Finite Group Databasebeta

← #30640021

Subgroups — #30640021

Order 16256.

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

normal subgroups only

This group has 282 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×#16861234C211yesmaximal, characteristic
2C2×#16861145C211yesmaximal, characteristic, complemented
3#22549877C211yesmaximal, characteristic, complemented
4C2×#16861131C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6#22549644C211yesmaximal, characteristic, complemented
7#408×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9#16861234—22no—
10#16861234—22no—
11C2×#14523235—21no—
12C2×#14523233—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C2×C4⋊Dic127C2211yescharacteristic
16#16861145—22no—
17#16861133—21no—
18#16861131—22no—
19#16861130—21no—
20#16861125—22no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23C22×C8×C127C2211yescharacteristic
24C22⋊C8×C127—22no—
25C23×D127—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508D412yes—
32C22×Dic127—21no—
33#14523235—24no—
34#14523233—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232D412yes—
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127—21no—
42C4⋊Dic127—21no—
43#41×C127—21no—
44#41×C127—21no—
45#41×C127—21no—
46#41×C127—21no—
47C2×Q8×C127D411yescharacteristic
48C2×D4×C127—21no—
49C2×D4×C127D411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52M16×C127—21no—
53M16×C127—21no—
54C2×C8×C127—21no—
55C2×C8×C127—21no—
56C2×C8×C127—21no—
57C2×C8×C127D412yes—
58C22×D127—22no—
59C22×D127—41no—
60C22×D127—41no—
61C22×D127—81no—
62C22×D127—81no—
63D508—22no—
64D508—22no—
65D508—41no—
66D508—41no—
67C2×Dic127—41no—
68C2×Dic127—81no—
69C127⋊C8—22no—
70C127⋊C8—41no—
71C23×C127—21no—
72C23×C127#4111yescharacteristic
73Q8×C127—21no—
74Q8×C127—21no—
75D4×C127—21no—
76D4×C127—21no—
77D4×C127—41no—
78D4×C127—41no—
79C2×C4×C127—21no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—22no—
89C8×C127—41no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—41no—
93C22×C127—41no—
94C22×C127—41no—
95C22×C127#12111yescharacteristic
96C22×C127#12311yescharacteristic
97C22×C127#12411yescharacteristic
98C2×D127—42no—
99C2×D127—82no—
100C2×D127—81no—
101C2×D127—81no—
102C2×D127—82no—
103C4×C127—41no—
104C4×C127#12011yescharacteristic
105C4×C127#12111yescharacteristic
106C4×C127#12111yescharacteristic
107C4×C127#12311yescharacteristic
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—81no—
112C2×C127#39411yescharacteristic
113C2×C127#47111yescharacteristic
114C2×C127#47211yescharacteristic
115D127—82no—
116#2051—1271nomaximal, Sylow
117C127#205111yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1271no—
119C2×#108—1271no—
120#417—1271no—
121C2×#102—1271no—
122C2×#100—1271no—
123#408D12711yescharacteristic, complemented
124#343—1271no—
125C2×#41C2×D12711yescharacteristic
126C22×D4—1271no—
127AGL(1,ℤ/8)—2542no—
128AGL(1,ℤ/8)—2542no—
129C2×SD16—2541no—
130C2×D8—2541no—
131C2×M16—1271no—
132C2×M16—1271no—
133C2×M16C2×D12711yescharacteristic
134C22×C8C2×D12711yescharacteristic
135C2×C4⋊C4—1271no—
136#108—2542no—
137#103—2541no—
138#102—2541no—
139#102—2542no—
140#100—2542no—
141C22⋊C8—22no—
142C24—2541no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8D50811yescharacteristic
148C2×D4—21no—
149C2×D4—1272no—
150C2×D4—2541no—
151C2×D4—2541no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4D50811yescharacteristic
156C22×C4—21no—
157C22×C4—2541no—
158C22×C4C22×D12711yescharacteristic
159SD16—2544no—
160D8—2544no—
161M16—21no—
162M16—21no—
163M16—1272no—
164M16—2541no—
165M16—2541no—
166M16—2541no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—2541no—
171C2×C8—2541no—
172C2×C8C22⋊D12712yes—
173C4⋊C4—2541no—
174C4⋊C4—2541no—
175C23—21no—
176C23—10161no—
177C23—10161no—
178C23—5081no—
179C23—5081no—
180C23—2542no—
181C23#1452324811yescharacteristic
182Q8—21no—
183Q8—21no—
184D4—21no—
185D4—21no—
186D4—41no—
187D4—41no—
188D4—5081no—
189D4—5081no—
190D4—2542no—
191D4—2542no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—10161no—
196C2×C4—5081no—
197C2×C4D4×D12711yescharacteristic
198C2×C4D4×D12711yescharacteristic
199C2×C4C2×C22⋊D12711yescharacteristic
200C2×C4C2×D50811yescharacteristic
201C2×C4#1452324711yescharacteristic
202C2×C4#1452324711yescharacteristic
203C8—22no—
204C8—41no—
205C8—5081no—
206C8—2542no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—10162no—
213C22—10162no—
214C22—10161no—
215C22—10161no—
216C22—5082no—
217C22#1686118311yescharacteristic
218C22#1686119111yescharacteristic
219C22#1686119511yescharacteristic
220C4—41no—
221C4—10161no—
222C4#1686117911yescharacteristic
223C4#1686118011yescharacteristic
224C4#1686118111yescharacteristic
225C4#1686119311yescharacteristic
226C2—21no—
227C2—21no—
228C2—81no—
229C2—10162no—
230C2#2254979211yescharacteristic
231C2#2255013911yescharacteristic
232C2#2255014611yescharacteristic
subgroupG/H
centerC22#16861183
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC22×C127#121
prime radicalC2×#16861234C2
hypercenter#408D127
hypocenterC127#2051
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#16861145C2semidirect
3. #22549877C2semidirect
4. C2×#16861131C2semidirect
5. C2×#16861125C2semidirect
6. #22549644C2semidirect
7. #408×C127C2semidirect
117. C127#2051semidirect
123. #408D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C2×#168611452
#225498772
C2×#168611312
C2×#168611252
#225496442
#408×C1272
#2051127