Finite Group Databasebeta

← #30639259

Subgroups — #30639259

Order 16256.

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

normal subgroups only

This group has 362 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
1#22550440C211yesmaximal, characteristic
2#22550350C211yesmaximal, characteristic, complemented
3#22549893C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic, complemented
5#22549633C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861205C412yescomplemented
11#16861198C412yescomplemented
12#16861183—22no—
13#16861177—22no—
14C4×D508—22no—
15#16861171—22no—
16#16861168C2211yescharacteristic
17#16861168C2211yescharacteristic
18C22⋊C4×D127—22no—
19#16861136—22no—
20#16861125—22no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#104×C127—22no—
25C23×D127D412yes—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508C2×C412yes—
31C2×D508D412yes—
32C2×C22⋊D127—24no—
33C2×C4×D127—24no—
34#14523232—21no—
35#14523232—21no—
36C2×C127⋊C8—21no—
37C4⋊Dic127—21no—
38C4⋊Dic127—21no—
39C22⋊Dic127—21no—
40C22⋊Dic127—21no—
41(C2×D127)⋊C4—24no—
42(C2×D127)⋊C4—24no—
43C4×Dic127C2×C412yescomplemented
44C4×Dic127D412yescomplemented
45#41×C127—21no—
46#41×C127—21no—
47#41×C127—21no—
48#41×C127—21no—
49C2×Q8×C127C2×C411yescharacteristic, complemented
50C2×D4×C127—21no—
51C2×D4×C127C2×C411yescharacteristic, complemented
52C22×C4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127—21no—
57C4⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C42×C127D412yescomplemented
60C22×D127—22no—
61C22×D127—22no—
62C22×D127—82no—
63C22×D127C22⋊C414yes—
64D508—22no—
65D508—22no—
66D508—22no—
67D508—22no—
68C22⋊D127—84no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C4×D127—84no—
72C127⋊C8—41no—
73C23×C127—21no—
74C23×C127C2×D411yescharacteristic
75Q8×C127—21no—
76Q8×C127—21no—
77D4×C127—21no—
78D4×C127—21no—
79D4×C127—41no—
80D4×C127—41no—
81C2×C4×C127—21no—
82C2×C4×C127—22no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127C22×C411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C2×C4×C127#4111yescharacteristic
91C8×C127—41no—
92C22×C127—41no—
93C22×C127—42no—
94C22×C127—41no—
95C22×C127C2×C22⋊C411yescharacteristic
96C22×C127#12011yescharacteristic
97C22×C127#12311yescharacteristic
98C2×D127—24no—
99C2×D127—44no—
100C2×D127—44no—
101C2×D127—82no—
102C4×C127—42no—
103C4×C127—41no—
104C4×C127C4×D411yescharacteristic
105C4×C127C4×D411yescharacteristic
106C4×C127#12111yescharacteristic
107C4×C127#12111yescharacteristic
108Dic127—42no—
109Dic127—42no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—21no—
113C2×C127—81no—
114C2×C127#38611yescharacteristic
115D127—44no—
116#1911—1271nomaximal, Sylow
117C127#191111yescharacteristic, complemented, Sylow
118#535—1271no—
119#518—1271no—
120#421—1271no—
121#421D12711yescharacteristic, complemented
122C2×#100—1271no—
123H(3,ℤ/4)—1271no—
124H(3,ℤ/4)—1271no—
125C2×#41C2×D12711yescharacteristic
126C22×D4—1271no—
127C2×M16—1271no—
128C2×M16C2×D12711yescharacteristic
129#127—1272no—
130#124—1272no—
131#121—2542no—
132#120—2542no—
133C4×D4—2542no—
134C4×D4—2542no—
135#117—1271no—
136#117—1271no—
137#117C2×D12711yescharacteristic
138C2×C22⋊C4—2542no—
139#104—22no—
140#104—2542no—
141#100—2542no—
142C24—1272no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C4×D12711yescharacteristic, complemented
148C2×D4—21no—
149C2×D4—1272no—
150C2×D4—1272no—
151C2×D4—2544no—
152C2×D4—2541no—
153C2×D4—2541no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4C4×D12711yescharacteristic, complemented
157C22×C4—21no—
158C22×C4—2544no—
159C22×C4C22×D12711yescharacteristic
160M16—21no—
161M16—21no—
162M16—2541no—
163M16—2541no—
164C2×C8—21no—
165C2×C8—2541no—
166C4⋊C4—21no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C22⋊C4—21no—
170C22⋊C4—2544no—
171C22⋊C4—2544no—
172C22⋊C4—2541no—
173C22⋊C4—2541no—
174C42—1272no—
175C42—1272no—
176C42C22⋊D12712yescomplemented
177C23—21no—
178C23—1274no—
179C23—10162no—
180C23—2542no—
181C23—2542no—
182C23D4×D12711yescharacteristic
183Q8—21no—
184Q8—21no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—10164no—
190D4—2542no—
191D4—2542no—
192D4—2542no—
193D4—2542no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—10164no—
199C2×C4—2542no—
200C2×C4—2542no—
201C2×C4C2×C4×D12711yescharacteristic
202C2×C4D4×D12711yescharacteristic
203C2×C4D4×D12711yescharacteristic
204C2×C4C2×C22⋊D12711yescharacteristic
205C2×C4#1452324711yescharacteristic
206C2×C4#1452324811yescharacteristic
207C8—41no—
208C8—5081no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—10162no—
213C22—5084no—
214C22—5084no—
215C22—2544no—
216C22C22⋊C4×D12711yescharacteristic
217C22#1686117711yescharacteristic
218C22#1686119111yescharacteristic
219C4—42no—
220C4—41no—
221C4—5082no—
222C4—5082no—
223C4(C22⋊D127)⋊C411yescharacteristic
224C4C4×C22⋊D12711yescharacteristic
225C4#1686118011yescharacteristic
226C4#1686118211yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—81no—
231C2—5084no—
232C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#386
prime radical#22550440C2
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1911
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #22550350C2semidirect
3. #22549893C2semidirect
4. C2×#16861125C2semidirect
5. #22549633C2semidirect
6. #22549632C2semidirect
7. #421×C127C2semidirect
10. #16861205C4semidirect
11. #16861198C4semidirect
43. C4×Dic127C2×C4semidirect
44. C4×Dic127D4semidirect
49. C2×Q8×C127C2×C4semidirect
51. C2×D4×C127C2×C4semidirect
59. C42×C127D4semidirect
117. C127#1911semidirect
121. #421D127semidirect
147. C2×Q8C4×D127semidirect
156. C2×D4C4×D127semidirect
176. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225504402
#225503502
#225498932
C2×#168611252
#225496332
#225496322
#421×C1272
#1911127