Finite Group Databasebeta

← #30639787

Subgroups — #30639787

Order 16256.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#16861239C211yesmaximal, characteristic, complemented
2#22550447C211yesmaximal, characteristic, complemented
3#22549893C211yesmaximal, characteristic, complemented
4#22549886C211yesmaximal, characteristic, complemented
5#22549850C211yesmaximal, characteristic, complemented
6#22549647C211yesmaximal, characteristic, complemented
7#575×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9#16861239—22no—
10#16861239—22no—
11C2×#14523238—21no—
12C2×#14523236—21no—
13C2×#14523232C2211yescharacteristic
14#16861207—21no—
15#16861196—21no—
16#16861190—22no—
17#16861188—22no—
18#16861168C2211yescharacteristic
19#16861168C2211yescharacteristic
20#16861136—22no—
21#16861134—22no—
22#16861124—22no—
23C2×#41×C127C2211yescharacteristic
24C2×#41×C127C2211yescharacteristic
25#137×C127—21no—
26#137×C127—21no—
27AGL(1,ℤ/8)×C127—21no—
28AGL(1,ℤ/8)×C127—21no—
29#135×C127—22no—
30C2×M16×C127C2211yescharacteristic
31C2×Dic254—21no—
32C2×Dic254—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254D412yes—
37C22×Dic127—21no—
38#14523238—24no—
39#14523236—24no—
40#14523232—21no—
41#14523232D412yes—
42C2×C127⋊C8—21no—
43C4⋊Dic127—21no—
44C4⋊Dic127—21no—
45Dic127⋊C4—24no—
46C22⋊Dic127—24no—
47C22⋊Dic127—21no—
48C22⋊Dic127D412yes—
49C4×Dic127—21no—
50C4×Dic127—21no—
51#41×C127—21no—
52#41×C127—21no—
53#41×C127—21no—
54#41×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57#41×C127—21no—
58#41×C127—21no—
59C2×Q8×C127D411yescharacteristic
60C2×Q8×C127D411yescharacteristic
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127D411yescharacteristic
64C2×D4×C127D411yescharacteristic
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127C2311yescharacteristic
68Q16×C127—22no—
69SD16×C127—22no—
70SD16×C127—22no—
71D8×C127—22no—
72M16×C127—21no—
73M16×C127—21no—
74C2×C8×C127D412yes—
75Dic254—22no—
76Dic254—22no—
77Dic254—41no—
78Dic254—41no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—41no—
82C2×Dic127—41no—
83C2×Dic127—41no—
84C127⋊C8—22no—
85C23×C127—21no—
86C23×C127—21no—
87C23×C127C2×D411yescharacteristic
88Q8×C127—21no—
89Q8×C127—21no—
90Q8×C127—21no—
91Q8×C127—21no—
92D4×C127—21no—
93D4×C127—21no—
94D4×C127—21no—
95D4×C127—21no—
96D4×C127—41no—
97D4×C127—41no—
98D4×C127—41no—
99D4×C127—41no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127C2×D411yescharacteristic
107C2×C4×C127C2×D411yescharacteristic
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127#4111yescharacteristic
112C8×C127—22no—
113C22×C127—41no—
114C22×C127—42no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—81no—
118C22×C127#12011yescharacteristic
119C22×C127#12111yescharacteristic
120C22×C127#12711yescharacteristic
121C4×C127—41no—
122C4×C127—41no—
123C4×C127#12011yescharacteristic
124C4×C127#12011yescharacteristic
125C4×C127#12111yescharacteristic
126C4×C127#12111yescharacteristic
127Dic127—42no—
128Dic127—42no—
129Dic127—81no—
130C2×C127—21no—
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—81no—
134C2×C127—81no—
135C2×C127#39211yescharacteristic
136#2009—1271nomaximal, Sylow
137C127#200911yescharacteristic, complemented, Sylow
138#575D12711yescharacteristic
139C2×#137—1271no—
140#536—1271no—
141#421—1271no—
142#419—1271no—
143#410—1271no—
144#344—1271no—
145C2×#41C2×D12711yescharacteristic
146C2×#41C2×D12711yescharacteristic
147C22×Q8—1271no—
148#137—21no—
149#137—21no—
150#137—2542no—
151#137—2542no—
152AGL(1,ℤ/8)—21no—
153AGL(1,ℤ/8)—21no—
154#135—22no—
155C2×Q16—2541no—
156C2×SD16—2541no—
157C2×M16—1271no—
158C2×M16C2×D12711yescharacteristic
159C4⋊Q8—2541no—
160#124—2541no—
161#123—2542no—
162C22⋊Q8—2542no—
163#117—1271no—
164#117—1271no—
165#104—2542no—
166#103—2542no—
167#99—2542no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174#41—21no—
175#41—21no—
176C2×Q8—1272no—
177C2×Q8—2541no—
178C2×Q8—2541no—
179C2×Q8—2541no—
180C2×Q8—2541no—
181C2×Q8—2541no—
182C2×Q8C22⋊D12711yescharacteristic
183C2×Q8C22⋊D12711yescharacteristic
184C2×D4—21no—
185C2×D4—21no—
186C2×D4C22⋊D12711yescharacteristic
187C2×D4C22⋊D12711yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192Q16—22no—
193Q16—2544no—
194SD16—22no—
195SD16—22no—
196SD16—2544no—
197D8—22no—
198M16—21no—
199M16—21no—
200M16—1272no—
201M16—2541no—
202C2×C8—2541no—
203C2×C8C22⋊D12712yes—
204C4⋊C4—2544no—
205C4⋊C4—2541no—
206C4⋊C4—2541no—
207C22⋊C4—1272no—
208C22⋊C4—2541no—
209C22⋊C4—2544no—
210C42—2541no—
211C42—2541no—
212C23—21no—
213C23—21no—
214C23D4×D12711yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—5081no—
220Q8—5081no—
221Q8—2542no—
222Q8—2542no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—5081no—
238C2×C4—5081no—
239C2×C4—5081no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4D4×D12711yescharacteristic
243C2×C4D4×D12711yescharacteristic
244C2×C4C2×C22⋊D12711yescharacteristic
245C2×C4C2×C22⋊D12711yescharacteristic
246C2×C4C2×C22⋊D12711yescharacteristic
247C2×C4#1452324711yescharacteristic
248C8—22no—
249C8—2542no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—81no—
255C22#1686117711yescharacteristic
256C22#1686118111yescharacteristic
257C22#1686120511yescharacteristic
258C4—41no—
259C4—41no—
260C4—10161no—
261C4—5082no—
262C4—5082no—
263C4#1686117711yescharacteristic
264C4#1686117811yescharacteristic
265C4#1686118011yescharacteristic
266C4#1686118111yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—81no—
272C2#2254978411yescharacteristic
subgroupG/H
centerC2#22549784
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#575×C127C2
socleC2×C127#392
hypercenter#575D127
hypocenterC127#2009
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. #22550447C2semidirect
3. #22549893C2semidirect
4. #22549886C2semidirect
5. #22549850C2semidirect
6. #22549647C2semidirect
137. C127#2009semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
#225504472
#225498932
#225498862
#225498502
#225496472
#575×C1272
#2009127