Finite Group Databasebeta

← #30638646

Subgroups — #30638646

Order 16256.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14523247C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic, complemented
3C2×#16861123C212yesmaximal, complemented
4#22549737C212yesmaximal, complemented
5C2×#99×C127C211yesmaximal, characteristic, complemented
6C2×#14523247—24no—
7C2×#14523247—24no—
8C2×#14523247—22no—
9C2×#14523247—22no—
10C22×Dic254C411yescharacteristic, complemented
11C22×C22⋊D127C411yescharacteristic, complemented
12C22×C22⋊D127C2211yescharacteristic
13C23×Dic127C411yescharacteristic, complemented
14C23×Dic127C2211yescharacteristic
15C22×C4×D127C411yescharacteristic, complemented
16C2×C22⋊Dic127C2212yes—
17#16861124—22no—
18#16861123—24no—
19C127⋊#95—24no—
20C22×D4×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2212yes—
22#99×C127—22no—
23C23×D127C2×C411yescharacteristic
24#14523247—216no—
25#14523247—216no—
26C2×Dic254—22no—
27C2×Dic254—22no—
28C2×Dic254—22no—
29C2×C22⋊D127—22no—
30C2×C22⋊D127—22no—
31C2×C22⋊D127—22no—
32C2×C22⋊D127—22no—
33C2×C22⋊D127—21no—
34C2×C22⋊D127—21no—
35C2×C22⋊D127D414yes—
36C22×Dic127—22no—
37C22×Dic127—22no—
38C22×Dic127—22no—
39C22×Dic127—22no—
40C22×Dic127—21no—
41C22×Dic127—21no—
42C22×Dic127—22no—
43C22×Dic127C2×C411yescharacteristic
44C22×Dic127C2×C411yescharacteristic
45C22×Dic127C2×C411yescharacteristic
46C22×Dic127D414yes—
47C2×C4×D127—22no—
48C2×C4×D127—22no—
49C2×C4×D127—22no—
50C22⋊Dic127—22no—
51C22⋊Dic127—22no—
52C24×C127C2×C411yescharacteristic
53C24×C127C2311yescharacteristic
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127D412yes—
57C2×D4×C127D412yes—
58C22×C4×C127—22no—
59C22×C4×C127C2×C411yescharacteristic
60C22⋊C4×C127—22no—
61C22⋊C4×C127—22no—
62C22×D127—21no—
63C22×D127—21no—
64C22×D127—42no—
65C22×D127C22⋊C412yes—
66Dic254—28no—
67C22⋊D127—28no—
68C22⋊D127—24no—
69C22⋊D127—24no—
70C2×Dic127—28no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—21no—
77C2×Dic127—24no—
78C2×Dic127—24no—
79C2×Dic127—44no—
80C2×Dic127—42no—
81C2×Dic127C22⋊C412yes—
82C2×Dic127C22⋊C412yes—
83C2×Dic127C22⋊C412yes—
84C4×D127—28no—
85C23×C127—21no—
86C23×C127—21no—
87C23×C127—42no—
88C23×C127—42no—
89C23×C127C22⋊C412yes—
90C23×C127C22×C411yescharacteristic
91C23×C127C2×D412yes—
92C23×C127C2×D412yes—
93C23×C127C2×D411yescharacteristic
94C23×C127C2×D411yescharacteristic
95D4×C127—24no—
96D4×C127—24no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—44no—
100C2×C4×C127—42no—
101C2×C4×C127C22⋊C412yes—
102C22×C127—24no—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—42no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127—42no—
111C22×C127C2×C22⋊C411yescharacteristic
112C22×C127C2×C22⋊C411yescharacteristic
113C22×C127C2×C22⋊C411yescharacteristic
114C22×C127#12012yes—
115C22×C127#12012yes—
116C2×D127—22no—
117C2×D127—42no—
118C2×D127—42no—
119C2×D127—42no—
120C4×C127—22no—
121C4×C127—82no—
122Dic127—22no—
123Dic127—22no—
124Dic127—22no—
125Dic127—82no—
126C2×C127—22no—
127C2×C127—22no—
128C2×C127—21no—
129C2×C127—21no—
130C2×C127—42no—
131C2×C127#37911yescharacteristic
132C2×C127#41012yes—
133D127—42no—
134#1782—1271nomaximal, Sylow
135C127#178211yescharacteristic, complemented, Sylow
136C22×#41—1271no—
137C2×#99—1271no—
138C2×#99—1272no—
139C2×#99D12711yescharacteristic, complemented
140#381—1272no—
141C2×#41—2544no—
142C2×#41—2544no—
143C2×#41—2542no—
144C2×#41—2542no—
145C22×Q8—1271no—
146C22×D4—1271no—
147C22×D4—1271no—
148C22×D4C2×D12711yescharacteristic
149C23×C4—1271no—
150C23×C4—1271no—
151C23×C4—1271no—
152C2×C22⋊C4—1272no—
153C2×C22⋊C4C2×D12712yes—
154#99—22no—
155#99—2544no—
156#99—2542no—
157#95—2544no—
158C24—1271no—
159C24C4×D12711yescharacteristic
160C24C22×D12711yescharacteristic
161#41—25416no—
162#41—25416no—
163C2×Q8—2542no—
164C2×Q8—2542no—
165C2×Q8—2542no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—1274no—
169C2×D4—2542no—
170C2×D4—2542no—
171C2×D4—2542no—
172C2×D4—2542no—
173C2×D4—2541no—
174C2×D4—2541no—
175C2×D4C22⋊D12712yes—
176C2×D4D50812yes—
177C22×C4—22no—
178C22×C4—1271no—
179C22×C4—1271no—
180C22×C4—1271no—
181C22×C4—1274no—
182C22×C4—2542no—
183C22×C4—2542no—
184C22×C4—2542no—
185C22×C4—2542no—
186C22×C4—2542no—
187C22×C4—2542no—
188C22×C4—2542no—
189C22×C4—2541no—
190C22×C4—2541no—
191C22×C4—2542no—
192C22×C4C4×D12711yescharacteristic
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2542no—
196C22⋊C4—2542no—
197C23—21no—
198C23—21no—
199C23—42no—
200C23—42no—
201C23—1272no—
202C23—5082no—
203C23—2541no—
204C23—2541no—
205C23(C2×D127)⋊C412yes—
206C23C2×C4×D12711yescharacteristic
207C23D4×D12712yes—
208C23D4×D12712yes—
209C23C2×C22⋊D12711yescharacteristic
210C23C2×D50811yescharacteristic
211Q8—2548no—
212D4—24no—
213D4—24no—
214D4—2548no—
215D4—2544no—
216D4—2544no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—1272no—
222C2×C4—1272no—
223C2×C4—1272no—
224C2×C4—5084no—
225C2×C4—5082no—
226C2×C4—2548no—
227C2×C4—2548no—
228C2×C4—2541no—
229C2×C4—2541no—
230C2×C4—2541no—
231C2×C4—2541no—
232C2×C4—2541no—
233C2×C4—2541no—
234C2×C4—2544no—
235C2×C4—2544no—
236C2×C4(C2×D127)⋊C412yes—
237C22—24no—
238C22—22no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—5082no—
247C22—5082no—
248C22—5082no—
249C22—2542no—
250C22C22⋊C4×D12711yescharacteristic
251C22C22⋊C4×D12711yescharacteristic
252C22C2×(C2×D127)⋊C411yescharacteristic
253C22#1686117712yes—
254C22#1686117912yes—
255C4—22no—
256C4—82no—
257C4—10162no—
258C4—2542no—
259C4—2542no—
260C4—2542no—
261C2—22no—
262C2—22no—
263C2—21no—
264C2—21no—
265C2—42no—
266C2—5082no—
267C2#2254973111yescharacteristic
268C2#2254984812yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC2×#99×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×#14523247C2
cosocleC24×C127C23
hypercenterC2×#99D127
hypocenterC127#1782
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16861124C2semidirect
3. C2×#16861123C2semidirect
4. #22549737C2semidirect
5. C2×#99×C127C2semidirect
10. C22×Dic254C4semidirect
11. C22×C22⋊D127C4semidirect
13. C23×Dic127C4semidirect
15. C22×C4×D127C4semidirect
135. C127#1782semidirect
139. C2×#99D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232472
C2×#168611242
C2×#168611232
#225497372
C2×#99×C1272
#1782127