Finite Group Databasebeta

← #30639264

Subgroups — #30639264

Order 16256.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#22550447C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic, complemented
3#22549893C211yesmaximal, characteristic
4C2×#16861127C211yesmaximal, characteristic, complemented
5#22549633C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861207C412yescomplemented
11#16861196C412yescomplemented
12#16861190—22no—
13#16861188—22no—
14C4×Dic254—22no—
15#16861174—22no—
16#16861168C2211yescharacteristic
17#16861168C2211yescharacteristic
18#16861165—22no—
19#16861136—22no—
20#16861127—22no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#104×C127—22no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C2×Dic254D412yes—
31C22×Dic127D412yes—
32#14523232—21no—
33#14523232—21no—
34C2×C127⋊C8—21no—
35C4⋊Dic127—21no—
36C4⋊Dic127—21no—
37Dic127⋊C4—24no—
38Dic127⋊C4—24no—
39C22⋊Dic127—24no—
40C22⋊Dic127—21no—
41C22⋊Dic127—21no—
42C4×Dic127—24no—
43C4×Dic127C2×C412yes—
44C4×Dic127D412yes—
45#41×C127—21no—
46#41×C127—21no—
47#41×C127—21no—
48#41×C127—21no—
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×C127D412yes—
60Dic254—22no—
61Dic254—22no—
62Dic254—22no—
63Dic254—22no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127C22⋊C414yes—
69C127⋊C8—41no—
70C23×C127—21no—
71C23×C127C2×D411yescharacteristic
72Q8×C127—21no—
73Q8×C127—21no—
74D4×C127—21no—
75D4×C127—21no—
76D4×C127—41no—
77D4×C127—41no—
78C2×C4×C127—21no—
79C2×C4×C127—22no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C8×C127—41no—
89C22×C127—41no—
90C22×C127—42no—
91C22×C127—41no—
92C22×C127C2×C22⋊C411yescharacteristic
93C22×C127#12011yescharacteristic
94C22×C127#12311yescharacteristic
95C4×C127—42no—
96C4×C127—41no—
97C4×C127C4×D411yescharacteristic
98C4×C127C4×D411yescharacteristic
99C4×C127#12111yescharacteristic
100C4×C127#12111yescharacteristic
101Dic127—24no—
102Dic127—42no—
103Dic127—42no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—81no—
108C2×C127#38611yescharacteristic
109#1912—1271nomaximal, Sylow
110C127#191211yescharacteristic, complemented, Sylow
111#536—1271no—
112#519—1271no—
113#421—1271no—
114#421D12711yescharacteristic
115C2×#101—1271no—
116H(3,ℤ/4)—1271no—
117H(3,ℤ/4)—1271no—
118C2×#41C2×D12711yescharacteristic
119C22×Q8—1271no—
120C2×M16—1271no—
121C2×M16C2×D12711yescharacteristic
122C4⋊Q8—1272no—
123#124—1272no—
124#123—2542no—
125C22⋊Q8—2542no—
126C4×Q8—2542no—
127C4×Q8—2542no—
128#117—1271no—
129#117—1271no—
130#117—2542no—
131#117C2×D12711yescharacteristic
132#104—22no—
133#104—2542no—
134#101—2542no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—1272no—
140C2×Q8—1272no—
141C2×Q8—2541no—
142C2×Q8—2541no—
143C2×Q8—2541no—
144C2×Q8—2541no—
145C2×Q8C4×D12711yescharacteristic
146C2×D4—21no—
147C2×D4C4×D12711yescharacteristic
148C22×C4—21no—
149C22×C4—1272no—
150C22×C4C22×D12711yescharacteristic
151M16—21no—
152M16—21no—
153M16—2541no—
154M16—2541no—
155C2×C8—21no—
156C2×C8—2541no—
157C4⋊C4—21no—
158C4⋊C4—2544no—
159C4⋊C4—2544no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C22⋊C4—21no—
163C22⋊C4—2544no—
164C22⋊C4—2541no—
165C22⋊C4—2541no—
166C42—1272no—
167C42—1272no—
168C42—2544no—
169C42C22⋊D12712yes—
170C23—21no—
171C23D4×D12711yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—2542no—
175Q8—2542no—
176Q8—2542no—
177Q8—2542no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1274no—
187C2×C4—2542no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4D4×D12711yescharacteristic
194C2×C4C2×C22⋊D12711yescharacteristic
195C2×C4#1452324711yescharacteristic
196C2×C4#1452324811yescharacteristic
197C8—41no—
198C8—5081no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D12711yescharacteristic
203C22#1686117711yescharacteristic
204C22#1686119111yescharacteristic
205C4—42no—
206C4—41no—
207C4—5082no—
208C4—5082no—
209C4—2544no—
210C4(C22⋊D127)⋊C411yescharacteristic
211C4C4×C22⋊D12711yescharacteristic
212C4#1686118011yescharacteristic
213C4#1686118211yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#386
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1912
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. #22550353C2semidirect
4. C2×#16861127C2semidirect
5. #22549633C2semidirect
6. #22549632C2semidirect
10. #16861207C4semidirect
11. #16861196C4semidirect
110. C127#1912semidirect

Maximal subgroups

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

subgroupindex
#225504472
#225503532
#225498932
C2×#168611272
#225496332
#225496322
#421×C1272
#1912127