Finite Group Databasebeta

← #30640026

Subgroups — #30640026

Order 16256.

262 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 71 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×#16861144C211yesmaximal, characteristic, complemented
2#22549877C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×#16861128C211yesmaximal, characteristic, complemented
5#22549843C211yesmaximal, characteristic
6#22549644C211yesmaximal, characteristic, complemented
7C2×#137×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12#16861144—22no—
13#16861133—22no—
14#16861133—21no—
15#16861130—21no—
16#16861128—22no—
17C22⋊(C127⋊C8)—22no—
18C2×#41×C127C2211yescharacteristic
19C22×Q8×C127C2211yescharacteristic
20#137×C127—22no—
21#137×C127—22no—
22C2×Q16×C127—21no—
23C2×SD16×C127—21no—
24C2×M16×C127C2211yescharacteristic
25C22×Dic127—21no—
26#14523232—21no—
27#14523232—21no—
28#14523232—21no—
29#14523232—21no—
30C2×C127⋊C8—21no—
31C2×C127⋊C8—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8D412yes—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37#41×C127—21no—
38#41×C127—21no—
39#41×C127—21no—
40#41×C127—21no—
41C2×Q8×C127—21no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127D411yescharacteristic
47C2×Q8×C127D412yes—
48C2×D4×C127—21no—
49C2×D4×C127D411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53Q16×C127—24no—
54SD16×C127—24no—
55M16×C127—21no—
56M16×C127D412yes—
57C2×C8×C127—21no—
58C2×Dic127—41no—
59C2×Dic127—81no—
60C127⋊C8—22no—
61C127⋊C8—41no—
62C127⋊C8—41no—
63C23×C127—21no—
64C23×C127#4111yescharacteristic
65Q8×C127—22no—
66Q8×C127—21no—
67Q8×C127—22no—
68Q8×C127—21no—
69Q8×C127—41no—
70Q8×C127—41no—
71D4×C127—21no—
72D4×C127—21no—
73D4×C127—41no—
74D4×C127—41no—
75C2×C4×C127—21no—
76C2×C4×C127—22no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C8×C127—22no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127#12111yescharacteristic
94C22×C127#12311yescharacteristic
95C22×C127#12411yescharacteristic
96C4×C127—41no—
97C4×C127—42no—
98C4×C127#12011yescharacteristic
99C4×C127#12111yescharacteristic
100C4×C127#12111yescharacteristic
101C4×C127#12311yescharacteristic
102Dic127—81no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127—81no—
106C2×C127#39411yescharacteristic
107C2×C127#47111yescharacteristic
108C2×C127#47311yescharacteristic
109#2052—1271nomaximal, Sylow
110C127#205211yescharacteristic, complemented, Sylow
111C2×#137D12711yescharacteristic
112C2×#108—1271no—
113#417—1271no—
114C2×#103—1271no—
115C2×#101—1271no—
116#408—1271no—
117#343—1271no—
118C2×#41C2×D12711yescharacteristic
119C22×Q8C2×D12711yescharacteristic
120#137—22no—
121#137—22no—
122C2×Q16—21no—
123C2×SD16—21no—
124C2×M16—1271no—
125C2×M16—1271no—
126C2×M16C2×D12711yescharacteristic
127C22×C8—1271no—
128C2×C4⋊C4—1271no—
129#108—2542no—
130#103—2542no—
131#103—2541no—
132#102—2541no—
133#101—2542no—
134C22⋊C8—2542no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8C22⋊D12712yes—
145C2×Q8C22⋊D12711yescharacteristic
146C2×D4—21no—
147C2×D4C22⋊D12711yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—2541no—
151C22×C4C22×D12711yescharacteristic
152Q16—24no—
153SD16—24no—
154M16—21no—
155M16—2541no—
156M16—2541no—
157M16—2541no—
158M16—2541no—
159M16C22⋊D12712yes—
160C2×C8—21no—
161C2×C8—1272no—
162C2×C8—2541no—
163C2×C8—2541no—
164C2×C8—2541no—
165C2×C8—2541no—
166C4⋊C4—2541no—
167C4⋊C4—2541no—
168C23—21no—
169C23#1452324711yescharacteristic
170Q8—21no—
171Q8—22no—
172Q8—21no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176D4—21no—
177D4—21no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—10161no—
187C2×C4—5081no—
188C2×C4D4×D12711yescharacteristic
189C2×C4C2×C22⋊D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4C2×C22⋊D12711yescharacteristic
192C2×C4#1452324711yescharacteristic
193C2×C4#1452324711yescharacteristic
194C8—22no—
195C8—5081no—
196C8—5081no—
197C8—2542no—
198C22—21no—
199C22—21no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22#1686118111yescharacteristic
204C22#1686119011yescharacteristic
205C22#1686119611yescharacteristic
206C4—41no—
207C4—42no—
208C4—10161no—
209C4#1686117811yescharacteristic
210C4#1686118011yescharacteristic
211C4#1686118111yescharacteristic
212C4#1686119011yescharacteristic
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2254979011yescharacteristic
217C2#2255014111yescharacteristic
218C2#2255014911yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#137×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
hypercenterC2×#137D127
hypocenterC127#2052
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16861144C2semidirect
3. C2×#16861133C2semidirect
4. C2×#16861128C2semidirect
6. #22549644C2semidirect
110. C127#2052semidirect

Maximal subgroups

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

subgroupindex
C2×#168611442
#225498772
C2×#168611332
C2×#168611282
#225498432
#225496442
C2×#137×C1272
#2052127