Finite Group Databasebeta

← #30639092

Subgroups — #30639092

Order 16256.

310 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550674C211yesmaximal, characteristic
2#22549892C211yesmaximal, characteristic, complemented
3#22549872C211yesmaximal, characteristic, complemented
4#22549844C211yesmaximal, characteristic, complemented
5#22549647C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7#568×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239C412yescomplemented
11#16861235—21no—
12#16861234C412yescomplemented
13#16861229—22no—
14C2×#14523232C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16#16861168C2211yescharacteristic
17#16861135—22no—
18#16861133—21no—
19#16861130—21no—
20(C2×D127)⋊C8—22no—
21C2×#41×C127C2211yescharacteristic
22#131×C127—22no—
23#131×C127—22no—
24C2×M16×C127—22no—
25C2×M16×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27#14523248—21no—
28#14523248—21no—
29#14523248—21no—
30#14523248—21no—
31C2×Dic254C2×C411yescharacteristic
32C2×D508C2×C411yescharacteristic
33C2×C22⋊D127—21no—
34C2×C4×D127—21no—
35#14523238—22no—
36#14523236—22no—
37#14523235—22no—
38#14523233—22no—
39#14523232—21no—
40#14523232C2×C412yes—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C2×C127⋊C8—21no—
44C2×C127⋊C8—21no—
45C4⋊Dic127—21no—
46C22⋊Dic127—21no—
47C4×Dic127—21no—
48#41×C127—21no—
49#41×C127—21no—
50#41×C127C2×C412yes—
51#41×C127D412yes—
52C2×Q8×C127D411yescharacteristic
53C2×D4×C127—21no—
54C2×D4×C127D411yescharacteristic
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57M16×C127—24no—
58M16×C127—21no—
59M16×C127—21no—
60M16×C127—21no—
61M16×C127D412yes—
62C2×C8×C127—24no—
63C2×C8×C127—21no—
64C2×C8×C127D412yes—
65C22×D127—21no—
66Dic254—21no—
67Dic254—21no—
68D508—21no—
69D508—21no—
70C22⋊D127—41no—
71C22⋊D127—41no—
72C2×Dic127—21no—
73C2×Dic127—41no—
74C4×D127—41no—
75C4×D127—41no—
76C127⋊C8—22no—
77C127⋊C8—41no—
78C23×C127—21no—
79C23×C127#4111yescharacteristic
80Q8×C127—21no—
81Q8×C127C22⋊C412yes—
82D4×C127—22no—
83D4×C127—22no—
84D4×C127—21no—
85D4×C127C22⋊C412yes—
86C2×C4×C127—21no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127C22×C411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C8×C127—22no—
96C8×C127—22no—
97C22×C127—22no—
98C22×C127—41no—
99C22×C127—81no—
100C22×C127C4×D411yescharacteristic
101C22×C127#12111yescharacteristic
102C22×C127#12311yescharacteristic
103C2×D127—42no—
104C2×D127—41no—
105C4×C127—22no—
106C4×C127C2×C22⋊C411yescharacteristic
107C4×C127C4×D411yescharacteristic
108C4×C127#12011yescharacteristic
109C4×C127#12111yescharacteristic
110Dic127—41no—
111Dic127—81no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—42no—
116C2×C127#38611yescharacteristic
117D127—81no—
118#1881—1271nomaximal, Sylow
119C127#188111yescharacteristic, complemented, Sylow
120#575—1271no—
121#568D12711yescharacteristic, complemented
122#421—1271no—
123#416—1271no—
124#408—1271no—
125#344—1271no—
126#341—1271no—
127C2×#41—1271no—
128C2×#41C2×D12711yescharacteristic
129#137—1272no—
130#137—2541no—
131AGL(1,ℤ/8)—1272no—
132AGL(1,ℤ/8)—2541no—
133#135—2542no—
134#131—22no—
135#131—22no—
136C2×M16—22no—
137C2×M16—1271no—
138C2×M16C2×D12711yescharacteristic
139C2×M16C2×D12711yescharacteristic
140C22×C8—1271no—
141#117—1271no—
142#104—2542no—
143#103—2541no—
144#102—2541no—
145C22⋊C8—2542no—
146#41—21no—
147#41—21no—
148#41—2541no—
149#41—2541no—
150#41—2541no—
151#41—2541no—
152#41C4×D12712yes—
153#41C22⋊D12712yes—
154C2×Q8—1271no—
155C2×Q8D50811yescharacteristic
156C2×D4—21no—
157C2×D4—1271no—
158C2×D4—2541no—
159C2×D4D50811yescharacteristic
160C22×C4—21no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163Q16—2542no—
164SD16—2542no—
165SD16—2542no—
166D8—2542no—
167M16—24no—
168M16—21no—
169M16—21no—
170M16—21no—
171M16—1272no—
172M16—2541no—
173M16C22⋊D12712yes—
174C2×C8—24no—
175C2×C8—21no—
176C2×C8—2541no—
177C2×C8—2541no—
178C2×C8—2541no—
179C2×C8—2541no—
180C2×C8C22⋊D12712yes—
181C4⋊C4—2541no—
182C22⋊C4—2541no—
183C42—2541no—
184C23—21no—
185C23—2541no—
186C23#1452324811yescharacteristic
187Q8—21no—
188Q8—2541no—
189Q8—2541no—
190Q8(C2×D127)⋊C412yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—5081no—
195D4—5081no—
196D4—2541no—
197D4—2541no—
198D4(C2×D127)⋊C412yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—5081no—
203C2×C4—5081no—
204C2×C4—5081no—
205C2×C4—2541no—
206C2×C4C2×C4×D12711yescharacteristic
207C2×C4C2×C22⋊D12711yescharacteristic
208C2×C4C2×C22⋊D12711yescharacteristic
209C2×C4C2×C22⋊D12711yescharacteristic
210C2×C4C2×D50811yescharacteristic
211C2×C4#1452324811yescharacteristic
212C8—22no—
213C8—22no—
214C8—5081no—
215C8—2542no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—5082no—
220C22—5081no—
221C22C4×C22⋊D12711yescharacteristic
222C22#1686118311yescharacteristic
223C22#1686119211yescharacteristic
224C4—22no—
225C4—10161no—
226C4—5081no—
227C4C2×(C2×D127)⋊C411yescharacteristic
228C4C4×C22⋊D12711yescharacteristic
229C4#1686117811yescharacteristic
230C4#1686118311yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—10161no—
236C2#2254975511yescharacteristic
subgroupG/H
centerC4#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#386
prime radical#22550674C2
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1881
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22549892C2semidirect
3. #22549872C2semidirect
4. #22549844C2semidirect
5. #22549647C2semidirect
6. #22549640C2semidirect
7. #568×C127C2semidirect
10. #16861239C4semidirect
12. #16861234C4semidirect
119. C127#1881semidirect
121. #568D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225498922
#225498722
#225498442
#225496472
#225496402
#568×C1272
#1881127