Finite Group Databasebeta

← #30648285

Subgroups — #30648285

Order 16256.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16861240C211yesmaximal, characteristic, complemented
2C2×#16861231C211yesmaximal, characteristic
3#22550450C211yesmaximal, characteristic, complemented
4#22550357C211yesmaximal, characteristic, complemented
5#22550101C212yesmaximal, complemented
6#22550094C212yesmaximal, complemented
7#22550085C212yesmaximal, complemented
8#22550080C212yesmaximal, complemented
9#22549902C211yesmaximal, characteristic, complemented
10#22549875C211yesmaximal, characteristic, complemented
11C2×#103×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C2×#14523248C2211yescharacteristic
14#16861240—24no—
15#16861231—24no—
16C2×Dic508C2211yescharacteristic
17C2×#14523238C2212yes—
18C2×#14523237C2211yescharacteristic
19C2×#14523237C2211yescharacteristic
20C2×#14523235C2212yes—
21C2×D1016C2211yescharacteristic
22C2×#14523232C2211yescharacteristic
23#16861207C2212yes—
24#16861198C2212yes—
25#16861192—22no—
26#16861185—21no—
27#16861185—21no—
28#16861174C2212yes—
29#16861171C2212yes—
30C4×C22⋊D127—22no—
31#16861168C2211yescharacteristic
32#16861167—21no—
33C4⋊C4×D127—21no—
34Dic127⋊C8C2212yes—
35Dic127⋊C8C2212yes—
36#16861132C2212yes—
37#16861129C2212yes—
38C22×Q8×C127C2211yescharacteristic
39C22×C8×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#103×C127C2212yes—
42#103×C127C2212yes—
43#14523248—22no—
44#14523248—22no—
45#14523248—22no—
46#14523248D414yes—
47C2×Dic254C2311yescharacteristic
48C2×Dic254C2311yescharacteristic
49C2×D508C2311yescharacteristic
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—21no—
52C2×C22⋊D127—21no—
53C2×C4×D127—22no—
54C2×C4×D127—21no—
55C2×C4×D127—21no—
56Dic508—22no—
57#14523238—24no—
58#14523237—22no—
59#14523237—22no—
60#14523235—24no—
61D1016—22no—
62#14523232—22no—
63C2×C127⋊C8C2312yes—
64C4⋊Dic127D411yescharacteristic
65C4⋊Dic127D411yescharacteristic
66Dic127⋊C4—22no—
67Dic127⋊C4—22no—
68Dic127⋊C4—22no—
69C22⋊Dic127D412yes—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—22no—
73C4×Dic127—22no—
74C4×Dic127C2312yes—
75C2×Q8×C127—21no—
76C2×Q8×C127—22no—
77C2×Q8×C127—22no—
78C2×Q8×C127C2312yes—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127C2311yescharacteristic
82C2×C8×C127—22no—
83C2×C8×C127C2311yescharacteristic
84C2×C8×C127C2311yescharacteristic
85C4⋊C4×C127—21no—
86C4⋊C4×C127C2312yes—
87C22×D127—21no—
88C22×D127—21no—
89Dic254—21no—
90Dic254—21no—
91Dic254—21no—
92Dic254C2×D412yes—
93D508—21no—
94D508—21no—
95D508—21no—
96D508C2×D412yes—
97C22⋊D127—24no—
98C22⋊D127—42no—
99C2×Dic127—21no—
100C2×Dic127—21no—
101C2×Dic127C2×D412yes—
102C2×Dic127C2×D412yes—
103C4×D127—24no—
104C4×D127—42no—
105C4×D127—82no—
106C127⋊C8—22no—
107C23×C127C2×D411yescharacteristic
108Q8×C127—22no—
109Q8×C127—22no—
110Q8×C127—42no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—42no—
114C2×C4×C127—81no—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D412yes—
117C2×C4×C127#4112yes—
118C2×C4×C127C2411yescharacteristic
119C8×C127—21no—
120C8×C127—21no—
121C22×C127—21no—
122C22×C127—21no—
123C22×C127#12112yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—41no—
127C2×D127—41no—
128C2×D127—41no—
129C2×D127—41no—
130C2×D127—41no—
131C4×C127—42no—
132C4×C127—42no—
133C4×C127#12112yes—
134C4×C127C22×D411yescharacteristic
135C4×C127C2×#4111yescharacteristic
136Dic127—22no—
137Dic127—22no—
138Dic127—22no—
139Dic127—41no—
140C2×C127—22no—
141C2×C127C2×#12111yescharacteristic
142C2×C127#57711yescharacteristic
143C2×C127#57811yescharacteristic
144D127—42no—
145D127—81no—
146#3046—1271nomaximal, Sylow
147C127#304611yescharacteristic, complemented, Sylow
148C2×#137—1271no—
149C2×#135—1271no—
150#536—1271no—
151#520—1271no—
152#464—1272no—
153#463—1272no—
154#462—1272no—
155#461—1272no—
156#424—1271no—
157#417—1271no—
158C2×#103D12711yescharacteristic, complemented
159C2×#41—1271no—
160C2×#41—1271no—
161C22×Q8C2×D12711yescharacteristic
162#137—2544no—
163#135—2544no—
164C2×Q16—1272no—
165C2×Q16—1271no—
166C2×SD16—1272no—
167C2×SD16—1271no—
168C2×SD16—1271no—
169C2×D8—1271no—
170C2×M16—1271no—
171C22×C8C2×D12711yescharacteristic
172C4⋊Q8—1272no—
173#124—1272no—
174#123—2542no—
175C22⋊Q8—2541no—
176C22⋊Q8—2541no—
177C4×Q8—1272no—
178C4×D4—1272no—
179C4×D4—2542no—
180#117—1271no—
181#117—2541no—
182C2×C4⋊C4—2541no—
183C2×C4⋊C4C2×D12711yescharacteristic
184C4⋊C8—1272no—
185C4⋊C8—1272no—
186#103—1272no—
187#103C2×D12712yes—
188#103C2×D12712yes—
189#102—1272no—
190#41—1274no—
191#41—2542no—
192#41—2542no—
193#41—2542no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—1271no—
198C2×Q8—1271no—
199C2×Q8C22×D12712yes—
200C2×D4—1271no—
201C2×D4—1271no—
202C2×D4—2541no—
203C2×D4—2541no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—2542no—
207C22×C4—2541no—
208C22×C4—2541no—
209C22×C4C22×D12711yescharacteristic
210Q16—2542no—
211Q16—2544no—
212SD16—2542no—
213SD16—2542no—
214SD16—2544no—
215D8—2542no—
216M16—2542no—
217C2×C8—22no—
218C2×C8—1272no—
219C2×C8C22×D12711yescharacteristic
220C2×C8C22×D12711yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—1271no—
223C4⋊C4—1271no—
224C4⋊C4—2542no—
225C4⋊C4—2542no—
226C4⋊C4—2542no—
227C4⋊C4C22×D12712yes—
228C22⋊C4—1272no—
229C22⋊C4—2542no—
230C22⋊C4—2542no—
231C22⋊C4—2542no—
232C42—1272no—
233C42—2542no—
234C23—2541no—
235C23—2541no—
236C23D4×D12711yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—1272no—
241Q8—2541no—
242Q8—2541no—
243Q8—2541no—
244D4—1272no—
245D4—5082no—
246D4—2544no—
247D4—2541no—
248D4—2541no—
249D4—2541no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—81no—
254C2×C4—1272no—
255C2×C4—1272no—
256C2×C4—10162no—
257C2×C4—5082no—
258C2×C4—2541no—
259C2×C4—2541no—
260C2×C4—2544no—
261C2×C4D4×D12712yes—
262C2×C4D4×D12711yescharacteristic
263C2×C4#1452324812yes—
264C2×C4C23×D12711yescharacteristic
265C8—21no—
266C8—21no—
267C8—2542no—
268C22—21no—
269C22—21no—
270C22—5081no—
271C22—5081no—
272C22—5081no—
273C22—5081no—
274C22—5081no—
275C22—2542no—
276C22#1686118212yes—
277C22C2×D4×D12711yescharacteristic
278C4—42no—
279C4—42no—
280C4—5081no—
281C4—2542no—
282C4—2542no—
283C4—2542no—
284C4#1686118212yes—
285C4C2×D4×D12711yescharacteristic
286C4C2×#1452324811yescharacteristic
287C2—22no—
288C2—10161no—
289C2—5082no—
290C2C2×#1686118211yescharacteristic
291C2#2255068211yescharacteristic
292C2#2255068911yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#103×C127C2
socleC22×C127C22×D4
prime radicalC2×#16861231C2
hypercenterC2×#103D127
hypocenterC127#3046
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C2×#16861240C2semidirect
3. #22550450C2semidirect
4. #22550357C2semidirect
5. #22550101C2semidirect
6. #22550094C2semidirect
7. #22550085C2semidirect
8. #22550080C2semidirect
9. #22549902C2semidirect
10. #22549875C2semidirect
11. C2×#103×C127C2semidirect
147. C127#3046semidirect
158. C2×#103D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612402
C2×#168612312
#225504502
#225503572
#225501012
#225500942
#225500852
#225500802
#225499022
#225498752
C2×#103×C1272
#3046127