Finite Group Databasebeta

← #30638954

Subgroups — #30638954

Order 16256.

302 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550359C211yesmaximal, characteristic
2#22549916C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic
4#22549745C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549637C211yesmaximal, characteristic, complemented
7#417×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9Q8×Dic127—21no—
10D4×Dic127—22no—
11D4×Dic127—21no—
12#16861168—21no—
13#16861168C2211yescharacteristic
14C2×C4⋊Dic127—21no—
15C2×C4⋊Dic127C2211yescharacteristic
16C2×Dic127⋊C4—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861142—21no—
19#16861140—21no—
20#16861136C414yescomplemented
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#103×C127—21no—
25#102×C127—21no—
26C22×Dic127—22no—
27C22×Dic127—21no—
28C22×Dic127D412yes—
29#14523232—21no—
30#14523232C2×C412yescomplemented
31C2×C127⋊C8—21no—
32C4⋊Dic127—22no—
33C4⋊Dic127—21no—
34C4⋊Dic127—21no—
35C4⋊Dic127D411yescharacteristic
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—44no—
38C22⋊Dic127—22no—
39C22⋊Dic127—21no—
40C4×Dic127—22no—
41C4×Dic127—21no—
42C4×Dic127—21no—
43C4×Dic127C2×C412yescomplemented
44#41×C127—21no—
45#41×C127—21no—
46#41×C127C2×C412yes—
47#41×C127Q812yes—
48C2×Q8×C127D411yescharacteristic
49C2×D4×C127—21no—
50C2×D4×C127D411yescharacteristic
51C22×C4×C127—21no—
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C4⋊C4×C127—21no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—21no—
63C2×Dic127—22no—
64C2×Dic127—21no—
65C2×Dic127—42no—
66C2×Dic127—82no—
67C127⋊C8—22no—
68C23×C127—21no—
69C23×C127C2×D411yescharacteristic
70Q8×C127—21no—
71Q8×C127C4⋊C412yescomplemented
72D4×C127—22no—
73D4×C127—22no—
74D4×C127—21no—
75D4×C127C4⋊C412yescomplemented
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—42no—
80C2×C4×C127—41no—
81C2×C4×C127C22×C411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×Q811yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C8×C127—41no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—22no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127C4×D411yescharacteristic
94C22×C127#12011yescharacteristic
95C22×C127C22⋊Q811yescharacteristic
96C4×C127—22no—
97C4×C127—81no—
98C4×C127C2×C4⋊C411yescharacteristic
99C4×C127C4×D411yescharacteristic
100C4×C127C22⋊Q811yescharacteristic
101C4×C127#12311yescharacteristic
102Dic127—24no—
103Dic127—42no—
104Dic127—41no—
105Dic127—41no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—42no—
109C2×C127#38511yescharacteristic
110C2×C127#45311yescharacteristic
111C2×C127#45611yescharacteristic
112#1859—1271nomaximal, Sylow
113C127#185911yescharacteristic, complemented, Sylow
114#520—1271no—
115#428—1271no—
116C2×#104—1271no—
117#417D12711yescharacteristic
118#384—1271no—
119#343—1271no—
120#339—1271no—
121C2×#41C2×D12711yescharacteristic
122C2×M16—1271no—
123C2×M16C2×D12711yescharacteristic
124C4×Q8—2541no—
125C4×D4—2542no—
126C4×D4—2541no—
127#117—1271no—
128#117—2541no—
129C2×C4⋊C4—1271no—
130C2×C4⋊C4—2541no—
131C2×C4⋊C4—2542no—
132C2×C4⋊C4C2×D12711yescharacteristic
133C2×C42—1271no—
134#107—2541no—
135#106—2541no—
136#104—1274no—
137#103—21no—
138#102—21no—
139#41—21no—
140#41—21no—
141#41C4×D12712yes—
142#41Dic25412yes—
143C2×Q8C22⋊D12711yescharacteristic
144C2×D4—21no—
145C2×D4C22⋊D12711yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—1272no—
149C22×C4—2542no—
150C22×C4—2541no—
151C22×C4C22×D12711yescharacteristic
152M16—21no—
153M16—21no—
154M16—1272no—
155M16—2541no—
156C2×C8—21no—
157C2×C8—2541no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—1271no—
161C4⋊C4—1271no—
162C4⋊C4—5084no—
163C4⋊C4—2542no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C22⋊C4—2542no—
167C22⋊C4—2541no—
168C42—1272no—
169C42—2542no—
170C42—2541no—
171C42—2541no—
172C23—21no—
173C23D4×D12711yescharacteristic
174Q8—21no—
175Q8Dic127⋊C412yescomplemented
176D4—22no—
177D4—22no—
178D4—21no—
179D4Dic127⋊C412yescomplemented
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—10162no—
186C2×C4—5082no—
187C2×C4—2541no—
188C2×C4—2541no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4C2×C4×D12711yescharacteristic
194C2×C4D4×D12711yescharacteristic
195C2×C4C2×C22⋊D12711yescharacteristic
196C2×C4C2×Dic25411yescharacteristic
197C2×C4#1452324711yescharacteristic
198C2×C4#1452324711yescharacteristic
199C8—41no—
200C8—2542no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22(C22⋊D127)⋊C411yescharacteristic
207C22#1686117711yescharacteristic
208C22#1686118611yescharacteristic
209C4—22no—
210C4—81no—
211C4—5081no—
212C4—5082no—
213C4—5081no—
214C4—2544no—
215C4C2×Dic127⋊C411yescharacteristic
216C4(C22⋊D127)⋊C411yescharacteristic
217C4#1686118611yescharacteristic
218C4#1686119011yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#2254974811yescharacteristic
223C2#2255004211yescharacteristic
224C2#2255005711yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#417×C127C2
socleC22×C127#120
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#417D127
hypocenterC127#1859
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #22549916C2semidirect
4. #22549745C2semidirect
5. #22549644C2semidirect
6. #22549637C2semidirect
20. #16861136C4semidirect
30. #14523232C2×C4semidirect
43. C4×Dic127C2×C4semidirect
71. Q8×C127C4⋊C4semidirect
75. D4×C127C4⋊C4semidirect
113. C127#1859semidirect
175. Q8Dic127⋊C4semidirect
179. D4Dic127⋊C4semidirect

Maximal subgroups

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

subgroupindex
#225503592
#225499162
C2×#168611362
#225497452
#225496442
#225496372
#417×C1272
#1859127