Finite Group Databasebeta

← #30648201

Subgroups — #30648201

Order 16256.

484 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16861242C211yesmaximal, characteristic, complemented
2C2×#16861239C211yesmaximal, characteristic, complemented
3#22550422C211yesmaximal, characteristic
4C2×#16861174C211yesmaximal, characteristic, complemented
5#22550101C212yesmaximal, complemented
6#22550097C212yesmaximal, complemented
7#22550085C212yesmaximal, complemented
8#22550075C212yesmaximal, complemented
9#22549899C211yesmaximal, characteristic, complemented
10#22549875C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14#16861242—24no—
15#16861239—24no—
16C2×Dic508C2212yes—
17C2×#14523238C2212yes—
18C2×#14523237C2212yes—
19C2×#14523236C2212yes—
20C2×#14523232C2211yescharacteristic
21#16861207C2211yescharacteristic
22#16861205C2211yescharacteristic
23#16861198C2211yescharacteristic
24#16861196C2211yescharacteristic
25#16861181—21no—
26#16861181—21no—
27#16861174—21no—
28#16861174—21no—
29#16861174C2212yes—
30#16861174C2212yes—
31C4×C22⋊D127—21no—
32C4×C22⋊D127—21no—
33C2×Dic127⋊C4—21no—
34C2×C4×Dic127—21no—
35C2×C4×Dic127C2211yescharacteristic
36Dic127⋊C8C2212yes—
37Dic127⋊C8C2212yes—
38#16861132C2212yes—
39#16861129C2212yes—
40C2×#41×C127C2211yescharacteristic
41C2×M16×C127C2211yescharacteristic
42C2×C4⋊C4×C127C2211yescharacteristic
43#103×C127C2212yes—
44#102×C127C2212yes—
45#14523248—22no—
46#14523248—22no—
47C2×Dic254—21no—
48C2×Dic254—22no—
49C2×Dic254D414yescomplemented
50C2×Dic254C2311yescharacteristic
51C2×Dic254C2312yes—
52C2×D508C2311yescharacteristic
53C2×C22⋊D127—21no—
54C2×C22⋊D127—21no—
55C2×C22⋊D127—21no—
56C22×Dic127—21no—
57C22×Dic127D411yescharacteristic
58C22×Dic127D411yescharacteristic
59C2×C4×D127—21no—
60Dic508—24no—
61#14523238—24no—
62#14523237—24no—
63#14523236—24no—
64#14523232—22no—
65C2×C127⋊C8C2312yes—
66Dic127⋊C4—21no—
67Dic127⋊C4—21no—
68Dic127⋊C4—22no—
69Dic127⋊C4—22no—
70C22⋊Dic127—21no—
71C22⋊Dic127—21no—
72(C2×D127)⋊C4—21no—
73(C2×D127)⋊C4—21no—
74C4×Dic127—22no—
75C4×Dic127—22no—
76C4×Dic127D411yescharacteristic
77C4×Dic127D411yescharacteristic
78C4×Dic127C2311yescharacteristic
79C4×Dic127C2311yescharacteristic
80#41×C127—22no—
81#41×C127—22no—
82C2×Q8×C127C2311yescharacteristic
83C2×D4×C127—21no—
84C2×D4×C127C2311yescharacteristic
85C22×C4×C127—21no—
86C22×C4×C127—21no—
87C22×C4×C127C2311yescharacteristic
88M16×C127—22no—
89C2×C8×C127C2312yes—
90C4⋊C4×C127—21no—
91C4⋊C4×C127C2312yes—
92C22×D127—21no—
93Dic254—24no—
94Dic254—22no—
95Dic254—21no—
96Dic254—21no—
97Dic254C2×D414yes—
98D508—21no—
99D508—21no—
100C22⋊D127—42no—
101C22⋊D127—42no—
102C22⋊D127—42no—
103C2×Dic127—24no—
104C2×Dic127—22no—
105C2×Dic127—22no—
106C2×Dic127—21no—
107C2×Dic127—41no—
108C2×Dic127C2×D411yescharacteristic
109C2×Dic127C2×D411yescharacteristic
110C2×Dic127C2×D411yescharacteristic
111C2×Dic127C2×D411yescharacteristic
112C4×D127—42no—
113C127⋊C8—22no—
114C23×C127—21no—
115C23×C127C2×D411yescharacteristic
116Q8×C127—21no—
117Q8×C127—21no—
118D4×C127—21no—
119D4×C127—21no—
120D4×C127—42no—
121C2×C4×C127—22no—
122C2×C4×C127—21no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127C2×D411yescharacteristic
126C2×C4×C127C2×D412yes—
127C2×C4×C127#4112yes—
128C2×C4×C127C2411yescharacteristic
129C8×C127—22no—
130C22×C127—21no—
131C22×C127—21no—
132C22×C127—41no—
133C22×C127—41no—
134C22×C127—41no—
135C22×C127#12112yes—
136C22×C127C22×D411yescharacteristic
137C2×D127—41no—
138C2×D127—41no—
139C2×D127—41no—
140C4×C127—41no—
141C4×C127—42no—
142C4×C127#12112yes—
143C4×C127C22×D411yescharacteristic
144C4×C127C2×#4111yescharacteristic
145Dic127—24no—
146Dic127—21no—
147Dic127—21no—
148Dic127—21no—
149Dic127—41no—
150Dic127#13712yescomplemented
151C2×C127—22no—
152C2×C127—81no—
153C2×C127C2×#12111yescharacteristic
154C2×C127C2×#13711yescharacteristic
155C2×C127#57511yescharacteristic
156D127—81no—
157#3037—1271nomaximal, Sylow
158C127#303711yescharacteristic, complemented, Sylow
159C2×#137—1271no—
160C2×#137—1271no—
161#532—1271no—
162C2×C4×Q8—1271no—
163#464—1272no—
164#464—1272no—
165#462—1272no—
166#460—1272no—
167#423—1271no—
168#417—1271no—
169#417D12711yescharacteristic, complemented
170C2×#41—1271no—
171C2×#41C2×D12711yescharacteristic
172C22×Q8—1271no—
173#137—2544no—
174#137—2544no—
175C2×Q16—1272no—
176C2×Q16—1272no—
177C2×SD16—1272no—
178C2×SD16—1272no—
179C2×M16—1271no—
180C2×M16C2×D12711yescharacteristic
181C4⋊Q8—1271no—
182#127—1271no—
183#124—1271no—
184#124—1271no—
185#121—2541no—
186#121—2541no—
187C4×Q8—1272no—
188C4×Q8—1272no—
189C4×Q8—2541no—
190C4×Q8—2541no—
191C4×D4—2541no—
192C4×D4—2541no—
193C2×C4⋊C4—2541no—
194C2×C4⋊C4C2×D12711yescharacteristic
195C2×C42—1271no—
196C2×C42—2541no—
197C4⋊C8—1272no—
198C4⋊C8—1272no—
199#103—1272no—
200#103C2×D12712yes—
201#102—1272no—
202#102C2×D12712yes—
203#41—22no—
204#41—22no—
205#41—2542no—
206#41—2542no—
207C2×Q8—1274no—
208C2×Q8—1272no—
209C2×Q8—1271no—
210C2×Q8—2541no—
211C2×Q8—2542no—
212C2×Q8C22×D12711yescharacteristic
213C2×D4—21no—
214C2×D4—1271no—
215C2×D4—2541no—
216C2×D4—2541no—
217C2×D4—2541no—
218C2×D4C22×D12711yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—1271no—
222C22×C4—1271no—
223C22×C4—2541no—
224C22×C4—2541no—
225C22×C4C22×D12711yescharacteristic
226Q16—2544no—
227Q16—2544no—
228SD16—2544no—
229SD16—2544no—
230M16—22no—
231M16—2542no—
232C2×C8—1272no—
233C2×C8C22×D12712yes—
234C4⋊C4—21no—
235C4⋊C4—2542no—
236C4⋊C4—2542no—
237C4⋊C4—2541no—
238C4⋊C4—2541no—
239C4⋊C4C22×D12712yes—
240C22⋊C4—2541no—
241C22⋊C4—2541no—
242C22⋊C4—2541no—
243C22⋊C4—2541no—
244C42—1271no—
245C42—1271no—
246C42—1271no—
247C42—1271no—
248C42—2542no—
249C42—2542no—
250C23—21no—
251C23—2541no—
252C23D4×D12711yescharacteristic
253Q8—21no—
254Q8—21no—
255Q8—1274no—
256Q8—2541no—
257Q8—2541no—
258Q8—2542no—
259Q8—2544no—
260D4—21no—
261D4—21no—
262D4—42no—
263D4—5082no—
264D4—5082no—
265D4—5082no—
266D4—2541no—
267D4—2541no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—1271no—
273C2×C4—1271no—
274C2×C4—1271no—
275C2×C4—1271no—
276C2×C4—5082no—
277C2×C4—5081no—
278C2×C4—2542no—
279C2×C4—2541no—
280C2×C4—2542no—
281C2×C4—2544no—
282C2×C4D4×D12712yes—
283C2×C4D4×D12711yescharacteristic
284C2×C4#1452324812yes—
285C2×C4C23×D12711yescharacteristic
286C8—22no—
287C8—2542no—
288C22—21no—
289C22—21no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—5081no—
294C22—5081no—
295C22—5081no—
296C22#1686118212yes—
297C22C2×D4×D12711yescharacteristic
298C4—41no—
299C4—42no—
300C4—1272no—
301C4—5081no—
302C4—2544no—
303C4—2541no—
304C4—2541no—
305C4—2541no—
306C4#1686118212yes—
307C4C2×D4×D12711yescharacteristic
308C4C2×#1452324811yescharacteristic
309C2—22no—
310C2—81no—
311C2—10161no—
312C2C2×#1686118211yescharacteristic
313C2#137×D12711yescharacteristic
314C2#2255067211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
prime radical#22550422C2
hypercenter#417D127
hypocenterC127#3037
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C2×#16861239C2semidirect
4. C2×#16861174C2semidirect
5. #22550101C2semidirect
6. #22550097C2semidirect
7. #22550085C2semidirect
8. #22550075C2semidirect
9. #22549899C2semidirect
10. #22549875C2semidirect
11. #417×C127C2semidirect
49. C2×Dic254D4semidirect
150. Dic127#137semidirect
158. C127#3037semidirect
169. #417D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612392
#225504222
C2×#168611742
#225501012
#225500972
#225500852
#225500752
#225498992
#225498752
#417×C1272
#3037127