Finite Group Databasebeta

← #30639208

Subgroups — #30639208

Order 16256.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 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
2#22550643C211yesmaximal, characteristic, complemented
3#22549880C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic, complemented
5#22549646C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9#16861234—22no—
10#16861234—22no—
11C2×#14523235C412yescomplemented
12C2×#14523233C412yescomplemented
13#16861216—22no—
14#16861216—22no—
15C2×#14523232—22no—
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18C2×#14523232C2211yescharacteristic
19#16861129—22no—
20#16861125—22no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#104×C127—22no—
25C23×D127—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508C2×C412yes—
32#14523235—24no—
33#14523233—24no—
34#14523232—24no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39#14523232—21no—
40#14523232D412yes—
41C2×C127⋊C8—24no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8C2×C412yes—
44C2×C127⋊C8D412yes—
45#41×C127—21no—
46#41×C127—21no—
47#41×C127D412yes—
48#41×C127D412yes—
49C2×Q8×C127C2×C411yescharacteristic
50C2×D4×C127—21no—
51C2×D4×C127C2×C411yescharacteristic
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C42×C127—21no—
60C22×D127—22no—
61C22×D127—41no—
62C22×D127—41no—
63C22×D127—82no—
64D508—22no—
65D508—22no—
66D508—41no—
67D508—41no—
68C127⋊C8—22no—
69C127⋊C8—22no—
70C127⋊C8—22no—
71C23×C127—21no—
72C23×C127#4111yescharacteristic
73Q8×C127—21no—
74Q8×C127C22⋊C412yes—
75D4×C127—22no—
76D4×C127—22no—
77D4×C127—21no—
78D4×C127C22⋊C412yes—
79C2×C4×C127—21no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—41no—
83C2×C4×C127C22×C411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C8×C127—41no—
90C22×C127—22no—
91C22×C127—41no—
92C22×C127—81no—
93C22×C127C4×D411yescharacteristic
94C22×C127#12111yescharacteristic
95C22×C127#12311yescharacteristic
96C2×D127—42no—
97C2×D127—82no—
98C2×D127—82no—
99C2×D127—82no—
100C4×C127—22no—
101C4×C127—81no—
102C4×C127C2×C22⋊C411yescharacteristic
103C4×C127C4×D411yescharacteristic
104C4×C127#12011yescharacteristic
105C4×C127#12111yescharacteristic
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—21no—
109C2×C127—42no—
110C2×C127#38611yescharacteristic
111D127—82no—
112#1903—1271nomaximal, Sylow
113C127#190311yescharacteristic, complemented, Sylow
114C2×AGL(1,ℤ/8)—1271no—
115#568—1271no—
116#421D12711yescharacteristic, complemented
117#418—1271no—
118C2×#100—1271no—
119#344—1271no—
120#338—1271no—
121C2×#41C2×D12711yescharacteristic
122C22×D4—1271no—
123AGL(1,ℤ/8)—2542no—
124AGL(1,ℤ/8)—2542no—
125C2×SD16—1272no—
126C2×D8—1272no—
127#131—2542no—
128#131—2542no—
129C2×M16—1271no—
130C2×M16—1271no—
131C2×M16—1271no—
132C2×M16—2542no—
133C2×M16C2×D12711yescharacteristic
134#117C2×D12711yescharacteristic
135#104—22no—
136#102—2542no—
137#100—2542no—
138C24—2541no—
139#41—21no—
140#41—21no—
141#41C22⋊D12712yes—
142#41D50812yes—
143C2×Q8C4×D12711yescharacteristic
144C2×D4—21no—
145C2×D4—1272no—
146C2×D4—2541no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4—2541no—
150C2×D4—2541no—
151C2×D4C4×D12711yescharacteristic
152C22×C4—21no—
153C22×C4C22×D12711yescharacteristic
154SD16—2544no—
155D8—2544no—
156M16—21no—
157M16—21no—
158M16—1272no—
159M16—2544no—
160M16—2541no—
161M16—2541no—
162M16—2541no—
163M16—2541no—
164M16—2541no—
165C2×C8—21no—
166C2×C8—1272no—
167C2×C8—1272no—
168C2×C8—2544no—
169C2×C8—2541no—
170C4⋊C4—21no—
171C22⋊C4—21no—
172C42—21no—
173C23—21no—
174C23—10162no—
175C23—5081no—
176C23—5081no—
177C23—2542no—
178C23#1452324711yescharacteristic
179Q8—21no—
180Q8(C2×D127)⋊C412yes—
181D4—22no—
182D4—22no—
183D4—21no—
184D4—5081no—
185D4—5081no—
186D4—2542no—
187D4—2542no—
188D4(C2×D127)⋊C412yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4C2×C4×D12711yescharacteristic
194C2×C4D4×D12711yescharacteristic
195C2×C4D4×D12711yescharacteristic
196C2×C4C2×C22⋊D12711yescharacteristic
197C2×C4C2×D50811yescharacteristic
198C2×C4#1452324711yescharacteristic
199C8—41no—
200C8—2542no—
201C8—2542no—
202C8—2542no—
203C22—22no—
204C22—41no—
205C22—81no—
206C22—10162no—
207C22—10162no—
208C22—10162no—
209C22—5082no—
210C22(C22⋊D127)⋊C411yescharacteristic
211C22#1686118111yescharacteristic
212C22#1686119311yescharacteristic
213C4—22no—
214C4—81no—
215C4C2×(C2×D127)⋊C411yescharacteristic
216C4(C22⋊D127)⋊C411yescharacteristic
217C4#1686117911yescharacteristic
218C4#1686118111yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—10162no—
224C2#2254975311yescharacteristic
subgroupG/H
centerC2#22549753
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#386
prime radicalC2×#16861234C2
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1903
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #22550643C2semidirect
3. #22549880C2semidirect
4. C2×#16861125C2semidirect
5. #22549646C2semidirect
6. C127⋊#338C2semidirect
7. #421×C127C2semidirect
11. C2×#14523235C4semidirect
12. C2×#14523233C4semidirect
113. C127#1903semidirect
116. #421D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
#225506432
#225498802
C2×#168611252
#225496462
C127⋊#3382
#421×C1272
#1903127