Finite Group Databasebeta

← #30648256

Subgroups — #30648256

Order 16256.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861239C211yesmaximal, characteristic
2C22×#14523238C211yesmaximal, characteristic, complemented
3#22550353C211yesmaximal, characteristic, complemented
4#22550095C214yesmaximal, complemented
5#22550083C214yesmaximal, complemented
6#22549903C211yesmaximal, characteristic, complemented
7#22549885C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic
9#536×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861239—24no—
12C2×#14523238—24no—
13C2×#14523238C2212yes—
14C2×#14523238C2214yes—
15C2×#14523236C2212yes—
16C2×#14523232C2211yescharacteristic
17C22×C127⋊C8C2211yescharacteristic
18C4×Dic254C2214yes—
19#16861174—22no—
20#16861168C2211yescharacteristic
21#16861165—22no—
22C4⋊(C127⋊C8)C2214yes—
23#16861133C2212yes—
24#16861133C2214yes—
25#16861130C2212yes—
26C2×#41×C127C2211yescharacteristic
27C22×Q8×C127C2211yescharacteristic
28C4⋊Q8×C127C2212yes—
29#124×C127C2212yes—
30#123×C127—22no—
31C22⋊Q8×C127—22no—
32#117×C127C2211yescharacteristic
33C2×Dic254—21no—
34C2×Dic254—22no—
35C2×Dic254D414yes—
36C2×Dic254C2312yes—
37C22×Dic127—21no—
38#14523238—28no—
39#14523238—24no—
40#14523236—24no—
41#14523232—22no—
42C2×C127⋊C8—22no—
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—24no—
47C22⋊Dic127—21no—
48C4×Dic127—24no—
49C4×Dic127—21no—
50#41×C127—22no—
51#41×C127—22no—
52C2×Q8×C127—21no—
53C2×Q8×C127—22no—
54C2×Q8×C127—22no—
55C2×Q8×C127C2312yes—
56C2×Q8×C127C2311yescharacteristic
57C2×D4×C127—21no—
58C2×D4×C127C2311yescharacteristic
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62C4⋊C4×C127—24no—
63C4⋊C4×C127D412yes—
64C22⋊C4×C127—24no—
65C22⋊C4×C127D412yes—
66C42×C127C2312yes—
67Dic254—24no—
68Dic254—22no—
69Dic254C2×D414yes—
70C2×Dic127—24no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C127⋊C8—22no—
74C127⋊C8—22no—
75C23×C127—21no—
76C23×C127C2×D411yescharacteristic
77Q8×C127—22no—
78Q8×C127—21no—
79Q8×C127—22no—
80Q8×C127—21no—
81Q8×C127—42no—
82D4×C127—21no—
83D4×C127—21no—
84D4×C127—42no—
85C2×C4×C127—22no—
86C2×C4×C127—21no—
87C2×C4×C127—42no—
88C2×C4×C127—42no—
89C2×C4×C127C2×D412yes—
90C2×C4×C127C2×D414yes—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127#4112yes—
93C2×C4×C127C2411yescharacteristic
94C22×C127—22no—
95C22×C127—41no—
96C22×C127—42no—
97C22×C127#12112yes—
98C22×C127C22×D411yescharacteristic
99C4×C127—24no—
100C4×C127—41no—
101C4×C127—42no—
102C4×C127#12112yes—
103C4×C127C22×D411yescharacteristic
104C4×C127C2×#4111yescharacteristic
105Dic127—24no—
106Dic127—42no—
107C2×C127—22no—
108C2×C127—81no—
109C2×C127C2×#12111yescharacteristic
110C2×C127#57812yes—
111#3044—1271nomaximal, Sylow
112C127#304411yescharacteristic, complemented, Sylow
113C2×#137—1271no—
114C22×Q16—1271no—
115#536D12711yescharacteristic
116#519—1271no—
117#464—1274no—
118#462—1274no—
119#424—1271no—
120#419—1271no—
121#416—1271no—
122C2×#41C2×D12711yescharacteristic
123C22×Q8—1271no—
124C22×Q8C2×D12711yescharacteristic
125#137—2544no—
126C2×Q16—1274no—
127C2×Q16—1272no—
128C2×Q16—2544no—
129C2×SD16—1272no—
130C2×M16—1271no—
131C22×C8—1271no—
132C4⋊Q8C2×D12712yes—
133#124C2×D12712yes—
134#123—22no—
135C22⋊Q8—22no—
136C4×Q8—1274no—
137C4×Q8—2542no—
138#117—1271no—
139#117—2542no—
140#117C2×D12711yescharacteristic
141C4⋊C8—1274no—
142#103—1272no—
143#103—1274no—
144#102—1272no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—1274no—
151C2×Q8—1272no—
152C2×Q8—2541no—
153C2×Q8—2542no—
154C2×Q8C22×D12712yes—
155C2×Q8C22×D12711yescharacteristic
156C2×D4—21no—
157C2×D4C22×D12711yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—2548no—
163Q16—2544no—
164SD16—2544no—
165M16—2542no—
166C2×C8—1272no—
167C2×C8—1272no—
168C2×C8—2542no—
169C4⋊C4—24no—
170C4⋊C4—1272no—
171C4⋊C4—2544no—
172C4⋊C4C22⋊D12712yes—
173C22⋊C4—24no—
174C22⋊C4—2541no—
175C22⋊C4C22⋊D12712yes—
176C42—2541no—
177C42—2544no—
178C42C22×D12712yes—
179C23—21no—
180C23C2×C22⋊D12711yescharacteristic
181Q8—22no—
182Q8—21no—
183Q8—22no—
184Q8—21no—
185Q8—42no—
186Q8—1274no—
187Q8—2542no—
188Q8—2544no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—2542no—
197C2×C4—2542no—
198C2×C4—2544no—
199C2×C4D4×D12712yes—
200C2×C4C2×C22⋊D12714yes—
201C2×C4C2×C22⋊D12711yescharacteristic
202C2×C4#1452324712yes—
203C2×C4C23×D12711yescharacteristic
204C8—2542no—
205C8—2542no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22#1686118012yes—
210C22C22×C22⋊D12711yescharacteristic
211C4—24no—
212C4—41no—
213C4—42no—
214C4—5082no—
215C4—2544no—
216C4#1686118012yes—
217C4C2×D4×D12711yescharacteristic
218C4C2×#1452324711yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1686118011yescharacteristic
222C2#2255069112yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#536×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×C127C22
hypercenter#536D127
hypocenterC127#3044
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C22×#14523238C2semidirect
3. #22550353C2semidirect
4. #22550095C2semidirect
5. #22550083C2semidirect
6. #22549903C2semidirect
7. #22549885C2semidirect
112. C127#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C22×#145232382
#225503532
#225500952
#225500832
#225499032
#225498852
#225498722
#536×C1272
#3044127