Finite Group Databasebeta

← #30649185

Subgroups — #30649185

Order 16256.

552 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16861240C211yesmaximal, characteristic, complemented
3C2×#16861239C211yesmaximal, characteristic, complemented
4#22550422C211yesmaximal, characteristic
5C2×#16861207C211yesmaximal, characteristic, complemented
6#22550269C212yesmaximal, complemented
7#22550267C212yesmaximal, complemented
8#22550254C212yesmaximal, complemented
9#22550247C212yesmaximal, complemented
10M16×Dic127C211yesmaximal, characteristic, complemented
11C2×#137×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14#16861242—24no—
15#16861240—24no—
16#16861239—24no—
17C2×Dic508C2212yes—
18C2×#14523238C2212yes—
19C2×#14523238C2212yes—
20C2×#14523237C2212yes—
21C2×#14523236C2212yes—
22C2×#14523235C2212yes—
23C2×#14523232C2211yescharacteristic
24#16861207—22no—
25#16861207C2212yes—
26#16861207C2212yes—
27#16861207C2211yescharacteristic
28#16861205C2211yescharacteristic
29#16861198C2211yescharacteristic
30#16861196C2211yescharacteristic
31#16861181—22no—
32C4×C22⋊D127—22no—
33C2×Dic127⋊C4—22no—
34C2×C4×Dic127C2211yescharacteristic
35#16861119C2212yes—
36C8×Dic127C2212yes—
37C2×#41×C127C2211yescharacteristic
38C22×Q8×C127C2211yescharacteristic
39#137×C127—24no—
40C2×Q16×C127C2212yes—
41C2×SD16×C127C2212yes—
42C2×M16×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254—22no—
48C2×Dic254C2311yescharacteristic
49C2×Dic254C2312yes—
50C2×D508C2311yescharacteristic
51C2×C22⋊D127—22no—
52C2×C22⋊D127—21no—
53C22×Dic127—21no—
54C22×Dic127D412yes—
55C2×C4×D127—21no—
56Dic508—24no—
57#14523238—24no—
58#14523238—24no—
59#14523237—24no—
60#14523236—24no—
61#14523235—24no—
62#14523232D414yescomplemented
63C2×C127⋊C8C2312yes—
64Dic127⋊C4—24no—
65Dic127⋊C4—24no—
66Dic127⋊C4—22no—
67C22⋊Dic127—22no—
68(C2×D127)⋊C4—22no—
69C4×Dic127D412yes—
70C4×Dic127C2311yescharacteristic
71C4×Dic127C2311yescharacteristic
72#41×C127—22no—
73#41×C127—22no—
74C2×Q8×C127—21no—
75C2×Q8×C127—22no—
76C2×Q8×C127—22no—
77C2×Q8×C127C2311yescharacteristic
78C2×Q8×C127C2312yes—
79C2×D4×C127—21no—
80C2×D4×C127C2311yescharacteristic
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127C2311yescharacteristic
84Q16×C127—24no—
85SD16×C127—24no—
86M16×C127D414yescomplemented
87C2×C8×C127C2312yes—
88C22×D127—21no—
89Dic254—22no—
90Dic254—22no—
91Dic254—21no—
92Dic254—21no—
93Dic254—42no—
94D508—21no—
95D508—21no—
96C22⋊D127—42no—
97C22⋊D127—44no—
98C2×Dic127—24no—
99C2×Dic127—21no—
100C2×Dic127—22no—
101C2×Dic127—42no—
102C2×Dic127C2×D412yes—
103C2×Dic127C2×D412yes—
104C4×D127—42no—
105C127⋊C8C2×D414yes—
106C23×C127—21no—
107C23×C127C2×D411yescharacteristic
108Q8×C127—22no—
109Q8×C127—21no—
110Q8×C127—22no—
111Q8×C127—21no—
112Q8×C127—42no—
113D4×C127—21no—
114D4×C127—21no—
115D4×C127—42no—
116C2×C4×C127—21no—
117C2×C4×C127—22no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D412yes—
123C2×C4×C127C2411yescharacteristic
124C8×C127C2×D414yes—
125C22×C127—22no—
126C22×C127—41no—
127C22×C127—42no—
128C22×C127#12712yes—
129C22×C127C22×D411yescharacteristic
130C2×D127—41no—
131C2×D127—42no—
132C4×C127—41no—
133C4×C127—42no—
134C4×C127#12712yes—
135C4×C127C22×D411yescharacteristic
136C4×C127C22×D411yescharacteristic
137Dic127—22no—
138Dic127—41no—
139Dic127—42no—
140Dic127#13714yescomplemented
141C2×C127—22no—
142C2×C127—81no—
143C2×C127C2×#12711yescharacteristic
144C2×C127C2×#13712yes—
145D127—81no—
146#3152—1271nomaximal, Sylow
147C127#315211yescharacteristic, complemented, Sylow
148C2×#137—1271no—
149C2×#137—1271no—
150C2×#137—1271no—
151C2×#137D12711yescharacteristic, complemented
152#532—1271no—
153C2×C4⋊Q8—1271no—
154#497—1272no—
155#497—1272no—
156#494—1272no—
157#492—1272no—
158C4×M16—1271no—
159C2×#41—1271no—
160C2×#41C2×D12711yescharacteristic
161C22×Q8—1271no—
162C22×Q8C2×D12711yescharacteristic
163#137—24no—
164#137—2544no—
165#137—2544no—
166#137—2544no—
167C2×Q16—1272no—
168C2×Q16—1272no—
169C2×Q16—1272no—
170C2×Q16C2×D12712yes—
171C2×SD16—1272no—
172C2×SD16—1272no—
173C2×SD16—1272no—
174C2×SD16C2×D12712yes—
175C2×M16—1271no—
176C2×M16C2×D12711yescharacteristic
177C4⋊Q8—1272no—
178C4⋊Q8—1272no—
179C4⋊Q8—1271no—
180C4⋊Q8—2542no—
181#127—1271no—
182#124—1271no—
183#124—1271no—
184#121—2542no—
185C4×D4—2542no—
186C2×C4⋊C4—2542no—
187C2×C42—1271no—
188#97—1272no—
189C4×C8—1272no—
190#41—22no—
191#41—22no—
192#41—2542no—
193#41—2542no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—1271no—
198C2×Q8—1272no—
199C2×Q8—2542no—
200C2×Q8—2542no—
201C2×Q8—2541no—
202C2×Q8C22×D12711yescharacteristic
203C2×Q8C22×D12712yes—
204C2×D4—21no—
205C2×D4—1271no—
206C2×D4—2542no—
207C2×D4—2541no—
208C2×D4C22×D12711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—1272no—
212C22×C4—2541no—
213C22×C4—2541no—
214C22×C4C22×D12711yescharacteristic
215Q16—24no—
216Q16—2544no—
217Q16—2544no—
218Q16—2544no—
219SD16—24no—
220SD16—2544no—
221SD16—2544no—
222SD16—2544no—
223M16—1274no—
224M16C22⋊D12714yescomplemented
225C2×C8—1272no—
226C2×C8C22×D12712yes—
227C4⋊C4—2544no—
228C4⋊C4—2544no—
229C4⋊C4—2542no—
230C22⋊C4—2542no—
231C22⋊C4—2542no—
232C42—1272no—
233C42—1271no—
234C42—1271no—
235C23—21no—
236C23—2541no—
237C23D4×D12711yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—22no—
241Q8—21no—
242Q8—42no—
243Q8—5082no—
244Q8—2542no—
245Q8—2541no—
246Q8—2542no—
247Q8—2541no—
248D4—21no—
249D4—21no—
250D4—42no—
251D4—5084no—
252D4—5082no—
253D4—2541no—
254D4—2541no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1272no—
260C2×C4—1272no—
261C2×C4—5082no—
262C2×C4—5082no—
263C2×C4—2544no—
264C2×C4—2541no—
265C2×C4—2542no—
266C2×C4D4×D12711yescharacteristic
267C2×C4D4×D12712yes—
268C2×C4C2×C22⋊D12712yes—
269C2×C4C23×D12711yescharacteristic
270C8—1274no—
271C8C2×C22⋊D12714yes—
272C22—22no—
273C22—42no—
274C22—41no—
275C22—5082no—
276C22—5081no—
277C22#1686120512yes—
278C22C2×D4×D12711yescharacteristic
279C4—41no—
280C4—42no—
281C4—1274no—
282C4—5081no—
283C4—5082no—
284C4—2542no—
285C4#1686120512yes—
286C4C2×D4×D12711yescharacteristic
287C4C22×C22⋊D12711yescharacteristic
288C2—22no—
289C2—81no—
290C2—10161no—
291C2C2×#1686120511yescharacteristic
292C2#137×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radical#22550422C2
hypercenterC2×#137D127
hypocenterC127#3152
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16861242C2semidirect
2. C2×#16861240C2semidirect
3. C2×#16861239C2semidirect
5. C2×#16861207C2semidirect
6. #22550269C2semidirect
7. #22550267C2semidirect
8. #22550254C2semidirect
9. #22550247C2semidirect
10. M16×Dic127C2semidirect
11. C2×#137×C127C2semidirect
62. #14523232D4semidirect
86. M16×C127D4semidirect
140. Dic127#137semidirect
147. C127#3152semidirect
151. C2×#137D127semidirect
224. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612402
C2×#168612392
#225504222
C2×#168612072
#225502692
#225502672
#225502542
#225502472
M16×Dic1272
C2×#137×C1272
#3152127