Finite Group Databasebeta

← #30640015

Subgroups — #30640015

Order 16256.

266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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#22549899C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549858C211yesmaximal, characteristic, complemented
6C127⋊#338C211yesmaximal, characteristic, complemented
7#413×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239—21no—
11#16861235—21no—
12#16861234—21no—
13#16861229—22no—
14C2×#14523232C2211yescharacteristic
15C2×#14523232C2211yescharacteristic
16C2×C4×Dic127C2211yescharacteristic
17Dic127⋊C8—22no—
18#16861136—21no—
19#16861136—21no—
20#16861135—21no—
21#16861135—21no—
22#16861127—21no—
23#16861125—21no—
24C2×#41×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27#101×C127—21no—
28#100×C127—21no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33C2×Dic254D411yescharacteristic
34C2×D508D411yescharacteristic
35C2×C22⋊D127—21no—
36C22×Dic127—21no—
37C2×C4×D127—21no—
38#14523238—22no—
39#14523236—22no—
40#14523235—22no—
41#14523233—22no—
42#14523232—21no—
43#14523232—21no—
44#14523232—21no—
45#14523232—21no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8D412yes—
48C4×Dic127—21no—
49C4×Dic127D411yescharacteristic
50C4×Dic127D411yescharacteristic
51#41×C127—21no—
52#41×C127—21no—
53#41×C127—21no—
54#41×C127—21no—
55C2×Q8×C127D411yescharacteristic
56C2×D4×C127—21no—
57C2×D4×C127D411yescharacteristic
58C22×C4×C127—21no—
59C22×C4×C127C2311yescharacteristic
60M16×C127—21no—
61M16×C127—21no—
62M16×C127—21no—
63M16×C127—21no—
64C2×C8×C127—21no—
65C2×C8×C127—21no—
66C22×D127—21no—
67Dic254—21no—
68Dic254—21no—
69D508—21no—
70D508—21no—
71C22⋊D127—41no—
72C22⋊D127—41no—
73C2×Dic127—21no—
74C2×Dic127—21no—
75C2×Dic127—21no—
76C2×Dic127—42no—
77C4×D127—41no—
78C4×D127—41no—
79C127⋊C8—22no—
80C127⋊C8—41no—
81C23×C127—21no—
82C23×C127#4111yescharacteristic
83Q8×C127—21no—
84Q8×C127—21no—
85D4×C127—21no—
86D4×C127—21no—
87D4×C127—41no—
88D4×C127—41no—
89C2×C4×C127—21no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127#4111yescharacteristic
97C2×C4×C127#4111yescharacteristic
98C8×C127—41no—
99C8×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—81no—
103C22×C127#12111yescharacteristic
104C22×C127#12311yescharacteristic
105C22×C127#12311yescharacteristic
106C2×D127—41no—
107C2×D127—81no—
108C4×C127—41no—
109C4×C127#12011yescharacteristic
110C4×C127#12111yescharacteristic
111C4×C127#12111yescharacteristic
112C4×C127#12411yescharacteristic
113Dic127—22no—
114Dic127—41no—
115Dic127—41no—
116C2×C127—21no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127—81no—
120C2×C127#39411yescharacteristic
121D127—81no—
122#2050—1271nomaximal, Sylow
123C127#205011yescharacteristic, complemented, Sylow
124#575—1271no—
125#423—1271no—
126C2×#104—1271no—
127C2×#104—1271no—
128#413—1271no—
129#413D12711yescharacteristic, complemented
130#338—1271no—
131C2×#41—1271no—
132C2×#41C2×D12711yescharacteristic
133#137—2541no—
134#137—2541no—
135AGL(1,ℤ/8)—2541no—
136AGL(1,ℤ/8)—2541no—
137#135—2542no—
138C2×M16—1271no—
139C2×M16—1271no—
140C2×M16C2×D12711yescharacteristic
141C2×M16C2×D12711yescharacteristic
142C2×C42—1271no—
143C4⋊C8—2542no—
144#104—2541no—
145#104—2541no—
146#104—2541no—
147#104—2541no—
148#101—21no—
149#101—2541no—
150#100—21no—
151#100—2541no—
152#41—21no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—2541no—
157#41—2541no—
158#41—2541no—
159#41—2541no—
160C2×Q8—1271no—
161C2×Q8C22⋊D12711yescharacteristic
162C2×D4—21no—
163C2×D4—1271no—
164C2×D4—2541no—
165C2×D4C22⋊D12711yescharacteristic
166C22×C4—21no—
167C22×C4—2541no—
168C22×C4—2541no—
169C22×C4C22×D12711yescharacteristic
170Q16—2542no—
171SD16—2542no—
172SD16—2542no—
173D8—2542no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16—21no—
178M16—2541no—
179M16—2541no—
180M16—2541no—
181M16—2541no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—1272no—
185C2×C8—2541no—
186C42—1271no—
187C42—1271no—
188C42—2541no—
189C23—21no—
190C23—2541no—
191C23#1452324811yescharacteristic
192Q8—21no—
193Q8—21no—
194Q8—2541no—
195Q8—2541no—
196D4—21no—
197D4—21no—
198D4—41no—
199D4—41no—
200D4—5081no—
201D4—5081no—
202D4—2541no—
203D4—2541no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—5082no—
208C2×C4—5081no—
209C2×C4—5081no—
210C2×C4—2541no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4D4×D12711yescharacteristic
214C2×C4D4×D12711yescharacteristic
215C2×C4D4×D12711yescharacteristic
216C2×C4C2×C22⋊D12711yescharacteristic
217C2×C4#1452324711yescharacteristic
218C2×C4#1452324811yescharacteristic
219C8—41no—
220C8—41no—
221C8—5081no—
222C8—2542no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—10161no—
227C22—5081no—
228C22#1686118211yescharacteristic
229C22#1686119111yescharacteristic
230C22#1686119211yescharacteristic
231C4—41no—
232C4—5081no—
233C4—5081no—
234C4—2542no—
235C4#1686117711yescharacteristic
236C4#1686118111yescharacteristic
237C4#1686118211yescharacteristic
238C4#1686119511yescharacteristic
239C2—21no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—10161no—
244C2#2254979111yescharacteristic
subgroupG/H
centerC4#16861195
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#413×C127C2
socleC2×C127#394
prime radical#22550674C2
hypercenter#413D127
hypocenterC127#2050
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
3. C2×#16861136C2semidirect
4. C2×#16861135C2semidirect
5. #22549858C2semidirect
6. C127⋊#338C2semidirect
7. #413×C127C2semidirect
123. C127#2050semidirect
129. #413D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225498992
C2×#168611362
C2×#168611352
#225498582
C127⋊#3382
#413×C1272
#2050127