Finite Group Databasebeta

← #30638704

Subgroups — #30638704

Order 16256.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861256C211yesmaximal, characteristic
2#22549893C211yesmaximal, characteristic, complemented
3#22549886C211yesmaximal, characteristic, complemented
4#22549884C211yesmaximal, characteristic, complemented
5#22549858C211yesmaximal, characteristic, complemented
6#22549849C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8#16861256—22no—
9#16861256—22no—
10#16861256—22no—
11#16861256—22no—
12C2×#14523248—21no—
13C2×#14523248C2211yescharacteristic
14C2×#14523247—22no—
15C2×#14523247C412yescomplemented
16C22×Dic254—21no—
17C22×Dic254C2211yescharacteristic
18C2×Q8×D127C412yescomplemented
19C2×#14523232C2211yescharacteristic
20#16861168C2211yescharacteristic
21#16861136—22no—
22#16861134—22no—
23#16861132—22no—
24#16861127—21no—
25#16861125—21no—
26#16861123—22no—
27C2×#41×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#104×C127—22no—
31#14523248—24no—
32#14523248—24no—
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36#14523248—21no—
37#14523247—28no—
38#14523247—28no—
39#14523247—24no—
40#14523247—24no—
41C2×Dic254—24no—
42C2×Dic254—24no—
43C2×Dic254—22no—
44C2×Dic254—21no—
45C2×Dic254—21no—
46C2×Dic254—21no—
47C2×Dic254C2×C412yes—
48C2×Dic254D412yes—
49C2×Dic254D412yes—
50C2×Dic254D411yescharacteristic
51Q8×D127—24no—
52Q8×D127—24no—
53C2×D508D411yescharacteristic
54C2×C22⋊D127—22no—
55C2×C22⋊D127D412yes—
56C22×Dic127—22no—
57C22×Dic127—21no—
58C2×C4×D127—22no—
59C2×C4×D127C2×C412yes—
60#14523232—21no—
61#14523232—21no—
62C2×C127⋊C8—21no—
63C4⋊Dic127—21no—
64C22⋊Dic127—21no—
65C4×Dic127—21no—
66#41×C127—21no—
67#41×C127—21no—
68#41×C127D412yes—
69#41×C127D412yes—
70C2×Q8×C127C2×C411yescharacteristic
71C2×D4×C127—21no—
72C2×D4×C127C2×C411yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127C2311yescharacteristic
75M16×C127—21no—
76M16×C127—21no—
77C2×C8×C127—21no—
78C4⋊C4×C127—21no—
79C22⋊C4×C127—21no—
80C42×C127—21no—
81C22×D127C22⋊C412yes—
82Dic254—28no—
83Dic254—22no—
84Dic254—22no—
85Dic254—21no—
86Dic254—22no—
87Dic254—21no—
88Dic254C22⋊C414yes—
89D508—21no—
90D508—21no—
91C22⋊D127—28no—
92C22⋊D127—22no—
93C22⋊D127—22no—
94C2×Dic127—28no—
95C2×Dic127—22no—
96C2×Dic127—22no—
97C2×Dic127—22no—
98C2×Dic127—41no—
99C2×Dic127C22⋊C412yes—
100C4×D127—28no—
101C4×D127—22no—
102C4×D127—22no—
103C127⋊C8—41no—
104C23×C127—21no—
105C23×C127C2×D411yescharacteristic
106Q8×C127—21no—
107Q8×C127C22⋊C412yes—
108D4×C127—22no—
109D4×C127—22no—
110D4×C127—21no—
111D4×C127C22⋊C412yes—
112C2×C4×C127—21no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—41no—
116C2×C4×C127C22×C411yescharacteristic
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D411yescharacteristic
122C8×C127—41no—
123C22×C127—22no—
124C22×C127—41no—
125C22×C127—42no—
126C22×C127C2×C22⋊C411yescharacteristic
127C22×C127#12011yescharacteristic
128C22×C127#12011yescharacteristic
129C2×D127—22no—
130C2×D127—42no—
131C4×C127—22no—
132C4×C127—81no—
133C4×C127C2×C22⋊C411yescharacteristic
134C4×C127C2×C22⋊C411yescharacteristic
135C4×C127#12011yescharacteristic
136C4×C127#12011yescharacteristic
137Dic127—24no—
138Dic127—22no—
139Dic127—81no—
140C2×C127—21no—
141C2×C127—21no—
142C2×C127—21no—
143C2×C127—42no—
144C2×C127#37911yescharacteristic
145D127—42no—
146#1796—1271nomaximal, Sylow
147C127#179611yescharacteristic, complemented, Sylow
148C2×2−1+4—1271no—
149#421—1271no—
150#421D12711yescharacteristic, complemented
151#419—1271no—
152#419—1271no—
153#413—1271no—
154#410—1271no—
1552−1+4—2542no—
1562−1+4—2542no—
1572−1+4—2542no—
1582−1+4—2542no—
159C2×#41—1271no—
160C2×#41—1272no—
161C2×#41—2541no—
162C2×#41—2542no—
163C2×#41C2×D12711yescharacteristic
164C22×Q8—1272no—
165C22×Q8—1271no—
166C22×Q8—2541no—
167C2×M16—1271no—
168C2×M16C2×D12711yescharacteristic
169#117—1271no—
170#117C2×D12711yescharacteristic
171#104—22no—
172#104—2542no—
173#103—2542no—
174#103—2542no—
175#101—2541no—
176#100—2541no—
177#99—2542no—
178#41—21no—
179#41—21no—
180#41—2544no—
181#41—2544no—
182#41—2544no—
183#41—2548no—
184#41—2548no—
185#41—2544no—
186#41—2541no—
187#41—2541no—
188#41—2541no—
189#41—2541no—
190#41C22⋊D12712yes—
191#41D50812yes—
192C2×Q8—1271no—
193C2×Q8—1272no—
194C2×Q8—1272no—
195C2×Q8—1272no—
196C2×Q8—2542no—
197C2×Q8—2544no—
198C2×Q8—2544no—
199C2×Q8—2544no—
200C2×Q8—2544no—
201C2×Q8—2541no—
202C2×Q8—2541no—
203C2×Q8—2541no—
204C2×Q8C4×D12711yescharacteristic
205C2×D4—21no—
206C2×D4—1272no—
207C2×D4—1271no—
208C2×D4—2542no—
209C2×D4C4×D12711yescharacteristic
210C22×C4—21no—
211C22×C4—1272no—
212C22×C4—2542no—
213C22×C4—2542no—
214C22×C4—2541no—
215C22×C4C22×D12711yescharacteristic
216M16—21no—
217M16—21no—
218M16—2541no—
219M16—2541no—
220C2×C8—21no—
221C2×C8—2541no—
222C4⋊C4—21no—
223C4⋊C4—2541no—
224C22⋊C4—21no—
225C22⋊C4—2541no—
226C42—21no—
227C42—2541no—
228C23—21no—
229C23—1272no—
230C23D4×D12711yescharacteristic
231Q8—21no—
232Q8—1274no—
233Q8—2548no—
234Q8—2542no—
235Q8—2541no—
236Q8—2542no—
237Q8—2541no—
238Q8—2542no—
239Q8(C2×D127)⋊C412yes—
240D4—22no—
241D4—22no—
242D4—21no—
243D4—2548no—
244D4—2542no—
245D4—2542no—
246D4—2541no—
247D4—2541no—
248D4(C2×D127)⋊C412yes—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—1272no—
254C2×C4—5081no—
255C2×C4—2548no—
256C2×C4—2548no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4—2542no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4C2×C4×D12711yescharacteristic
263C2×C4D4×D12711yescharacteristic
264C2×C4D4×D12711yescharacteristic
265C2×C4D4×D12711yescharacteristic
266C2×C4C2×C22⋊D12711yescharacteristic
267C2×C4C2×D50811yescharacteristic
268C8—41no—
269C8—5081no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—5082no—
274C22—2542no—
275C22C22⋊C4×D12711yescharacteristic
276C22#1686117711yescharacteristic
277C22#1686117911yescharacteristic
278C4—22no—
279C4—81no—
280C4—10161no—
281C4—2542no—
282C4—2544no—
283C4C22⋊C4×D12711yescharacteristic
284C4C2×(C2×D127)⋊C411yescharacteristic
285C4#1686117711yescharacteristic
286C4#1686117911yescharacteristic
287C2—21no—
288C2—21no—
289C2—21no—
290C2—42no—
291C2—5082no—
292C2#2254973111yescharacteristic
subgroupG/H
centerC2#22549731
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#379
prime radicalC2×#16861256C2
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1796
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. #22549893C2semidirect
3. #22549886C2semidirect
4. #22549884C2semidirect
5. #22549858C2semidirect
6. #22549849C2semidirect
7. #421×C127C2semidirect
15. C2×#14523247C4semidirect
18. C2×Q8×D127C4semidirect
147. C127#1796semidirect
150. #421D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
#225498932
#225498862
#225498842
#225498582
#225498492
#421×C1272
#1796127