Finite Group Databasebeta

← #30637694

Subgroups — #30637694

Order 16256.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550471C211yesmaximal, characteristic
2#22550034C211yesmaximal, characteristic, complemented
3#22550014C211yesmaximal, characteristic
4#22549797C211yesmaximal, characteristic, complemented
5#22549612C211yesmaximal, characteristic, complemented
6#22549601C211yesmaximal, characteristic, complemented
7#327×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×Dic508—21no—
10C2×#14523236—21no—
11#16861202—21no—
12#16861199—21no—
13#16861190—21no—
14#16861190—21no—
15#16861188C2211yescharacteristic
16#16861186—21no—
17D4×Dic127—21no—
18C2×C4⋊Dic127—21no—
19C2×Dic127⋊C4C2211yescharacteristic
20#16861134—21no—
21#16861132—21no—
22C22⋊(C127⋊C8)C2211yescharacteristic
23C127⋊#95—21no—
24#121×C127C2211yescharacteristic
25C22⋊C8×C127C2211yescharacteristic
26#95×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—42no—
30C2×Dic254D411yescharacteristic
31C22×Dic127—21no—
32C22×Dic127—21no—
33C22×Dic127D411yescharacteristic
34C22×Dic127D411yescharacteristic
35Dic508—42no—
36#14523236—42no—
37C2×C127⋊C8—21no—
38C4⋊Dic127—21no—
39C4⋊Dic127—42no—
40C4⋊Dic127D411yescharacteristic
41Dic127⋊C4—21no—
42Dic127⋊C4—21no—
43Dic127⋊C4—21no—
44Dic127⋊C4—21no—
45Dic127⋊C4—21no—
46C22⋊Dic127—21no—
47C22⋊Dic127—21no—
48C22⋊Dic127—21no—
49C22⋊Dic127—21no—
50C4×Dic127—21no—
51C2×D4×C127—21no—
52C2×D4×C127D411yescharacteristic
53C22×C4×C127—21no—
54C22×C4×C127C2311yescharacteristic
55C2×C8×C127—21no—
56C4⋊C4×C127D411yescharacteristic
57C22⋊C4×C127—21no—
58Dic254—22no—
59Dic254—22no—
60Dic254—41no—
61Dic254—41no—
62Dic254—41no—
63C2×Dic127—22no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127—81no—
74C127⋊C8—41no—
75C23×C127—21no—
76C23×C127C2×D411yescharacteristic
77D4×C127—41no—
78D4×C127—41no—
79D4×C127—41no—
80C2×C4×C127—22no—
81C2×C4×C127—21no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C8×C127—41no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127#12011yescharacteristic
94C22×C127#13511yescharacteristic
95C22×C127#13711yescharacteristic
96C4×C127—21no—
97C4×C127—21no—
98C4×C127—42no—
99C4×C127—41no—
100Dic127—41no—
101Dic127—41no—
102Dic127—42no—
103Dic127—41no—
104Dic127—41no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—81no—
108C2×C127#45211yescharacteristic
109C2×C127#45611yescharacteristic
110C2×C127B(4,𝔽2)11yescharacteristic
111#1603—1271nomaximal, Sylow
112C127#160311yescharacteristic, complemented, Sylow
113#539—1271no—
114#451—1271no—
115#448—1271no—
116#395—1271no—
117#328—1271no—
118#327D12711yescharacteristic
119#324—1271no—
120C22×Q8—1271no—
121C2×Q16—2541no—
122C2×SD16—2541no—
123#126—2541no—
124#125—2541no—
125#123—2541no—
126#123—2541no—
127C22⋊Q8—1271no—
128C22⋊Q8—2541no—
129#121C2×D12711yescharacteristic
130C4×D4—2541no—
131C2×C4⋊C4—1271no—
132C2×C4⋊C4—2541no—
133#103—2541no—
134#103—2541no—
135C22⋊C8—1271no—
136C22⋊C8C2×D12711yescharacteristic
137#95—2541no—
138#95C2×D12711yescharacteristic
139C2×Q8—1271no—
140C2×Q8—5082no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C2×D4—21no—
144C2×D4D50811yescharacteristic
145C22×C4—21no—
146C22×C4—1271no—
147C22×C4—1271no—
148C22×C4—2541no—
149C22×C4—2541no—
150C22×C4C22×D12711yescharacteristic
151Q16—5082no—
152SD16—5082no—
153C2×C8—21no—
154C2×C8—2541no—
155C4⋊C4—1271no—
156C4⋊C4—5082no—
157C4⋊C4—2541no—
158C4⋊C4—2541no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C4⋊C4D50811yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C22⋊C4—2541no—
169C42—2541no—
170C23—21no—
171C23D4×D12711yescharacteristic
172Q8—5081no—
173Q8—5081no—
174Q8—5081no—
175Q8—2542no—
176Q8—2542no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—10161no—
186C2×C4—5081no—
187C2×C4—5081no—
188C2×C4—5081no—
189C2×C4—2542no—
190C2×C4—2541no—
191C2×C4—2541no—
192C2×C4—2541no—
193C2×C4—2541no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4D4×D12711yescharacteristic
197C2×C4C2×D50811yescharacteristic
198C8—41no—
199C8—5081no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#1686117911yescharacteristic
206C22#1686122811yescharacteristic
207C22#1686123811yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—5081no—
213C4—5081no—
214C4—5081no—
215C4—5081no—
216C4—5082no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#2255003911yescharacteristic
221C2#2255006011yescharacteristic
222C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#327×C127C2
socleC22×C127#120
prime radical#121×C127C22
hypercenter#327D127
hypocenterC127#1603
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22550034C2semidirect
4. #22549797C2semidirect
5. #22549612C2semidirect
6. #22549601C2semidirect
112. C127#1603semidirect

Maximal subgroups

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

subgroupindex
#225504712
#225500342
#225500142
#225497972
#225496122
#225496012
#327×C1272
#1603127