Finite Group Databasebeta

← #30640030

Subgroups — #30640030

Order 16256.

262 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 69 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
2C2×#16861145C211yesmaximal, characteristic, complemented
3#22549875C211yesmaximal, characteristic
4C2×#16861128C211yesmaximal, characteristic, complemented
5#22549842C211yesmaximal, characteristic
6#22549645C211yesmaximal, characteristic, complemented
7C2×#103×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—22no—
10#16861240—22no—
11C2×#14523238—21no—
12C2×#14523235—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C2×#14523232C2211yescharacteristic
16#16861145—22no—
17#16861132—21no—
18#16861129—21no—
19#16861128—22no—
20(C2×D127)⋊C8—22no—
21C22×Q8×C127C2211yescharacteristic
22C22×C8×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24#103×C127—22no—
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254D411yescharacteristic
30C2×D508D411yescharacteristic
31C2×C22⋊D127—21no—
32C2×C4×D127—21no—
33#14523238—24no—
34#14523235—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39#14523232—21no—
40#14523232D412yes—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127—21no—
49C2×Q8×C127D412yes—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127C2311yescharacteristic
53C2×C8×C127—21no—
54C2×C8×C127—21no—
55C2×C8×C127D412yes—
56C4⋊C4×C127—21no—
57C4⋊C4×C127—21no—
58C22×D127—21no—
59Dic254—21no—
60Dic254—21no—
61D508—21no—
62D508—21no—
63C22⋊D127—41no—
64C22⋊D127—41no—
65C2×Dic127—21no—
66C4×D127—41no—
67C4×D127—41no—
68C127⋊C8—22no—
69C127⋊C8—41no—
70C127⋊C8—41no—
71C23×C127#4111yescharacteristic
72Q8×C127—22no—
73Q8×C127—22no—
74Q8×C127—41no—
75Q8×C127—41no—
76C2×C4×C127—22no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—81no—
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#12111yescharacteristic
91C22×C127#12311yescharacteristic
92C22×C127#12411yescharacteristic
93C2×D127—41no—
94C2×D127—41no—
95C2×D127—41no—
96C4×C127—42no—
97C4×C127—81no—
98C4×C127#12011yescharacteristic
99C4×C127#12111yescharacteristic
100C4×C127#12111yescharacteristic
101C4×C127#12311yescharacteristic
102Dic127—41no—
103C2×C127—21no—
104C2×C127—21no—
105C2×C127#39411yescharacteristic
106C2×C127#47111yescharacteristic
107C2×C127#47311yescharacteristic
108D127—81no—
109#2052—1271nomaximal, Sylow
110C127#205211yescharacteristic, complemented, Sylow
111C2×#137—1271no—
112C2×#108—1271no—
113#417—1271no—
114C2×#103D12711yescharacteristic, complemented
115C2×#101—1271no—
116#408—1271no—
117#343—1271no—
118C2×#41—1271no—
119C22×Q8C2×D12711yescharacteristic
120#137—2542no—
121#137—2542no—
122C2×Q16—2541no—
123C2×SD16—2541no—
124C2×M16—1271no—
125C2×M16—1271no—
126C2×M16—1271no—
127C22×C8C2×D12711yescharacteristic
128C2×C4⋊C4C2×D12711yescharacteristic
129#108—2542no—
130#103—22no—
131#103—2541no—
132#102—2541no—
133#101—2542no—
134C22⋊C8—2542no—
135#41—2541no—
136#41—2541no—
137#41—2541no—
138#41—2541no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—1271no—
145C2×Q8D50812yes—
146C2×D4—1271no—
147C2×D4—2541no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—2541no—
151C22×C4C22×D12711yescharacteristic
152Q16—2544no—
153SD16—2544no—
154M16—1272no—
155M16—2541no—
156M16—2541no—
157M16—2541no—
158M16—2541no—
159M16—2541no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—2541no—
163C2×C8—2541no—
164C2×C8—2541no—
165C2×C8C22⋊D12712yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C23—2541no—
169C23#1452324711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—41no—
173Q8—41no—
174Q8—2541no—
175Q8—2541no—
176D4—5081no—
177D4—5081no—
178D4—2541no—
179D4—2541no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—5081no—
186C2×C4—5081no—
187C2×C4—2541no—
188C2×C4D4×D12711yescharacteristic
189C2×C4D4×D12711yescharacteristic
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4C2×D50811yescharacteristic
192C2×C4#1452324711yescharacteristic
193C2×C4#1452324811yescharacteristic
194C8—22no—
195C8—5081no—
196C8—5081no—
197C8—2542no—
198C22—21no—
199C22—21no—
200C22—5081no—
201C22—5081no—
202C22—5081no—
203C22#1686118011yescharacteristic
204C22#1686119311yescharacteristic
205C22#1686119511yescharacteristic
206C4—42no—
207C4—81no—
208C4—5081no—
209C4#1686117911yescharacteristic
210C4#1686118111yescharacteristic
211C4#1686118311yescharacteristic
212C4#1686119111yescharacteristic
213C2—21no—
214C2—21no—
215C2—10161no—
216C2#2254979211yescharacteristic
217C2#2255013611yescharacteristic
218C2#2255014811yescharacteristic
subgroupG/H
centerC22#16861180
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#103×C127C2
socleC22×C127#121
prime radicalC2×#14523248C22
hypercenterC2×#103D127
hypocenterC127#2052
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. C2×#16861145C2semidirect
4. C2×#16861128C2semidirect
6. #22549645C2semidirect
7. C2×#103×C127C2semidirect
110. C127#2052semidirect
114. C2×#103D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168611452
#225498752
C2×#168611282
#225498422
#225496452
C2×#103×C1272
#2052127