Finite Group Databasebeta

← #30648021

Subgroups — #30648021

Order 16256.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861257C211yesmaximal, characteristic
2C2×#16861240C211yesmaximal, characteristic, complemented
3#22550450C211yesmaximal, characteristic, complemented
4#22550037C214yesmaximal, complemented
5#22550030C214yesmaximal, complemented
6#22549885C211yesmaximal, characteristic, complemented
7#22549873C211yesmaximal, characteristic, complemented
8#22549842C211yesmaximal, characteristic, complemented
9C22×Q16×C127C211yesmaximal, characteristic, complemented
10#16861257—24no—
11#16861257—24no—
12C2×#14523249—21no—
13C2×#14523249C2212yes—
14C2×#14523248—22no—
15C2×#14523248C2211yescharacteristic
16C2×#14523248C2211yescharacteristic
17C2×Q8×D127—21no—
18C2×Q8×D127C2212yes—
19#16861240—24no—
20C2×#14523238C2212yes—
21C2×#14523235C2212yes—
22C2×#14523232C2211yescharacteristic
23#16861207—21no—
24#16861198—21no—
25#16861192—22no—
26#16861185C2214yes—
27#16861168C2211yescharacteristic
28#16861134C2212yes—
29#16861133C2214yes—
30#16861131C2212yes—
31(C2×D127)⋊C8C2214yes—
32C22×Q8×C127C2211yescharacteristic
33C22×Q8×C127C2211yescharacteristic
34C2×Q16×C127—24no—
35C2×Q16×C127C2214yes—
36C22×C8×C127C2211yescharacteristic
37#14523249—28no—
38#14523249—24no—
39#14523249—24no—
40#14523248—216no—
41#14523248—22no—
42#14523248—22no—
43#14523248—22no—
44#14523248D414yes—
45C2×Dic254—22no—
46C2×Dic254D411yescharacteristic
47C2×Dic254C2311yescharacteristic
48Q8×D127—28no—
49Q8×D127—24no—
50Q8×D127—24no—
51C2×D508—22no—
52C2×D508D411yescharacteristic
53C2×D508C2311yescharacteristic
54C2×C22⋊D127—21no—
55C2×C22⋊D127D412yes—
56C2×C4×D127—22no—
57C2×C4×D127—22no—
58C2×C4×D127—21no—
59C2×C4×D127C2312yes—
60#14523238—24no—
61#14523235—24no—
62#14523232—22no—
63C2×C127⋊C8C2312yes—
64C4⋊Dic127C2312yes—
65Dic127⋊C4—24no—
66C22⋊Dic127—21no—
67(C2×D127)⋊C4—24no—
68C4×Dic127—21no—
69C2×Q8×C127—21no—
70C2×Q8×C127—22no—
71C2×Q8×C127—21no—
72C2×Q8×C127—22no—
73C2×Q8×C127—22no—
74C2×Q8×C127D414yes—
75C2×Q8×C127C2312yes—
76C2×Q8×C127C2312yes—
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127C2311yescharacteristic
80Q16×C127—28no—
81C2×C8×C127—22no—
82C2×C8×C127C2312yes—
83C22×D127—21no—
84C22×D127C2×D412yes—
85Dic254—28no—
86Dic254—22no—
87Dic254—21no—
88Dic254C2×D412yes—
89D508—28no—
90D508—22no—
91D508—21no—
92D508C2×D412yes—
93C22⋊D127—24no—
94C22⋊D127—24no—
95C2×Dic127—21no—
96C2×Dic127—22no—
97C2×Dic127C2×D412yes—
98C4×D127—28no—
99C4×D127—28no—
100C4×D127—24no—
101C4×D127—22no—
102C4×D127—22no—
103C127⋊C8—22no—
104C23×C127C2×D411yescharacteristic
105Q8×C127—24no—
106Q8×C127—22no—
107Q8×C127—22no—
108Q8×C127—22no—
109Q8×C127—42no—
110Q8×C127C2×D414yes—
111C2×C4×C127—24no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—42no—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2411yescharacteristic
119C8×C127—22no—
120C22×C127—22no—
121C22×C127#12012yes—
122C22×C127C22×D411yescharacteristic
123C2×D127—22no—
124C2×D127—22no—
125C2×D127—24no—
126C2×D127—42no—
127C4×C127—24no—
128C4×C127—42no—
129C4×C127#12012yes—
130C4×C127C22×D411yescharacteristic
131C4×C127C22×D411yescharacteristic
132Dic127—22no—
133Dic127—22no—
134Dic127—42no—
135C2×C127—22no—
136C2×C127C2×#12011yescharacteristic
137C2×C127#57812yes—
138D127—42no—
139D127—42no—
140#3011—1271nomaximal, Sylow
141C127#301111yescharacteristic, complemented, Sylow
142C2×2−1+4—1271no—
143C2×#137—1271no—
144C22×Q16D12711yescharacteristic, complemented
145#536—1271no—
146#452—1274no—
147#451—1274no—
148#419—1271no—
149#416—1271no—
150#408—1271no—
1512−1+4—2544no—
1522−1+4—2544no—
153C2×#41—1271no—
154C2×#41—1272no—
155C2×#41—1271no—
156C2×#41—2542no—
157C2×#41—2541no—
158C22×Q8—1272no—
159C22×Q8—2541no—
160C22×Q8C2×D12711yescharacteristic
161C22×Q8C2×D12711yescharacteristic
162#137—2544no—
163C2×Q16—24no—
164C2×Q16—1272no—
165C2×Q16C2×D12714yes—
166C2×SD16—1272no—
167C2×M16—1271no—
168C22×C8C2×D12711yescharacteristic
169C4⋊Q8—2541no—
170#124—2541no—
171#123—2542no—
172C22⋊Q8—1274no—
173#117—1271no—
174#103—1274no—
175#103—1272no—
176#102—1272no—
177C22⋊C8—1274no—
178#41—1274no—
179#41—25416no—
180#41—2544no—
181#41—2548no—
182#41—2544no—
183#41—2542no—
184#41—2542no—
185#41—2542no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—1271no—
192C2×Q8—1271no—
193C2×Q8—2542no—
194C2×Q8—2544no—
195C2×Q8—2548no—
196C2×Q8—2544no—
197C2×Q8C22⋊D12714yes—
198C2×Q8C22×D12712yes—
199C2×Q8C22×D12712yes—
200C2×D4—1272no—
201C2×D4—1271no—
202C2×D4—1271no—
203C2×D4—2542no—
204C2×D4—2541no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—1272no—
208C22×C4—2542no—
209C22×C4—2542no—
210C22×C4—2541no—
211C22×C4C22×D12711yescharacteristic
212Q16—28no—
213Q16—2544no—
214SD16—2544no—
215M16—2542no—
216C2×C8—22no—
217C2×C8—1272no—
218C2×C8C22×D12712yes—
219C4⋊C4—1272no—
220C4⋊C4—2544no—
221C22⋊C4—2544no—
222C22⋊C4—2541no—
223C42—2541no—
224C23—1272no—
225C23—2541no—
226C23D4×D12711yescharacteristic
227Q8—22no—
228Q8—24no—
229Q8—22no—
230Q8—22no—
231Q8—42no—
232Q8—1272no—
233Q8—2548no—
234Q8—2541no—
235Q8—2542no—
236Q8C2×C22⋊D12714yes—
237D4—1272no—
238D4—2548no—
239D4—2544no—
240D4—2544no—
241D4—2541no—
242D4—2542no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—1272no—
248C2×C4—2548no—
249C2×C4—2548no—
250C2×C4—2542no—
251C2×C4—2541no—
252C2×C4—2542no—
253C2×C4—2542no—
254C2×C4—2544no—
255C2×C4D4×D12712yes—
256C2×C4D4×D12711yescharacteristic
257C2×C4C2×C22⋊D12712yes—
258C2×C4C23×D12711yescharacteristic
259C8—22no—
260C8—2542no—
261C22—22no—
262C22—5082no—
263C22—2542no—
264C22—2542no—
265C22—2544no—
266C22#1686117712yes—
267C22C2×D4×D12711yescharacteristic
268C4—24no—
269C4—42no—
270C4—5082no—
271C4—2542no—
272C4—2542no—
273C4#1686117712yes—
274C4C2×D4×D12711yescharacteristic
275C4C22×C22⋊D12711yescharacteristic
276C2—22no—
277C2—5082no—
278C2—5082no—
279C2C2×#1686117711yescharacteristic
280C2#2255068912yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×Q16×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861257C2
hypercenterC22×Q16D127
hypocenterC127#3011
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22550450C2semidirect
4. #22550037C2semidirect
5. #22550030C2semidirect
6. #22549885C2semidirect
7. #22549873C2semidirect
8. #22549842C2semidirect
9. C22×Q16×C127C2semidirect
141. C127#3011semidirect
144. C22×Q16D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612572
C2×#168612402
#225504502
#225500372
#225500302
#225498852
#225498732
#225498422
C22×Q16×C1272
#3011127