Finite Group Databasebeta

← #30639587

Subgroups — #30639587

Order 16256.

348 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#14523235C211yesmaximal, characteristic
2C2×#16861140C211yesmaximal, characteristic, complemented
3C2×#16861133C211yesmaximal, characteristic, complemented
4C2×#16861129C211yesmaximal, characteristic
5C2×C4×C127⋊C8C211yesmaximal, characteristic, complemented
6#22549778C211yesmaximal, characteristic
7#391×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523235—22no—
10C2×#14523235—22no—
11C2×#14523235C414yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14C2×C4⋊Dic127C2211yescharacteristic
15C2×(C2×D127)⋊C4—22no—
16#16861140—22no—
17#16861133—22no—
18#16861129—22no—
19C4×C127⋊C8—22no—
20C22×Q8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C42×C127C2211yescharacteristic
23#95×C127—22no—
24C23×D127—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508C2×C412yes—
31C22×Dic127—21no—
32#14523235—28no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8C2×C412yes—
36C2×C127⋊C8D412yes—
37C2×C127⋊C8D412yes—
38C2×C127⋊C8D412yes—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41(C2×D127)⋊C4—44no—
42C2×Q8×C127—21no—
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127C2×C412yes—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127D412yes—
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C42×C127—21no—
55C42×C127—21no—
56C22×D127—22no—
57C22×D127—41no—
58C22×D127—41no—
59C22×D127—42no—
60C22×D127—42no—
61D508—22no—
62D508—22no—
63D508—41no—
64D508—41no—
65C2×Dic127—41no—
66C2×Dic127—42no—
67C127⋊C8—22no—
68C127⋊C8C22⋊C414yescomplemented
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—22no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—42no—
81C2×C4×C127SD1614yescomplemented
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C22×C127C4×D411yescharacteristic
89C22×C127#12111yescharacteristic
90C22×C127#12711yescharacteristic
91C22×C127C2×SD1612yes—
92C22×C127#13512yes—
93C2×D127—42no—
94C2×D127—44no—
95C2×D127—82no—
96C2×D127—81no—
97C2×D127—81no—
98C4×C127—24no—
99C4×C127—42no—
100C4×C127—81no—
101C4×C127C2×C22⋊C411yescharacteristic
102C4×C127C4×D411yescharacteristic
103C4×C127#12111yescharacteristic
104C4×C127#12411yescharacteristic
105Dic127—81no—
106C2×C127#39011yescharacteristic
107C2×C127C4×SD1612yes—
108C2×C127#46512yes—
109C2×C127#49211yescharacteristic
110C2×C127#49511yescharacteristic
111D127—82no—
112#1968—1271nomaximal, Sylow
113C127#196811yescharacteristic, complemented, Sylow
114C22×SD16—1271no—
115C2×#106—1271no—
116C2×#103—1271no—
117C2×#102—1271no—
118C2×C4×C8—1271no—
119#391D12711yescharacteristic, complemented
120#390—1271no—
121C22×Q8C2×D12711yescharacteristic
122C22×D4—1271no—
123C2×SD16—1274no—
124C2×SD16—2542no—
125C2×SD16—2542no—
126C22×C8—1271no—
127C22×C8—1271no—
128C2×C4⋊C4—1271no—
129C2×C4⋊C4C2×D12711yescharacteristic
130C2×C22⋊C4—2542no—
131C2×C42C2×D12711yescharacteristic
132#106—2542no—
133#103—2542no—
134#102—2542no—
135C4×C8—2542no—
136#95—22no—
137C24—2541no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C4×D12712yes—
144C2×D4—1272no—
145C2×D4—2541no—
146C2×D4—2541no—
147C2×D4—2541no—
148C2×D4—2541no—
149C2×D4—2541no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—2541no—
153C22×C4C22⋊D12712yes—
154C22×C4C22×D12711yescharacteristic
155SD16—2548no—
156C2×C8—1272no—
157C2×C8—1272no—
158C2×C8—1272no—
159C2×C8—1272no—
160C2×C8—2541no—
161C2×C8—2541no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C22⋊C4—5084no—
167C42—21no—
168C42—21no—
169C23—5081no—
170C23—5082no—
171C23—5082no—
172C23—5081no—
173C23—2542no—
174C23C2×C22⋊D12711yescharacteristic
175Q8—22no—
176Q8—22no—
177Q8—41no—
178Q8—41no—
179D4—5081no—
180D4—5081no—
181D4—2542no—
182D4—2542no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—5081no—
191C2×C4—5082no—
192C2×C4#1452323514yescomplemented
193C2×C4C2×C4×D12711yescharacteristic
194C2×C4D4×D12711yescharacteristic
195C2×C4D4×D12711yescharacteristic
196C2×C4D4×D12711yescharacteristic
197C2×C4#1452324711yescharacteristic
198C2×C4#1452324811yescharacteristic
199C8—1274no—
200C8—2542no—
201C22—10162no—
202C22—10161no—
203C22—10161no—
204C22—5082no—
205C22—5084no—
206C22C4×C22⋊D12711yescharacteristic
207C22#1686118111yescharacteristic
208C22#1686120511yescharacteristic
209C22C2×#1452323512yes—
210C22#1686122912yes—
211C4—24no—
212C4—42no—
213C4—81no—
214C4—10161no—
215C4C22⋊C4×D12711yescharacteristic
216C4(C22⋊D127)⋊C411yescharacteristic
217C4#1686118211yescharacteristic
218C4#1686119511yescharacteristic
219C2—10162no—
220C2#2254977611yescharacteristic
221C2C4×#1452323512yes—
222C2#2255010712yes—
223C2#2255025011yescharacteristic
224C2#2255025811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#391D127
hypocenterC127#1968
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#16861140C2semidirect
3. C2×#16861133C2semidirect
5. C2×C4×C127⋊C8C2semidirect
7. #391×C127C2semidirect
11. C2×#14523235C4semidirect
68. C127⋊C8C22⋊C4semidirect
81. C2×C4×C127SD16semidirect
113. C127#1968semidirect
119. #391D127semidirect
192. C2×C4#14523235semidirect

Maximal subgroups

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

subgroupindex
C22×#145232352
C2×#168611402
C2×#168611332
C2×#168611292
C2×C4×C127⋊C82
#225497782
#391×C1272
#1968127