Finite Group Databasebeta

← #30639227

Subgroups — #30639227

Order 16256.

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

normal subgroups only

This group has 310 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
1C2×#16861239C211yesmaximal, characteristic
2#22550353C211yesmaximal, characteristic, complemented
3#22549893C211yesmaximal, characteristic
4#22549884C211yesmaximal, characteristic, complemented
5#22549646C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#410×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9#16861239—22no—
10#16861239—22no—
11C2×#14523238C412yescomplemented
12C2×#14523236C412yescomplemented
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C4×Dic254—22no—
16#16861174—22no—
17#16861168C2211yescharacteristic
18#16861165—22no—
19#16861136—22no—
20#16861132—22no—
21C2×#41×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#99×C127—22no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254C2×C412yes—
29C2×Dic254D412yes—
30C2×Dic254D412yes—
31C22×Dic127—21no—
32#14523238—24no—
33#14523236—24no—
34#14523232—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8C2×C412yes—
40C4⋊Dic127—21no—
41Dic127⋊C4—24no—
42C22⋊Dic127—21no—
43C4×Dic127—24no—
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
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127D412yes—
58C42×C127—21no—
59C42×C127—21no—
60Dic254—22no—
61Dic254—22no—
62Dic254—22no—
63Dic254C22⋊C414yes—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C127⋊C8—22no—
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×C127—41no—
83C2×C4×C127C22×C411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C22×C127—41no—
90C22×C127—41no—
91C22×C127—81no—
92C22×C127C4×D411yescharacteristic
93C22×C127#12011yescharacteristic
94C22×C127#12111yescharacteristic
95C4×C127—42no—
96C4×C127—41no—
97C4×C127—81no—
98C4×C127C2×C22⋊C411yescharacteristic
99C4×C127C4×D411yescharacteristic
100C4×C127#12111yescharacteristic
101C4×C127#12311yescharacteristic
102Dic127—24no—
103Dic127—42no—
104C2×C127—21no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—81no—
108C2×C127#38611yescharacteristic
109#1905—1271nomaximal, Sylow
110C127#190511yescharacteristic, complemented, Sylow
111C2×#137—1271no—
112#519—1271no—
113#421—1271no—
114#419—1271no—
115#410D12711yescharacteristic
116#344—1271no—
117H(3,ℤ/4)—1271no—
118C2×#41C2×D12711yescharacteristic
119C22×Q8—1271no—
120#137—2542no—
121#137—2542no—
122C2×Q16—1272no—
123C2×SD16—1272no—
124C2×M16—1271no—
125C2×M16—1271no—
126C4×Q8—2542no—
127C4×Q8—2542no—
128#117—1271no—
129#117—2542no—
130#117C2×D12711yescharacteristic
131#117C2×D12711yescharacteristic
132#104—2542no—
133#103—2542no—
134#99—22no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—1272no—
140C2×Q8—1272no—
141C2×Q8—1272no—
142C2×Q8—2541no—
143C2×Q8—2541no—
144C2×Q8—2541no—
145C2×Q8C4×D12711yescharacteristic
146C2×D4—21no—
147C2×D4C4×D12711yescharacteristic
148C22×C4—21no—
149C22×C4—2541no—
150C22×C4C22×D12711yescharacteristic
151Q16—2544no—
152SD16—2544no—
153M16—2541no—
154M16—2541no—
155M16—2541no—
156M16—2541no—
157C2×C8—1272no—
158C2×C8—2541no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—2544no—
162C4⋊C4—2541no—
163C22⋊C4—21no—
164C22⋊C4—2541no—
165C22⋊C4C22⋊D12712yes—
166C42—21no—
167C42—21no—
168C42—1272no—
169C42—2544no—
170C23—21no—
171C23C2×C22⋊D12711yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—1274no—
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—41no—
187C2×C4—2542no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4C2×C4×D12711yescharacteristic
192C2×C4D4×D12711yescharacteristic
193C2×C4D4×D12711yescharacteristic
194C2×C4D4×D12711yescharacteristic
195C2×C4#1452324711yescharacteristic
196C2×C4#1452324811yescharacteristic
197C8—5081no—
198C8—2542no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22C4×C22⋊D12711yescharacteristic
203C22#1686117711yescharacteristic
204C22#1686118011yescharacteristic
205C4—42no—
206C4—41no—
207C4—81no—
208C4—5082no—
209C4—2544no—
210C4C22⋊C4×D12711yescharacteristic
211C4(C22⋊D127)⋊C411yescharacteristic
212C4#1686118211yescharacteristic
213C4#1686119111yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#386
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#410D127
hypocenterC127#1905
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22550353C2semidirect
4. #22549884C2semidirect
5. #22549646C2semidirect
6. #22549632C2semidirect
11. C2×#14523238C4semidirect
12. C2×#14523236C4semidirect
110. C127#1905semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
#225503532
#225498932
#225498842
#225496462
#225496322
#410×C1272
#1905127