Finite Group Databasebeta

← #30637691

Subgroups — #30637691

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#22550473C211yesmaximal, characteristic
2#22550032C211yesmaximal, characteristic, complemented
3#22550015C211yesmaximal, characteristic
4#22549612C211yesmaximal, characteristic, complemented
5#22549609C211yesmaximal, characteristic
6#22549602C211yesmaximal, characteristic, complemented
7#395×C127C211yesmaximal, characteristic, complemented
8C2×#14523238—21no—
9C2×#14523235—21no—
10#16861204—21no—
11#16861199—21no—
12#16861194—21no—
13#16861192—21no—
14#16861188C2211yescharacteristic
15#16861187—21no—
16#16861183C2211yescharacteristic
17C4×C22⋊D127—21no—
18#16861133—21no—
19#16861133—21no—
20C22⋊(C127⋊C8)C2211yescharacteristic
21C22⋊(C127⋊C8)C2211yescharacteristic
22C22×Q8×C127C2211yescharacteristic
23C2×C4⋊C4×C127—21no—
24C2×C4⋊C4×C127C2211yescharacteristic
25#95×C127—21no—
26#95×C127C2211yescharacteristic
27C2×Dic254D411yescharacteristic
28C2×D508D411yescharacteristic
29C2×C22⋊D127—21no—
30C2×C4×D127—21no—
31#14523238—42no—
32#14523235—42no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127D411yescharacteristic
36C4⋊Dic127D411yescharacteristic
37Dic127⋊C4—21no—
38Dic127⋊C4—21no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42(C2×D127)⋊C4—21no—
43(C2×D127)⋊C4—21no—
44(C2×D127)⋊C4—21no—
45(C2×D127)⋊C4—21no—
46C4×Dic127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127—42no—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127D411yescharacteristic
53C22×C4×C127D411yescharacteristic
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127—42no—
58C22×D127—21no—
59Dic254—41no—
60D508—41no—
61C22⋊D127—41no—
62C22⋊D127—41no—
63C2×Dic127—21no—
64C2×Dic127—21no—
65C2×Dic127—21no—
66C4×D127—81no—
67C127⋊C8—41no—
68C127⋊C8—41no—
69C23×C127C2×D411yescharacteristic
70Q8×C127—22no—
71Q8×C127—22no—
72Q8×C127—41no—
73Q8×C127—41no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—41no—
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C22×C127—21no—
89C22×C127—21no—
90C22×C127#12011yescharacteristic
91C22×C127#13511yescharacteristic
92C22×C127#13711yescharacteristic
93C2×D127—41no—
94C2×D127—41no—
95C2×D127—41no—
96C4×C127—21no—
97C4×C127—21no—
98C4×C127—42no—
99C4×C127—41no—
100C4×C127—42no—
101C4×C127—41no—
102Dic127—41no—
103Dic127—41no—
104Dic127—41no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127#45211yescharacteristic
108C2×C127#45611yescharacteristic
109C2×C127B(4,𝔽2)11yescharacteristic
110D127—81no—
111#1603—1271nomaximal, Sylow
112C127#160311yescharacteristic, complemented, Sylow
113#539—1271no—
114#451—1271no—
115#448—1271no—
116#395D12711yescharacteristic, complemented
117#328—1271no—
118#327—1271no—
119#324—1271no—
120C22×Q8C2×D12711yescharacteristic
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#121—1271no—
130C4×D4—2541no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2×D12711yescharacteristic
133#103—2541no—
134#103—2541no—
135C22⋊C8—1271no—
136C22⋊C8—1271no—
137#95—21no—
138#95C2×D12711yescharacteristic
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8—1271no—
143C2×D4—1271no—
144C2×D4—2541no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—2541no—
148C22×C4C22⋊D12711yescharacteristic
149C22×C4C22⋊D12711yescharacteristic
150C22×C4C22×D12711yescharacteristic
151Q16—5082no—
152SD16—5082no—
153C2×C8—2541no—
154C2×C8—2541no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—42no—
158C4⋊C4—1271no—
159C4⋊C4—1271no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C4⋊C4—2541no—
163C4⋊C4—2541no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C22⋊C4—2541no—
169C42—2541no—
170C23—2541no—
171C23C2×C22⋊D12711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—5081no—
177D4—5081no—
178D4—5081no—
179D4—5081no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—10161no—
193C2×C4—2541no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4D4×D12711yescharacteristic
197C2×C4D4×D12711yescharacteristic
198C8—5081no—
199C8—5081no—
200C22—21no—
201C22—21no—
202C22—5081no—
203C22—5081no—
204C22—5081no—
205C22#1686117711yescharacteristic
206C22#1686122911yescharacteristic
207C22#1686124011yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—42no—
213C4—41no—
214C4—5081no—
215C4—5081no—
216C4—5081no—
217C2—21no—
218C2—21no—
219C2—10161no—
220C2#2255003611yescharacteristic
221C2#2255005811yescharacteristic
222C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#395×C127C2
socleC22×C127#120
prime radical#16861183C22
hypercenter#395D127
hypocenterC127#1603
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
2. #22550032C2semidirect
4. #22549612C2semidirect
6. #22549602C2semidirect
7. #395×C127C2semidirect
112. C127#1603semidirect
116. #395D127semidirect

Maximal subgroups

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

subgroupindex
#225504732
#225500322
#225500152
#225496122
#225496092
#225496022
#395×C1272
#1603127