximal, characteristic, complemented
2#22550387C211yesmaximal, characteristic
3C2×#16861137C212yesmaximal, complemented
4#22549636C212yesmaximal, complemented
5C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2211yescharacteristic
7C2×#14523232—24no—
8C2×#14523232—21no—
9C2×#14523232C2212yes—
10C22×C127⋊C8—21no—
11#16861192—21no—
12#16861188C411yescharacteristic, complemented
13#16861188C411yescharacteristic, complemented
14#16861183C411yescharacteristic, complemented
15#16861183C411yescharacteristic, complemented
16#16861178—21no—
17C4×C22⋊D127—21no—
18C4×C22⋊D127—21no—
19#16861168C2211yescharacteristic
20#16861137—24no—
21C23×C4×C127C2211yescharacteristic
22C2×C22⋊C4×C127—21no—
23C2×C42×C127C2212yes—
24#95×C127—21no—
25#14523248—22no—
26#14523248—22no—
27C2×Dic254C2×C411yescharacteristic
28C2×D508C2×C411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523232—24no—
34#14523232—22no—
35#14523232D414yescomplemented
36C2×C127⋊C8—24no—
37C2×C127⋊C8—22no—
38C4⋊Dic127C2×C411yescharacteristic
39C4⋊Dic127C2×C411yescharacteristic
40Dic127⋊C4—21no—
41Dic127⋊C4—21no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C22⋊Dic127—21no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47C4×Dic127—21no—
48C24×C127D411yescharacteristic
49C22×C4×C127—22no—
50C22×C4×C127—22no—
51C22×C4×C127—22no—
52C22×C4×C127C2×C411yescharacteristic
53C22×C4×C127C2×C411yescharacteristic
54C22×C4×C127D411yescharacteristic
55C22×C4×C127D411yescharacteristic
56C22×C4×C127D411yescharacteristic
57C22×C4×C127C2311yescharacteristic
58C22⋊C4×C127—24no—
59C42×C127—22no—
60C42×C127—22no—
61C22×D127—21no—
62Dic254—22no—
63D508—22no—
64C22⋊D127—42no—
65C22⋊D127—42no—
66C22⋊D127—42no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C4×D127—41no—
71C4×D127—41no—
72C127⋊C8—24no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—41no—
76C23×C127C22⋊C411yescharacteristic
77C23×C127C22⋊C411yescharacteristic
78C23×C127C2×D411yescharacteristic
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—24no—
87C2×C4×C127—42no—
88C2×C4×C127—42no—
89C2×C4×C127C22⋊C411yescharacteristic
90C2×C4×C127C22⋊C411yescharacteristic
91C2×C4×C127C22×C411yescharacteristic
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4112yes—
95C22×C127—21no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—21no—
99C22×C127—21no—
100C22×C127—21no—
101C22×C127—21no—
102C22×C127—44no—
103C22×C127#10412yescomplemented
104C22×C127C2×C22⋊C411yescharacteristic
105C22×C127#12112yes—
106C2×D127—41no—
107C2×D127—41no—
108C2×D127—41no—
109C4×C127—21no—
110C4×C127—21no—
111C4×C127—44no—
112C4×C127C4×D412yes—
113C4×C127#12011yescharacteristic
114C4×C127#12311yescharacteristic
115Dic127—41no—
116Dic127—41no—
117Dic127—41no—
118C2×C127—22no—
119C2×C127—22no—
120C2×C127—41no—
121C2×C127#38611yescharacteristic
122C2×C127C2×#10411yescharacteristic
123C2×C127#42111yescharacteristic
124D127—81no—
125#1871—1271nomaximal, Sylow
126C127#187111yescharacteristic, complemented, Sylow
127C22×M16—1271no—
128#525—1271no—
129C2×#104—1272no—
130C4×C22⋊C4D12711yescharacteristic, complemented
131#339—1272no—
132C2×#41—1271no—
133C23×C4C2×D12711yescharacteristic
134C2×M16—1272no—
135C2×M16—2544no—
136C2×M16—2541no—
137C22×C8—2541no—
138#123—2541no—
139C22⋊Q8—1271no—
140C22⋊Q8—1271no—
141#121—1271no—
142#121—1271no—
143#120—2541no—
144C4×D4—2541no—
145C4×D4—2541no—
146#117—1271no—
147C2×C22⋊C4—21no—
148C2×C42C2×D12712yes—
149#104—2544no—
150#95—21no—
151C24C22⋊D12711yescharacteristic
152#41—2542no—
153#41—2542no—
154C2×Q8—1271no—
155C2×D4—1271no—
156C2×D4—2541no—
157C2×D4—2541no—
158C2×D4—2541no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—2541no—
163C22×C4C4×D12711yescharacteristic
164C22×C4C4×D12711yescharacteristic
165C22×C4C22⋊D12711yescharacteristic
166C22×C4D50811yescharacteristic
167C22×C4D50811yescharacteristic
168C22×C4C22×D12711yescharacteristic
169M16—1274no—
170M16—2544no—
171M16—2542no—
172C2×C8—2544no—
173C2×C8—2542no—
174C4⋊C4—1271no—
175C4⋊C4—1271no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C22⋊C4—24no—
179C22⋊C4—2541no—
180C22⋊C4—2541no—
181C22⋊C4—2541no—
182C22⋊C4—2541no—
183C22⋊C4—2541no—
184C42—22no—
185C42—22no—
186C42—2541no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—2541no—
191C23(C2×D127)⋊C411yescharacteristic
192C23(C2×D127)⋊C411yescharacteristic
193C23C2×C22⋊D12711yescharacteristic
194Q8—2542no—
195D4—5082no—
196D4—5082no—
197D4—5082no—
198D4—2542no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—5081no—
210C2×C4—5081no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4—2541no—
214C2×C4(C2×D127)⋊C411yescharacteristic
215C2×C4(C2×D127)⋊C411yescharacteristic
216C2×C4C2×C4×D12711yescharacteristic
217C2×C4D4×D12712yes—
218C2×C4C2×D50811yescharacteristic
219C2×C4#1452324712yes—
220C8—2544no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—5081no—
230C22—5081no—
231C22—5081no—
232C22#1686113712yescomplemented
233C22C2×(C2×D127)⋊C411yescharacteristic
234C22#1686118112yes—
235C4—21no—
236C4—21no—
237C4—44no—
238C4—5081no—
239C4—5081no—
240C4—5081no—
241C4(C22⋊D127)⋊C412yes—
242C4#1686117911yescharacteristic
243C4#1686119311yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—10161no—
248C2#2254975311yescharacteristic
249C2C2×#1686113711yescharacteristic
250C2#2254989411yescharacteristic
subgroupG/H
centerC2×C4C2×D508
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC4×C22⋊C4×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550387C2
cosocleC22×C4×C127C23
hypercenterC4×C22⋊C4D127
hypocenterC127#1871
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#14523232C2semidirect
3. C2×#16861137C2semidirect
4. #22549636C2semidirect
5. C4×C22⋊C4×C127C2semidirect
12. #16861188C4semidirect
13. #16861188C4semidirect
14. #16861183C4semidirect
15. #16861183C4semidirect
35. #14523232D4semidirect
103. C22×C127#104semidirect
126. C127#1871semidirect
130. C4×C22⋊C4D127semidirect
232. C22#16861137semidirect

Maximal subgroups

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

subgroupindex
C22×#145232322
#225503872
C2×#168611372
#225496362
C4×C22⋊C4×C1272
#1871127