Finite Group Databasebeta

← #30648193

Subgroups — #30648193

Order 16256.

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

normal subgroups only

This group has 548 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×#16861237C211yesmaximal, characteristic, complemented
2C2×#16861234C211yesmaximal, characteristic, complemented
3#22550422C211yesmaximal, characteristic, complemented
4C2×#16861171C211yesmaximal, characteristic, complemented
5#22550094C212yesmaximal, complemented
6#22550090C212yesmaximal, complemented
7#22550080C212yesmaximal, complemented
8#22550070C212yesmaximal, complemented
9#22549899C211yesmaximal, characteristic, complemented
10#22549875C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×D508C2211yescharacteristic
14C22×C4×D127—21no—
15#16861237—24no—
16#16861234—24no—
17C2×#14523237C2212yes—
18C2×#14523235C2212yes—
19C2×D1016C2212yes—
20C2×#14523233C2212yes—
21C2×#14523232C2211yescharacteristic
22#16861207C2211yescharacteristic, complemented
23#16861205C2211yescharacteristic, complemented
24#16861198C2211yescharacteristic, complemented
25#16861196C2211yescharacteristic, complemented
26#16861181—21no—
27#16861181—21no—
28#16861171—21no—
29#16861171—21no—
30#16861171C2212yes—
31#16861171C2212yes—
32C4×C22⋊D127—21no—
33C4×C22⋊D127—21no—
34C2×(C2×D127)⋊C4—21no—
35C2×C4×Dic127C2211yescharacteristic
36Dic127⋊C8C2212yescomplemented
37Dic127⋊C8C2212yescomplemented
38#16861132C2212yescomplemented
39#16861129C2212yescomplemented
40C2×#41×C127C2211yescharacteristic
41C2×M16×C127C2211yescharacteristic
42C2×C4⋊C4×C127C2211yescharacteristic
43#103×C127C2212yescomplemented
44#102×C127C2212yescomplemented
45C23×D127—21no—
46#14523248—22no—
47#14523248—22no—
48C2×Dic254C2311yescharacteristic
49C2×D508—21no—
50C2×D508—22no—
51C2×D508D414yescomplemented
52C2×D508C2311yescharacteristic
53C2×D508C2312yes—
54C2×C22⋊D127—21no—
55C2×C22⋊D127—21no—
56C2×C22⋊D127—21no—
57C22×Dic127D411yescharacteristic
58C22×Dic127D411yescharacteristic
59C2×C4×D127—22no—
60C2×C4×D127—22no—
61C2×C4×D127—21no—
62C2×C4×D127—44no—
63#14523237—24no—
64#14523235—24no—
65D1016—24no—
66#14523233—24no—
67#14523232—22no—
68C2×C127⋊C8C2312yes—
69Dic127⋊C4—21no—
70Dic127⋊C4—21no—
71C22⋊Dic127—21no—
72C22⋊Dic127—21no—
73(C2×D127)⋊C4—21no—
74(C2×D127)⋊C4—21no—
75(C2×D127)⋊C4—22no—
76(C2×D127)⋊C4—22no—
77C4×Dic127D411yescharacteristic
78C4×Dic127D411yescharacteristic
79C4×Dic127C2311yescharacteristic
80C4×Dic127C2311yescharacteristic
81#41×C127—22no—
82#41×C127—22no—
83C2×Q8×C127C2311yescharacteristic
84C2×D4×C127—21no—
85C2×D4×C127C2311yescharacteristic
86C22×C4×C127—21no—
87C22×C4×C127—21no—
88C22×C4×C127C2311yescharacteristic
89M16×C127—22no—
90C2×C8×C127C2312yes—
91C4⋊C4×C127—21no—
92C4⋊C4×C127C2312yes—
93C22×D127—24no—
94C22×D127—22no—
95C22×D127—21no—
96C22×D127—42no—
97C22×D127—42no—
98Dic254—21no—
99Dic254—21no—
100D508—24no—
101D508—21no—
102D508—22no—
103D508—21no—
104D508C2×D414yes—
105C22⋊D127—42no—
106C22⋊D127—42no—
107C22⋊D127—42no—
108C2×Dic127—22no—
109C2×Dic127—21no—
110C2×Dic127—41no—
111C2×Dic127C2×D411yescharacteristic
112C2×Dic127C2×D411yescharacteristic
113C2×Dic127C2×D411yescharacteristic
114C2×Dic127C2×D411yescharacteristic
115C4×D127—44no—
116C4×D127—42no—
117C4×D127—44no—
118C127⋊C8—22no—
119C23×C127—21no—
120C23×C127C2×D411yescharacteristic
121Q8×C127—21no—
122Q8×C127—21no—
123D4×C127—21no—
124D4×C127—21no—
125D4×C127—42no—
126C2×C4×C127—22no—
127C2×C4×C127—21no—
128C2×C4×C127—42no—
129C2×C4×C127—42no—
130C2×C4×C127C2×D411yescharacteristic
131C2×C4×C127C2×D412yes—
132C2×C4×C127#4112yes—
133C2×C4×C127C2411yescharacteristic
134C8×C127—22no—
135C22×C127—21no—
136C22×C127—21no—
137C22×C127—41no—
138C22×C127—41no—
139C22×C127—41no—
140C22×C127#12112yes—
141C22×C127C22×D411yescharacteristic
142C2×D127—24no—
143C2×D127—48no—
144C2×D127—42no—
145C2×D127—42no—
146C2×D127—41no—
147C2×D127—41no—
148C2×D127—41no—
149C4×C127—41no—
150C4×C127—42no—
151C4×C127#12112yes—
152C4×C127C22×D411yescharacteristic
153C4×C127C2×#4111yescharacteristic
154Dic127—21no—
155Dic127—21no—
156Dic127—21no—
157Dic127—41no—
158Dic127AGL(1,ℤ/8)12yescomplemented
159C2×C127—22no—
160C2×C127—81no—
161C2×C127C2×#12111yescharacteristic
162C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
163C2×C127#57511yescharacteristic
164D127—44no—
165D127—81no—
166#3036—1271nomaximal, Sylow
167C127#303611yescharacteristic, complemented, Sylow
168C2×AGL(1,ℤ/8)—1271no—
169C2×AGL(1,ℤ/8)—1271no—
170#532—1271no—
171C2×C4×D4—1271no—
172#463—1272no—
173#463—1272no—
174#461—1272no—
175#459—1272no—
176#423—1271no—
177#417—1271no—
178#417D12711yescharacteristic, complemented
179C2×#41—1271no—
180C2×#41C2×D12711yescharacteristic
181C22×D4—1271no—
182C23×C4—2541no—
183AGL(1,ℤ/8)—2544no—
184AGL(1,ℤ/8)—2544no—
185C2×SD16—1272no—
186C2×SD16—1272no—
187C2×D8—1272no—
188C2×D8—1272no—
189C2×M16—1271no—
190C2×M16C2×D12711yescharacteristic
191C4⋊Q8—1271no—
192#127—1271no—
193#124—1271no—
194#124—1271no—
195#121—2541no—
196#121—2541no—
197C4×D4—1272no—
198C4×D4—1272no—
199C4×D4—2541no—
200C4×D4—2541no—
201C4×D4—2541no—
202C4×D4—2541no—
203C2×C4⋊C4C2×D12711yescharacteristic
204C2×C22⋊C4—2541no—
205C2×C42—1271no—
206C4⋊C8—1272no—
207C4⋊C8—1272no—
208#103—1272no—
209#103C2×D12712yescomplemented
210#102—1272no—
211#102C2×D12712yescomplemented
212C24—2541no—
213#41—22no—
214#41—22no—
215#41—2542no—
216#41—2542no—
217C2×Q8—1271no—
218C2×Q8C22×D12711yescharacteristic
219C2×D4—21no—
220C2×D4—1274no—
221C2×D4—1272no—
222C2×D4—1271no—
223C2×D4—2541no—
224C2×D4—2541no—
225C2×D4—2541no—
226C2×D4—2541no—
227C2×D4—2542no—
228C2×D4C22×D12711yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—1271no—
232C22×C4—1271no—
233C22×C4—5084no—
234C22×C4—2542no—
235C22×C4—2542no—
236C22×C4—2541no—
237C22×C4C22×D12711yescharacteristic
238SD16—2544no—
239SD16—2544no—
240D8—2544no—
241D8—2544no—
242M16—22no—
243M16—2542no—
244C2×C8—1272no—
245C2×C8C22×D12712yes—
246C4⋊C4—21no—
247C4⋊C4—2541no—
248C4⋊C4—2541no—
249C4⋊C4C22×D12712yes—
250C22⋊C4—2542no—
251C22⋊C4—2542no—
252C22⋊C4—2541no—
253C22⋊C4—2541no—
254C22⋊C4—2541no—
255C22⋊C4—2541no—
256C42—1271no—
257C42—1271no—
258C42—1271no—
259C42—1271no—
260C23—21no—
261C23—5082no—
262C23—5082no—
263C23—2541no—
264C23—2542no—
265C23—2544no—
266C23D4×D12711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2541no—
270Q8—2541no—
271D4—21no—
272D4—21no—
273D4—42no—
274D4—1274no—
275D4—5082no—
276D4—5082no—
277D4—5082no—
278D4—2541no—
279D4—2541no—
280D4—2542no—
281D4—2544no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—1271no—
287C2×C4—1271no—
288C2×C4—1271no—
289C2×C4—1271no—
290C2×C4—5084no—
291C2×C4—5084no—
292C2×C4—5082no—
293C2×C4—5081no—
294C2×C4—2542no—
295C2×C4—2541no—
296C2×C4D4×D12712yes—
297C2×C4D4×D12711yescharacteristic
298C2×C4#1452324812yes—
299C2×C4C23×D12711yescharacteristic
300C8—22no—
301C8—2542no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—5088no—
308C22—5081no—
309C22—5081no—
310C22—5081no—
311C22—5082no—
312C22—5082no—
313C22—2544no—
314C22#1686118212yes—
315C22C2×D4×D12711yescharacteristic
316C4—41no—
317C4—42no—
318C4—1272no—
319C4—5081no—
320C4—2541no—
321C4—2541no—
322C4—2541no—
323C4#1686118212yes—
324C4C2×D4×D12711yescharacteristic
325C4C2×#1452324811yescharacteristic
326C2—22no—
327C2—81no—
328C2—10161no—
329C2—5084no—
330C2C2×#1686118211yescharacteristic
331C2AGL(1,ℤ/8)×D12711yescharacteristic
332C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
hypercenter#417D127
hypocenterC127#3036
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×#16861237C2semidirect
2. C2×#16861234C2semidirect
3. #22550422C2semidirect
4. C2×#16861171C2semidirect
5. #22550094C2semidirect
6. #22550090C2semidirect
7. #22550080C2semidirect
8. #22550070C2semidirect
9. #22549899C2semidirect
10. #22549875C2semidirect
11. #417×C127C2semidirect
22. #16861207C22semidirect
23. #16861205C22semidirect
24. #16861198C22semidirect
25. #16861196C22semidirect
36. Dic127⋊C8C22semidirect
37. Dic127⋊C8C22semidirect
38. #16861132C22semidirect
39. #16861129C22semidirect
43. #103×C127C22semidirect
44. #102×C127C22semidirect
51. C2×D508D4semidirect
158. Dic127AGL(1,ℤ/8)semidirect
167. C127#3036semidirect
178. #417D127semidirect
209. #103C2×D127semidirect
211. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
C2×#168612342
#225504222
C2×#168611712
#225500942
#225500902
#225500802
#225500702
#225498992
#225498752
#417×C1272
#3036127