Finite Group Databasebeta

← #30649280

Subgroups — #30649280

Order 16256.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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#22550429C211yesmaximal, characteristic
2C2×#16861200C211yesmaximal, characteristic
3#22550284C212yesmaximal, complemented
4#22550283C212yesmaximal, complemented
5#22550277C212yesmaximal, complemented
6#22550276C212yesmaximal, complemented
7#22549919C211yesmaximal, characteristic
8#22549918C211yesmaximal, characteristic
9#22549916C211yesmaximal, characteristic
10M16×Dic127C211yesmaximal, characteristic
11#428×C127C211yesmaximal, characteristic
12C2×#14523232C2211yescharacteristic
13#16861208C2211yescharacteristic
14#16861208C2211yescharacteristic
15#16861200—21no—
16#16861200—21no—
17#16861200C2211yescharacteristic
18#16861200C2212yes—
19#16861200C2211yescharacteristic
20#16861200C2212yes—
21#16861186—21no—
22#16861186—21no—
23C4×Dic254—21no—
24C4×Dic254—21no—
25#16861168C2211yescharacteristic
26C2×C4⋊Dic127—21no—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×Dic127⋊C4—21no—
29C2×C4×Dic127C2211yescharacteristic
30#16861143C2212yes—
31#16861142C2212yes—
32#16861142C2212yes—
33#16861141C2212yes—
34#16861140C2212yes—
35#16861140C2212yes—
36#16861119C2212yes—
37C8×Dic127C2212yes—
38C2×M16×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#107×C127C2212yes—
42#106×C127C2212yes—
43C2×Dic254—21no—
44C2×Dic254—21no—
45C22×Dic127—21no—
46C22×Dic127D411yescharacteristic
47C22×Dic127D411yescharacteristic
48#14523232Q814yescomplemented
49C2×C127⋊C8C2312yes—
50C4⋊Dic127—21no—
51C4⋊Dic127—21no—
52C4⋊Dic127—22no—
53C4⋊Dic127—22no—
54C4⋊Dic127—21no—
55C4⋊Dic127C2312yes—
56C4⋊Dic127C2311yescharacteristic
57C4⋊Dic127C2311yescharacteristic
58Dic127⋊C4—21no—
59Dic127⋊C4—21no—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62Dic127⋊C4—22no—
63Dic127⋊C4—22no—
64C22⋊Dic127—21no—
65C4×Dic127—21no—
66C4×Dic127D411yescharacteristic
67C4×Dic127D411yescharacteristic
68C4×Dic127C2311yescharacteristic
69C4×Dic127C2311yescharacteristic
70C22×C4×C127—21no—
71C22×C4×C127C2311yescharacteristic
72M16×C127Q814yescomplemented
73C2×C8×C127C2312yes—
74C4⋊C4×C127—21no—
75C4⋊C4×C127C2312yes—
76C4⋊C4×C127C2311yescharacteristic
77C4⋊C4×C127C2311yescharacteristic
78C22⋊C4×C127—21no—
79C42×C127—21no—
80Dic254—42no—
81Dic254—42no—
82C2×Dic127—21no—
83C2×Dic127—21no—
84C2×Dic127—22no—
85C2×Dic127—41no—
86C2×Dic127—41no—
87C2×Dic127—42no—
88C2×Dic127C2×D411yescharacteristic
89C2×Dic127C2×D411yescharacteristic
90C2×Dic127C2×D411yescharacteristic
91C2×Dic127C2×D411yescharacteristic
92C127⋊C8C2×Q814yes—
93C23×C127C2×D411yescharacteristic
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—42no—
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×Q812yes—
100C2×C4×C127C2×Q812yes—
101C2×C4×C127C2411yescharacteristic
102C8×C127C2×Q814yes—
103C22×C127—21no—
104C22×C127—21no—
105C22×C127C4⋊Q812yes—
106C22×C127C22×D411yescharacteristic
107C4×C127—42no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127C4⋊Q812yes—
111C4×C127C22×Q811yescharacteristic
112C4×C127C22×Q811yescharacteristic
113Dic127—21no—
114Dic127—21no—
115Dic127—21no—
116Dic127—21no—
117Dic127—42no—
118Dic127—41no—
119Dic127—41no—
120C2×C127—22no—
121C2×C127C2×C4⋊Q811yescharacteristic
122C2×C127#57511yescharacteristic
123C2×C127#57511yescharacteristic
124#3166—1271nomaximal, Sylow
125C127#316611yescharacteristic, complemented, Sylow
126#533—1271no—
127C2×#125—1271no—
128#501—1272no—
129#501—1272no—
130#499—1272no—
131#499—1272no—
132#428—1271no—
133#428—1271no—
134#428—1271no—
135#428D12711yescharacteristic
136C4×M16—1271no—
137C2×M16—1271no—
138C2×M16C2×D12711yescharacteristic
139C4⋊Q8—1271no—
140C4⋊Q8—1271no—
141#125—1272no—
142#125—1272no—
143#125—1271no—
144#125—1271no—
145#125—2541no—
146#125—2541no—
147C22⋊Q8—2541no—
148C22⋊Q8—2541no—
149C4×Q8—2541no—
150C4×Q8—2541no—
151#117—1271no—
152#117C2×D12711yescharacteristic
153C2×C4⋊C4—1271no—
154C2×C4⋊C4—2541no—
155C2×C4⋊C4—2541no—
156C2×C4⋊C4C2×D12711yescharacteristic
157C2×C42—1271no—
158#107—1272no—
159#107—1272no—
160#107—1272no—
161#107C2×D12712yes—
162#106—1272no—
163#106—1272no—
164#106—1272no—
165#106C2×D12712yes—
166#97—1272no—
167C4×C8—1272no—
168C2×Q8—2541no—
169C2×Q8—2541no—
170C22×C4—21no—
171C22×C4—1271no—
172C22×C4—1271no—
173C22×C4—2541no—
174C22×C4C22×D12711yescharacteristic
175M16—1274no—
176M16Dic25414yescomplemented
177C2×C8—1272no—
178C2×C8C22×D12712yes—
179C4⋊C4—21no—
180C4⋊C4—1272no—
181C4⋊C4—1271no—
182C4⋊C4—1271no—
183C4⋊C4—2542no—
184C4⋊C4—2542no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C4⋊C4—2541no—
188C4⋊C4—2541no—
189C4⋊C4—2542no—
190C4⋊C4—2542no—
191C4⋊C4—2541no—
192C4⋊C4—2541no—
193C4⋊C4—2541no—
194C4⋊C4C22×D12711yescharacteristic
195C4⋊C4C22×D12712yes—
196C4⋊C4C22×D12711yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—2541no—
199C42—21no—
200C42—1271no—
201C42—1271no—
202C42—1271no—
203C42—1271no—
204C42—2541no—
205C23D4×D12711yescharacteristic
206Q8—5082no—
207Q8—5082no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—1271no—
213C2×C4—1271no—
214C2×C4—1271no—
215C2×C4—1271no—
216C2×C4—5082no—
217C2×C4—5081no—
218C2×C4—5081no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2542no—
222C2×C4D4×D12711yescharacteristic
223C2×C4Q8×D12712yes—
224C2×C4C2×Dic25412yes—
225C2×C4C23×D12711yescharacteristic
226C8—1274no—
227C8C2×Dic25414yes—
228C22—21no—
229C22—21no—
230C22#1686120812yes—
231C22C2×D4×D12711yescharacteristic
232C4—42no—
233C4—41no—
234C4—41no—
235C4—5081no—
236C4—5082no—
237C4—5081no—
238C4—2541no—
239C4—2541no—
240C4—2541no—
241C4—2541no—
242C4#1686120812yes—
243C4C2×Q8×D12711yescharacteristic
244C4C22×Dic25411yescharacteristic
245C2—22no—
246C2C2×#1686120811yescharacteristic
247C2#2255067211yescharacteristic
248C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radicalC23×C127C2×D4
hypercenter#428D127
hypocenterC127#3166
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #22550284C2semidirect
4. #22550283C2semidirect
5. #22550277C2semidirect
6. #22550276C2semidirect
48. #14523232Q8semidirect
72. M16×C127Q8semidirect
125. C127#3166semidirect
176. M16Dic254semidirect

Maximal subgroups

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

subgroupindex
#225504292
C2×#168612002
#225502842
#225502832
#225502772
#225502762
#225499192
#225499182
#225499162
M16×Dic1272
#428×C1272
#3166127