Finite Group Databasebeta

← #30651356

Subgroups — #30651356

Order 16256.

354 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 123 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#22550591C211yesmaximal, characteristic
2#22550279C211yesmaximal, characteristic, complemented
3#22550214C212yesmaximal, complemented
4#22550189C212yesmaximal, complemented
5#22550171C211yesmaximal, characteristic, complemented
6#22550163C211yesmaximal, characteristic, complemented
7#22550124C212yesmaximal, complemented
8C4×#14523235C211yesmaximal, characteristic, complemented
9#22549954C211yesmaximal, characteristic, complemented
10#22549916C212yesmaximal, complemented
11#554×C127C211yesmaximal, characteristic, complemented
12C2×#14523235C2211yescharacteristic
13C2×#14523232C2212yes—
14#16861209C2211yescharacteristic
15#16861195—22no—
16#16861183C2212yes—
17#16861180—21no—
18C4×D508C2211yescharacteristic
19#16861168C2212yes—
20#16861167—21no—
21#16861142C2212yes—
22#16861142C2212yes—
23#16861142C2212yes—
24#16861140C2212yes—
25#16861140C2211yescharacteristic
26C4⋊(C127⋊C8)C2211yescharacteristic
27#16861133C2212yes—
28#16861133C2211yescharacteristic
29#16861129C2212yes—
30#16861129C2211yescharacteristic
31C22⋊(C127⋊C8)C2212yes—
32C4×C127⋊C8C2211yescharacteristic
33C4⋊Q8×C127C2211yescharacteristic
34#125×C127—21no—
35C22⋊Q8×C127—22no—
36C22⋊Q8×C127C2212yes—
37C4×Q8×C127C2211yescharacteristic
38C4×D4×C127—21no—
39C4×D4×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2212yes—
41C2×Dic254—21no—
42C2×D508C2311yescharacteristic
43C2×C22⋊D127—22no—
44C2×C4×D127—21no—
45#14523235—22no—
46#14523232—24no—
47C2×C127⋊C8C2312yes—
48C2×C127⋊C8C2311yescharacteristic
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127—21no—
51C4⋊Dic127C2312yes—
52C4⋊Dic127C2311yescharacteristic
53C22⋊Dic127—22no—
54(C2×D127)⋊C4—21no—
55(C2×D127)⋊C4—22no—
56C4×Dic127—22no—
57C2×Q8×C127—21no—
58C2×Q8×C127C2311yescharacteristic
59C2×D4×C127D411yescharacteristic
60C22×C4×C127—22no—
61C22×C4×C127C2312yes—
62C4⋊C4×C127—22no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127—21no—
65C4⋊C4×C127—22no—
66C4⋊C4×C127D411yescharacteristic
67C4⋊C4×C127C2311yescharacteristic
68C4⋊C4×C127C2312yes—
69C22⋊C4×C127—22no—
70C22⋊C4×C127D412yes—
71C42×C127—21no—
72C42×C127C2311yescharacteristic
73C22×D127—21no—
74Dic254—81no—
75D508—21no—
76D508—21no—
77C22⋊D127—82no—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C4×D127—42no—
81C127⋊C8—22no—
82C127⋊C8—21no—
83C127⋊C8#4112yes—
84C23×C127C2×D412yes—
85Q8×C127—21no—
86Q8×C127—21no—
87Q8×C127—81no—
88D4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—22no—
94C2×C4×C127—82no—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2×D412yes—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2411yescharacteristic
99C22×C127—22no—
100C22×C127—22no—
101C22×C127—22no—
102C22×C127C22×D411yescharacteristic
103C2×D127—41no—
104C2×D127—41no—
105C2×D127—41no—
106C4×C127—22no—
107C4×C127—21no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127—42no—
111C4×C127#13712yes—
112C4×C127C2×#4111yescharacteristic
113C4×C1272−1+411yescharacteristic
114Dic127—41no—
115Dic127—42no—
116C2×C127—42no—
117C2×C127#54711yescharacteristic
118C2×C127C2×#13711yescharacteristic
119C2×C127#57611yescharacteristic
120D127—81no—
121#3357—1271nomaximal, Sylow
122C127#335711yescharacteristic, complemented, Sylow
123#555—1271no—
124#554D12711yescharacteristic, complemented
125#500—1271no—
126#485—1272no—
127#480—1272no—
128#477—1271no—
129#476—1271no—
130#468—1272no—
131C4×SD16—1271no—
132#436—1271no—
133#428—1272no—
134C2×SD16—1271no—
135C2×M16—1272no—
136C4⋊Q8—1271no—
137C4⋊Q8C2×D12711yescharacteristic
138#125—21no—
139#124—2542no—
140C22⋊Q8—22no—
141C22⋊Q8C2×D12712yes—
142#121—1272no—
143#121—2541no—
144C4×Q8C2×D12711yescharacteristic
145C4×D4—21no—
146C4×D4—1271no—
147C4×D4C2×D12711yescharacteristic
148#117—1272no—
149#117—2541no—
150C2×C4⋊C4C2×D12712yes—
151#107—1272no—
152#107—1272no—
153#107—1272no—
154#106—1271no—
155#106—1272no—
156C4⋊C8—1271no—
157#103—1271no—
158#103—1272no—
159#102—1271no—
160#102—1272no—
161C22⋊C8—1272no—
162C4×C8—1271no—
163C2×Q8—21no—
164C2×Q8—2541no—
165C2×Q8C22×D12711yescharacteristic
166C2×D4—1271no—
167C2×D4—2542no—
168C2×D4C22⋊D12711yescharacteristic
169C22×C4—22no—
170C22×C4—2541no—
171C22×C4C22×D12712yes—
172SD16—2542no—
173M16—2544no—
174C2×C8—1271no—
175C2×C8—1271no—
176C2×C8—1272no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—1271no—
182C4⋊C4—1272no—
183C4⋊C4—2541no—
184C4⋊C4C22⋊D12711yescharacteristic
185C4⋊C4C22×D12711yescharacteristic
186C4⋊C4C22×D12712yes—
187C22⋊C4—22no—
188C22⋊C4—2542no—
189C22⋊C4—2542no—
190C22⋊C4—2541no—
191C22⋊C4C22⋊D12712yes—
192C42—21no—
193C42—2542no—
194C42C22×D12711yescharacteristic
195C23—2541no—
196C23C2×C22⋊D12712yes—
197Q8—21no—
198Q8—21no—
199Q8—81no—
200Q8—10161no—
201D4—22no—
202D4—10162no—
203D4—2541no—
204D4—2541no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—82no—
211C2×C4—5082no—
212C2×C4—2541no—
213C2×C4—2542no—
214C2×C4C2×C22⋊D12712yes—
215C2×C4C2×C22⋊D12711yescharacteristic
216C2×C4C2×C22⋊D12711yescharacteristic
217C2×C4C23×D12711yescharacteristic
218C8—1272no—
219C8—2542no—
220C8—2541no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—5081no—
225C22—5081no—
226C22—5081no—
227C22C22×C22⋊D12711yescharacteristic
228C4—21no—
229C4—22no—
230C4—41no—
231C4—42no—
232C4—41no—
233C4—5081no—
234C4—5082no—
235C4#1686124012yes—
236C4#41×D12711yescharacteristic
237C4#1686125611yescharacteristic
238C2—42no—
239C2—10161no—
240C2#2255052511yescharacteristic
241C2C2×#1686124011yescharacteristic
242C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#554×C127C2
socleC22×C127C22×D4
prime radical#22550591C2
hypercenter#554D127
hypocenterC127#3357
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #22550279C2semidirect
3. #22550214C2semidirect
4. #22550189C2semidirect
5. #22550171C2semidirect
6. #22550163C2semidirect
7. #22550124C2semidirect
8. C4×#14523235C2semidirect
9. #22549954C2semidirect
10. #22549916C2semidirect
11. #554×C127C2semidirect
122. C127#3357semidirect
124. #554D127semidirect

Maximal subgroups

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

subgroupindex
#225505912
#225502792
#225502142
#225501892
#225501712
#225501632
#225501242
C4×#145232352
#225499542
#225499162
#554×C1272
#3357127