Finite Group Databasebeta

← #30648397

Subgroups — #30648397

Order 16256.

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

normal subgroups only

This group has 484 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×#16861229C211yesmaximal, characteristic, complemented
2C2×#16861216C211yesmaximal, characteristic, complemented
3#22550454C211yesmaximal, characteristic, complemented
4#22550124C212yesmaximal, complemented
5#22550120C212yesmaximal, complemented
6#22550115C212yesmaximal, complemented
7#22550110C212yesmaximal, complemented
8C2×#16861142C211yesmaximal, characteristic, complemented
9#22549877C211yesmaximal, characteristic, complemented
10#22549875C211yesmaximal, characteristic, complemented
11#537×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861229—24no—
14C2×#14523238C2211yescharacteristic
15C2×#14523236C2211yescharacteristic
16C2×#14523235C2211yescharacteristic
17C2×#14523233C2211yescharacteristic
18#16861216—24no—
19C2×#14523232C2212yes—
20C2×#14523232C2211yescharacteristic
21C22×C127⋊C8C2212yes—
22C22×C127⋊C8C2211yescharacteristic
23#16861188C2212yes—
24#16861185—21no—
25#16861183C2212yes—
26#16861181—22no—
27#16861180—21no—
28C2×C4⋊Dic127C2211yescharacteristic
29#16861142C2212yes—
30#16861142C2212yes—
31#16861133C2212yes—
32#16861132C2212yes—
33#16861130C2212yes—
34#16861129C2212yes—
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
43#14523248—22no—
44#14523248—22no—
45C2×Dic254C2311yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C2×C22⋊D127—22no—
49C22×Dic127—21no—
50C2×C4×D127—21no—
51#14523238—22no—
52#14523236—22no—
53#14523235—22no—
54#14523233—22no—
55#14523232—24no—
56#14523232—22no—
57C2×C127⋊C8—24no—
58C2×C127⋊C8D414yescomplemented
59C2×C127⋊C8C2312yes—
60C2×C127⋊C8C2311yescharacteristic
61C2×C127⋊C8C2311yescharacteristic
62C4⋊Dic127—21no—
63C4⋊Dic127C2312yes—
64Dic127⋊C4—22no—
65C22⋊Dic127—22no—
66(C2×D127)⋊C4—22no—
67#41×C127—22no—
68#41×C127—22no—
69#41×C127—22no—
70#41×C127—22no—
71C2×Q8×C127D411yescharacteristic
72C2×Q8×C127C2311yescharacteristic
73C2×D4×C127—21no—
74C2×D4×C127—22no—
75C2×D4×C127D411yescharacteristic
76C2×D4×C127D412yes—
77C2×D4×C127C2311yescharacteristic
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127C2312yes—
81C22×C4×C127C2311yescharacteristic
82C4⋊C4×C127—21no—
83C4⋊C4×C127—22no—
84C4⋊C4×C127C2312yes—
85C22⋊C4×C127—22no—
86C22⋊C4×C127—22no—
87C22×D127—21no—
88Dic254—21no—
89Dic254—21no—
90D508—21no—
91D508—21no—
92C22⋊D127—42no—
93C22⋊D127—82no—
94C2×Dic127—22no—
95C2×Dic127—21no—
96C2×Dic127—81no—
97C4×D127—42no—
98C127⋊C8—22no—
99C127⋊C8C2×D412yes—
100C127⋊C8C2×D412yes—
101C23×C127—21no—
102C23×C127C2×D412yes—
103C23×C127C2×D411yescharacteristic
104Q8×C127—22no—
105Q8×C127—21no—
106Q8×C127—21no—
107D4×C127—24no—
108D4×C127—22no—
109D4×C127—21no—
110D4×C127—21no—
111D4×C127—42no—
112D4×C127—82no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—42no—
118C2×C4×C127—81no—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D412yes—
121C2×C4×C127C2×D411yescharacteristic
122C2×C4×C127#4112yes—
123C2×C4×C127C2411yescharacteristic
124C22×C127—21no—
125C22×C127—21no—
126C22×C127—22no—
127C22×C127—22no—
128C22×C127—22no—
129C22×C127—41no—
130C22×C127—41no—
131C22×C127—41no—
132C22×C127#12112yes—
133C22×C127C22×D411yescharacteristic
134C2×D127—41no—
135C2×D127—41no—
136C2×D127—41no—
137C4×C127—22no—
138C4×C127—41no—
139C4×C127—42no—
140C4×C127#12112yes—
141C4×C127C22×D411yescharacteristic
142C4×C127C2×#4111yescharacteristic
143Dic127—41no—
144Dic127—42no—
145C2×C127—22no—
146C2×C127—42no—
147C2×C127—81no—
148C2×C127C2×#12111yescharacteristic
149C2×C127#57611yescharacteristic
150C2×C127#57811yescharacteristic
151D127—81no—
152#3061—1271nomaximal, Sylow
153C127#306111yescharacteristic, complemented, Sylow
154C2×#135—1271no—
155C2×#131—1271no—
156#537—1271no—
157#537D12711yescharacteristic, complemented
158#468—1272no—
159#468—1272no—
160#467—1272no—
161#466—1272no—
162C2×#107—1271no—
163#417—1271no—
164#417—1271no—
165C2×#41—1271no—
166C2×#41C2×D12711yescharacteristic
167C2×#41C2×D12711yescharacteristic
168#135—2544no—
169C2×Q16—1271no—
170C2×SD16—1271no—
171C2×SD16—1271no—
172C2×D8—1271no—
173#131—2544no—
174C2×M16—1271no—
175C2×M16—1272no—
176C22×C8—1271no—
177C22×C8—1272no—
178C22⋊Q8—21no—
179C22⋊Q8—1272no—
180C22⋊Q8—2541no—
181C22⋊Q8C2×D12712yes—
182#121—21no—
183#121—22no—
184#121—1272no—
185#121—2541no—
186#121—2542no—
187#121C2×D12712yes—
188C2×C4⋊C4—1271no—
189C2×C4⋊C4C2×D12711yescharacteristic
190#107—1272no—
191#107—1272no—
192#103—1272no—
193#103—1272no—
194#102—1272no—
195#102—1272no—
196#41—22no—
197#41—22no—
198#41—22no—
199#41—22no—
200#41—2542no—
201#41—2542no—
202C2×Q8—1271no—
203C2×Q8C22⋊D12711yescharacteristic
204C2×Q8C22×D12711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—1271no—
208C2×D4—2542no—
209C2×D4—2541no—
210C2×D4C22⋊D12712yes—
211C2×D4C22⋊D12711yescharacteristic
212C2×D4C22×D12711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2541no—
216C22×C4—2541no—
217C22×C4C22×D12712yes—
218C22×C4C22×D12711yescharacteristic
219Q16—2542no—
220SD16—2542no—
221SD16—2542no—
222D8—2542no—
223M16—2544no—
224M16—2542no—
225C2×C8—1274no—
226C2×C8—1271no—
227C2×C8—1271no—
228C2×C8—1272no—
229C2×C8—2544no—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—1272no—
233C4⋊C4—2542no—
234C4⋊C4—2541no—
235C4⋊C4C22×D12712yes—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—2542no—
239C22⋊C4—2542no—
240C23—21no—
241C23—2541no—
242C23C2×C22⋊D12712yes—
243C23C2×C22⋊D12711yescharacteristic
244Q8—22no—
245Q8—21no—
246Q8—21no—
247Q8—2541no—
248Q8—2541no—
249D4—24no—
250D4—21no—
251D4—21no—
252D4—22no—
253D4—42no—
254D4—82no—
255D4—10162no—
256D4—5082no—
257D4—2541no—
258D4—2541no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—10161no—
266C2×C4—5082no—
267C2×C4—2542no—
268C2×C4—2541no—
269C2×C4D4×D12712yes—
270C2×C4C2×C22⋊D12712yes—
271C2×C4C2×C22⋊D12711yescharacteristic
272C2×C4#1452324712yes—
273C2×C4C23×D12711yescharacteristic
274C8—1272no—
275C8—1272no—
276C8—2542no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—5081no—
286C22—5081no—
287C22—5081no—
288C22#1686118112yes—
289C22C22×C22⋊D12711yescharacteristic
290C4—22no—
291C4—41no—
292C4—42no—
293C4—5081no—
294C4—5082no—
295C4#1686118112yes—
296C4C2×D4×D12711yescharacteristic
297C4C2×#1452324711yescharacteristic
298C2—22no—
299C2—42no—
300C2—81no—
301C2—10161no—
302C2C2×#1686118111yescharacteristic
303C2#2255067911yescharacteristic
304C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#537×C127C2
socleC22×C127C22×D4
hypercenter#537D127
hypocenterC127#3061
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×#16861229C2semidirect
2. C2×#16861216C2semidirect
3. #22550454C2semidirect
4. #22550124C2semidirect
5. #22550120C2semidirect
6. #22550115C2semidirect
7. #22550110C2semidirect
8. C2×#16861142C2semidirect
9. #22549877C2semidirect
10. #22549875C2semidirect
11. #537×C127C2semidirect
58. C2×C127⋊C8D4semidirect
153. C127#3061semidirect
157. #537D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
C2×#168612162
#225504542
#225501242
#225501202
#225501152
#225501102
C2×#168611422
#225498772
#225498752
#537×C1272
#3061127