Finite Group Databasebeta

← #30648257

Subgroups — #30648257

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×#16861240C211yesmaximal, characteristic
2#22550450C211yesmaximal, characteristic
3#22550354C211yesmaximal, characteristic, complemented
4#22550100C214yesmaximal, complemented
5#22550084C214yesmaximal, complemented
6#22549902C211yesmaximal, characteristic, complemented
7#22549885C211yesmaximal, characteristic, complemented
8#22549873C211yesmaximal, characteristic
9C22×Q16×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11#16861240—24no—
12C2×#14523238C2212yes—
13C2×#14523235C2212yes—
14C2×#14523232C2211yescharacteristic
15#16861207C2212yes—
16#16861198C2212yes—
17#16861192—22no—
18#16861185—22no—
19Q8×Dic127—22no—
20Q8×Dic127C2214yes—
21#16861168—22no—
22#16861168C2211yescharacteristic
23#16861168C2211yescharacteristic
24Dic127⋊C8C2214yes—
25#16861134C2212yes—
26#16861133C2214yes—
27#16861131C2212yes—
28C22×Q8×C127C2211yescharacteristic
29C22×Q8×C127C2211yescharacteristic
30C2×Q16×C127—24no—
31C2×Q16×C127C2214yes—
32C22×C8×C127C2211yescharacteristic
33#14523248—22no—
34#14523248—22no—
35C2×Dic254C2311yescharacteristic
36C2×D508C2311yescharacteristic
37C2×C22⋊D127—21no—
38C2×C4×D127—21no—
39#14523238—24no—
40#14523235—24no—
41#14523232—22no—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127—24no—
44C4⋊Dic127D412yes—
45C4⋊Dic127C2312yes—
46Dic127⋊C4—24no—
47C22⋊Dic127—21no—
48C22⋊Dic127D412yes—
49(C2×D127)⋊C4—24no—
50C4×Dic127—24no—
51C4×Dic127—21no—
52C4×Dic127C2312yes—
53C2×Q8×C127—21no—
54C2×Q8×C127—21no—
55C2×Q8×C127—22no—
56C2×Q8×C127—22no—
57C2×Q8×C127—22no—
58C2×Q8×C127D414yes—
59C2×Q8×C127C2312yes—
60C2×Q8×C127C2312yes—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127C2311yescharacteristic
64Q16×C127—28no—
65C2×C8×C127—22no—
66C2×C8×C127C2312yes—
67C22×D127—21no—
68Dic254—21no—
69Dic254—21no—
70D508—21no—
71D508—21no—
72C22⋊D127—42no—
73C2×Dic127—22no—
74C2×Dic127—21no—
75C2×Dic127C2×D414yes—
76C4×D127—42no—
77C127⋊C8—22no—
78C23×C127C2×D411yescharacteristic
79Q8×C127—24no—
80Q8×C127—22no—
81Q8×C127—22no—
82Q8×C127—22no—
83Q8×C127—42no—
84Q8×C127C2×D414yes—
85C2×C4×C127—24no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—42no—
89C2×C4×C127C2×D411yescharacteristic
90C2×C4×C127C2×D412yes—
91C2×C4×C127#4112yes—
92C2×C4×C127C2411yescharacteristic
93C8×C127—22no—
94C22×C127—22no—
95C22×C127#12112yes—
96C22×C127C22×D411yescharacteristic
97C2×D127—41no—
98C2×D127—42no—
99C4×C127—24no—
100C4×C127—42no—
101C4×C127#12112yes—
102C4×C127C22×D411yescharacteristic
103C4×C127C2×#4111yescharacteristic
104Dic127—24no—
105Dic127—41no—
106Dic127—42no—
107C2×C127—22no—
108C2×C127C2×#12111yescharacteristic
109C2×C127#57812yes—
110D127—81no—
111#3044—1271nomaximal, Sylow
112C127#304411yescharacteristic, complemented, Sylow
113C2×#137—1271no—
114C22×Q16D12711yescharacteristic, complemented
115#536—1271no—
116#519—1271no—
117#464—1274no—
118#462—1274no—
119#424—1271no—
120#419—1271no—
121#416—1271no—
122C2×#41—1271no—
123C22×Q8C2×D12711yescharacteristic
124C22×Q8C2×D12711yescharacteristic
125#137—2544no—
126C2×Q16—24no—
127C2×Q16—1272no—
128C2×Q16C2×D12714yes—
129C2×SD16—1272no—
130C2×M16—1271no—
131C22×C8C2×D12711yescharacteristic
132C4⋊Q8—1272no—
133#124—1272no—
134#123—2542no—
135C22⋊Q8—2542no—
136C4×Q8—1274no—
137C4×Q8—2542no—
138#117—1271no—
139#117—1271no—
140#117—2542no—
141C4⋊C8—1274no—
142#103—1274no—
143#103—1272no—
144#102—1272no—
145#41—2542no—
146#41—2542no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—1271no—
153C2×Q8C22⋊D12714yes—
154C2×Q8C22×D12712yes—
155C2×Q8C22×D12712yes—
156C2×D4—1271no—
157C2×D4—2541no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2541no—
161C22×C4C22×D12711yescharacteristic
162Q16—28no—
163Q16—2544no—
164SD16—2544no—
165M16—2542no—
166C2×C8—22no—
167C2×C8—1272no—
168C2×C8C22×D12712yes—
169C4⋊C4—1272no—
170C4⋊C4—1272no—
171C4⋊C4—2544no—
172C4⋊C4—2544no—
173C22⋊C4—1272no—
174C22⋊C4—2544no—
175C22⋊C4—2541no—
176C42—1272no—
177C42—2544no—
178C42—2541no—
179C23—2541no—
180C23D4×D12711yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—22no—
184Q8—22no—
185Q8—42no—
186Q8—2541no—
187Q8—2541no—
188Q8C2×C22⋊D12714yes—
189D4—5082no—
190D4—2541no—
191D4—2541no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—1274no—
197C2×C4—5082no—
198C2×C4—2542no—
199C2×C4—2541no—
200C2×C4D4×D12711yescharacteristic
201C2×C4C2×C22⋊D12712yes—
202C2×C4#1452324712yes—
203C2×C4C23×D12711yescharacteristic
204C8—22no—
205C8—2542no—
206C22—22no—
207C22—5081no—
208C22—5082no—
209C22#1686118112yes—
210C22C2×D4×D12711yescharacteristic
211C4—24no—
212C4—42no—
213C4—5081no—
214C4—5082no—
215C4—2544no—
216C4#1686118112yes—
217C4C22×C22⋊D12711yescharacteristic
218C4C2×#1452324711yescharacteristic
219C2—22no—
220C2—10161no—
221C2C2×#1686118111yescharacteristic
222C2#2255068912yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×Q16×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenterC22×Q16D127
hypocenterC127#3044
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #22550354C2semidirect
4. #22550100C2semidirect
5. #22550084C2semidirect
6. #22549902C2semidirect
7. #22549885C2semidirect
9. C22×Q16×C127C2semidirect
112. C127#3044semidirect
114. C22×Q16D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
#225504502
#225503542
#225501002
#225500842
#225499022
#225498852
#225498732
C22×Q16×C1272
#3044127