Finite Group Databasebeta

← #30639254

Subgroups — #30639254

Order 16256.

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

normal subgroups only

This group has 344 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#22550642C211yesmaximal, characteristic, complemented
2#22550443C211yesmaximal, characteristic
3#22549881C211yesmaximal, characteristic, complemented
4#22549849C211yesmaximal, characteristic, complemented
5#22549633C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861217—22no—
10#16861215—22no—
11C2×#14523232C2211yescharacteristic
12M16×D127—22no—
13#16861205C412yescomplemented
14#16861196C412yescomplemented
15#16861180—22no—
16#16861178—22no—
17#16861168C2211yescharacteristic
18#16861168C2211yescharacteristic
19#16861130—22no—
20#16861123—22no—
21C22×D4×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#102×C127—22no—
25#14523248—21no—
26#14523248—21no—
27#14523248—21no—
28#14523248—21no—
29C2×Dic254C2×C411yescharacteristic
30C2×D508C2×C411yescharacteristic
31C2×C22⋊D127—24no—
32C2×C22⋊D127D412yes—
33C2×C4×D127D412yes—
34#14523232—21no—
35#14523232—21no—
36#14523231—24no—
37C2×C127⋊C8D412yescomplemented
38C8×D127—24no—
39C4⋊Dic127—21no—
40C4⋊Dic127—21no—
41C22⋊Dic127—24no—
42C22⋊Dic127—21no—
43C22⋊Dic127—21no—
44C4×Dic127—21no—
45C4×Dic127C2×C412yes—
46C24×C127—21no—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127C2×C412yes—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56C2×C8×C127D412yescomplemented
57C4⋊C4×C127—21no—
58C22⋊C4×C127—21no—
59C42×C127—21no—
60C22×D127C22⋊C412yes—
61Dic254—21no—
62Dic254—21no—
63D508—21no—
64D508—21no—
65C22⋊D127—22no—
66C22⋊D127—22no—
67C22⋊D127—84no—
68C2×Dic127—22no—
69C2×Dic127—41no—
70C2×Dic127C22⋊C412yes—
71C4×D127—22no—
72C4×D127—22no—
73C127⋊C8—22no—
74C23×C127—22no—
75C23×C127—41no—
76C23×C127—41no—
77C23×C127—82no—
78C23×C127C2×D411yescharacteristic
79D4×C127—22no—
80D4×C127—22no—
81D4×C127—41no—
82D4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C8×C127—22no—
91C22×C127—41no—
92C22×C127—44no—
93C22×C127—42no—
94C22×C127—82no—
95C22×C127—82no—
96C22×C127C2×C22⋊C411yescharacteristic
97C22×C127#12111yescharacteristic
98C22×C127#12111yescharacteristic
99C2×D127—22no—
100C2×D127—42no—
101C4×C127—81no—
102C4×C127C4×D411yescharacteristic
103C4×C127C4×D411yescharacteristic
104C4×C127#12011yescharacteristic
105C4×C127#12311yescharacteristic
106Dic127—22no—
107Dic127—42no—
108Dic127—81no—
109C2×C127—21no—
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—82no—
113C2×C127#38611yescharacteristic
114D127—42no—
115#1910—1271nomaximal, Sylow
116C127#191011yescharacteristic, complemented, Sylow
117#568—1271no—
118#535—1271no—
119#418—1271no—
120#418D12711yescharacteristic, complemented
121#410—1271no—
122H(3,ℤ/4)—1271no—
123H(3,ℤ/4)—1271no—
124C2×#41—1271no—
125C22×D4C2×D12711yescharacteristic
126#131—2542no—
127#131—2542no—
128C2×M16—1271no—
129C2×M16—2542no—
130C2×M16C2×D12711yescharacteristic
131#127—1272no—
132#124—1272no—
133#121—2542no—
134#120—2542no—
135#117—1271no—
136#117—1271no—
137#117C2×D12711yescharacteristic
138#102—22no—
139#102—2542no—
140#99—2542no—
141C24—21no—
142#41—2541no—
143#41—2541no—
144#41—2541no—
145#41—2541no—
146C2×Q8—1271no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—1272no—
153C2×D4—1271no—
154C2×D4—2544no—
155C2×D4C4×D12712yes—
156C22×C4—1272no—
157C22×C4C22×D12711yescharacteristic
158M16—21no—
159M16—21no—
160M16—2544no—
161M16—2541no—
162M16—2541no—
163C2×C8—1272no—
164C2×C8—2544no—
165C2×C8C22⋊D12712yescomplemented
166C4⋊C4—21no—
167C4⋊C4—2541no—
168C4⋊C4—2541no—
169C22⋊C4—21no—
170C22⋊C4—2544no—
171C22⋊C4—2541no—
172C22⋊C4—2541no—
173C42—21no—
174C42—1272no—
175C42—2541no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—82no—
180C23—1272no—
181C23D4×D12711yescharacteristic
182Q8—2541no—
183Q8—2541no—
184D4—22no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4—10164no—
189D4—2542no—
190D4—2542no—
191D4—2541no—
192D4—2541no—
193C2×C4—41no—
194C2×C4—1272no—
195C2×C4—5081no—
196C2×C4—2542no—
197C2×C4—2542no—
198C2×C4—2542no—
199C2×C4C2×C4×D12711yescharacteristic
200C2×C4D4×D12711yescharacteristic
201C2×C4D4×D12711yescharacteristic
202C2×C4C2×C22⋊D12711yescharacteristic
203C2×C4#1452324711yescharacteristic
204C2×C4#1452324811yescharacteristic
205C8—22no—
206C8—2542no—
207C22—41no—
208C22—44no—
209C22—42no—
210C22—82no—
211C22—82no—
212C22—5082no—
213C22—2542no—
214C22C22⋊C4×D12711yescharacteristic
215C22#1686118011yescharacteristic
216C22#1686118211yescharacteristic
217C4—81no—
218C4—10161no—
219C4—5082no—
220C4—2542no—
221C4(C22⋊D127)⋊C411yescharacteristic
222C4C4×C22⋊D12711yescharacteristic
223C4#1686117711yescharacteristic
224C4#1686119111yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—82no—
229C2—5082no—
230C2#2254975211yescharacteristic
subgroupG/H
centerC2#22549752
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC2×C127#386
prime radical#22550443C2
cosocleC22×C4×C127C23
hypercenter#418D127
hypocenterC127#1910
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #22550642C2semidirect
3. #22549881C2semidirect
4. #22549849C2semidirect
5. #22549633C2semidirect
6. #22549632C2semidirect
7. #418×C127C2semidirect
13. #16861205C4semidirect
14. #16861196C4semidirect
37. C2×C127⋊C8D4semidirect
56. C2×C8×C127D4semidirect
116. C127#1910semidirect
120. #418D127semidirect
165. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506422
#225504432
#225498812
#225498492
#225496332
#225496322
#418×C1272
#1910127