Finite Group Databasebeta

← #30638955

Subgroups — #30638955

Order 16256.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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#22550357C211yesmaximal, characteristic
2#22549916C211yesmaximal, characteristic, complemented
3C2×#16861137C211yesmaximal, characteristic
4#22549875C211yesmaximal, characteristic
5#22549645C211yesmaximal, characteristic, complemented
6#22549636C211yesmaximal, characteristic, complemented
7#384×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C2×#14523232C2211yescharacteristic
11#16861174—21no—
12#16861171—21no—
13C4×C22⋊D127—22no—
14#16861168C2211yescharacteristic
15#16861167—21no—
16C4⋊C4×D127—21no—
17#16861142—21no—
18#16861140—21no—
19#16861137C414yescomplemented
20#16861132—21no—
21#16861129—21no—
22C2×C4⋊C4×C127—22no—
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26#14523248—21no—
27#14523248—21no—
28#14523248C2×C412yes—
29#14523248Q812yes—
30C2×Dic254D411yescharacteristic
31C2×D508D411yescharacteristic
32C2×C22⋊D127—21no—
33C2×C4×D127—22no—
34C2×C4×D127—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232C2×C412yescomplemented
39C2×C127⋊C8—21no—
40C2×C127⋊C8—21no—
41C4⋊Dic127D411yescharacteristic
42C4⋊Dic127D411yescharacteristic
43Dic127⋊C4—22no—
44C22⋊Dic127—21no—
45(C2×D127)⋊C4—22no—
46C4×Dic127—22no—
47C4×Dic127—21no—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127D412yes—
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127—44no—
57C42×C127—21no—
58C42×C127C2×C412yescomplemented
59C22×D127—21no—
60Dic254—21no—
61Dic254C4⋊C412yescomplemented
62D508—21no—
63D508C4⋊C412yescomplemented
64C22⋊D127—22no—
65C22⋊D127—22no—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C4×D127—22no—
70C4×D127—22no—
71C4×D127—82no—
72C127⋊C8—22no—
73C127⋊C8—41no—
74C23×C127C2×D411yescharacteristic
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—42no—
80C2×C4×C127—42no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×Q811yescharacteristic
86C2×C4×C127#4111yescharacteristic
87C2×C4×C127#4111yescharacteristic
88C22×C127—21no—
89C22×C127—21no—
90C22×C127C4×D411yescharacteristic
91C22×C127#12011yescharacteristic
92C22×C127C22⋊Q811yescharacteristic
93C2×D127—22no—
94C2×D127—41no—
95C2×D127—41no—
96C4×C127—24no—
97C4×C127—42no—
98C4×C127—81no—
99C4×C127C2×C4⋊C411yescharacteristic
100C4×C127C4×D411yescharacteristic
101C4×C127C22⋊Q811yescharacteristic
102C4×C127#12311yescharacteristic
103Dic127—22no—
104Dic127—41no—
105Dic127—41no—
106C2×C127—21no—
107C2×C127—21no—
108C2×C127#38511yescharacteristic
109C2×C127#45311yescharacteristic
110C2×C127#45611yescharacteristic
111D127—42no—
112#1859—1271nomaximal, Sylow
113C127#185911yescharacteristic, complemented, Sylow
114#520—1271no—
115#428—1271no—
116C2×#104—1271no—
117#417—1271no—
118#384D12711yescharacteristic, complemented
119#343—1271no—
120#339—1271no—
121C2×#41—1271no—
122C2×M16—1271no—
123C2×M16—1271no—
124C4×Q8—2541no—
125C4×D4—2541no—
126C4×D4—2542no—
127#117—1271no—
128#117—2541no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—2541no—
131C2×C4⋊C4C2×D12711yescharacteristic
132C2×C4⋊C4C2×D12711yescharacteristic
133C2×C42C2×D12711yescharacteristic
134#107—2541no—
135#106—2541no—
136#104—1274no—
137#103—2541no—
138#102—2541no—
139#41—1272no—
140#41—1272no—
141#41—2541no—
142#41—2541no—
143C2×Q8—1271no—
144C2×D4—1271no—
145C2×D4—2541no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—2542no—
149C22×C4—2541no—
150C22×C4C22⋊D12712yes—
151C22×C4C22×D12711yescharacteristic
152M16—1272no—
153M16—2541no—
154M16—2541no—
155M16—2541no—
156C2×C8—2541no—
157C2×C8—2541no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—44no—
163C4⋊C4—1271no—
164C4⋊C4—1271no—
165C4⋊C4—2542no—
166C22⋊C4—2542no—
167C22⋊C4—2541no—
168C42—21no—
169C42—2542no—
170C42—2541no—
171C42C4×D12712yescomplemented
172C23—2541no—
173C23C2×C22⋊D12711yescharacteristic
174Q8—1272no—
175Q8—2541no—
176D4—1272no—
177D4—2542no—
178D4—2542no—
179D4—2541no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—10162no—
188C2×C4—2541no—
189C2×C4—2541no—
190C2×C4—2541no—
191C2×C4—2542no—
192C2×C4—2542no—
193C2×C4C2×C4×D12711yescharacteristic
194C2×C4D4×D12711yescharacteristic
195C2×C4D4×D12711yescharacteristic
196C2×C4Q8×D12711yescharacteristic
197C2×C4#1452324811yescharacteristic
198C2×C4#1452324911yescharacteristic
199C8—5081no—
200C8—2542no—
201C22—21no—
202C22—21no—
203C22—5081no—
204C22—5081no—
205C22—2542no—
206C22C4×C22⋊D12711yescharacteristic
207C22#1686117711yescharacteristic
208C22#1686118511yescharacteristic
209C4—24no—
210C4—42no—
211C4—81no—
212C4—5081no—
213C4—5081no—
214C4—2542no—
215C4C4⋊C4×D12711yescharacteristic
216C4#1686117111yescharacteristic
217C4#1686118711yescharacteristic
218C4#1686119411yescharacteristic
219C2—21no—
220C2—21no—
221C2—5082no—
222C2#2254974711yescharacteristic
223C2#2255004311yescharacteristic
224C2#2255005811yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#384×C127C2
socleC22×C127#120
prime radicalC2×#14523248C22
cosocleC22×C4×C127C23
hypercenter#384D127
hypocenterC127#1859
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #22549916C2semidirect
5. #22549645C2semidirect
6. #22549636C2semidirect
7. #384×C127C2semidirect
19. #16861137C4semidirect
38. #14523232C2×C4semidirect
58. C42×C127C2×C4semidirect
61. Dic254C4⋊C4semidirect
63. D508C4⋊C4semidirect
113. C127#1859semidirect
118. #384D127semidirect
171. C42C4×D127semidirect

Maximal subgroups

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

subgroupindex
#225503572
#225499162
C2×#168611372
#225498752
#225496452
#225496362
#384×C1272
#1859127