Finite Group Databasebeta

← #30651469

Subgroups — #30651469

Order 16256.

338 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550606C211yesmaximal, characteristic, complemented
2#22550596C211yesmaximal, characteristic
3#22550234C212yesmaximal
4#22550228C211yesmaximal, characteristic, complemented
5#22550160C212yesmaximal, complemented
6#22550097C212yesmaximal
7#22550087C211yesmaximal, characteristic, complemented
8#22550003C211yesmaximal, characteristic, complemented
9#22549982C212yesmaximal, complemented
10C4×#14523236C211yesmaximal, characteristic
11#476×C127C211yesmaximal, characteristic
12C2×Dic508C2212yes—
13C2×#14523236C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861208—21no—
16#16861208C2212yes—
17#16861207—22no—
18#16861202—22no—
19#16861196C2212yes—
20#16861188—21no—
21C4×Dic254C2211yescharacteristic
22C4×Dic254C2211yescharacteristic
23#16861174—21no—
24#16861174C2212yes—
25#16861174C2212yes—
26#16861168—21no—
27#16861140C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861134C2212yes—
30#16861132C2212yes—
31#16861132C2211yescharacteristic
32#16861132C2212yes—
33#16861130C2211yescharacteristic
34#16861119C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C4⋊Q8×C127C2211yescharacteristic
37C4×D4×C127C2211yescharacteristic
38#106×C127C2212yes—
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41C2×Dic254—22no—
42C2×Dic254D411yescharacteristic
43C2×Dic254C2312yes—
44C2×Dic254C2311yescharacteristic
45Dic508—24no—
46#14523236—22no—
47C2×C127⋊C8C2311yescharacteristic
48C2×C127⋊C8C2311yescharacteristic
49C4⋊Dic127—21no—
50C4⋊Dic127—21no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55Dic127⋊C4—22no—
56Dic127⋊C4—21no—
57Dic127⋊C4D412yes—
58C22⋊Dic127—22no—
59C22⋊Dic127—22no—
60C4×Dic127—22no—
61C4×Dic127—22no—
62C4×Dic127—21no—
63C4×Dic127C2312yes—
64C2×Q8×C127—21no—
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—21no—
67C2×C8×C127C2312yes—
68C4⋊C4×C127—21no—
69C4⋊C4×C127C2312yes—
70C4⋊C4×C127C2311yescharacteristic
71C22⋊C4×C127—21no—
72C42×C127C2311yescharacteristic
73Dic254—22no—
74Dic254—21no—
75Dic254—22no—
76Dic254—22no—
77Dic254—82no—
78Dic254#4112yes—
79C2×Dic127—21no—
80C2×Dic127—22no—
81C2×Dic127—21no—
82C2×Dic127C2×D412yes—
83C2×Dic127C2×D412yes—
84C127⋊C8—21no—
85C127⋊C8—21no—
86C23×C127—21no—
87Q8×C127—81no—
88D4×C127—21no—
89D4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2411yescharacteristic
96C8×C127—22no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127C22×D411yescharacteristic
101C4×C127—21no—
102C4×C127—41no—
103C4×C127—42no—
104C4×C127#13712yes—
105C4×C127C2×#4111yescharacteristic
106C4×C1272−1+411yescharacteristic
107Dic127—22no—
108Dic127—22no—
109Dic127—22no—
110Dic127—42no—
111Dic127—41no—
112C2×C127—81no—
113C2×C127#54811yescharacteristic
114C2×C127C2×#13711yescharacteristic
115C2×C127#57711yescharacteristic
116#3369—1271nomaximal, Sylow
117C127#336911yescharacteristic, complemented, Sylow
118Q82—1271no—
119#556—1271no—
120#489—1272no—
121#487—1271no—
122#476D12711yescharacteristic
123#475—1272no—
124#464—1272no—
125#462—1271no—
126#446—1271no—
127#441—1272no—
128C4×SD16—1271no—
129C2×Q16—1272no—
130C2×SD16—1271no—
131C4⋊Q8—1272no—
132C4⋊Q8—1271no—
133C4⋊Q8—2541no—
134C4⋊Q8—2542no—
135C4⋊Q8C2×D12711yescharacteristic
136#126—2542no—
137#124—1272no—
138C22⋊Q8—2541no—
139C4×Q8—1271no—
140C4×Q8—1272no—
141C4×Q8—1272no—
142C4×Q8—1271no—
143C4×Q8—2541no—
144C4×D4C2×D12711yescharacteristic
145#117—2541no—
146#106—1271no—
147#106C2×D12712yes—
148C4⋊C8—1272no—
149C4⋊C8C2×D12711yescharacteristic
150#103—1272no—
151#103—1271no—
152#103—1272no—
153#103—1272no—
154#102—1271no—
155#102C2×D12712yes—
156#97—1272no—
157C4×C8—1271no—
158C2×Q8—21no—
159C2×Q8—1271no—
160C2×Q8—1271no—
161C2×Q8—1272no—
162C2×Q8—2542no—
163C2×D4C22×D12711yescharacteristic
164C22×C4—21no—
165Q16—2544no—
166SD16—2542no—
167C2×C8—1271no—
168C2×C8—1271no—
169C2×C8C22×D12712yes—
170C4⋊C4—21no—
171C4⋊C4—1272no—
172C4⋊C4—1271no—
173C4⋊C4—1271no—
174C4⋊C4—2542no—
175C4⋊C4—2542no—
176C4⋊C4—2542no—
177C4⋊C4—2541no—
178C4⋊C4—2541no—
179C4⋊C4—2541no—
180C4⋊C4C22×D12712yes—
181C4⋊C4C22×D12711yescharacteristic
182C22⋊C4—21no—
183C22⋊C4—2542no—
184C22⋊C4—2542no—
185C42—1272no—
186C42—2542no—
187C42—2542no—
188C42—2541no—
189C42C22×D12711yescharacteristic
190C23—21no—
191Q8—81no—
192Q8—1272no—
193Q8—10162no—
194Q8—2542no—
195Q8—2542no—
196Q8—2542no—
197Q8—2541no—
198D4—21no—
199D4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—1272no—
204C2×C4—1272no—
205C2×C4—2542no—
206C2×C4—2541no—
207C2×C4—2541no—
208C2×C4D4×D12711yescharacteristic
209C2×C4D4×D12711yescharacteristic
210C2×C4C23×D12711yescharacteristic
211C8—22no—
212C8—2541no—
213C8—2541no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22C2×D4×D12711yescharacteristic
218C4—21no—
219C4—41no—
220C4—42no—
221C4—5082no—
222C4—5081no—
223C4—2542no—
224C4—2542no—
225C4—2542no—
226C4#1686123812yes—
227C4#41×D12711yescharacteristic
228C4#1686125711yescharacteristic
229C2—81no—
230C2#2255053611yescharacteristic
231C2C2×#1686123811yescharacteristic
232C2#2255068611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#476×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×C127C23
hypercenter#476D127
hypocenterC127#3369
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #22550606C2semidirect
4. #22550228C2semidirect
5. #22550160C2semidirect
7. #22550087C2semidirect
8. #22550003C2semidirect
9. #22549982C2semidirect
117. C127#3369semidirect

Maximal subgroups

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

subgroupindex
#225506062
#225505962
#225502342
#225502282
#225501602
#225500972
#225500872
#225500032
#225499822
C4×#145232362
#476×C1272
#3369127