Finite Group Databasebeta

← #30650531

Subgroups — #30650531

Order 16256.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550692C212yesmaximal, complemented
2#22550691C212yesmaximal, complemented
3#22550255C211yesmaximal, characteristic, complemented
4#22550150C212yesmaximal, complemented
5#22550149C212yesmaximal, complemented
6#22550060C212yesmaximal, complemented
7#22550058C211yesmaximal, characteristic, complemented
8#22549990C212yesmaximal, complemented
9#456×C127C211yesmaximal, characteristic, complemented
10#16861257C2211yescharacteristic
11#16861256C2212yes—
12#16861242C2212yes—
13#16861241—22no—
14#16861240—22no—
15#16861240C2212yes—
16#16861239—22no—
17#16861239C2212yes—
18#16861238—22no—
19#16861230—22no—
20#16861229—22no—
21#16861229C2212yes—
22C2×#14523238—22no—
23C2×#14523238C2212yes—
24C2×#14523238C2212yes—
25Q16×D127—22no—
26#16861217C2212yes—
27#16861216C2212yes—
28#16861209C2211yescharacteristic
29#16861144C2212yes—
30#16861137C2212yes—
31#16861128C2211yescharacteristic
32#16861127C2212yes—
33C4×C127⋊C8C2211yescharacteristic
342−1+4×C127C2211yescharacteristic
35#137×C127C2212yes—
36C4⋊Q8×C127C2211yescharacteristic
37#104×C127C2212yes—
38#101×C127C2211yescharacteristic
39#14523249—22no—
40#14523248—22no—
41#14523248—41no—
42#14523248C2312yes—
43#14523247—22no—
44#14523247—42no—
45C2×Dic254—22no—
46C2×Dic254C2312yes—
47Q8×D127—22no—
48Q8×D127—22no—
49Dic508—22no—
50#14523238—22no—
51#14523238—22no—
52#14523238—22no—
53#14523238—22no—
54#14523238—22no—
55#14523238—22no—
56#14523238D412yes—
57#14523237—22no—
58#14523236—22no—
59#14523236D412yes—
60#14523235—22no—
61#14523235D412yes—
62#14523233D412yes—
63#14523232—22no—
64#14523232C2312yes—
65#14523231—22no—
66C2×C127⋊C8—22no—
67C2×C127⋊C8C2312yes—
68C8×D127—22no—
69C4⋊Dic127—41no—
70#41×C127—22no—
71#41×C127—41no—
72#41×C127C2312yes—
73C2×Q8×C127—22no—
74C2×Q8×C127C2311yescharacteristic
75C2×Q8×C127C2311yescharacteristic
76Q16×C127—22no—
77SD16×C127—22no—
78M16×C127C2312yes—
79C4⋊C4×C127—41no—
80C42×C127C2311yescharacteristic
81Dic254—22no—
82Dic254—22no—
83Dic254—41no—
84Dic254—42no—
85Dic254C2×D412yes—
86D508—41no—
87D508C2×D412yes—
88C22⋊D127—22no—
89C22⋊D127—42no—
90C2×Dic127—22no—
91C2×Dic127—42no—
92C4×D127—22no—
93C4×D127—42no—
94C4×D127—42no—
95C127⋊C8—22no—
96C127⋊C8C2×D412yes—
97C127⋊C8C2×D412yes—
98Q8×C127—22no—
99Q8×C127—22no—
100Q8×C127—41no—
101Q8×C127C2×D412yes—
102D4×C127—22no—
103D4×C127—41no—
104D4×C127C2×D412yes—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—42no—
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C22×C127—22no—
113C22×C127H(5,𝔽2)11yescharacteristic
114C2×D127—22no—
115C4×C127—22no—
116C4×C127—22no—
117C4×C127—41no—
118C4×C127—41no—
119C4×C127C22×D412yes—
120Dic127—22no—
121Dic127—42no—
122C2×C127—21no—
123C2×C127—42no—
124C2×C127D4211yescharacteristic
125D127—42no—
126#3293—1271nomaximal, Sylow
127C127#329311yescharacteristic, complemented, Sylow
128#578—1272no—
129#578—1272no—
130#494—1271no—
131#473—1272no—
132#473—1272no—
133#456—1271no—
134#456—1272no—
135#456D12711yescharacteristic, complemented
136#443—1272no—
1372−1+4—1272no—
1382−1+4—1271no—
1392−1+4C2×D12711yescharacteristic
140#137—1272no—
141#137—1272no—
142#137—1272no—
143#137—2542no—
144#137—2542no—
145#137—2542no—
146#137—2542no—
147#137C2×D12712yes—
148#135—1272no—
149#135—2542no—
150#135—2542no—
151C2×Q16—1272no—
152C2×Q16—1272no—
153C2×Q16—2542no—
154C2×Q16—2542no—
155#131—1272no—
156#131—1272no—
157C4⋊Q8—1271no—
158C4⋊Q8C2×D12711yescharacteristic
159#108—1272no—
160#104—1272no—
161#104C2×D12712yes—
162#101—1271no—
163#101—1272no—
164#101C2×D12711yescharacteristic
165C4×C8—1271no—
166#41—22no—
167#41—41no—
168#41—1272no—
169#41—5081no—
170#41—5082no—
171#41—2542no—
172#41—2542no—
173#41—2542no—
174#41C22×D12712yes—
175C2×Q8—22no—
176C2×Q8—1272no—
177C2×Q8—2542no—
178C2×Q8—2542no—
179C2×Q8—2542no—
180C2×Q8C22×D12711yescharacteristic
181C2×Q8C22×D12711yescharacteristic
182Q16—22no—
183Q16—1272no—
184Q16—2542no—
185Q16—2542no—
186Q16—2542no—
187Q16—2542no—
188Q16—2542no—
189Q16—2542no—
190Q16—2542no—
191SD16—22no—
192SD16—1272no—
193SD16—1272no—
194SD16—2542no—
195SD16—2542no—
196SD16—2542no—
197D8—1272no—
198M16—1272no—
199M16—2542no—
200M16—2542no—
201M16C22×D12712yes—
202C2×C8—1272no—
203C2×C8—2542no—
204C2×C8—2542no—
205C4⋊C4—41no—
206C4⋊C4—5081no—
207C42C22×D12711yescharacteristic
208Q8—22no—
209Q8—22no—
210Q8—41no—
211Q8—1272no—
212Q8—5082no—
213Q8—5081no—
214Q8—2542no—
215Q8—2542no—
216Q8D4×D12712yes—
217D4—22no—
218D4—41no—
219D4—1272no—
220D4—5081no—
221D4—5082no—
222D4—2542no—
223D4D4×D12712yes—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—5082no—
230C2×C4—5082no—
231C2×C4—5082no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4C23×D12711yescharacteristic
235C8—22no—
236C8—1272no—
237C8—1272no—
238C8—2542no—
239C22—22no—
240C22—2542no—
241C22#1686125411yescharacteristic
242C4—22no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—5082no—
247C4—2542no—
248C4C2×D4×D12712yes—
249C2—21no—
250C2—42no—
251C2—5082no—
252C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#456×C127C2
socleC2×C127D42
hypercenter#456D127
hypocenterC127#3293
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #22550692C2semidirect
2. #22550691C2semidirect
3. #22550255C2semidirect
4. #22550150C2semidirect
5. #22550149C2semidirect
6. #22550060C2semidirect
7. #22550058C2semidirect
8. #22549990C2semidirect
9. #456×C127C2semidirect
127. C127#3293semidirect
135. #456D127semidirect

Maximal subgroups

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

subgroupindex
#225506922
#225506912
#225502552
#225501502
#225501492
#225500602
#225500582
#225499902
#456×C1272
#3293127