Finite Group Databasebeta

← #30648423

Subgroups — #30648423

Order 16256.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C2×#16861216C211yesmaximal, characteristic, complemented
3#22550447C211yesmaximal, characteristic, complemented
4#22550132C212yesmaximal, complemented
5#22550120C212yesmaximal, complemented
6#22550115C212yesmaximal, complemented
7#22550103C212yesmaximal, complemented
8#22549916C211yesmaximal, characteristic, complemented
9#22549872C211yesmaximal, characteristic, complemented
10C2×#16861132C211yesmaximal, characteristic, complemented
11#537×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13#16861239—24no—
14C2×#14523238C2212yes—
15C2×#14523236C2212yes—
16#16861216—24no—
17C2×#14523232C2212yes—
18C2×#14523232C2211yescharacteristic
19C22×C127⋊C8C2212yes—
20C22×C127⋊C8C2211yescharacteristic
21#16861207—21no—
22#16861196—21no—
23#16861190—22no—
24#16861188C2212yes—
25#16861188C2212yes—
26#16861168C2211yescharacteristic
27#16861142C2212yes—
28#16861140C2212yes—
29#16861133C2211yescharacteristic
30#16861133C2211yescharacteristic
31#16861132C2212yes—
32#16861132C2212yes—
33#16861130C2211yescharacteristic
34#16861130C2211yescharacteristic
35C2×#41×C127C2211yescharacteristic
36C2×#41×C127C2211yescharacteristic
37C22⋊Q8×C127—21no—
38C22⋊Q8×C127C2212yes—
39#121×C127—21no—
40#121×C127—22no—
41#121×C127C2212yes—
42C2×C4⋊C4×C127C2211yescharacteristic
43C2×Dic254—21no—
44C2×Dic254—22no—
45C2×Dic254—22no—
46C2×Dic254C2312yes—
47C22×Dic127—21no—
48#14523238—24no—
49#14523236—24no—
50#14523232—24no—
51#14523232D414yescomplemented
52C2×C127⋊C8—24no—
53C2×C127⋊C8—22no—
54C2×C127⋊C8C2311yescharacteristic
55C2×C127⋊C8C2311yescharacteristic
56C2×C127⋊C8C2312yes—
57C4⋊Dic127C2311yescharacteristic
58C4⋊Dic127C2311yescharacteristic
59Dic127⋊C4—22no—
60Dic127⋊C4—22no—
61C22⋊Dic127—22no—
62C22⋊Dic127—22no—
63C22⋊Dic127—21no—
64C4×Dic127—21no—
65#41×C127—22no—
66#41×C127—22no—
67#41×C127—22no—
68#41×C127—22no—
69C2×Q8×C127D411yescharacteristic
70C2×Q8×C127C2311yescharacteristic
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127D412yes—
74C2×D4×C127D411yescharacteristic
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127C2312yes—
79C22×C4×C127C2311yescharacteristic
80C4⋊C4×C127—21no—
81C4⋊C4×C127—22no—
82C4⋊C4×C127C2312yes—
83C22⋊C4×C127—22no—
84C22⋊C4×C127—22no—
85Dic254—22no—
86Dic254—22no—
87Dic254—42no—
88C2×Dic127—21no—
89C2×Dic127—21no—
90C2×Dic127—22no—
91C2×Dic127—42no—
92C127⋊C8—21no—
93C127⋊C8—21no—
94C127⋊C8C2×D414yes—
95C23×C127—21no—
96C23×C127C2×D412yes—
97C23×C127C2×D411yescharacteristic
98Q8×C127—22no—
99Q8×C127—21no—
100Q8×C127—21no—
101D4×C127—24no—
102D4×C127—22no—
103D4×C127—21no—
104D4×C127—21no—
105D4×C127—42no—
106D4×C127—82no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—21no—
111C2×C4×C127—42no—
112C2×C4×C127—81no—
113C2×C4×C127C2×D412yes—
114C2×C4×C127C2×D412yes—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127#4112yes—
117C2×C4×C127C2411yescharacteristic
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—41no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127#12112yes—
127C22×C127C22×D411yescharacteristic
128C4×C127—22no—
129C4×C127—41no—
130C4×C127—42no—
131C4×C127#12112yes—
132C4×C127C22×D411yescharacteristic
133C4×C127C2×#4111yescharacteristic
134Dic127—42no—
135Dic127—41no—
136Dic127—41no—
137C2×C127—22no—
138C2×C127—42no—
139C2×C127—81no—
140C2×C127C2×#12111yescharacteristic
141C2×C127#57711yescharacteristic
142C2×C127#57811yescharacteristic
143#3063—1271nomaximal, Sylow
144C127#306311yescharacteristic, complemented, Sylow
145C2×#137—1271no—
146C2×#131—1271no—
147#537D12711yescharacteristic
148#536—1271no—
149#470—1272no—
150#468—1272no—
151#467—1272no—
152#465—1272no—
153#428—1271no—
154#416—1271no—
155C2×#103—1271no—
156C2×#41C2×D12711yescharacteristic
157C2×#41C2×D12711yescharacteristic
158C22×Q8—1271no—
159#137—2544no—
160C2×Q16—1272no—
161C2×SD16—1272no—
162#131—2544no—
163C2×M16—1271no—
164C2×M16—1272no—
165C22×C8—1271no—
166C22×C8—1272no—
167C4⋊Q8—2541no—
168#124—2541no—
169#123—2542no—
170C22⋊Q8—21no—
171C22⋊Q8—1272no—
172C22⋊Q8—1272no—
173C22⋊Q8C2×D12712yes—
174#121—21no—
175#121—22no—
176#121C2×D12712yes—
177#117—1271no—
178C2×C4⋊C4C2×D12711yescharacteristic
179#107—1272no—
180#106—1272no—
181#103—1271no—
182#103—1271no—
183#103—1272no—
184#103—1272no—
185#102—1271no—
186#102—1271no—
187#41—22no—
188#41—22no—
189#41—22no—
190#41—22no—
191C2×Q8—1272no—
192C2×Q8—2542no—
193C2×Q8—2542no—
194C2×Q8—2541no—
195C2×Q8C22⋊D12711yescharacteristic
196C2×Q8C22×D12711yescharacteristic
197C2×D4—21no—
198C2×D4—22no—
199C2×D4C22⋊D12712yes—
200C2×D4C22⋊D12711yescharacteristic
201C2×D4C22×D12711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—2541no—
205C22×C4C22×D12712yes—
206C22×C4C22×D12711yescharacteristic
207Q16—2544no—
208SD16—2544no—
209M16—1274no—
210M16—2544no—
211C2×C8—1272no—
212C2×C8—1271no—
213C2×C8—1271no—
214C2×C8—2544no—
215C2×C8—2542no—
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—1271no—
219C4⋊C4—1271no—
220C4⋊C4—2542no—
221C4⋊C4—2542no—
222C4⋊C4C22×D12712yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—2541no—
226C22⋊C4—2542no—
227C22⋊C4—2542no—
228C42—2541no—
229C23—21no—
230C23C2×C22⋊D12712yes—
231C23C2×C22⋊D12711yescharacteristic
232Q8—22no—
233Q8—21no—
234Q8—21no—
235Q8—5082no—
236Q8—2542no—
237Q8—2542no—
238D4—24no—
239D4—21no—
240D4—21no—
241D4—22no—
242D4—42no—
243D4—82no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—81no—
250C2×C4—5082no—
251C2×C4—2541no—
252C2×C4—2541no—
253C2×C4—2542no—
254C2×C4D4×D12712yes—
255C2×C4C2×C22⋊D12712yes—
256C2×C4C2×C22⋊D12711yescharacteristic
257C2×C4#1452324712yes—
258C2×C4C23×D12711yescharacteristic
259C8—1274no—
260C8—2541no—
261C8—2541no—
262C22—21no—
263C22—21no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22#1686118112yes—
271C22C22×C22⋊D12711yescharacteristic
272C4—22no—
273C4—41no—
274C4—42no—
275C4—5082no—
276C4—5081no—
277C4—5081no—
278C4#1686118112yes—
279C4C2×D4×D12711yescharacteristic
280C4C2×#1452324711yescharacteristic
281C2—22no—
282C2—42no—
283C2—81no—
284C2C2×#1686118111yescharacteristic
285C2#2255068711yescharacteristic
286C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radical#537×C127C2
socleC22×C127C22×D4
hypercenter#537D127
hypocenterC127#3063
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861216C2semidirect
3. #22550447C2semidirect
4. #22550132C2semidirect
5. #22550120C2semidirect
6. #22550115C2semidirect
7. #22550103C2semidirect
8. #22549916C2semidirect
9. #22549872C2semidirect
10. C2×#16861132C2semidirect
51. #14523232D4semidirect
144. C127#3063semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168612162
#225504472
#225501322
#225501202
#225501152
#225501032
#225499162
#225498722
C2×#168611322
#537×C1272
#3063127