Finite Group Databasebeta

← #30638677

Subgroups — #30638677

Order 16256.

670 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861254C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic, complemented
3C2×#16861123C212yesmaximal, complemented
4#22549755C212yesmaximal, complemented
5#386×C127C211yesmaximal, characteristic, complemented
6#16861254—24no—
7#16861254—22no—
8#16861254C414yescomplemented
9C2×#14523248—21no—
10C2×#14523247—22no—
11C22×C22⋊D127—21no—
12C22×C22⋊D127C2212yes—
13C2×D4×D127—22no—
14C2×C22⋊Dic127C2212yescomplemented
15C2×(C2×D127)⋊C4—22no—
16#16861124—22no—
17#16861123—24no—
18C127⋊#95—22no—
19C22×D4×C127C2211yescharacteristic
20C23×C4×C127C2211yescharacteristic, complemented
21C2×C22⋊C4×C127—21no—
22C2×C22⋊C4×C127C2211yescharacteristic, complemented
23#95×C127—21no—
24C23×D127—21no—
25C23×D127D412yes—
26#14523248—24no—
27#14523248—24no—
28#14523247—28no—
29#14523247—28no—
30C2×Dic254—21no—
31C2×D508—21no—
32C2×C22⋊D127—22no—
33C2×C22⋊D127—24no—
34C2×C22⋊D127—24no—
35C2×C22⋊D127—22no—
36C2×C22⋊D127—22no—
37C2×C22⋊D127—22no—
38C2×C22⋊D127—22no—
39C2×C22⋊D127C2×C414yes—
40C2×C22⋊D127D414yes—
41D4×D127—28no—
42D4×D127—28no—
43C22×Dic127—21no—
44C22×Dic127—22no—
45C22×Dic127D412yes—
46C2×C4×D127—22no—
47C22⋊Dic127—22no—
48C22⋊Dic127—22no—
49(C2×D127)⋊C4—44no—
50C24×C127D411yescharacteristic
51C24×C127C2311yescharacteristic
52C2×D4×C127—22no—
53C2×D4×C127—22no—
54C2×D4×C127C2×C412yes—
55C2×D4×C127D412yes—
56C22×C4×C127—21no—
57C22×C4×C127—22no—
58C22×C4×C127—21no—
59C22×C4×C127—42no—
60C22×C4×C127D411yescharacteristic
61C22⋊C4×C127—22no—
62C22⋊C4×C127—42no—
63C22×D127—22no—
64C22×D127—24no—
65C22×D127—22no—
66C22×D127—42no—
67C22×D127—42no—
68C22×D127—81no—
69C22×D127—81no—
70C22×D127C22⋊C414yescomplemented
71Dic254—24no—
72D508—24no—
73C22⋊D127—28no—
74C22⋊D127—24no—
75C22⋊D127—24no—
76C22⋊D127—24no—
77C22⋊D127—24no—
78C2×Dic127—24no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—42no—
82C2×Dic127—42no—
83C2×Dic127—42no—
84C2×Dic127C22⋊C414yescomplemented
85C4×D127—28no—
86C23×C127—21no—
87C23×C127—21no—
88C23×C127—41no—
89C23×C127—41no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127C22⋊C412yescomplemented
93C23×C127C22×C411yescharacteristic
94C23×C127C2×D412yes—
95C23×C127C2×D412yes—
96C23×C127C2×D411yescharacteristic
97C23×C127C2×D411yescharacteristic
98D4×C127—24no—
99D4×C127—24no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—22no—
103C2×C4×C127—24no—
104C2×C4×C127—41no—
105C2×C4×C127—42no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127C22⋊C412yescomplemented
111C22×C127—21no—
112C22×C127—22no—
113C22×C127—21no—
114C22×C127—22no—
115C22×C127—22no—
116C22×C127—41no—
117C22×C127—42no—
118C22×C127—41no—
119C22×C127—42no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127—81no—
123C22×C127C2×C22⋊C412yes—
124C22×C127C2×C22⋊C411yescharacteristic
125C22×C127#12011yescharacteristic
126C22×C127#12011yescharacteristic
127C22×C127#12012yes—
128C2×D127—24no—
129C2×D127—42no—
130C2×D127—42no—
131C2×D127—44no—
132C2×D127—82no—
133C2×D127—84no—
134C4×C127—22no—
135C4×C127—42no—
136C4×C127—81no—
137Dic127—24no—
138Dic127—82no—
139C2×C127—22no—
140C2×C127—21no—
141C2×C127—22no—
142C2×C127—21no—
143C2×C127—42no—
144C2×C127#37911yescharacteristic
145C2×C127B(4,𝔽2)11yescharacteristic
146C2×C127#45811yescharacteristic
147D127—44no—
148#1790—1271nomaximal, Sylow
149C127#179011yescharacteristic, complemented, Sylow
150C2×H(5,𝔽2)—1271no—
151C2×#99—1271no—
152C2×#99—1272no—
153#386—1272no—
154#386D12711yescharacteristic, complemented
155H(5,𝔽2)—1274no—
156H(5,𝔽2)—2544no—
157H(5,𝔽2)—2542no—
158C2×#41—2542no—
159C2×#41—2541no—
160C22×D4—1272no—
161C22×D4—2542no—
162C22×D4—2541no—
163C22×D4C2×D12711yescharacteristic
164C23×C4C2×D12711yescharacteristic, complemented
165C2×C22⋊C4—21no—
166C2×C22⋊C4—1272no—
167C2×C22⋊C4—2542no—
168C2×C22⋊C4C2×D12711yescharacteristic, complemented
169#99—2544no—
170#99—2542no—
171#95—21no—
172#95—2542no—
173C24—1272no—
174C24—2541no—
175C24C22⋊D12711yescharacteristic
176C24C22×D12711yescharacteristic
177#41—2544no—
178#41—2548no—
179#41—2548no—
180#41—2544no—
181C2×Q8—2541no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—1274no—
185C2×D4—1274no—
186C2×D4—2541no—
187C2×D4—2548no—
188C2×D4—2548no—
189C2×D4—2542no—
190C2×D4—2544no—
191C2×D4—2544no—
192C2×D4—2542no—
193C2×D4—2542no—
194C2×D4—2542no—
195C2×D4—2542no—
196C2×D4C4×D12712yes—
197C2×D4D50812yes—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—42no—
202C22×C4—1272no—
203C22×C4—2542no—
204C22×C4—2542no—
205C22×C4—2541no—
206C22×C4C22⋊D12711yescharacteristic
207C22⋊C4—22no—
208C22⋊C4—42no—
209C22⋊C4—5084no—
210C22⋊C4—2542no—
211C22⋊C4—2542no—
212C23—21no—
213C23—21no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—1274no—
219C23—10161no—
220C23—10161no—
221C23—5082no—
222C23—5082no—
223C23—2544no—
224C23—2542no—
225C23—2542no—
226C23(C2×D127)⋊C412yescomplemented
227C23C2×C4×D12711yescharacteristic
228C23D4×D12712yes—
229C23D4×D12712yes—
230C23C2×C22⋊D12711yescharacteristic
231C23C2×D50811yescharacteristic
232Q8—2544no—
233D4—24no—
234D4—24no—
235D4—2544no—
236D4—2548no—
237D4—2544no—
238D4—2544no—
239D4—2544no—
240D4—2544no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—1274no—
252C2×C4—5082no—
253C2×C4—5082no—
254C2×C4—5082no—
255C2×C4—2548no—
256C2×C4—2544no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4(C2×D127)⋊C412yescomplemented
260C22—21no—
261C22—22no—
262C22—21no—
263C22—22no—
264C22—22no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—81no—
272C22—10162no—
273C22—10164no—
274C22—5084no—
275C22—5082no—
276C22—5082no—
277C22—2544no—
278C22C22⋊C4×D12712yes—
279C22C2×(C2×D127)⋊C411yescharacteristic
280C22#1686117711yescharacteristic
281C22#1686117711yescharacteristic
282C22#1686117912yes—
283C4—22no—
284C4—42no—
285C4—81no—
286C4—10162no—
287C4—2544no—
288C2—21no—
289C2—22no—
290C2—21no—
291C2—22no—
292C2—42no—
293C2—5084no—
294C2#2254973111yescharacteristic
295C2#2255006211yescharacteristic
296C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#386×C127C2
socleC22×C127#120
prime radicalC2×#16861254C2
cosocleC24×C127C23
hypercenter#386D127
hypocenterC127#1790
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#16861124C2semidirect
3. C2×#16861123C2semidirect
4. #22549755C2semidirect
5. #386×C127C2semidirect
8. #16861254C4semidirect
14. C2×C22⋊Dic127C22semidirect
20. C23×C4×C127C22semidirect
22. C2×C22⋊C4×C127C22semidirect
70. C22×D127C22⋊C4semidirect
84. C2×Dic127C22⋊C4semidirect
92. C23×C127C22⋊C4semidirect
110. C2×C4×C127C22⋊C4semidirect
149. C127#1790semidirect
154. #386D127semidirect
164. C23×C4C2×D127semidirect
168. C2×C22⋊C4C2×D127semidirect
226. C23(C2×D127)⋊C4semidirect
259. C2×C4(C2×D127)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168611242
C2×#168611232
#225497552
#386×C1272
#1790127